Search is not available for this dataset
text
stringlengths 1
134k
| id
stringlengths 11
16
| article_id
stringlengths 8
10
| section_title
stringlengths 1
1.35k
| educational_score
float64 0.52
5.16
| domain
stringclasses 3
values | document_type
stringclasses 4
values | domain_scores
listlengths 3
3
| document_type_scores
listlengths 4
4
| language
stringclasses 38
values | language_score
float64 0
1
|
|---|---|---|---|---|---|---|---|---|---|---|
Among the 55 patients receiving adjuvant chemotherapy, 29 were treated with Capecitabine and 26 with CAPOX/FOLFOX. The median time from surgery to initiation of adjuvant chemotherapy was comparable between the groups: 5 weeks (IQR: 3–7) for Capecitabine and 4 weeks (IQR: 2.5–6.5) for CAPOX/FOLFOX. Most patients completed the recommended 5–6 cycles of chemotherapy, with 75.9% in the Capecitabine group and 80.7% in the CAPOX/FOLFOX group. A small percentage received fewer cycles, with 20.7% in the Capecitabine group completing 3–4 cycles compared to 7.7% in the CAPOX/FOLFOX group. Only one patient (3.4%) in the Capecitabine group completed 1–2 cycles, while none in the CAPOX/FOLFOX group had fewer than 3 cycles. These findings as illustrated in Table 2 indicate that the majority of patients adhered well to the chemotherapy regimens, with minimal delays in initiating treatment after surgery. Table 2 Adjuvant chemotherapy treatment characteristics Characteristic Adjuvant Chemotherapy ( n = 55) Capecitabine ( n = 29) CAPOX/FOLFOX ( n = 26) The time between surgery and the start of chemotherapy in weeks (median, IQR) 4.71 (3–7) 5 (3–7) 4 (2.5–6.5) 1–2 cycles (%) 1 (1.8%) 1 (3.4%) 0 (0%) 3–4 cycles (%) 8 (14.5%) 6 (20.7%) 2 (7.7%) 5–6 cycles (%) 43 (78.2%) 22 (75.9%) 21 (80.7%)
|
39751895_p17
|
39751895
|
Treatment characteristics
| 4.110595 |
biomedical
|
Study
|
[
0.9260625839233398,
0.0728248655796051,
0.0011125713353976607
] |
[
0.9708971977233887,
0.024692842736840248,
0.0016974874306470156,
0.002712548477575183
] |
en
| 0.999998 |
The incidence of local recurrence and distant metastasis based on adjuvant chemotherapy shows distinct trends as shown in Table 3 . Local recurrence occurred in 3 patients (3.5%) overall, with a lower rate in the adjuvant chemotherapy group (1.8%) versus the no-adjuvant group (6.7%). Distant metastasis was observed in 9 patients (10.6%), with similar rates between the adjuvant chemotherapy group (10.9%) and the no-adjuvant group (10.0%). Liver metastasis occurred in 5 patients (5.9%) overall, with comparable rates between the adjuvant group (5.4%) and the no-adjuvant group (6.7%). Similarly, lung metastasis affected 3 patients (3.5%), with nearly identical rates in both the adjuvant (3.6%) and no-adjuvant groups (3.3%). Table 3 Recurrence and metastasis rates by adjuvant chemotherapy status Characteristic Total ( n = 85) Adjuvant Chemotherapy ( n = 55) No Adjuvant Chemotherapy ( n = 30) Local recurrence (%) 3 (3.5%) 1 (1.8%) 2 (6.7%) Distant metastasis (%) 9 (10.6%) 6 (10.9%) 3 (10.0%) Liver metastasis (%) 5 (5.9%) 3 (5.4%) 2 (6.7%) Lung metastasis (%) 3 (3.5%) 2 (3.6%) 1 (3.3%)
|
39751895_p18
|
39751895
|
Local recurrence/distant metastasis
| 4.149589 |
biomedical
|
Study
|
[
0.9905747771263123,
0.009006001986563206,
0.0004191971675027162
] |
[
0.9970189332962036,
0.0018054996617138386,
0.0007588327280245721,
0.00041673972737044096
] |
en
| 0.999998 |
Kaplan–Meier analysis showed mean local recurrence-free survival (LRFS) times of 87.56 ± 1.43 months in the adjuvant chemotherapy group and 79.70 ± 5.56 months in the no-adjuvant group ( P = 0.115). Similarly, mean distant metastasis-free survival (DMFS) times were 81.24 ± 3.28 months and 77.00 ± 5.92 months, respectively ( P = 0.479). Although adjuvant chemotherapy showed a trend toward improved survival, these differences were not statistically significant as shown in Fig. 1 Fig. 1 Kaplan–Meier analysis of LRFS and DMFS in ypT0-2 N0 rectal cancer: adjuvant chemotherapy vs. no chemotherapy
|
39751895_p19
|
39751895
|
Local recurrence/distant metastasis
| 4.138166 |
biomedical
|
Study
|
[
0.9968597888946533,
0.002854245714843273,
0.0002858924272004515
] |
[
0.9983017444610596,
0.0010001551127061248,
0.0004886659444309771,
0.00020940681861247867
] |
en
| 0.999999 |
The analysis of mortality based on adjuvant chemotherapy status shows notable differences. Overall mortality was observed in 11 patients (12.9%) across the cohort, with a significantly lower rate in the adjuvant chemotherapy group (7.3%) compared to the no-adjuvant chemotherapy group (23.3%). Cancer-related mortality occurred in 7 patients (8.2%), with 2 patients (3.6%) in the adjuvant group and 5 patients (16.7%) in the no-adjuvant group. Non-cancer-related mortality was comparable between the two groups, affecting 4 patients (4.7%) in total, with 2 patients (3.6%) from the adjuvant group and 2 patients (6.7%) from the no-adjuvant group. These findings, as shown in Table 4 , suggest that adjuvant chemotherapy may significantly reduce overall and cancer-related mortality in rectal cancer patients. Table 4 Mortality based on adjuvant chemotherapy status Characteristic Total ( n = 85) Adjuvant Chemotherapy ( n = 55) No Adjuvant Chemotherapy ( n = 30) Overall mortality (%) 11 (12.9%) 4 (7.3%) 7 (23.3%) Cancer-related mortality (%) 7 (8.2%) 2 (3.6%) 5 (16.7%) Non-cancer-related mortality (%) 4 (4.7%) 2 (3.6%) 2 (6.7%)
|
39751895_p20
|
39751895
|
Mortality
| 4.151703 |
biomedical
|
Study
|
[
0.9946280121803284,
0.005009007640182972,
0.0003629211860243231
] |
[
0.9980011582374573,
0.0011533905053511262,
0.0005884352722205222,
0.000257044070167467
] |
en
| 0.999995 |
The survival analysis compared DFS and OS between the two cohorts. The mean DFS was 69.8 ± 7.16 months in the no-adjuvant chemotherapy group and 79.9 ± 3.46 months in the adjuvant chemotherapy group as shown in Fig. 2 . Notably, the median DFS was not reached in either group because more than 50% of patients remained disease-free after the follow-up, indicating favorable outcomes. At the 3-year mark, 81.9% of patients in the no-adjuvant chemotherapy group were disease-free, compared to 89.5% of patients in the adjuvant chemotherapy group. While these numbers suggest a trend toward improved short-term DFS in the adjuvant chemotherapy group, this did not reach statistical significance ( P = 0.153). Fig. 2 Kaplan–Meier analysis of DFS in ypT0-2 N0 rectal cancer: adjuvant chemotherapy vs. no chemotherapy
|
39751895_p21
|
39751895
|
Survival analysis
| 4.143447 |
biomedical
|
Study
|
[
0.9935271739959717,
0.006147871725261211,
0.00032497153733856976
] |
[
0.9971928000450134,
0.001781551749445498,
0.0006265392294153571,
0.0003991262929048389
] |
en
| 0.999998 |
Similar trends were observed for overall survival (OS). The mean OS was 78.8 ± 5.96 months with adjuvant chemotherapy and 82.4 ± 3.73 months in the group without adjuvant treatment. Like DFS, the median OS was not reached in either group, reflecting that more than half of the population were still alive after the follow-up period. At 3 years, OS was 84.6% in the no-adjuvant chemotherapy group compared to 88.1% in the adjuvant chemotherapy group. Although there was a slight improvement in survival rates with the use of adjuvant chemotherapy, the difference between the two groups was not statistically significant, as demonstrated by the log-rank test ( P = 0.654) . Fig. 3 Kaplan–Meier analysis of OS in ypT0-2 N0 rectal cancer with and without adjuvant chemotherapy
|
39751895_p22
|
39751895
|
Survival analysis
| 4.142092 |
biomedical
|
Study
|
[
0.994598388671875,
0.0051072402857244015,
0.0002944551524706185
] |
[
0.9976950287818909,
0.0013626634608954191,
0.0006105417851358652,
0.00033174612326547503
] |
en
| 0.999997 |
As presented in Table 5 , the univariate and multivariate analyses conducted in this study revealed that none of the assessed clinicopathological factors, including age, gender, tumor location, grade, initial CEA levels, clinical T and N stages, type of surgery, surgical approach, or the use of adjuvant chemotherapy, demonstrated a statistically significant association with disease-free survival (DFS) or overall survival (OS). Notably, adjuvant chemotherapy, hypothesized to improve outcomes, showed no significant effect on DFS ( P = 0.166 in univariate and P = 0.674 in multivariate analyses) or OS ( P = 0.656 in univariate and P = 0.843 in multivariate analyses). Similarly, other factors, such as the higher proportion of T3 tumors and the greater number of patients undergoing abdominoperineal resection (APR) in the adjuvant chemotherapy group, did not significantly influence DFS or OS. These findings suggest that in this cohort, the examined variables, including the receipt of adjuvant chemotherapy, were not significant prognostic factors for survival outcomes. Table 5 Factors influencing OS and DFS: a univariate and multivariate approach DFS OS Univariate Analysis Multivariate analysis Univariate Analysis Multivariate analysis P P P P Age (> 50 vs ≤ 50 years) 0.726 0.335 0.466 0.609 Male vs Female 0.331 0.891 0.178 0.876 Lower third vs middle-upper two-thirds tumors 0.116 0.619 0.917 0.943 Grade (II vs III) 0.909 0. 709 0.851 0.749 Initial CEA (≤ 5 vs > 5) 0.968 0.286 0.825 0.520 Initial T status (T2 vs T3 and T4) 0.323 0.847 0.998 0.973 Initial N stage (N0 vs N1 and N2) 0.545 0.483 0.364 1.000 Abdominoperineal vs Lower Anterior Resection 0.969 0.684 0.528 0.546 Open vs. Laparoscopic Surgery 0.372 0.526 0.514 0.643 Adjuvant chemotherapy (Yes vs. No) 0.166 0.674 0.656 0.843
|
39751895_p23
|
39751895
|
Univariate and multivariate analysis
| 4.171257 |
biomedical
|
Study
|
[
0.9984660148620605,
0.0012435847893357277,
0.00029044257826171815
] |
[
0.9988802075386047,
0.00032972588087432086,
0.0006591660203412175,
0.0001308710197918117
] |
en
| 0.999996 |
The purpose of this study was to evaluate the effect of adjuvant chemotherapy on disease-free survival (DFS) and overall survival (OS) in patients with downstaged rectal cancer (ypT0-2 N0) following neoadjuvant CRT and radical resection at Shefa Al Orman Hospital. In this group of patients, we found no evidence that adjuvant chemotherapy improved overall survival, disease-free survival, or recurrence rates. Although patients who underwent adjuvant chemotherapy tended toward better survival outcomes, these changes lacked statistical significance. Furthermore, no significant difference was seen between patients who got adjuvant chemotherapy and those who did not in terms of local recurrence-free survival or distant metastasis-free survival. These results support an increasing amount of evidence that downstaged rectal cancer patients with ypT0-2 N0 status may be at risk of overtreatment, and they cast doubt on the use of adjuvant chemotherapy in these patients.
|
39751895_p24
|
39751895
|
Discussion
| 4.117695 |
biomedical
|
Study
|
[
0.9949192404747009,
0.004752663429826498,
0.0003280562232248485
] |
[
0.996929943561554,
0.0013207696611061692,
0.0013895358424633741,
0.0003597413015086204
] |
en
| 0.999997 |
Patients with ypT0N0 are known to have better outcomes than those with residual disease. In our study, 37.5% of patients achieved ypT0N0, representing a substantial proportion of the cohort. Excluding this subgroup would significantly limit the sample size and the generalizability of our findings. Furthermore, including ypT0N0 patients allows us to explore the heterogeneity of response to neoadjuvant chemoradiotherapy and the potential role of adjuvant chemotherapy in different subgroups, contributing to the ongoing debate on optimizing treatment strategies for rectal cancer.
|
39751895_p25
|
39751895
|
Discussion
| 4.027046 |
biomedical
|
Study
|
[
0.9987819790840149,
0.0009524941560812294,
0.0002655586285982281
] |
[
0.9989444613456726,
0.0005119825946167111,
0.00044982944382354617,
0.00009381672134622931
] |
en
| 0.999999 |
Previous research has also shown no survival benefit from adjuvant chemotherapy in rectal cancer patients who had good histological responses after neoadjuvant treatment, which is in line with our findings. For instance, the European Organization for Research and Treatment of Cancer (EORTC) trial 22921, a landmark randomized phase III trial, assessed the effects of postoperative chemotherapy on patients with rectal cancer who had received preoperative radiotherapy, either with or without chemotherapy. Although the trial did not show a substantial improvement in 10-year OS or DFS, particularly in patients with favorable pathological downstaging, the addition of chemotherapy did improve local control . In our study, patients who achieved ypT0-2 N0 status did not demonstrate a significant survival advantage from adjuvant chemotherapy, indicating that this additional treatment may not be required for patients with minimal residual disease after neoadjuvant CRT.
|
39751895_p26
|
39751895
|
Discussion
| 4.111366 |
biomedical
|
Study
|
[
0.9986152648925781,
0.0011447699507698417,
0.0002399728400632739
] |
[
0.9982940554618835,
0.0007132862228900194,
0.0008724323124624789,
0.00012020943540846929
] |
en
| 0.999998 |
Similar results have been observed in other trials as well. According to Sainato et al. , in the randomized Italian study, which compared postoperative adjuvant chemotherapy with observation in patients treated with preoperative chemoradiotherapy, there was no significant difference in recurrence rates or overall survival between the two groups. Similarly, the Dutch colorectal PROCTOR/SCRIPT trials, which assessed postoperative fluorouracil/leucovorin or capecitabine in patients with stage II or III rectal cancer, failed to reach full accrual and did not demonstrate a statistically significant difference in survival outcomes between patients and non-recipients for adjuvant chemotherapy . Another study that found no significant increase in survival outcomes with adjuvant chemotherapy was the phase III Chronicle trial in the United Kingdom. This trial compared capecitabine/oxaliplatin with no adjuvant treatment in rectal cancer patients . These results provide more evidence that adjuvant chemotherapy might not help individuals with rectal cancer who had downstaging and ypT0-2 N0 illness.
|
39751895_p27
|
39751895
|
Discussion
| 3.993234 |
biomedical
|
Study
|
[
0.9985198378562927,
0.0009457811829634011,
0.0005343582597561181
] |
[
0.6904500722885132,
0.0018893844680860639,
0.3071814179420471,
0.0004791461688000709
] |
en
| 0.999996 |
A meta-analysis of individual patient data from four main trials also determined that fluorouracil-based adjuvant chemotherapy did not enhance OS, DFS, or reduce distant recurrence rates in patients with rectal cancer . Notably, this analysis determined that patients with tumors situated 10 to 15 cm above the anal verge experienced a substantially improved DFS with adjuvant chemotherapy; however, this advantage did not extend to the general rectal cancer population. While we did not find a statistically significant difference in survival rates according to tumor location in our study, this could mean that adjuvant chemotherapy is only helpful for specific subsets of rectal cancer patients, like those with higher tumor locations or worse pathological stages.
|
39751895_p28
|
39751895
|
Discussion
| 4.084316 |
biomedical
|
Study
|
[
0.9994015693664551,
0.0004197925445623696,
0.0001785970525816083
] |
[
0.9978072047233582,
0.00036501954309642315,
0.0017299936152994633,
0.00009781803237274289
] |
en
| 0.999996 |
Our research is also consistent with the results of the ADORE trial, which assessed the efficacy of oxaliplatin-containing adjuvant chemotherapy in patients with rectal cancer who had undergone neoadjuvant chemoradiotherapy and surgery . In the ADORE trial, patients with ypT3-4 N0 or ypN + disease, who are more advanced than the patients in our research, had a substantially higher 6-year DFS rate in the FOLFOX group. However, our patient type was not included in this benefit. The absence of a survival benefit in our cohort, which consisted solely of patients with ypT0-2 N0 status, implies that adjuvant chemotherapy may not be required in patients who have attained a favorable pathological downstaging following neoadjuvant therapy.
|
39751895_p29
|
39751895
|
Discussion
| 4.109888 |
biomedical
|
Study
|
[
0.9977295994758606,
0.0019970678258687258,
0.00027328310534358025
] |
[
0.9979947805404663,
0.0013639499666169286,
0.00046111393021419644,
0.00018012727377936244
] |
en
| 0.999997 |
Several studies have demonstrated the prognostic value of pathological response to neoadjuvant CRT in rectal cancer. Park et al. showed that regardless of response category (full, partial, or poor), tumor response to neoadjuvant therapy is a vital early predictor of long-term outcomes, with 5-year recurrence-free survival rates varying significantly. Patients who have complete or almost complete responses, such as those with ypT0-2 N0 disease, typically have excellent long-term results, raising doubts about the necessity of extra adjuvant chemotherapy in these individuals. Our findings reinforce this concept, as patients with ypT0-2 N0 status in our study exhibited favorable survival outcomes without a substantial survival benefit from adjuvant chemotherapy.
|
39751895_p30
|
39751895
|
Discussion
| 4.0525 |
biomedical
|
Study
|
[
0.9991490840911865,
0.0006449443753808737,
0.0002059118269244209
] |
[
0.9969455599784851,
0.0003379867412149906,
0.002595694735646248,
0.00012077728752046824
] |
en
| 0.999996 |
Additionally, the value of adjuvant chemotherapy in patients with downstaged rectal cancer has been questioned by several retrospective studies and meta-analyses. In Taiwan, 720 patients with rectal cancer underwent a comprehensive retrospective countrywide investigation by Kuo et al. , and they discovered no statistically significant difference in 5-year OS or DFS between patients who received adjuvant chemotherapy and those who did not. Furthermore, no protective benefit of adjuvant chemotherapy was found in stratified analyses, even in patients with more advanced clinical T or N classifications. Liao and his colleagues reported comparable results, following 110 patients with ypT0-2 N0 rectal cancer for a median of 60 months and finding no statistically significant difference in the survival outcomes between patients receiving adjuvant chemotherapy and those not.
|
39751895_p31
|
39751895
|
Discussion
| 4.012572 |
biomedical
|
Study
|
[
0.9990622401237488,
0.0005349458078853786,
0.00040283595444634557
] |
[
0.9077718257904053,
0.0007284936727955937,
0.09123747795820236,
0.0002621831663418561
] |
en
| 0.999997 |
Nevertheless, certain studies have demonstrated the potential advantages of adjuvant chemotherapy, particularly in cases of more advanced rectal cancer. Another study offered an intriguing viewpoint, demonstrating that adjuvant chemotherapy was considerably beneficial for patients with ypT2 ypN0, as seen by better OS and DFS compared to no adjuvant treatment. However, Adjuvant chemotherapy did not offer significant survival advantages for patients with ypT0-1 N0 disease. These results indicate that patients with ypT0-1 N0 status may not require adjuvant chemotherapy; however, those with ypT2 N0 may still derive substantial advantages from additional treatment .
|
39751895_p32
|
39751895
|
Discussion
| 3.907421 |
biomedical
|
Study
|
[
0.9984982013702393,
0.0008250079117715359,
0.0006768651655875146
] |
[
0.5505962371826172,
0.0016000403556972742,
0.4473649561405182,
0.00043877787538804114
] |
en
| 0.999996 |
Seventeen nonrandomized studies involving 4,747 patients with ypT0-2N0 rectal cancer were reviewed in the meta-analysis . According to their findings, patients with ypT0N0 or ypT1-2N0 status did not experience a significant difference in OS, DFS, local recurrence, or distant recurrence following adjuvant chemotherapy. According to the odds ratios for OS (1.53, 95% CI: 0.86–2.72) and DFS (1.22, 95% CI: 0.61–2.42), adjuvant chemotherapy did not significantly improve outcomes. The absence of statistical significance in the recurrence and survival outcomes raises the possibility that routine adjuvant chemotherapy usage in this patient group is not warranted. This meta-analysis supports the idea that patients with rectal cancer who show good pathology responses after neoadjuvant therapy would benefit more from a targeted strategy to adjuvant therapy as opposed to a standard one .
|
39751895_p33
|
39751895
|
Discussion
| 4.028416 |
biomedical
|
Study
|
[
0.9991145730018616,
0.0006190703134052455,
0.0002663469349499792
] |
[
0.9630566835403442,
0.0008075741352513433,
0.03585446998476982,
0.00028121870127506554
] |
en
| 0.999995 |
Our study has important clinical implications. The lack of a clear survival benefit from adjuvant chemotherapy in patients with ypT0-2 N0 rectal cancer suggests that routine use of adjuvant chemotherapy in this population may be unnecessary. By avoiding overtreatment, we can reduce the toxicity and side effects associated with chemotherapy, which can have a significant impact on patients' quality of life. Furthermore, avoiding unnecessary adjuvant chemotherapy has important economic implications, as it allows healthcare resources to be allocated more efficiently to patients who are more likely to benefit from additional treatment. These findings support a more individualized approach to adjuvant therapy, where treatment decisions are based on the patient's pathological response to neoadjuvant therapy and their overall risk of recurrence.
|
39751895_p34
|
39751895
|
Discussion
| 4.10572 |
biomedical
|
Study
|
[
0.9838323593139648,
0.01566997542977333,
0.0004977388889528811
] |
[
0.9908046722412109,
0.005220718216150999,
0.003056607209146023,
0.0009180028573609889
] |
en
| 0.999996 |
This study has several limitations that must be acknowledged. First, the retrospective design limits control over confounding variables, and the relatively small sample size reduces statistical power. Second, imbalances in baseline characteristics, such as a higher proportion of T3 tumors and APR in the adjuvant chemotherapy group, may have influenced the results. Third, important prognostic factors, including extramural venous invasion (EMVI), tumor deposits, and circumferential margin (CRM) status, were not consistently documented, limiting the ability to fully assess recurrence and survival risks. Fourth, the lack of randomization and the variability in adjuvant chemotherapy regimens introduce selection bias and treatment heterogeneity. Finally, while follow-up was sufficient for observing trends in survival, longer-term outcomes remain unknown. These limitations highlight the need for larger, prospective, randomized studies to validate these findings.
|
39751895_p35
|
39751895
|
Discussion
| 4.104061 |
biomedical
|
Study
|
[
0.9990203380584717,
0.0007813358679413795,
0.00019833246187772602
] |
[
0.9984034895896912,
0.0003611598804127425,
0.0011081902775913477,
0.00012715951015707105
] |
en
| 0.999997 |
In conclusion, our study suggests that adjuvant chemotherapy does not provide a significant survival benefit in patients with ypT0-2 N0 rectal cancer following neoadjuvant CRT and surgery, indicating that routine use may not be necessary. Avoiding overtreatment could reduce chemotherapy-related toxicity and improve resource allocation. While our study provides valuable insights into the role of adjuvant chemotherapy in downstaged rectal cancer patients, its retrospective design, monocentric, small sample size, and lack of data on certain prognostic factors underscore the need for prospective studies to confirm these results. Future research should focus on larger, multicenter trials incorporating molecular profiling and biomarkers to better tailor adjuvant therapy decisions.
|
39751895_p36
|
39751895
|
Discussion
| 4.07551 |
biomedical
|
Study
|
[
0.9986642599105835,
0.0011357187759131193,
0.00019995209004264325
] |
[
0.9960777163505554,
0.0006957852165214717,
0.0030315041076391935,
0.00019497025641612709
] |
en
| 0.999996 |
Below is the link to the electronic supplementary material. Supplementary file1 (PDF 112 KB)
|
39751895_p37
|
39751895
|
Supplementary Information
| 1.006511 |
other
|
Other
|
[
0.21873579919338226,
0.003099254798144102,
0.7781649231910706
] |
[
0.012934827245771885,
0.984826922416687,
0.0014815115137025714,
0.0007567983120679855
] |
en
| 0.999996 |
This study was conducted and reported in accordance with the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) and the Meta-analysis Of Observational Studies in Epidemiology (MOOSE) guidelines. 9 , 10
|
39068325_p0
|
39068325
|
Methods
| 3.633225 |
biomedical
|
Study
|
[
0.9993053674697876,
0.0003338634269312024,
0.0003607570251915604
] |
[
0.9983933568000793,
0.0009728869772516191,
0.0005412192549556494,
0.0000925676358747296
] |
en
| 0.999996 |
We systematically searched the MEDLINE, Embase, PubMed, Web of Science, and Cochrane Library databases to capture potentially eligible articles from the date of inception until March 12, 2023. In brief, a combination of key words for “esophageal cancer,” “esophagectomy or surgery or operation,” and “dumping syndrome” were searched in title or abstract. The specific search strategies are illustrated in detail in the Supplementary Materials . To identify additional studies that might have been missed in search in electric databases, we also reviewed the reference lists of the included studies and relevant review articles.
|
39068325_p1
|
39068325
|
Literature Search
| 3.974515 |
biomedical
|
Study
|
[
0.999357283115387,
0.0003185754467267543,
0.0003241006634198129
] |
[
0.9955596923828125,
0.00032161554554477334,
0.004037184175103903,
0.00008151023939717561
] |
en
| 0.999996 |
Titles and abstracts of the identified articles were reviewed in the first round of screening, and full text reports were further referred to for studies considered as potentially relevant, after which eligibility was assessed according to the predetermined inclusion and exclusion criteria. We included original and independent cohort studies and clinical trials, with access to full text, followed up to investigate the occurrence of dumping syndrome in esophageal cancer patients who underwent curative intent surgery. Articles meeting any of the following were excluded: (1) being written in languages other than English; (2) reviews, comments, protocols, editorials, letters, case reports, or animal studies; (3) duplicate publications; and 4) studies without data sufficient for calculating a proportion and its 95% confidence interval (CI). If multiple studies from the same cohort were identified, the most recent study was selected for analysis. Two authors (Y.L. and H.W.) independently screened each article for eligible studies. Uncertainties were resolved through joint reevaluation and verified by a senior investigator (S.H.X.).
|
39068325_p2
|
39068325
|
Selection Criteria
| 4.060743 |
biomedical
|
Study
|
[
0.9990642666816711,
0.0006930093513801694,
0.00024266204854939133
] |
[
0.9963958859443665,
0.0004186833102721721,
0.0030759628862142563,
0.00010955358447972685
] |
en
| 0.999997 |
We assessed the quality of observational studies in terms of risk of selection bias, information misclassification and confounding, with the help of the Newcastle-Ottawa Scale (NOS) for assessing the quality of nonrandomized studies in meta-analyses. 11 The NOS scoring system rates the study quality of nonrandomized epidemiological studies on the three major methodological aspects, i.e., risk of selection bias, comparability between groups, and information bias (assessment of exposure or outcome), which are assessed by eight specific items. An overall score ranging from 0 to 9 was given to each original study; higher scores indicated higher study quality. We assessed methodological quality and risk of bias in randomized controlled trials by using the Cochrane tool for assessing risk of bias, version 2 (RoB 2). 12 The RoB 2 tool assess the quality of randomized controlled trials in terms of the following five sources of bias: whether randomization is appropriate, the effect of assignment and adhering to the intervention, loss to follow-up, misclassification of outcomes, and the analysis and selection of the reported data; the overall risk of bias from an individual study will be judged as “low,” “some concerns,” or “high” risk of bias. The assessment of study quality was processed independently by two authors (Y.L. and H.W.), and any discrepancies were dealt with by discussion or consultation with a senior investigator (S.H.X.).
|
39068325_p3
|
39068325
|
Study Quality Assessment
| 4.140139 |
biomedical
|
Study
|
[
0.9993869066238403,
0.00040092412382364273,
0.00021214723528828472
] |
[
0.9973297119140625,
0.0004166063154116273,
0.0021577044390141964,
0.00009587319800630212
] |
en
| 0.999997 |
The following data were extracted from the included studies: first author, year of publication, type of study design, country, number of participants, age and gender distribution of participants, length of follow-up, completeness of follow-up, measuring method of dumping syndrome, surgical approach (transthoracic, transhiatal, or minimally invasive esophagectomy), anastomosis approach (cervical or intrathoracic), preoperative neoadjuvant therapy (yes or no), preoperative adjuvant therapy (yes or no), and number of patients with dumping syndrome. Two authors (Y.L. and Y.Q.) separately extracted the data and compared the consistency together; disagreement between the two sets of data was solved by discussion, with involvement of a senior investigator ((S.H.X.) whenever necessary.
|
39068325_p4
|
39068325
|
Data Extraction
| 4.06246 |
biomedical
|
Study
|
[
0.998978853225708,
0.0007479397463612258,
0.00027320990920998156
] |
[
0.9983945488929749,
0.00040916993748396635,
0.0010985255939885974,
0.00009768472227733582
] |
en
| 0.999997 |
Because of the considerate heterogeneity across individual studies, we conducted random-effects meta-analysis to estimate the pooled prevalence (proportion) of dumping syndrome after esophageal cancer surgery and its 95% CI. Heterogeneity across studies was quantitatively assessed by using the Cochran’s Q test and I 2 statistic. 13 We assessed potential publication bias with the help of funnel plots, supplemented by quantitative evaluation by the Begg’s and Egger’s tests. 14 , 15 Subgroup analyses were conducted by the following factors to assess the sources of heterogeneity between individual studies: by measuring method of dumping syndrome (using specialized questionnaires, or not), published year , study population (Western or Eastern Asian), length of follow-up (≤ 6 months, ≤ 12 months, or > 12 months) and completeness of follow-up (loss to follow-up ≤20%, or >20%). We conducted sensitivity analyses by dropping individual primary studies one by one to examine the fluctuation of the pooled prevalence of dumping syndrome.
|
39068325_p5
|
39068325
|
Statistical Analysis
| 4.094304 |
biomedical
|
Study
|
[
0.9992585778236389,
0.0005078366957604885,
0.00023356953170150518
] |
[
0.9983125925064087,
0.000344921019859612,
0.0012470120564103127,
0.000095469004008919
] |
en
| 0.999998 |
All analyses were performed using the software R (version 4.2.3) and the meta package (version 6.2-1). 16 All statistical tests were two-sided, P value < 0.05 was considered to be statistically significant.
|
39068325_p6
|
39068325
|
Statistical Analysis
| 2.403723 |
biomedical
|
Study
|
[
0.9967890977859497,
0.0004862358036916703,
0.0027247315738350153
] |
[
0.9231711626052856,
0.07412020862102509,
0.002040812512859702,
0.0006678480422124267
] |
en
| 0.999996 |
The detailed procedure of identification and selection of articles is presented in Supplementary Fig. S1. The literature search identified a total of 2256 studies after duplicates were removed. Among these, 10 studies fulfilled the eligibility criteria. Six additional studies were identified through reviewing reference lists of relevant articles. Thus, a total of 16 studies, including 15 cohort studies and one randomized clinical trial, were enlisted in this systematic review. 17 – 32 These studies included 2240 patients who underwent curative surgery for esophageal cancer. Detailed characteristics of each study are shown in Tables 1 and 2 . The overall quality of the 15 cohort studies was rated either 5 or 6 in the NOS scoring system (Supplementary Table S1), and the included randomized trial showed “some concerns” in methodological quality according to the RoB 2 tool (Supplementary Table S2). Table 1 Characteristics of included studies References Country Study period Male (%) Age (years) No. patients Follow-up (months) Loss to follow-up Assessment of dumping syndrome No. events (%) Cohort studies Mannell et al. 17 South Africa Unspecified 66.70% Range 35–60 15 1.25–48 0.0% “Clinical assessment" 2 (13.0%) King et al. 18 United States 1980–1982 81.00% Mean 60.6 (range 33–91) 95 2–74.4 5.3% Medical records 5 (5.0%) Collard et al. 19 Belgium 1984–1987 76.50% Range 23–66 17 36–76 0.0% Questionnaire 3 (6.0%) Wang et al. 20 China 1974-1984 96.50% Mean 61.6 (range 29–82) 76 >12 79.4% Questionnaire 10 (13.2%) Kuwano et al. 21 Japan 1986–1990 86.00% Mean 59.86 50 60 2-year 28.0%, 3-year 60.0%, 4-year 80.0%, 5-year 92.0% Unspecified 6 (12.0%) Orringer et al. 22 United States 1976–unspecified 79.60% Average 63 (range 29–92) 377 1–181 9.6% Unspecified 182 (48.0%) Finley et al. 23 Canada 1980-1994 79.30% Mean 64 (range 16–86) 169 3 5.1% Postprandial lightheadedness or diarrhea 10 (6.0%) McLarty et al. 24 United States 1972–1990 75.70% Median 62 (range 30–81) 107 >60 32.7% Mail survey 53 (50.0%) Aghajanzadeh et al. 25 Iran 1993–2003 69.80% Mean 48 (range 22–75) 192 12–48 20.0% Self-administered health questionnaire 79 (46.0%) Antonoff et al. 26 United States 2007-2012 84.60% Mean 61.8 ± SD 0.6 140 >12 6-month, 6.8%, 12-month 43.0%, >12 months 52.2% Patient reported symptoms 25 (17.9%) Anandavadivelan et al. 27 Sweden 2013–2018 86.70% Average 66 ± SD 8.6) 188 0–18 1-year 15.3%, 1.5- year 33.1% Sigstad’s score and the Arts dumping questionnaire 129 (69.0%) Klevebro et al. 28 United States 1995–2017 80.70% Median 66.2 (range 30-90) 159 3–277 7.0% Dumping Symptom Rating Scale 97 (61.0%) Yoshida et al. 29 Japan 2008–2019 84.00% Mean 67.6 ± SD 9.1 300 1–60 52.3% Unpecified 7 (2.3%) Bennett et al. 30 Ireland 2017–2019 84.00% Mean 63.3 35 6–12 6-month 12.0%, 12-month 50.7% Sigstad’s score 26 (74.3%) Chen et al. 31 China 2020–2021 85.00% Mean 62.6 ± SD 7.1 20 – 0.0% Unspecified 0 (0.0%) Randomized controlled trial Li et al. 32 China 2015–2017 86.7% Range 18–75 300 1–30 0.0% Unspecified 0 (0.0%) SD Standard deviation Table 2 Surgical approaches and other treatments in the included studies References Pyloric procedure Surgical approach Graft position Conduit Site of anastomosis Neoadjuvant therapy Adjuvant therapy Cohort studies Mannell et al. 17 – TTE – G CE (53%), IT (47%) – – King et al. 18 PP (56%), PM (39%) TTE PM G IT – – Collard et al. 19 PP (100%) TTE (65%), THE (18%), Other (17%) PM (65%), RS (35%) G (82%), C (18%) CE – – Wang et al. 20 PP (24%) TTE(93%), THE (7%) – G CE – – Kuwano et al. 21 PP (100%) TTE PM (34%), RS (48%), SC (18%) G (96%), C (4%) CE (66%), IT (34%) – – Orringer et al. 22 PM (routine) TTE (3%), THE (97%) PM (96%), RS (3%), Other (1%) G (95%), C (4%), Other (1%) CE Yes – Finley et al. 23 PP (6.4%), PM (80.0%) TTE (26%), THE (74%) – G CE (93%), IT (7%) Yes – McLarty et al. 24 PP (34%), PM (49%) THE ( 85%), TTE (4 %), Other (11%) – G (93%), C (3%), SB (4%) CE (19%), IT (81%) Yes – Aghajanzadeh et al. 25 PP (22%), PM (47%) TTE (26%), THE (74%) – G (80%), C (15%), SB (5%) CE (90%), IT (10%) – Yes Antonoff et al. 26 PP (12.3%), PM (54.9%), FB (2.7%), FB+BI (15.0%) TTE (56%), THE (44%) – G (98.6%) – – – Anandavadivelan et al. 27 – MIE (30%), TTE (30%), Other (30%) – – – Yes – Klevebro et al. 28 PP (2.3%) TTE (88%), THE (2%), Other (10%) – – IT (55%), CE (45%) Yes – Yoshida et al. 29 – TTE PM (84%), RS (16%) G CE Yes – Bennett et al. 30 PP THE (27%), Other (73%) – C – Yes – Chen et al. 31 – MIE – – – – – Randomized controlled trial Li et al. 32 – THE – SB – – – BI botulinum injection; C colon; CE cervical; E esophagus; FB finger bougie; G gastric; IT intrathoracic; MIE minimally invasive esophagectomy; PM (for graft position) posterior mediastinum; PM (for pyloric procedure) pyloromyotomy; PP pyloroplasty; RS retrosternal; SB small bowel; SC subcutaneous; THE transhiatal esophagectomy; TTE transthoracic esophagectomy
|
39068325_p7
|
39068325
|
Literature Search and Study Characteristics
| 4.225482 |
biomedical
|
Review
|
[
0.9960683584213257,
0.002344437874853611,
0.0015871780924499035
] |
[
0.07183000445365906,
0.0013787101488560438,
0.9261083602905273,
0.0006828587502241135
] |
en
| 0.999998 |
The prevalence of dumping syndrome ranged 0–74% in the 16 individual studies, and the meta-analysis estimated a pooled prevalence of 27% (95% CI 14–39%) . The I 2 statistics and Cochran's Q test showed high heterogeneity across the included studies ( I 2 = 99%, P < 0.01 in Cochran’s Q test). Possible publication bias was revealed by visual inspection of funnel plots, i.e., seemingly more reports of relatively low proportion of dumping syndrome in small-size studies ; the Begg’s and Egger’s tests showed P values of 0.5285 and 0.0010, respectively. Fig. 1 Forest plots for meta-analysis of 16 studies on the prevalence of dumping syndrome after esophageal cancer surgery
|
39068325_p8
|
39068325
|
Meta-analysis
| 4.125143 |
biomedical
|
Study
|
[
0.9992306232452393,
0.0005170473014004529,
0.00025234222994185984
] |
[
0.9980753660202026,
0.00027268496342003345,
0.001556921168230474,
0.00009499445150140673
] |
en
| 0.999996 |
The detailed results of subgroup analyses by measuring method of dumping syndrome, year of publication, study population, and length of follow-up are presented in Table 3 , Fig. 2 , and Supplementary Fig. S3. Only three of 16 studies used specialized questionnaires for measuring dumping syndrome. 27 , 28 , 30 The pooled prevalence of dumping syndrome in these three studies using specialized questionnaires was higher than that generated from the remaining studies (67%, 95% CI 60–73% vs. 17%, 95% CI 7–27%), with reduced heterogeneity across studies ( I 2 = 43%, P < 0.01 in Cochran’s Q test) . Table 3 Subgroup meta-analyses on prevalence of dumping syndrome after esophageal cancer surgery Subgroups No. studies Pooled prevalence (95% confidence interval) P Heterogeneity I 2 (%) Begg’s test Egger’s test Z P t P Use of professional questionnaires Yes 3 0.67 (0.60–0.73) 0.17 43 0.52 0.6015 0.51 0.7021 No 13 0.17 (0.07–0.27) < 0.01 98 0.49 0.6255 3.25 0.0077 Year of publication Before 2000 8 0.21 (0.08–0.34) < 0.01 97 1.48 0.1376 0.64 0.5486 2000 or later 8 0.33 (0.11–0.55) < 0.01 99 1.73 0.0833 3.64 0.0109 Study population Western 9 0.39 (0.21–0.56) < 0.01 99 1.04 0.2971 1.83 0.1107 Eastern Asian 5 0.05 (0.00–0.10) < 0.01 84 1.47 0.1416 2.68 0.0750 Length of follow-up (months) ≤6 4 0.22 (0.00–0.51) < 0.01 98 2.04 0.0415 6.10 0.0259 ≤12 4 0.41 (0.07–0.75) < 0.01 98 0.68 0.4969 0.29 0.8006 >12 8 0.27 (0.09–0.44) < 0.01 97 0.99 0.3223 0.98 0.3644 Loss to follow-up rate ≤20% 9 0.21 (0.06–0.37) < 0.01 99 0.42 0.6767 2.57 0.0371 >20% 7 0.34 (0.12–0.55) < 0.01 99 1.05 0.2931 2.66 0.0449 Fig. 2 Forest plots for meta-analyses on the prevalence of dumping syndrome after esophageal cancer surgery for in studies using A specialized questionnaires B and not using
|
39068325_p9
|
39068325
|
Subgroup Analyses
| 4.20259 |
biomedical
|
Study
|
[
0.9989250302314758,
0.0007179389940574765,
0.0003570060944184661
] |
[
0.998200535774231,
0.00029921249370090663,
0.001407018513418734,
0.00009320618846686557
] |
en
| 0.999997 |
The pooled prevalence of dumping syndrome in meta-analysis of the eight studies published before 2000 ranged was 21% (95% CI 8–34%), whereas the pooled prevalence in the other eight studies in 2000 or later was 33% (95% CI 11–55%) . Among the 16 included studies, nine studies were from Western populations, 18 , 19 , 22 – 24 , 26 – 28 , 30 five studies from Eastern Asian population, 20 , 21 , 28 , 31 , 32 one study from South Africa, 17 and one study was from Iran. 25 The pooled prevalence of dumping syndrome in studies in Western populations was 39% (95% CI 21–56%) in meta-analysis, which was higher than that in Eastern Asian populations (5%, 95% CI 0–10%) . Two of 15 patients in the study in South Africa had dumping syndrome, whereas 79 out of 192 patients in the study in Iran reported dumping syndrome. The pooled prevalence of dumping syndrome varied by length of follow-up, i.e., 22% (95% CI 0–51%) within 6 months, 41% (95% CI 7–75%) within 6–12 months, and 27% (95% CI 9–44%) in 12 months or longer after esophageal cancer surgery . Meta-analysis of the nine studies with relatively complete follow-up (loss to follow-up ≤ 20%) estimated a prevalence of 21% (95% CI 6–37%) after esophageal cancer surgery. 17 – 19 , 22 , 23 , 25 , 28 , 31 , 32 Of the other seven studies with a loss to follow-up, rates >20% estimated a prevalence of 34% (95% CI 12–55%) . 20 , 21 , 24 , 26 , 27 , 29 , 30 High heterogeneity across studies was indicated in most of the subgroup analyses, and no obvious publication bias was observed except for studies assessing prevalence within 6 months of follow-up (Table 3 ).
|
39068325_p10
|
39068325
|
Subgroup Analyses
| 4.170151 |
biomedical
|
Study
|
[
0.998955488204956,
0.000671931600663811,
0.00037259573582559824
] |
[
0.9965527057647705,
0.00028258541715331376,
0.0030517708510160446,
0.00011291946429992095
] |
en
| 0.999996 |
Sensitivity analyses by omitting one individual study at a time showed no substantial changes in the pooled estimates .
|
39068325_p11
|
39068325
|
Sensitivity Analyses
| 2.691952 |
biomedical
|
Study
|
[
0.9929781556129456,
0.0012661617947742343,
0.005755747202783823
] |
[
0.9890590906143188,
0.006247684359550476,
0.004398873075842857,
0.0002942322171293199
] |
en
| 0.999997 |
This systematic literature review with meta-analysis showed varying prevalence of postoperative dumping syndrome in esophageal cancer patients across previous studies. The pooled prevalence was higher in studies using specialized questionnaires for measuring dumping syndrome with reduced heterogeneity across studies. The prevalence of dumping syndrome also was higher in studies published in 2000 or later than in earlier studies, was higher in Western populations than in Eastern Asian populations, and varied by length and completeness of follow-up.
|
39068325_p12
|
39068325
|
Discussion
| 4.033577 |
biomedical
|
Review
|
[
0.9921432733535767,
0.004532624036073685,
0.003324040211737156
] |
[
0.013519695028662682,
0.0009181982604786754,
0.9851295948028564,
0.0004325089103076607
] |
en
| 0.999998 |
The relatively high prevalence of dumping syndrome and reduced heterogeneity in studies using specialized questionnaires, indicating that the varying prevalence of dumping syndrome across previous studies may be largely due to differences in measuring method of dumping syndrome. Use of unspecific questionnaire or unstandardized “clinical assessment” might have underestimated the occurrence of dumping syndrome in the remaining studies. Therefore, use of specific symptom-based questionnaires, preferably after external validation, should be encouraged in future investigations on dumping syndrome after esophageal cancer surgery. An international consensus on the diagnosis and management of dumping syndrome in 2020 highlighted the need for uniform diagnosis of dumping syndrome, particularly development and validation of specific patient-reported outcome questionnaires. 33 Three of the 16 included studies used questionnaires specifically designed for assessing dumping syndrome, i.e., the Sigstad’s score, the Dumping Syndrome Rating Scale by Laurenius, or the Arts dumping questionnaire. These questionnaires cover a range of symptoms including borborygmus, nausea, headaches, and dizziness. The Sigstad’s score and the Dumping Syndrome Rating Scale were initially developed for assessing dumping syndrome after peptic ulcer surgery and gastric bypass surgery, respectively, and have been applied in patients who have undergone other types of gastroesophageal surgery, such as gastric and esophageal cancer surgery and bariatric surgery. The Sigstad’s score is based on a total of 16 symptoms of dumping syndrome to which different scores are given, and a diagnostic index is calculated according to the total score. It has been proposed that a score > 7 indicates dumping syndrome and a score < 4 suggests diagnoses other than dumping syndrome. 34 However, the diagnostic accuracy of the Sigstad’s score, particularly that for such cutoff points, has not been well validated for patients undergoing cancer surgery. The Dumping Syndrome Rating Scale is a questionnaire based on nine symptoms related to early dumping syndrome: one associated with fluid intake and one related to sweetened drinks intake, generating a summary score. The validity of the Dumping Syndrome Rating Scale has been tested in patients undergoing gastric bypass surgery but remains to be further tested in cancer patients. 35 The Arts dumping questionnaire assesses the severity of eight symptoms of early syndrome and six symptoms of late dumping syndrome, using a 4-point Likert scale where 0 indicates absent, 1 mild, 2 relevant, and 3 indicates severe. However, the Arts dumping questionnaire has not been formally validated. 7
|
39068325_p13
|
39068325
|
Discussion
| 4.219124 |
biomedical
|
Study
|
[
0.9986363053321838,
0.0007509777205996215,
0.000612784584518522
] |
[
0.9113816022872925,
0.0007239251281134784,
0.08757296204566956,
0.00032148160971701145
] |
en
| 0.999997 |
Apart from symptom-based diagnosis, a modified oral glucose tolerance test is believed to be useful in the diagnosis of dumping syndrome. Specifically, according to an international consensus, a 3% increase in hematocrit or an 10 bpm increase in pulse rate in the first 30 minutes after glucose intake suggests early dumping syndrome, and hypoglycemia after 2 to 3 h suggests late dumping syndrome. 5 , 6 Single plasma glucose measurements can be conducted during postoperative clinical visits; however, it has shown limited diagnostic value for dumping syndrome. Nevertheless, continuous monitoring of glucose levels is potentially beneficial for managing complex cases with dumping syndrome. 6
|
39068325_p14
|
39068325
|
Discussion
| 3.912712 |
biomedical
|
Review
|
[
0.9935818910598755,
0.005060428753495216,
0.0013576839119195938
] |
[
0.0215182825922966,
0.030293775722384453,
0.9471124410629272,
0.00107555091381073
] |
en
| 0.999997 |
Surgical factors, such as surgical approach, also may influence the occurrence of dumping syndrome after esophageal cancer surgery. However, only one study compared the prevalence of dumping syndrome in patients who had undergone surgery by different approaches and showed a slightly higher prevalence after open esophagectomy (63%) than minimally invasive surgery (54%) or hybrid thorascopic/laparoscopic surgery (55%). 27 Differences in patients’ characteristics may explain the high heterogeneity across studies to some extent, but this has not been specifically explored in previous studies.
|
39068325_p15
|
39068325
|
Discussion
| 3.937607 |
biomedical
|
Study
|
[
0.9994007349014282,
0.00036566663766279817,
0.00023365221568383276
] |
[
0.9866728782653809,
0.0006181625649333,
0.012564119882881641,
0.00014478577941190451
] |
en
| 0.999998 |
The considerably high prevalence of dumping syndrome after esophageal cancer surgery, particularly as reported in studies using specific symptom-based questionnaires, should draw attention from medical personnel and caregivers. Dumping syndrome is mainly associated with the vagus nerve division inevitably involved in esophagectomy to allow radical tumor resection and may not be prevented in most cases. However, increased awareness of dumping syndrome is needed to help patients understand what their dumping symptoms represent and more readily report to healthcare for interventions, and it also is valuable for healthcare workers to know the considerably high frequency of dumping syndrome when planning follow-up appointments, because severe dumping symptoms need more intensive medical follow-up and care support. Various measures often are required before the optimal interventions are found for each individual patient. Modifications with resting after meals, dietary changes with meal sizes, timing and contents (e.g., less high-sugar foods), and medical interventions often are used in various combinations to manage dumping symptoms. Timely recognition of the occurrence of dumping syndrome, particularly in patients at high risk of severe dumping symptoms, would depend on a better understanding of risk factors for dumping symptoms. Therefore, more well-designed studies are needed to identify the risk factors for dumping syndrome after esophageal cancer surgery, explore the physical and functional reasons behind dumping syndrome, and assess strategies preventing its occurrence and improving management of dumping syndrome.
|
39068325_p16
|
39068325
|
Discussion
| 4.083673 |
biomedical
|
Review
|
[
0.9956530332565308,
0.0029864206444472075,
0.001360580907203257
] |
[
0.06899725645780563,
0.004265766590833664,
0.9260427951812744,
0.0006941857282072306
] |
en
| 0.999996 |
There are several limitations in this systematic review and meta-analysis. First, considering the high heterogeneity across studies as elaborated above, the quantitative synthesize of previous studies, i.e., meta-analysis, should be interpreted with caution and probably for reference only. Second, because of the limited number of included studies, estimates in subgroup analyses were lacking accuracy and stratifications by other factors, e.g., patients’ characteristics was not possible. Third, few studies have provided detailed data on severity or spectrum of specific symptoms, or separated early and late dumping syndrome, for which detailed analysis were not possible. Fourth, the quality of the included studies was fair and has limited the strengths of conclusions to some extent. Finally, we only included articles published in English, although it has been increasingly recognized that restriction to English-language publications has little impact on the conclusions of systematic reviews. 36 , 37
|
39068325_p17
|
39068325
|
Discussion
| 4.056405 |
biomedical
|
Review
|
[
0.9924749732017517,
0.003534498158842325,
0.003990607801824808
] |
[
0.024067848920822144,
0.0011124217417091131,
0.9744189381599426,
0.00040073678246699274
] |
en
| 0.999997 |
This systematic review and meta-analysis comprehensively synthesizing the existing evidence suggests considerably high prevalence of postoperative dumping syndrome in esophageal cancer patients. Previous studies have shown high heterogeneity, which is probably explained by differences in definition and measuring method of dumping syndrome, surgical factors, and patients’ characteristics. More specifically designed studies using validated assessment tools, including specific patient-reported outcome questionnaires, are warranted to investigate the occurrence and risk factors of dumping syndrome after esophageal cancer surgery in the future.
|
39068325_p18
|
39068325
|
Conclusions
| 4.01564 |
biomedical
|
Review
|
[
0.9887210726737976,
0.007020408287644386,
0.004258539993315935
] |
[
0.008204631507396698,
0.0012697090860456228,
0.9899729490280151,
0.0005526239983737469
] |
en
| 0.999998 |
Below is the link to the electronic supplementary material. Supplementary file1 (PDF 894 kb)
|
39068325_p19
|
39068325
|
Supplementary Information
| 0.999671 |
other
|
Other
|
[
0.20396332442760468,
0.0029300705064088106,
0.7931065559387207
] |
[
0.010339726693928242,
0.9878721237182617,
0.0011855725897476077,
0.0006026092451065779
] |
en
| 0.999997 |
Alignment in Biology Alignment of particles in a system is a phenomenon that can be observed in many contexts. In biology, this ranges from the large scale e.g. schools of fish , down to the microscopic scale e.g. cells and bacteria . This alignment of particles, especially when it occurs collectively, can play key roles in their migration e.g. hydrodynamic benefits as a result of collective alignment can aid migration in schools of fish and alignment of fibroblasts can affect key mechanical properties of the tissue in which they are found .
|
39751668_p0
|
39751668
|
Introduction
| 3.663706 |
biomedical
|
Other
|
[
0.9944075345993042,
0.0005856815259903669,
0.005006765481084585
] |
[
0.0884784385561943,
0.883289635181427,
0.02750261127948761,
0.0007293187663890421
] |
en
| 0.999997 |
Motivation and Context of this Work Many collective alignment models include self-propulsion and some kind of overlap avoidance or volume exclusion as model ingredients, suggesting that these are key components needed for collective alignment between particles to occur. This can be understood intuitively, since overlap avoidance provides a way for cells to change their orientation in reaction to other cells while self-propulsion ensures that cells continue to interact with each other, allowing for alignment to propagate through the population. Motivated by experiments on the alignment of fibroblasts in Kenny et al. , an agent-based model was recently developed in Leech et al. to investigate the mechanism behind the collective alignment of self-propelled interacting particles. The model is set in two spatial dimensions and describes a collective of ellipse-shaped cells with fixed area. These cells move in the direction of their orientation and change their position, orientation and shape in order to avoid overlap. In Leech et al. and also in this work, we use the term overlap avoidance rather than volume exclusion to emphasis that overlap is allowed in principle, but punished by a tunable potential. Cell overlap then corresponds to cells being positioned partly on top of each other, an observed phenomena for cells crawling on surfaces . Through computational analysis of the model, it is found that these model components lead to collective alignment, with the amount and spatial scale of the alignment depending on model parameters.
|
39751668_p1
|
39751668
|
Introduction
| 4.226799 |
biomedical
|
Study
|
[
0.9990841150283813,
0.00028509582625702024,
0.0006307522999122739
] |
[
0.9979655742645264,
0.0002372527524130419,
0.0017343114595860243,
0.0000629442511126399
] |
en
| 0.999998 |
Limitations of this Work There are several limitations of this work. Firstly, it does not capture collective effects, however by comparing to Leech et al. we can learn which phenomena are likely consequences of the underlying alignment mechanisms and which are driven by collectivity. Secondly, the symmertry condition imposed in this work limits alignment to “velocity alignment” (where cells move in the same direction) and cannot capture “nematic alignment” (where cells might also move in opposite directions). Also, not all model ingredients of Leech et al. are included in this analysis, most notably we omitted cell-cell adhesions and any cytoskeletal forces transmitted through them (see also the discussion at the end of this work). Finally, the model of Leech et al. itself already omits various biological mechanisms that have been shown to affect alignment, at least in some situations. Examples are feedback with the substrate (Wang et al. ), the effect of a surrounding fluid (Ng and Swartz ) or more complicated cell signalling.
|
39751668_p2
|
39751668
|
Introduction
| 4.013556 |
biomedical
|
Study
|
[
0.9989720582962036,
0.00014383481175173074,
0.000884098291862756
] |
[
0.9981870055198669,
0.0003934498527087271,
0.0013734321109950542,
0.000046159413614077494
] |
en
| 0.999995 |
The Challenge of Analysing Agent-Based Models The most common approach for the analysis of agent-based models tends to be computational (as was the case in Leech et al. ), since there are fewer analytic tools for analysing agent-based models. For an overview of different approaches to modelling and analysing pattern formation, see e.g. Deutsch and Dormann . One method is to take the mean-field limit to obtain an equivalent continuum model, where the positions and orientations of cells are translated into cell density and mean orientation across continuous space . This has been done for the Vicsek model , but is mathematically challenging and poses challenges for discontinuous coefficients, such as those that arise in Leech et al. . Instead of considering the large-cell-number limit, in this work we look in the other direction and consider two interacting cells, with the goal of making analytic progress. Since the agent-based model in Leech et al. requires computing the points of intersection between the boundaries of overlapping ellipses, it is helpful to impose some symmetry in relative ellipse orientation and position to facilitate analysis. Specifically, this allows us to analytically determine the overlap points of the two overlapping ellipses, which allows us to write down explicit governing equations. We are able to make significant analytic progress in understanding the non-trivial dynamic interaction between two interacting cells, and how the different aspects of the model contribute to alignment between two cells.
|
39751668_p3
|
39751668
|
Introduction
| 4.21025 |
biomedical
|
Study
|
[
0.9988601207733154,
0.00024184321227949113,
0.0008980472339317203
] |
[
0.9990866184234619,
0.0002542389265727252,
0.0006162756471894681,
0.00004285710019757971
] |
en
| 0.999995 |
Structure of this Work In this work, we will mathematically analyse the model derived in Leech et al. by considering two interacting cells with some symmetry imposed. We begin by introducing the full model, and then derive the analytical framework which leads to a coupled dynamical system for three time-dependent scalar quantities: distance between the cells, relative cell orientation and cell aspect ratio (Sect. 2 ). Analysis of this three dimensional (in variable space) dynamical system is done in Sect. 3 . We then reduce the system to two dimensions (in variable space) by taking the limit of rigid-cell-shapes (i.e. fixed aspect ratio), which allows us to fully understand the effect of the self-propulsion speed on alignment, as well as to quantify the dependence of alignment strength on various model parameters (Sect. 4 ).
|
39751668_p4
|
39751668
|
Introduction
| 4.131787 |
biomedical
|
Study
|
[
0.9992905855178833,
0.00019891903502866626,
0.0005104988813400269
] |
[
0.9990278482437134,
0.00030256365425884724,
0.0006298137013800442,
0.00003977376036345959
] |
en
| 0.999997 |
In Leech et al. , an energy minimisation approach was used to derive a system of governing equations that describe the behaviour of a collective of self-propelled ellipse-shaped cells, moving in two spatial dimensions, that strive to avoid cell overlap upon interacting with one another. Full details of the derivation and equations for cell collectives can be found in Leech et al. . Here we only summarise the most important aspects and provide equations for the interaction of two cells. Fig. 1 A (Non-dimensional) cell geometry. B Naming of intersection points. C , D Equations and schematic for interaction of two cells for changes in position & orientation ( C ) and aspect ratio ( D )
|
39751668_p5
|
39751668
|
Full Model Summary
| 4.005264 |
biomedical
|
Study
|
[
0.9993095397949219,
0.00012841817806474864,
0.0005620783776976168
] |
[
0.994309663772583,
0.0011263727210462093,
0.004479956813156605,
0.00008390360017074272
] |
en
| 0.999995 |
Description of the Cells In the following all quantities are non-dimensionalised with respect to a reference length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{A/\pi }$$\end{document} A / π and reference time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\eta /\sigma $$\end{document} A η / σ , where A is the cell area, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} η is the strength of friction that a cell experiences with the substrate, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} σ is the strength of overlap avoidance. A cell is then characterised by its (non-dimensional) position \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}\in \mathbb {R}^2$$\end{document} X ∈ R 2 , its orientation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α , and its aspect ratio r , see Fig. 1 A. Each material point inside the elliptic cell is described by the parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in $$\end{document} s ∈ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in [0,2\pi )$$\end{document} θ ∈ [ 0 , 2 π ) by 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{z}(s,\theta )=\textbf{X}+s\textbf{R}(\alpha )\textbf{k}(r,\theta ), \end{aligned}$$\end{document} z ( s , θ ) = X + s R ( α ) k ( r , θ ) , where the rotation matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{R}$$\end{document} R and the shape vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{k}$$\end{document} k are given by 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{R}(\alpha ) = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix},\quad \textbf{k}(r,\theta ) = \begin{pmatrix} \sqrt{r}\cos \theta \\ \frac{1}{\sqrt{r}}\sin \theta \end{pmatrix}. \end{aligned}$$\end{document} R ( α ) = cos α - sin α sin α cos α , k ( r , θ ) = r cos θ 1 r sin θ . Note that this parametrisation leads to an area element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\, d(s,\theta )$$\end{document} s d ( s , θ ) , which is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} θ and r . This implies that changes in aspect ratio do not affect the (assumed) homogeneous density of material points inside the cell.
|
39751668_p6
|
39751668
|
Full Model Summary
| 4.208062 |
biomedical
|
Study
|
[
0.9927892088890076,
0.0002335727185709402,
0.006977164186537266
] |
[
0.9764003753662109,
0.020451989024877548,
0.002986426930874586,
0.00016127899289131165
] |
en
| 0.999997 |
Model Equations To obtain equations that describe how cells change their position, orientation and aspect ratio (while keeping their area constant), we assume their movement minimises a potential, which models friction, overlap avoidance, relaxation to a preferred aspect ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ and self-propulsion. The governing equations for two interacting cells then are 3a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{\textrm{d}\textbf{X}}{\textrm{d}t} = - \sum _{k=1}^K(\textbf{P}_{2k-1} - \textbf{P}_{2k})^{\perp } + \nu \textbf{e}(\alpha ), \end{aligned}$$\end{document} d X d t = - ∑ k = 1 K ( P 2 k - 1 - P 2 k ) ⊥ + ν e ( α ) , 3b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{\textrm{d}\alpha }{\textrm{d}t} = \frac{2r}{r^2 + 1}\sum _{k=1}^K(|\textbf{X} - \textbf{P}_{2k}|^2 - |\textbf{X} - \textbf{P}_{2k-1}|^2), \end{aligned}$$\end{document} d α d t = 2 r r 2 + 1 ∑ k = 1 K ( | X - P 2 k | 2 - | X - P 2 k - 1 | 2 ) , 3c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{\textrm{d}r}{\textrm{d}t} = \frac{4r^2}{r^2 + 1}\sum _{k=1}^K\left( \sin (2\theta _{2k-1}) - \sin (2\theta _{2k})\right) + \frac{16}{\gamma }\frac{\bar{r} r^3+1}{r^2+1}\left( 1 - \frac{r}{\bar{r}}\right) , \end{aligned}$$\end{document} d r d t = 4 r 2 r 2 + 1 ∑ k = 1 K sin ( 2 θ 2 k - 1 ) - sin ( 2 θ 2 k ) + 16 γ r ¯ r 3 + 1 r 2 + 1 1 - r r ¯ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{e}(\alpha ) = (\cos \alpha , \sin \alpha )^T$$\end{document} e ( α ) = ( cos α , sin α ) T , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\perp $$\end{document} ⊥ denotes the left-turned normal vector and 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu =\frac{v\eta }{\sigma }\sqrt{\pi A},\qquad \gamma =\frac{A\sigma }{\pi g}. \end{aligned}$$\end{document} ν = v η σ π A , γ = A σ π g . The non-dimensional quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ defined in ( 4 ) depend on the self-propulsion speed v , the cell area A , the strength of overlap avoidance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} σ , the strength of friction with the substrate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} η , and the strength of relaxation to the preferred aspect ratio g . The quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν can be interpreted as the ratio of the self-propulsion speed to the strength of overlap avoidance in the presence of friction. A larger value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν means a faster self-propulsion, or a smaller overlap avoidance strength. The quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ can be interpreted as the ratio between the strength of overlap avoidance and the strength of shape restoration. A larger value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ means that the strength of relaxation to the preferred aspect ratio g is smaller and hence cell aspect ratios away from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ will be punished less. The pairs of points where the boundaries of the two cells intersect are given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \textbf{P}_{2k-1}, \textbf{P}_{2k}\right) $$\end{document} P 2 k - 1 , P 2 k , with . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=0$$\end{document} K = 0 , 1 or 2 indicates the number of intersection point pairs (the case of one or three intersection points can be reduced to the case of zero and two intersection points respectively). The ordering of intersection points is shown in Fig. 1 B. The angles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} θ j in ( 3c ) correspond to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} θ values that parameterise the intersection points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_j$$\end{document} P j , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_j=\textbf{z}(1,\theta _j)$$\end{document} P j = z ( 1 , θ j ) in ( 1 ).
|
39751668_p7
|
39751668
|
Full Model Summary
| 4.432335 |
biomedical
|
Study
|
[
0.9979591369628906,
0.00031702007981948555,
0.0017238467698916793
] |
[
0.9970506429672241,
0.002070177346467972,
0.0008064154535531998,
0.00007269326306413859
] |
en
| 0.999997 |
Interpretation For ease of interpretation we refer to the case of only one pair of intersection points, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=1$$\end{document} K = 1 . This is the situation depicted in Fig. 1 C, D. From equation ( 3a ), we see that the centre of the cell, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}$$\end{document} X , is being pushed perpendicular to the vector connecting the points of overlap \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_1$$\end{document} P 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_2$$\end{document} P 2 . We can see from ( 3b ) that the cell is rotated by an amount proportional to the difference in the square of the lengths of the lines connecting the cell centre and the intersection points, thus turning the cell in the direction from the shorter to the longer line, Fig. 1 C. The first term in ( 3c ) leads to cells shortening if the cell overlap is near the ends of the cells and lengthening if the overlap is along the sides. This will happen at a faster rate for larger values of r . The final term on the right-hand side of ( 3c ) acts to restore the cell’s aspect ratio to the preferred aspect ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ . In this work we mostly focus on “long” cells ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 ), but will also consider “wide” cells ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 ) in the stability analysis of Sect. 3 .
|
39751668_p8
|
39751668
|
Full Model Summary
| 3.957689 |
biomedical
|
Study
|
[
0.7923469543457031,
0.000563278968911618,
0.20708976686000824
] |
[
0.9935423135757446,
0.005853486713021994,
0.0005272214184515178,
0.00007696317334193736
] |
en
| 0.999997 |
To further analyse and understand system ( 3 ), we impose a symmetry condition on the two interacting cells. This allows us to obtain explicit expressions for the points of intersection, and therefore analytically tractable equations. We consider cell 1 with centre ( x ( t ), y ( t )), orientation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)$$\end{document} α ( t ) and aspect ratio r ( t ) interacting with cell 2 with centre \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x(t),-y(t))$$\end{document} ( x ( t ) , - y ( t ) ) , orientation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\alpha (t)$$\end{document} - α ( t ) , and aspect ratio r ( t ) as shown in Fig. 2 C. Both cells have equal self-propulsion parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν . The points belonging to the two cells are then parameterised as in ( 1 ) by 5a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{z}_1(s,\theta ;t)&= \begin{pmatrix} x(t)\\ y(t) \end{pmatrix} + s \textbf{R}(\alpha (t))\textbf{k}(r(t),\theta ),\quad s \in , \theta \in [0,2\pi ), \end{aligned}$$\end{document} z 1 ( s , θ ; t ) = x ( t ) y ( t ) + s R ( α ( t ) ) k ( r ( t ) , θ ) , s ∈ , θ ∈ [ 0 , 2 π ) , 5b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{z}_2(s,\theta ;t)&= \begin{pmatrix} x(t)\\ -y(t) \end{pmatrix} + s \textbf{R}(-\alpha (t))\textbf{k}(r(t),\theta ), \quad s \in , \theta \in [0,2\pi ). \end{aligned}$$\end{document} z 2 ( s , θ ; t ) = x ( t ) - y ( t ) + s R ( - α ( t ) ) k ( r ( t ) , θ ) , s ∈ , θ ∈ [ 0 , 2 π ) .
|
39751668_p9
|
39751668
|
Symmetric Cells—The Analytical Framework
| 4.241546 |
biomedical
|
Study
|
[
0.9966198205947876,
0.000379828066797927,
0.003000392345711589
] |
[
0.9995070695877075,
0.00024456731625832617,
0.00020961406698916107,
0.00003873892637784593
] |
en
| 0.999994 |
By considering where the cell boundaries \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \mapsto \textbf{z}_1(1,\theta ;t)$$\end{document} θ ↦ z 1 ( 1 , θ ; t ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \mapsto \textbf{z}_2(1,\theta ;t)$$\end{document} θ ↦ z 2 ( 1 , θ ; t ) intersect (neglecting borderline cases), the two cells can have zero, two or four points of intersection. We obtain the following points of intersection (in Cartesian coordinates, relative to the cells’ x -position), see Fig. 2 C: 6a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & {\textbf{P}}^\pm _B =\frac{1}{\gamma _1^2}\left( -y\frac{(r^2-1)\sin \alpha \cos \alpha }{r} \pm \sqrt{\gamma _1^2 - y^2},0\right) , \end{aligned}$$\end{document} P B ± = 1 γ 1 2 - y ( r 2 - 1 ) sin α cos α r ± γ 1 2 - y 2 , 0 , 6b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & {\textbf{P}}^\pm _C =\frac{r}{(r^2-1)\sin \alpha \cos \alpha }\left( -y\gamma _2^2, \pm \sqrt{\left( \frac{r^2-1}{r}\frac{\sin \alpha \cos \alpha }{\gamma _2}\right) ^2-y^2}\right) , \end{aligned}$$\end{document} P C ± = r ( r 2 - 1 ) sin α cos α - y γ 2 2 , ± r 2 - 1 r sin α cos α γ 2 2 - y 2 , where we have defined 7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma _1^2 = r\sin ^2 \alpha +\frac{1}{r} \cos ^2 \alpha , \quad \gamma _2^2 = \frac{1}{r}\sin ^2 \alpha + r\cos ^2 \alpha . \end{aligned}$$\end{document} γ 1 2 = r sin 2 α + 1 r cos 2 α , γ 2 2 = 1 r sin 2 α + r cos 2 α . Given the square roots in ( 6a ) and ( 6b ), these points of intersection only exist in certain regions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y,r)$$\end{document} ( α , y , r ) -space. We denote these regions by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}, \mathcal {B}$$\end{document} A , B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C , see Fig. 2 A, B: In region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A there are no points of intersection, in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B there are two points of intersection, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{P}}^\pm _B$$\end{document} P B ± , and in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C there are four points of intersection, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}^\pm _B$$\end{document} P B ± and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{P}}^\pm _C$$\end{document} P C ± . The boundaries between these regions occur when the square root terms in ( 6a ) and ( 6b ) vanish, which allows us to write these boundaries using the curves 8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma _{\mathcal {A}\mathcal {B}}(\alpha )=\gamma _1(\alpha ),\quad \Gamma _{\mathcal {B}\mathcal {C}}(\alpha )=\frac{(r^2-1)\sin \alpha \cos \alpha }{r\gamma _2(\alpha )}. \end{aligned}$$\end{document} Γ A B ( α ) = γ 1 ( α ) , Γ B C ( α ) = ( r 2 - 1 ) sin α cos α r γ 2 ( α ) . We can now formally define the regions via 9a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathcal {A}=\left\{ (\alpha ,y,r) \mid \Gamma _{\mathcal {A}\mathcal {B}}^2(\alpha )<y^2\right\} , \end{aligned}$$\end{document} A = ( α , y , r ) ∣ Γ A B 2 ( α ) < y 2 , 9b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathcal {B}=\left\{ (\alpha ,y,r) \mid \Gamma _{\mathcal {B}\mathcal {C}}^2(\alpha )<y^2<\Gamma ^2_{\mathcal {A}\mathcal {B}}(\alpha )\right\} , \end{aligned}$$\end{document} B = ( α , y , r ) ∣ Γ B C 2 ( α ) < y 2 < Γ A B 2 ( α ) , 9c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathcal {C}=\left\{ (\alpha ,y,r) \mid y^2<\Gamma _{\mathcal {B}\mathcal {C}}^2(\alpha )\right\} . \end{aligned}$$\end{document} C = ( α , y , r ) ∣ y 2 < Γ B C 2 ( α ) .
|
39751668_p10
|
39751668
|
Symmetric Cells—The Analytical Framework
| 4.339565 |
biomedical
|
Study
|
[
0.9982282519340515,
0.00027105066692456603,
0.0015006553148850799
] |
[
0.9991029500961304,
0.0005969316116534173,
0.00025386750348843634,
0.00004618943057721481
] |
en
| 0.999998 |
With ( 9 ) we can substitute the points of overlap in ( 6a ) and ( 6b ) into the two-cell versions of the full governing equations ( 3 ) to formulate the explicit two-cell governing equations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) \in \mathbb {R} \times \mathbb {R} \times \mathbb {R}^{+}$$\end{document} ( α , y , r ) ∈ R × R × R + with given initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha (0), y(0), r(0)) = (\alpha _0, y_0, r_0)$$\end{document} ( α ( 0 ) , y ( 0 ) , r ( 0 ) ) = ( α 0 , y 0 , r 0 ) . 10a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{x}= & \nu \cos \alpha , \text {for } (\alpha ,y,r)\in \mathcal {A}, \mathcal {B}, \mathcal {C} \end{aligned}$$\end{document} x ˙ = ν cos α , for ( α , y , r ) ∈ A , B , C 10b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{y}= & \nu \sin \alpha + {\left\{ \begin{array}{ll} 0, & \text {for } (\alpha ,y,r)\in \mathcal {A}\\ 2\frac{\sqrt{\gamma _1^2 - y^2}}{\gamma _1^2}\text {sign}(y), & \text {for } (\alpha ,y,r)\in \mathcal {B}\\ 2\frac{r}{\gamma _1^2{|(r^2-1)\sin \alpha \cos \alpha |}}y, & \text {for } (\alpha ,y,r)\in \mathcal {C}\end{array}\right. } \end{aligned}$$\end{document} y ˙ = ν sin α + 0 , for ( α , y , r ) ∈ A 2 γ 1 2 - y 2 γ 1 2 sign ( y ) , for ( α , y , r ) ∈ B 2 r γ 1 2 | ( r 2 - 1 ) sin α cos α | y , for ( α , y , r ) ∈ C 10c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{\alpha }= & {\left\{ \begin{array}{ll} 0, & \text {for } (\alpha ,y,r)\in \mathcal {A}\\ -8\frac{r^2-1}{r^2+1}\sin \alpha \cos \alpha \frac{\sqrt{\gamma _1^2 - y^2}}{\gamma _1^4}|y|, & \text {for } (\alpha ,y,r)\in \mathcal {B}\\ \frac{4r\textrm{sign}({(r-1)}\sin \alpha \cos \alpha )}{r^2+1}\left[ y^2\left( \big (\frac{r^2-1}{r}\big )^2\frac{(\sin \alpha \cos \alpha )^2}{\gamma _1^4}\right. \right. \\ \left. \left. - \big (\frac{r}{r^2-1}\big )^2\frac{(\gamma _2^4 - 1)}{(\sin \alpha \cos \alpha )^2}- \frac{1}{\gamma _1^4} \right) +\frac{\gamma _2^2-\gamma _1^2}{\gamma _1^2\gamma _2^2}\right] , & \text {for } (\alpha ,y,r)\in \mathcal {C}\end{array}\right. } \end{aligned}$$\end{document} α ˙ = 0 , for ( α , y , r ) ∈ A - 8 r 2 - 1 r 2 + 1 sin α cos α γ 1 2 - y 2 γ 1 4 | y | , for ( α , y , r ) ∈ B 4 r sign ( ( r - 1 ) sin α cos α ) r 2 + 1 y 2 ( r 2 - 1 r ) 2 ( sin α cos α ) 2 γ 1 4 - ( r r 2 - 1 ) 2 ( γ 2 4 - 1 ) ( sin α cos α ) 2 - 1 γ 1 4 + γ 2 2 - γ 1 2 γ 1 2 γ 2 2 , for ( α , y , r ) ∈ C 10d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{r}= & - \frac{1}{\gamma }\frac{16( 1 + \bar{r} r^3)}{\bar{r} (r^2 + 1)}(r - \bar{r}) \nonumber \\ & + {\left\{ \begin{array}{ll} 0, & \text {for } (\alpha ,y,r)\in \mathcal {A}\\ \frac{16r}{r^2+1}(\cos ^2\alpha - r^2 \sin ^2 \alpha )\frac{\sqrt{\gamma _1^2 - y^2}}{\gamma _1^4}|y|, & \text {for } (\alpha ,y,r)\in \mathcal {B}\\ 4r \sin \alpha \cos \alpha {\dot{\alpha }} + \frac{8r^2y(\dot{y} - \nu \sin \alpha )}{1+r^2}(\sin ^2\alpha - \cos ^2\alpha ), \hspace{2.5mm} & \quad \text {for } (\alpha ,y,r)\in \mathcal {C}\end{array}\right. } \nonumber \\ \end{aligned}$$\end{document} r ˙ = - 1 γ 16 ( 1 + r ¯ r 3 ) r ¯ ( r 2 + 1 ) ( r - r ¯ ) + 0 , for ( α , y , r ) ∈ A 16 r r 2 + 1 ( cos 2 α - r 2 sin 2 α ) γ 1 2 - y 2 γ 1 4 | y | , for ( α , y , r ) ∈ B 4 r sin α cos α α ˙ + 8 r 2 y ( y ˙ - ν sin α ) 1 + r 2 ( sin 2 α - cos 2 α ) , for ( α , y , r ) ∈ C Fig. 2 A The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y)$$\end{document} ( α , y ) -plane for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = 2$$\end{document} r = 2 with rotational and reflexive symmetries marked. Numbers 1–5 show example cell configurations on region boundaries. Regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C are coloured in purple, green and yellow respectively. B Zoom into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,\pi /2]$$\end{document} α ∈ [ 0 , π / 2 ] with regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C marked and typical cell configurations depicted. C Notation for cell centres, orientation and intersection points
|
39751668_p11
|
39751668
|
Symmetric Cells—The Analytical Framework
| 4.141451 |
biomedical
|
Study
|
[
0.9098660349845886,
0.0005709213437512517,
0.08956298232078552
] |
[
0.9891006350517273,
0.010189143009483814,
0.0006205925601534545,
0.00008970290946308523
] |
en
| 0.999997 |
This is a rather complicated system of coupled non-linear differential equations, however the following sections will show that we can make significant (analytical) progress in understanding its behaviour. We start by discussing the general behaviour of the system ( 10 ) to gain initial insight, before presenting several analytical results in Sects. 3 and 4 . We note that the system is invariant to the transformation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) \rightarrow (-\alpha , -y)$$\end{document} ( α , y ) → ( - α , - y ) , i.e. there is rotational symmetry about the line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = y = 0$$\end{document} α = y = 0 . We also note that the system has reflectional symmetry about \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \pi /2$$\end{document} α = π / 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = -\pi /2$$\end{document} α = - π / 2 , see Fig. 2 A. We can consequently restrict our analysis to the region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,\pi /2]$$\end{document} α ∈ [ 0 , π / 2 ] , see Fig. 2 B, since the results can be extended to the full \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α -range via symmetry arguments. To aid interpretation of the dynamical system, we indicate various physical configurations of the cells in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y)$$\end{document} ( α , y ) -space in Fig. 2 .
|
39751668_p12
|
39751668
|
Symmetric Cells—The Analytical Framework
| 3.946229 |
biomedical
|
Study
|
[
0.7054587602615356,
0.000892848358489573,
0.2936484217643738
] |
[
0.9830467104911804,
0.015288086608052254,
0.0014943155692890286,
0.00017093487258534878
] |
en
| 0.999996 |
Interpretation of Equations With a better understanding of the phase space, and how this corresponds to cell configuration, we now discuss the equations in ( 10 ). Equation ( 10a ) is decoupled from ( 10b ), ( 10c ) and ( 10d ), hence it suffices to analyse the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y,r)$$\end{document} ( α , y , r ) -system. The first terms in ( 10a ) and ( 10b ) represent self-propulsion that leads to movement in the direction of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\cos \alpha , \sin \alpha )$$\end{document} ( cos α , sin α ) . This is proportional to the non-dimensional parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν that has been defined in ( 4 ). Inspecting ( 10b ) further, we note that if cell 1 is lower than cell 2 ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<0$$\end{document} y < 0 ), then the second term in ( 10b ) is negative and cell 1 will move downwards, and vice versa for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y>0$$\end{document} y > 0 . This is a result of overlap avoidance pushing the cells apart. Equation ( 10c ) describes how the cell orientation changes over time. If we consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>1,$$\end{document} r > 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\pi /2)$$\end{document} α ∈ ( 0 , π / 2 ) , we see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\alpha }} <0$$\end{document} α ˙ < 0 in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . This means that overlap avoidance in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B causes a clockwise rotation and drives the system towards \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0$$\end{document} α = 0 , i.e. towards velocity alignment. Conversely, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r<1$$\end{document} r < 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\alpha }} >0$$\end{document} α ˙ > 0 in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and the cells become less aligned. We will revisit this dependence of the behaviour on r when inspecting the stability of steady states, see also Fig. 5 . In region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C , both signs of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\alpha }}$$\end{document} α ˙ are possible. The first term of ( 10d ) describes how the cells will relax back to their preferred aspect ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{r} > 0$$\end{document} r ˙ > 0 if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r < \bar{r}$$\end{document} r < r ¯ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{r} < 0$$\end{document} r ˙ < 0 if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r > \bar{r}$$\end{document} r > r ¯ . We see that in regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C , overlap avoidance causes the aspect ratio r to change. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos ^2\alpha - r^2\sin ^2\alpha > 0$$\end{document} cos 2 α - r 2 sin 2 α > 0 then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{r} > 0$$\end{document} r ˙ > 0 . This will happen for example when the cells are side-by-side ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \approx 0$$\end{document} α ≈ 0 ) and hence strive to increase their aspect ratio (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>1$$\end{document} r > 1 this would mean elongation) to avoid overlap. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos ^2\alpha - r^2\sin ^2\alpha < 0$$\end{document} cos 2 α - r 2 sin 2 α < 0 then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{r} < 0$$\end{document} r ˙ < 0 , which would for example occur when cells are head-to-head ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \approx \pi /2)$$\end{document} α ≈ π / 2 ) . In this case the cells will decrease their aspect ratio (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>1$$\end{document} r > 1 this would correspond to shortening) to avoid overlap. The behaviour in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C is more complex in general.
|
39751668_p13
|
39751668
|
Symmetric Cells—The Analytical Framework
| 4.309606 |
biomedical
|
Study
|
[
0.9980043768882751,
0.00027869699988514185,
0.0017168396152555943
] |
[
0.999173104763031,
0.0004688079352490604,
0.00031463635968975723,
0.000043410771468188614
] |
en
| 0.999996 |
We first explore the full shape-change model ( 10 ) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma > 0$$\end{document} γ > 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = O(1)$$\end{document} ν = O ( 1 ) . This involves explicitly accounting for the restoration time of the aspect ratio r to its preferred value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ can be thought of as the timescale of shape restoration. The full model is a 3D (in variable space) dynamical system where y , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α and r vary in time in response to cell overlap, self-propulsion and shape restoration. Ignoring movement in the x -direction, system ( 10 ) has steady states at the points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, \pm 1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , ± 1 / r ¯ , r ¯ ) . This corresponds to the cells having their preferred aspect ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ and being positioned side-by-side, with their orientations parallel and their boundaries just touching, illustrated in Fig. 2 A, examples 1 and 2. We note that there are additional steady states, but only consider the two listed above in the subsequent analysis since these correspond to cell alignment, which is the focus of this work. To determine the stability of these steady states, we perturb the system around the points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, \pm 1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , ± 1 / r ¯ , r ¯ ) . Importantly, a standard linear stability analysis would not be sufficient to determine their stability. In fact, such an analysis would be degenerate, and determining the stability of these states is non-trivial, as we will see below.
|
39751668_p14
|
39751668
|
Deformable Cells: \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\gamma > 0$$\end{document} γ > 0
| 4.18138 |
biomedical
|
Study
|
[
0.8831051588058472,
0.0006275224732235074,
0.11626728624105453
] |
[
0.9973948001861572,
0.0018388335593044758,
0.0007096037734299898,
0.000056793644034769386
] |
en
| 0.999998 |
We consider the steady point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, -1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , - 1 / r ¯ , r ¯ ) , noting that the analysis of the steady point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, 1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , 1 / r ¯ , r ¯ ) will follow via the rotational symmetry of the system. In order to perform the stability analysis, we perturb the steady point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, -1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , - 1 / r ¯ , r ¯ ) by a small amount (to be defined below) and calculate the subsequent dynamics of the system. This steady point is degenerate, so the scalings of our perturbation and its subsequent dynamics are non-standard.
|
39751668_p15
|
39751668
|
Stability Analysis
| 4.168055 |
biomedical
|
Study
|
[
0.900780439376831,
0.0007124635158106685,
0.09850708395242691
] |
[
0.9967555403709412,
0.0026136634405702353,
0.000565940688829869,
0.00006487409700639546
] |
en
| 0.999997 |
Defining the Perturbations To capture the richest dynamics, we consider the distinguished asymptotic limit in which as many mechanisms as possible balance at the same time. If we define the (small) perturbation in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α to be of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon )$$\end{document} O ( ε ) , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll 1$$\end{document} ε ≪ 1 , then with the benefit of hindsight and justified a posteriori , distinguished asymptotic limits occur when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) and when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(1/\varepsilon )$$\end{document} γ = O ( 1 / ε ) . The former is the physically relevant case, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(1/\varepsilon )$$\end{document} γ = O ( 1 / ε ) would allow for larger deformations to the aspect ratio r , which are not observed biologically. We henceforth focus on the distinguished limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) and therefore scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \varepsilon \Gamma $$\end{document} γ = ε Γ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma = O(1)$$\end{document} Γ = O ( 1 ) . In this case, the appropriate asymptotic scalings for the perturbations (justified a posteriori ) are 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \alpha (t) = \varepsilon A(t), \quad y(t) = -\frac{1}{\sqrt{\bar{r}}} + \varepsilon ^2 B(t), \quad r(t) = \bar{r} + \varepsilon ^2 C(t), \end{aligned}$$\end{document} α ( t ) = ε A ( t ) , y ( t ) = - 1 r ¯ + ε 2 B ( t ) , r ( t ) = r ¯ + ε 2 C ( t ) , where A , B , and C are perturbations in their respective variables, and are functions of time that we will calculate. Understanding their dynamical behaviours will determine the stability of the steady point. To get an idea of the asymptotic structure of the solution before we go into the details, it is helpful to note that there are two distinguished timescales of interest in the system: the ‘early time’ where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) and the ‘late time’ where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) . Over the early timescale, we will show below that A ( t ) remains unchanged, B ( t ) is driven by overlap avoidance and C ( t ) is driven by overlap avoidance and restoration to aspect ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ . Using the early time results, we will then show that over the late timescale, A ( t ) is affected by overlap avoidance and, along with B ( t ) and C ( t ), decays to zero algebraically, demonstrating that the system is stable.
|
39751668_p16
|
39751668
|
Stability Analysis
| 4.206437 |
biomedical
|
Study
|
[
0.9733651876449585,
0.0003867271007038653,
0.026248127222061157
] |
[
0.9965740442276001,
0.0026440422516316175,
0.0007270031492225826,
0.00005485599103849381
] |
en
| 0.999996 |
The Dynamics of the Perturbations We now substitute ( 11 ) into ( 10 ) and note that since the steady state lies at the boundary of region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A and region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B , the perturbations could push the system in either of those two regions. We obtain 12a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\textrm{d}A}{\textrm{d}t} = -8\varepsilon \frac{\bar{r}^2-1}{\bar{r}^2+1}\bar{r}A\sqrt{D}\cdot \mathbb {I}_{D>0} + O(\varepsilon ^3), & A(0) = a, \end{aligned}$$\end{document} d A d t = - 8 ε r ¯ 2 - 1 r ¯ 2 + 1 r ¯ A D · I D > 0 + O ( ε 3 ) , A ( 0 ) = a , 12b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{\textrm{d}B}{\textrm{d}t} = \nu A - 2 \sqrt{\bar{r}} \sqrt{D}\cdot \mathbb {I}_{D>0} + O(\varepsilon ^2), & B(0) = b, \end{aligned}$$\end{document} ε d B d t = ν A - 2 r ¯ D · I D > 0 + O ( ε 2 ) , B ( 0 ) = b , 12c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{\textrm{d}C}{\textrm{d}t} = - \frac{1}{\Gamma }\frac{16(1+\bar{r}^4)}{\bar{r}(1 + \bar{r}^2)}C + \frac{16\bar{r}^2}{1 + \bar{r}^2}\sqrt{D}\cdot \mathbb {I}_{D>0} + O(\varepsilon ^2), & C(0) = c. \end{aligned}$$\end{document} ε d C d t = - 1 Γ 16 ( 1 + r ¯ 4 ) r ¯ ( 1 + r ¯ 2 ) C + 16 r ¯ 2 1 + r ¯ 2 D · I D > 0 + O ( ε 2 ) , C ( 0 ) = c . where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {I}_{D>0}=1$$\end{document} I D > 0 = 1 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D>0$$\end{document} D > 0 and zero otherwise, and D defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D(t):=(\bar{r}^2-1)A^2 + 2\sqrt{\bar{r}}B - \frac{C}{\bar{r}}. \end{aligned}$$\end{document} D ( t ) : = ( r ¯ 2 - 1 ) A 2 + 2 r ¯ B - C r ¯ . The sign of D determines whether we are in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A or region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and hence whether overlap avoidance takes effect. We now analyse the system ( 12 ), starting with the early time. Fig. 3 Early time dynamics: Phase portrait for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\hat{D},\hat{C})$$\end{document} ( D ^ , C ^ ) -system ( 14c ), ( 15 ) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 ( A ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 ( B ). Nullclines are marked in dashed-blue ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\text {d}\hat{D}}{\text {d}\tau }=0$$\end{document} d D ^ d τ = 0 ) and dotted-red ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\text {d}\hat{C}}{\text {d}\tau }=0$$\end{document} d C ^ d τ = 0 ). An example solution trajectory is shown in solid-black. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r} = 2$$\end{document} r ¯ = 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 1.5$$\end{document} ν = 1.5
|
39751668_p17
|
39751668
|
Stability Analysis
| 4.23068 |
biomedical
|
Study
|
[
0.9657825827598572,
0.00045544328168034554,
0.033761974424123764
] |
[
0.9972901344299316,
0.002278780099004507,
0.0003687269927468151,
0.00006237834895728156
] |
en
| 0.999996 |
We start our analysis under the early timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = O(1)$$\end{document} τ = O ( 1 ) , defined via \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = \varepsilon \tau $$\end{document} t = ε τ . We indicate the early timescale variables with overhats, and therefore write 13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(t) = \hat{A}(\tau ), \quad B(t) = \hat{B}(\tau ), \quad C(t) = \hat{C}(\tau ), \quad t = \varepsilon \tau . \end{aligned}$$\end{document} A ( t ) = A ^ ( τ ) , B ( t ) = B ^ ( τ ) , C ( t ) = C ^ ( τ ) , t = ε τ . On substituting ( 13 ) into our governing equations ( 10 ) and taking the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} ε → 0 , we obtain the following leading-order equations. 14a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\textrm{d}\hat{A}}{\textrm{d}\tau } = 0, & \hat{A}(0) = a, \end{aligned}$$\end{document} d A ^ d τ = 0 , A ^ ( 0 ) = a , 14b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\textrm{d}\hat{B}}{\textrm{d}\tau } = \nu \hat{A} - 2\sqrt{\bar{r}}\sqrt{\hat{D}}\cdot \mathbb {I}_{\hat{D}>0}, & \hat{B}(0) = b, \end{aligned}$$\end{document} d B ^ d τ = ν A ^ - 2 r ¯ D ^ · I D ^ > 0 , B ^ ( 0 ) = b , 14c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\textrm{d}\hat{C}}{\textrm{d}\tau } = - \frac{1}{\Gamma } \frac{16(1 + \bar{r}^4)}{\bar{r} (1 + \bar{r}^2)}\hat{C} + \frac{16\bar{r}^2}{1 + \bar{r}^2}\sqrt{\hat{D}}\cdot \mathbb {I}_{\hat{D}>0}, & \hat{C}(0) = c. \end{aligned}$$\end{document} d C ^ d τ = - 1 Γ 16 ( 1 + r ¯ 4 ) r ¯ ( 1 + r ¯ 2 ) C ^ + 16 r ¯ 2 1 + r ¯ 2 D ^ · I D ^ > 0 , C ^ ( 0 ) = c . where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}(\tau )=D(t)$$\end{document} D ^ ( τ ) = D ( t ) .
|
39751668_p18
|
39751668
|
Early Time
| 4.104429 |
biomedical
|
Study
|
[
0.816148042678833,
0.0006413733353838325,
0.1832105666399002
] |
[
0.9929636120796204,
0.006335967220366001,
0.0006131813279353082,
0.00008731847628951073
] |
en
| 0.999997 |
It is straightforward to use ( 14a ) to determine that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{A}(\tau ) = a$$\end{document} A ^ ( τ ) = a over the early time, and hence to deduce that the orientation is not affected over this timescale. The remaining system ( 14b , 14c ) governs the dynamics of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{B}(\tau )$$\end{document} B ^ ( τ ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{C}(\tau )$$\end{document} C ^ ( τ ) , and can be solved computationally. The first term on the right-hand side of ( 14b ) indicates that self-propulsion causes a change in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{B}$$\end{document} B ^ over the early time depending on the sign of a . This will be an increase if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 since this means that the cell is inclined slightly upwards. The second term on the right-hand side of ( 14b ) represents overlap avoidance, suggesting that cells will move apart to avoid overlap, consequently causing a decrease in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{B}$$\end{document} B ^ . The relative size of these two terms will determine whether \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\textrm{d}\hat{B}}{\textrm{d}\tau }$$\end{document} d B ^ d τ is initially positive or negative. The first term on the right-hand side of ( 14c ) leads to a decrease in magnitude of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{C}$$\end{document} C ^ , essentially restoring r to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ . The second term on the right-hand side of ( 14c ) is positive, corresponding to an increase in aspect ratio to avoid overlap. This forcing occurs because the cells are close to a side-by-side configuration near the steady state, and therefore increasing the aspect ratio reduces overlap. We also note that the non-trivial dynamics over this early timescale justify the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} ε scalings we initially imposed in ( 11 ).
|
39751668_p19
|
39751668
|
Early Time
| 4.249667 |
biomedical
|
Study
|
[
0.9324697852134705,
0.0007267207838594913,
0.06680351495742798
] |
[
0.9977262616157532,
0.001719169900752604,
0.0004900790518149734,
0.00006458178540924564
] |
en
| 0.999998 |
Early-Time Overlap Dynamics To determine in which region, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B the early time solution lies, it is useful to inspect the change in time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}$$\end{document} D ^ , since its sign determines whether there is overlap ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}>0$$\end{document} D ^ > 0 ) or not ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}\le 0$$\end{document} D ^ ≤ 0 ). We obtain 15 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}\hat{D}}{\textrm{d}\tau }=2\sqrt{\bar{r}}\nu a-\frac{1}{\bar{r}}\frac{1}{\Gamma } \frac{16(1 + \bar{r}^4)}{\bar{r} (1 + \bar{r}^2)}\hat{C}-4\bar{r} \frac{5+\bar{r}^2}{1+\bar{r}^2}\sqrt{\hat{D}}\cdot \mathbb {I}_{\hat{D}>0}, \end{aligned}$$\end{document} d D ^ d τ = 2 r ¯ ν a - 1 r ¯ 1 Γ 16 ( 1 + r ¯ 4 ) r ¯ ( 1 + r ¯ 2 ) C ^ - 4 r ¯ 5 + r ¯ 2 1 + r ¯ 2 D ^ · I D ^ > 0 , which, together with ( 14c ) forms a nonlinear system of two autonomous ODEs. Phase plane analysis reveals that there is a qualitative difference for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 , see Fig. 3 . We see that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 , i.e. a slightly upward inclined cell 1, the dynamics will lead to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}>0$$\end{document} D ^ > 0 , i.e. we end up in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and the two cells will eventually interact, even if they initially did not. Interestingly it is possible that cells initially interact, then stop interacting for a short time, and then interact again. Such intermediate short non-interaction periods can be caused by the relaxation to the preferred aspect ratio, see example trajectory in Fig. 3 A. If on the other hand \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{D}$$\end{document} D ^ will eventually become negative and cells will stop interacting, irrespective of whether they did so initially. We additionally note that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 it is possible that cells that did not interact initially, then interact for a short amount of time due to shape relaxation, before ceasing to interact again, see example trajectory in Fig. 3 B.
|
39751668_p20
|
39751668
|
Early Time
| 3.810105 |
biomedical
|
Study
|
[
0.7931935787200928,
0.0006023302557878196,
0.2062041014432907
] |
[
0.9759368896484375,
0.023132234811782837,
0.0007766911876387894,
0.00015422790602315217
] |
en
| 0.999996 |
Early-Time Limiting Behaviour For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 , cells will eventually stop to interact and move apart, hence the steady state is unstable for such perturbations, compare Fig. 5 B, example 2. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 , system ( 14 ) becomes independent of early time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} τ in the large- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} τ limit. We can calculate the specific constants to which the solutions tend by setting the left-hand sides of ( 14b , 14c ) to zero and solving the resulting algebraic equations. This procedure yields the following \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow \infty $$\end{document} τ → ∞ results 16 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{A}(\tau ) = a, \quad \hat{B}(\tau ) \rightarrow \frac{\nu ^2 a^2}{8 \bar{r}^{3/2}} - \frac{(\bar{r}^2 - 1)a^2}{2 \sqrt{\bar{r}}} + \frac{\Gamma \nu a \bar{r}}{4(1 + \bar{r}^4)}, \quad \hat{C}(\tau ) \rightarrow \frac{\Gamma \nu a \bar{r}^{5/2}}{2(1 + \bar{r}^4)}. \end{aligned}$$\end{document} A ^ ( τ ) = a , B ^ ( τ ) → ν 2 a 2 8 r ¯ 3 / 2 - ( r ¯ 2 - 1 ) a 2 2 r ¯ + Γ ν a r ¯ 4 ( 1 + r ¯ 4 ) , C ^ ( τ ) → Γ ν a r ¯ 5 / 2 2 ( 1 + r ¯ 4 ) . The early time ‘far-field’ conditions ( 16 ) will be required to asymptotically match into the late-time dynamics we consider next. Note that for these limiting values, the cells are still in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B .
|
39751668_p21
|
39751668
|
Early Time
| 4.274534 |
biomedical
|
Study
|
[
0.9918786287307739,
0.0003581131750252098,
0.00776323489844799
] |
[
0.9988322854042053,
0.0007974985637702048,
0.00032689786166884005,
0.00004334582990850322
] |
en
| 0.999995 |
Since we have already shown instability for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 , we only consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 here. Further, based on the discussion above we can assume that solutions are in overlap region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . System ( 12 ) behaves as the early time far-field ( 16 ) until \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) , when we have a new distinguished timescale that we refer to as the ‘late time’. By inspecting ( 12 ), we note that this is the timescale over which the dynamics of A become relevant. Formally, the late timescale is defined by the new variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = O(1)$$\end{document} T = O ( 1 ) , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = T/\varepsilon $$\end{document} t = T / ε . We retain the perturbation scalings ( 11 ), but now use tildes to denote late-time variables, defining 17 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(t) = \tilde{A}(T), \quad B(t) = \tilde{B}(T), \quad C(t) = \tilde{C}(T), \quad t = \frac{T}{\varepsilon }. \end{aligned}$$\end{document} A ( t ) = A ~ ( T ) , B ( t ) = B ~ ( T ) , C ( t ) = C ~ ( T ) , t = T ε . Over this timescale, in the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} ε → 0 , our governing equations ( 12 ) become 18a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}\tilde{A}}{\textrm{d}T}&= - 8 \frac{\bar{r}^2 - 1}{\bar{r}^2 + 1}\bar{r} \tilde{A} \sqrt{\tilde{D}}, \end{aligned}$$\end{document} d A ~ d T = - 8 r ¯ 2 - 1 r ¯ 2 + 1 r ¯ A ~ D ~ , 18b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= \nu \tilde{A} - 2\sqrt{\bar{r}}\sqrt{\tilde{D}}, \end{aligned}$$\end{document} 0 = ν A ~ - 2 r ¯ D ~ , 18c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= - \dfrac{16(1 + \bar{r}^4)}{\Gamma \bar{r} (1 + \bar{r}^2)}\tilde{C} + \frac{16 \bar{r}^2}{1 + \bar{r}^2}\sqrt{\tilde{D}}, \end{aligned}$$\end{document} 0 = - 16 ( 1 + r ¯ 4 ) Γ r ¯ ( 1 + r ¯ 2 ) C ~ + 16 r ¯ 2 1 + r ¯ 2 D ~ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{D}:=(\bar{r}^2 - 1)\tilde{A}^2 + 2 \sqrt{\bar{r}}\tilde{B} - \frac{\tilde{C}}{\bar{r}}. \end{aligned}$$\end{document} D ~ : = ( r ¯ 2 - 1 ) A ~ 2 + 2 r ¯ B ~ - C ~ r ¯ . The ‘initial’ conditions are obtained by matching with the early-time far-field conditions ( 16 ), 19a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{T\rightarrow 0^+} \tilde{A}(T) = a \end{aligned}$$\end{document} lim T → 0 + A ~ ( T ) = a 19b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{T\rightarrow 0^+} \tilde{B}(T) = \frac{\nu ^2 a^2}{8 \bar{r}^{3/2}} - \frac{(\bar{r}^2 - 1)a^2}{2 \sqrt{\bar{r}}} + \frac{\Gamma \nu a \bar{r}}{4(1 + \bar{r}^4)} \end{aligned}$$\end{document} lim T → 0 + B ~ ( T ) = ν 2 a 2 8 r ¯ 3 / 2 - ( r ¯ 2 - 1 ) a 2 2 r ¯ + Γ ν a r ¯ 4 ( 1 + r ¯ 4 ) 19c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{T\rightarrow 0^+} \tilde{C}(T) = \frac{\Gamma \nu a \bar{r}^{5/2}}{2(1 + \bar{r}^4)} \end{aligned}$$\end{document} lim T → 0 + C ~ ( T ) = Γ ν a r ¯ 5 / 2 2 ( 1 + r ¯ 4 ) Although the differential-algebraic system ( 18 )–( 19 ) is nonlinear, we can simplify the nonlinearity in the differential equation ( 18a ) using the algebraic equation ( 18b ). This procedure reduces ( 18a ) to the following nonlinear but separable differential equation in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{A}$$\end{document} A ~ 20 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}\tilde{A}}{\textrm{d}T}&= - 4\nu \sqrt{\bar{r}} \frac{\bar{r}^2 - 1}{\bar{r}^2 + 1} \tilde{A}^2. \end{aligned}$$\end{document} d A ~ d T = - 4 ν r ¯ r ¯ 2 - 1 r ¯ 2 + 1 A ~ 2 . Solving ( 20 ), and substituting into the remaining algebraic equations ( 18b )–( 18c ), we obtain the late-time solutions: 21a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{A}(T)&= \frac{a}{4a\nu \sqrt{\bar{r}}\frac{\bar{r}^2 - 1}{\bar{r}^2+1}T + 1}, \end{aligned}$$\end{document} A ~ ( T ) = a 4 a ν r ¯ r ¯ 2 - 1 r ¯ 2 + 1 T + 1 , 21b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{B}(T)&= \left( \frac{\nu ^2 + 4\bar{r}(1 - \bar{r}^2)}{8\bar{r}^{3/2}}\right) \tilde{A}^2(T) + \frac{\Gamma \nu \bar{r}}{4 (\bar{r}^4 + 1)}\tilde{A}(T), \end{aligned}$$\end{document} B ~ ( T ) = ν 2 + 4 r ¯ ( 1 - r ¯ 2 ) 8 r ¯ 3 / 2 A ~ 2 ( T ) + Γ ν r ¯ 4 ( r ¯ 4 + 1 ) A ~ ( T ) , 21c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{C}(T)&= \frac{\Gamma \nu \bar{r}^{5/2}}{2(\bar{r}^4 + 1)}\tilde{A}(T). \end{aligned}$$\end{document} C ~ ( T ) = Γ ν r ¯ 5 / 2 2 ( r ¯ 4 + 1 ) A ~ ( T ) . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 , ( 21 ) shows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{A}(T), \tilde{B}(T)$$\end{document} A ~ ( T ) , B ~ ( T ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{C}(T)$$\end{document} C ~ ( T ) all decay to zero algebraically as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \rightarrow \infty $$\end{document} T → ∞ , compare Fig. 5 A, B, example 1. The factor in front of T in ( 21a ) is an increasing function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 ), suggesting that larger preferred aspect ratios will lead to faster decay and hence faster alignment. If on the other hand \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{A}$$\end{document} A ~ will blow up in finite time, indicating an unstable situation, compare Fig. 5 C, D. Fig. 4 A , B Plots of the exact solution of system ( 10 ) and the approximate solution given in ( 22 ) against time until \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=100$$\end{document} t = 100 ( A ) and until \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=6$$\end{document} t = 6 in a log-log plot ( B ), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0.1$$\end{document} ε = 0.1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0.01, 0.1, 1$$\end{document} γ = 0.01 , 0.1 , 1 . Legend applies to all plots. Other parameters: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r} = 2$$\end{document} r ¯ = 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 2$$\end{document} ν = 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a = b = c = 1$$\end{document} a = b = c = 1
|
39751668_p22
|
39751668
|
Late Time
| 3.834469 |
biomedical
|
Study
|
[
0.6333989500999451,
0.0008191344677470624,
0.3657819330692291
] |
[
0.987034261226654,
0.011910238303244114,
0.0009128194651566446,
0.00014263071352615952
] |
en
| 0.999997 |
Combining the solutions ( 14 ) and ( 21 ) in both timescales, we can obtain a uniformly regular (additive) composite asymptotic solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y, \alpha $$\end{document} y , α and r in terms of t . 22a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \alpha (t)&\sim \varepsilon \tilde{A}(\varepsilon t), \end{aligned}$$\end{document} α ( t ) ∼ ε A ~ ( ε t ) , 22b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y(t) + \frac{1}{\sqrt{\bar{r}}}&\sim \varepsilon ^2 \hat{B}\left( \frac{t}{\varepsilon }\right) + \varepsilon ^2\tilde{B}(\varepsilon t) - \varepsilon ^2 \left( \frac{\nu ^2 + 4\bar{r} - 4 \bar{r}^3}{8\bar{r}^{3/2}}\right) a^2 - \varepsilon \frac{\gamma \nu \bar{r} a}{4 (\bar{r}^4 + 1)}, \end{aligned}$$\end{document} y ( t ) + 1 r ¯ ∼ ε 2 B ^ t ε + ε 2 B ~ ( ε t ) - ε 2 ν 2 + 4 r ¯ - 4 r ¯ 3 8 r ¯ 3 / 2 a 2 - ε γ ν r ¯ a 4 ( r ¯ 4 + 1 ) , 22c \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r(t) - \bar{r}&\sim \varepsilon ^2 \hat{C}\left( \frac{t}{\varepsilon }\right) + \varepsilon ^2\tilde{C}(\varepsilon t) - \varepsilon \frac{\gamma \nu \bar{r}^{5/2} a}{2(\bar{r}^4 + 1)}, \end{aligned}$$\end{document} r ( t ) - r ¯ ∼ ε 2 C ^ t ε + ε 2 C ~ ( ε t ) - ε γ ν r ¯ 5 / 2 a 2 ( r ¯ 4 + 1 ) , where the hatted (early-time) variables are defined in ( 14 ) and the tilded (late-time) variables are defined in ( 21 ). In Fig. 4 we verify that the quantities A , B and C all decay to zero as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow \infty $$\end{document} t → ∞ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 , demonstrating that the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y, r) = (0, -\frac{1}{\sqrt{\bar{r}}}, \bar{r})$$\end{document} ( α , y , r ) = ( 0 , - 1 r ¯ , r ¯ ) is stable in this case. On comparing our analytical solution for the stability analysis with the computational full solution, solved using ode15s in MATLAB, we see that we have a good agreement between the two for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon = 0.1$$\end{document} ε = 0.1 and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0.01, 0.1, 1$$\end{document} γ = 0.01 , 0.1 , 1 . The fact that the approximation remains good also for larger values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ is expected from our analysis since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) is a distinguished limit of the system. Hence we expect our analysis to hold in its sublimits until a new distinguished limit is reached. Specifically for this problem, our above analysis allows us to straightforwardly understand the subcases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ll \varepsilon $$\end{document} γ ≪ ε and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll \gamma \ll 1/\varepsilon $$\end{document} ε ≪ γ ≪ 1 / ε as regular sublimits of our analysis. We also note that the solution for A ( t ) is unaffected by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ , suggesting that having shape change in the model does not affect the change in orientation that two cells will experience in response to overlap avoidance. Since a change in orientation is needed for cells to align with each other, this suggests that varying the non-dimensional shape change parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ , at least if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = O(1)$$\end{document} ν = O ( 1 ) , has little effect on cell alignment. Fig. 5 A , C Typical phase portrait (omitting the r -direction) in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y)$$\end{document} ( α , y ) -space for long cells, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 , ( A ) and wide cells, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 , ( B ) with the analysed steady state marked with a blue star. Example trajectories (omitting the r -component) are shown in blue, numbers correspond to plots in B and D. B, D: Cell shapes at various time points corresponding to the trajectories marked in A and C. Other parameters: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =2$$\end{document} ν = 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document} γ = 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}=2$$\end{document} r ¯ = 2 ( A , B ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}=0.5$$\end{document} r ¯ = 0.5 ( C , D )
|
39751668_p23
|
39751668
|
Composite Solution
| 4.206765 |
biomedical
|
Study
|
[
0.9108949303627014,
0.0006053498364053667,
0.0884997546672821
] |
[
0.9950060248374939,
0.004257726948708296,
0.0006596710300073028,
0.00007662691496079788
] |
en
| 0.999998 |
Together we have shown that the steady state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0, -1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( 0 , - 1 / r ¯ , r ¯ ) is half-stable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 : If the point is perturbed in the positive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α -direction, then the perturbation will decay. If it is perturbed in the negative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α -direction, perturbations will grow and cells will eventually stop interacting and move away from each other, see Fig. 5 A,B. We also emphasize that the solutions ( 21 ) yield an algebraic decay for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 , rather than an exponential decay that might arise from a standard linear stability analysis. This a posteriori justifies our claim that a non-standard stability analysis is required to determine the system stability. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 the point is always unstable: For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 cells again eventually stop interacting and move away from each other. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 , we can see that the orientation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α moves away from away from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 0$$\end{document} α = 0 and increases. While we cannot use our approximate solutions to determine the limiting behaviour in this case, numerical results suggest that the solution converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y,r)=\left( \pi /2,-\sqrt{\hat{r}-\frac{\nu ^2 \hat{r}^2}{4}},\hat{r}\right) $$\end{document} ( α , y , r ) = π / 2 , - r ^ - ν 2 r ^ 2 4 , r ^ for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{r}$$\end{document} r ^ , such that the limit is another steady state of ( 10b , 10c ). In this situation the cells face each other and self-propulsion, overlap avoidance and shape relaxation balance such that their distance stays constant, see Fig. 5 C, D. Since this steady state is less biologically relevant, we do not systematically investigate its stability, but we expect it to be stable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 .
|
39751668_p24
|
39751668
|
Summary and Discussion of Stability Results
| 4.151974 |
biomedical
|
Study
|
[
0.9636480212211609,
0.0005278998287394643,
0.03582414612174034
] |
[
0.9973783493041992,
0.002136089140549302,
0.00042659821338020265,
0.000058963621995644644
] |
en
| 0.999998 |
Using the rotational symmetry of our phase space, we obtain that the other steady state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( 0 , 1 / r ¯ , r ¯ ) is also half stable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 and unstable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 .
|
39751668_p25
|
39751668
|
Summary and Discussion of Stability Results
| 2.870954 |
biomedical
|
Study
|
[
0.5436541438102722,
0.001095289015211165,
0.45525065064430237
] |
[
0.8770679831504822,
0.12107162177562714,
0.001324332901276648,
0.0005361359799280763
] |
en
| 0.999997 |
From the above stability analysis, we can understand why \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) is a distinguished asymptotic limit of the system (i.e. a ‘least degenerate’ limit where as many processes as possible occur over the same timescale). Specifically, the natural overlap avoidance timescale is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) , and the aspect ratio restoration timescale is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) . The distinguished limit therefore arises because the timescales of these different processes coincide when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) . Since there is an additional natural timescale in the system of orientation response when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) (the ‘late’ time in our analysis above), we can deduce that there is a different (additional) distinguished limit when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(1/\varepsilon )$$\end{document} γ = O ( 1 / ε ) . This additional distinguished limit would correspond to extremely “squishy” cells where shape change is punished very little, generating cells with aspect ratios far away from the natural shape \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ as a result of overlap avoidance. As this is less biologically relevant, we do not consider this case further.
|
39751668_p26
|
39751668
|
Asymptotic Sublimits
| 4.18469 |
biomedical
|
Study
|
[
0.8524712920188904,
0.0008521481067873538,
0.14667661488056183
] |
[
0.9939953684806824,
0.004683758597820997,
0.0012313415063545108,
0.00008952303323894739
] |
en
| 0.999997 |
Since distinguished limits correspond to maximal interaction of the natural processes, our asymptotic results for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) above will also hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ll \varepsilon $$\end{document} γ ≪ ε and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \gg \varepsilon $$\end{document} γ ≫ ε (i.e. ‘sublimits’ of the distinguished asymptotic limit), until new distinguished limits of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} γ are reached. It is instructive to briefly summarize what happens in the sublimits of the distinguished limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(\varepsilon )$$\end{document} γ = O ( ε ) we analysed above.
|
39751668_p27
|
39751668
|
Asymptotic Sublimits
| 4.078948 |
biomedical
|
Study
|
[
0.7523841857910156,
0.000706513412296772,
0.24690935015678406
] |
[
0.9942029118537903,
0.004785899072885513,
0.0009257091442123055,
0.00008549420454073697
] |
en
| 0.999996 |
Sublimit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll \gamma \ll 1/\varepsilon $$\end{document} ε ≪ γ ≪ 1 / ε In the sublimit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll \gamma \ll 1/\varepsilon $$\end{document} ε ≪ γ ≪ 1 / ε , equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \ll \Gamma \ll 1/\varepsilon ^2$$\end{document} 1 ≪ Γ ≪ 1 / ε 2 (the upper limits of which correspond to the different distinguished limit noted above), the different balancing mechanisms over the early timescale diverge into two separate timescales: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) . Over the standard early timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) , only overlap avoidance affects C . Over this timescale, B and C grow without a restoring force. The timescale over which this restoring force becomes important is then a new intermediate timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) , which kicks in when C gets large enough for the overlap avoidance term to balance the aspect-ratio-restoring term. Over this intermediate timescale, B and C reach the equivalent steady states to those in the full behaviour we determined above. Finally, the late timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) is essentially unchanged from the full analysis in this sublimit; it is this timescale over which A ( t ) varies. The full analysis for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = O(1)$$\end{document} γ = O ( 1 ) is outlined in Appendix A.1 .
|
39751668_p28
|
39751668
|
Asymptotic Sublimits
| 4.239077 |
biomedical
|
Study
|
[
0.8604386448860168,
0.0009042511228471994,
0.13865713775157928
] |
[
0.9930918216705322,
0.005809496156871319,
0.0010049378033727407,
0.00009368819883093238
] |
en
| 0.999997 |
Sublimit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ll \varepsilon $$\end{document} γ ≪ ε The opposite sublimit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ll \varepsilon $$\end{document} γ ≪ ε (equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \ll 1$$\end{document} Γ ≪ 1 ), corresponds to cells that restore their natural aspect ratio very quickly (over a timescale of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) ), and then behave as non-deformable cells with distinct early ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) ) and late ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) ) time behaviours. This sublimit is tractable for a more general dynamic analysis, not just near the steady points, and is of biological interest. Therefore, we examine this case in more detail in Sect. 4 .
|
39751668_p29
|
39751668
|
Asymptotic Sublimits
| 4.374904 |
biomedical
|
Study
|
[
0.9878250956535339,
0.0004197825619485229,
0.011755077168345451
] |
[
0.995585560798645,
0.0033149346709251404,
0.0010350530501455069,
0.00006439701974159107
] |
en
| 0.999994 |
Alignment Stability of “Long” Cells The points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\pm 1/\sqrt{\bar{r}}, \bar{r})$$\end{document} ( 0 , ± 1 / r ¯ , r ¯ ) represent perfect alignment of two cells. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 (“long” cells), the fact that these points are half-stable shows that alignment is sensitive to some perturbations, and suggests that overlap avoidance drives alignment, but that perfect alignment is fleeting in practice. In the many-cell version of the model in Leech et al. , interactions with a third cell could perturb the perfect alignment between two cells in the unstable direction, i.e. such that they point away from each other (corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 in the above analysis), in which case they would cease to interact and alignment would be broken. This could also occur if cell orientation is subject to noise, as would be the case in many realistic biological systems. This could explain why we see pockets of alignment, but no global alignment in the simulations in Leech et al. .
|
39751668_p30
|
39751668
|
Lessons for (Collective) Cell Alignment
| 4.373225 |
biomedical
|
Study
|
[
0.9980998635292053,
0.0002570942451711744,
0.0016430235700681806
] |
[
0.9981474876403809,
0.0008300390327349305,
0.0009554032003507018,
0.00006714588380418718
] |
en
| 0.999996 |
The Role of the Aspect Ratio and Deformability We found that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 the decay to perfect alignment is faster for larger \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ , indicating that larger preferred aspect ratios lead to faster alignment. We will revisit the question of how the strength of alignment depends on the aspect ratio in Sect. 4 . The stability analysis also showed that, near the steady state, the change in orientation is independent of the shape change parameter, suggesting that deformability does not aid alignment. The latter result is in contrast to the collective dynamics results of Leech et al. , where deformability lead to more collective alignment. This indicates that the increased collective alignment due to deformability might be a consequence of the many-cell system, not of the alignment mechanism itself. The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 describes “wide” cells that move orthogonal to their long axis, such as keratocytes (fish fibroblasts). Here the perfect-alignment steady state is always unstable. This could indicate that we do not expect to see velocity alignment in such cells. However, it is also possible that the imposed symmetry condition in this work is not appropriate in this case.
|
39751668_p31
|
39751668
|
Lessons for (Collective) Cell Alignment
| 4.292732 |
biomedical
|
Study
|
[
0.9981057643890381,
0.0003315220819786191,
0.001562766614370048
] |
[
0.9992132186889648,
0.00022514178999699652,
0.0005125683965161443,
0.00004903263834421523
] |
en
| 0.999998 |
In this section, we consider the rigid-cell-shape limit, taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow 0$$\end{document} γ → 0 . Further, based on the findings in Sect. 3 we focus on “long” cells and hence assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}>1$$\end{document} r ¯ > 1 throughout this section. As noted in the previous section, there is a very fast timescale of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) over which the cells restore and then maintain their preferred aspect ratio of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = \bar{r}$$\end{document} r = r ¯ . We will proceed by taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0$$\end{document} γ = 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \equiv \bar{r}$$\end{document} r ≡ r ¯ , essentially analysing ( 10 ) after the very fast \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) timescale. This reduces ( 10 ) to a 2D (in variable space) dynamical system in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α and y , ( 10b ), ( 10c ). This reduction is both biologically relevant and more straightforward to analyse and interpret. In the rest of this section, we will denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}$$\end{document} r ¯ by r for notational convenience, where r no longer has a time dependence. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y)$$\end{document} ( α , y ) state space is split into regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}, \mathcal {B}$$\end{document} A , B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C , see Fig. 2 B. We start our investigation by confirming that we obtain the same stability result from Sect. 3 before exploring the resulting 2D dynamical system computationally. Then, to gain deeper insight into the precise effect of self-propulsion, which has dimensionless strength \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν , we consider the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 before using asymptotic methods to understand the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \ll 1$$\end{document} ν ≪ 1 , noting that the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \rightarrow 0$$\end{document} ν → 0 is singular.
|
39751668_p32
|
39751668
|
Non-deformable Cells, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\gamma = 0$$\end{document} γ = 0 , \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$r>1$$\end{document} r > 1
| 4.313691 |
biomedical
|
Study
|
[
0.9942177534103394,
0.00031207819120027125,
0.005470114294439554
] |
[
0.9985937476158142,
0.0005502784624695778,
0.0008106138557195663,
0.00004541644739219919
] |
en
| 0.999994 |
Stability Analysis For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu > 0$$\end{document} ν > 0 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = O(1)$$\end{document} ν = O ( 1 ) , the stability of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y) = (0, - \frac{1}{\sqrt{r}})$$\end{document} ( α , y ) = ( 0 , - 1 r ) is inherited directly as a sublimit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \rightarrow 0$$\end{document} Γ → 0 of the full system stability analysis we conducted for deformable cells in Sect. 3 above. We assume the initial conditions are such that we stay in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B (see discussion in Sect. 3 ). Specifically for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document} a > 0 , and if we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow 0$$\end{document} γ → 0 in ( 10d ), or equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \rightarrow 0$$\end{document} Γ → 0 in ( 14c ), we find that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t) \rightarrow 0$$\end{document} C ( t ) → 0 over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\gamma )$$\end{document} t = O ( γ ) , which is much quicker compared to the other two timescales of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon )$$\end{document} O ( ε ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\varepsilon )$$\end{document} O ( 1 / ε ) . Then, over the fast timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(\varepsilon )$$\end{document} t = O ( ε ) , ( 14 ) reduces to 23 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{A}(\tau ) = a,\quad \dfrac{\textrm{d}\hat{B}}{\textrm{d} \tau } = \nu a - 2\sqrt{ r}\sqrt{(r^2 - 1)a^2 + 2\sqrt{r}B}, \end{aligned}$$\end{document} A ^ ( τ ) = a , d B ^ d τ = ν a - 2 r ( r 2 - 1 ) a 2 + 2 r B , and over the slow timescale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = O(1/\varepsilon )$$\end{document} t = O ( 1 / ε ) , ( 18 ) becomes 24 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{A}(T) = \frac{a(r^2+1)}{4a\nu \sqrt{r}( r^2 - 1)T + r^2 + 1},\quad \tilde{B}(T) = \left( \frac{\nu ^2 + 4 r - 4 r^3}{8\bar{r}^{3/2}}\right) \tilde{A}^2(T). \end{aligned}$$\end{document} A ~ ( T ) = a ( r 2 + 1 ) 4 a ν r ( r 2 - 1 ) T + r 2 + 1 , B ~ ( T ) = ν 2 + 4 r - 4 r 3 8 r ¯ 3 / 2 A ~ 2 ( T ) . We can see that since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r > 1$$\end{document} r > 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{A}, \bar{B} \rightarrow 0$$\end{document} A ¯ , B ¯ → 0 algebraically as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \rightarrow \infty $$\end{document} T → ∞ . Since perturbations with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document} a < 0 will not decay, we therefore showed that, as for the system for deformable cells, the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0, -\frac{1}{\sqrt{r}})$$\end{document} ( 0 , - 1 r ) is half-stable. Fig. 6 A Phase portrait for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 2$$\end{document} ν = 2 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = 2$$\end{document} r = 2 showing nullclines (dotted pink and green), boundaries between regions (red), the separatrix (solid-blue) and example trajectories (dashed-blue). B Cell shape snapshots of the system over time for two example trajectories labelled (1) and (2) in A with a 3D side view included for (2). Note that which cell is above and which is below is arbitrary. The cell in blue corresponds to the trajectory in blue
|
39751668_p33
|
39751668
|
General Behaviour
| 4.124862 |
biomedical
|
Study
|
[
0.8737888336181641,
0.000663142476696521,
0.12554804980754852
] |
[
0.9929836988449097,
0.00637006713077426,
0.0005731056444346905,
0.00007305153121706098
] |
en
| 0.999997 |
Phase Portrait Next we explore the system more generally by computationally generating a phase portrait with some example trajectories in Fig. 6 . Nullclines corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y} = 0$$\end{document} y ˙ = 0 (pink) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\alpha }} = 0$$\end{document} α ˙ = 0 (green) are shown as dotted lines. We see that in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A (purple) the arrows all point vertically upwards. This is because this is the region of no overlap and hence the cells will only change their position (as a result of self-propulsion) and not their orientation. The arrows are pointed upwards since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\pi /2)$$\end{document} α ∈ ( 0 , π / 2 ) and hence this leads to an increase in y since the cell is inclined upwards. If we focus our attention on the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -1/\sqrt{r})$$\end{document} ( α , y ) = ( 0 , - 1 / r ) , we see that all arrows in the surrounding area are directed towards this point, illustrating that this point is an attractor from the side where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} α > 0 . Example trajectories are shown in blue and these indicate that there are two outcomes depending on the initial conditions: (1) the trajectory moves towards the half-stable point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -\frac{1}{\sqrt{r}})$$\end{document} ( α , y ) = ( 0 , - 1 r ) , or (2) the trajectory crosses region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C and ends up leaving the overlap region with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y>0$$\end{document} y > 0 . This is a situation where cells crawl over each other, then stop interacting and continue to move away from each other. The line separating these two outcomes is a non-trivial separatrix, indicated by the solid blue line in Fig. 6 B. We can calculate this separatrix computationally, by starting near the separatrix and solving the system ( 10 ) backwards in time .
|
39751668_p34
|
39751668
|
General Behaviour
| 4.161408 |
biomedical
|
Study
|
[
0.9967413544654846,
0.00026175720267929137,
0.002996895695105195
] |
[
0.9992855191230774,
0.00046543203643523157,
0.00021463190205395222,
0.00003443768946453929
] |
en
| 0.999996 |
Quantification of Alignment Next, we want to quantify how favourable different sets of model parameters are for alignment. To do this we test initial conditions with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\sqrt{r}<y_0<0$$\end{document} - r < y 0 < 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha _0<\pi /2$$\end{document} 0 < α 0 < π / 2 (i.e. cell 1 underneath cell 2 and cells moving towards each other) and record the resulting post-interaction angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\text {final}(\alpha _0,y_0)$$\end{document} α final ( α 0 , y 0 ) . This will either be the angle obtained once the interaction has ended after finite time, or the limiting angle for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \infty $$\end{document} t → ∞ , in the case of infinitely long interactions. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\text {final}(\alpha _0,y_0)$$\end{document} α final ( α 0 , y 0 ) will lie between 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi /2$$\end{document} π / 2 , and 0 corresponds to perfect velocity alignment, we define the interaction strength for a particular parameter set by 25 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_\text {align}=\frac{1}{\pi /2\sqrt{r}}\int _{-\sqrt{r}}^0\int _0^{\pi /2}\!\frac{\pi /2-\alpha _\text {final}(\alpha _0,y_0)}{\pi /2}\,\text {d}\alpha _0\text {d}y_0. \end{aligned}$$\end{document} S align = 1 π / 2 r ∫ - r 0 ∫ 0 π / 2 π / 2 - α final ( α 0 , y 0 ) π / 2 d α 0 d y 0 . The alignment strength will be between 0 and 1, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\text {align}=1$$\end{document} S align = 1 corresponding to perfect alignment for all used initial conditions. We determined \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\text {align}$$\end{document} S align computationally by discretising the integral. Figure 7 A shows a non-monotonous dependence of alignment strength on the self-propulsion speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν : For non-zero speeds, larger speeds lead to less alignment. Figure 7 E, F suggest this is because for larger \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν fewer initial conditions lead to cells getting trapped in the region underneath the separatrix, where the interaction leads to perfect alignment. This is because larger speeds will more often allow cells to leave the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<0$$\end{document} y < 0 region, where they will stop interacting in finite time. However \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 is not favourable for alignment. We will discuss this case in Sect. 4.2 . The dependence of alignment strength on the aspect ratio r is also non-monotonic. The general picture is that both too round and too elongated cells shapes lead to less alignment. This can be understood by comparing Figs. 7 C, F. On the one hand, larger aspect ratios lead to fewer initial conditions being in the perfect-alignment region underneath the separatrix. On the other hand, larger aspect ratios cause more post-interaction alignment (smaller values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\text {final}(\alpha _0,y_0)$$\end{document} α final ( α 0 , y 0 ) ) for cells outside the perfect-alignment region. Fig. 7 A , B Alignment strength as defined in ( 25 ) as functions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν ( A ) and r ( B ). C – F Examples of interaction outcomes for different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν and r . Colour shows angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α after interaction has ended for the initial conditions at that position. Region boundaries are shown in red, the separatrix in white
|
39751668_p35
|
39751668
|
General Behaviour
| 4.214608 |
biomedical
|
Study
|
[
0.9968382120132446,
0.0003078082809224725,
0.002854008227586746
] |
[
0.9993637204170227,
0.00041503726970404387,
0.00018767327128443867,
0.0000335618169629015
] |
en
| 0.999995 |
To investigate the effect of propulsion on ( 10 ) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0$$\end{document} γ = 0 , we now analyse the singular change in the system in the limit of small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν . Specifically, we are interested in the strong difference in system behaviour for zero self-propulsion ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 ) and small self-propulsion ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \nu \ll 1$$\end{document} 0 < ν ≪ 1 ) suggested by Fig. 7 A, D. We start by considering zero self-propulsion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 . The governing equations ( 10 ) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 are 26a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \dot{y} = {\left\{ \begin{array}{ll} 0, & \text {for } (\alpha ,y)\in \mathcal {A}\\ 2\frac{\sqrt{\gamma _1^2 - y^2}}{\gamma _1^2}\text {sign}(y), & \text {for } (\alpha ,y)\in \mathcal {B}\\ 2\frac{r}{\gamma _1^2|(r^2-1)\sin \alpha \cos \alpha |}y, \hspace{8.1cm} & \text {for } (\alpha ,y)\in \mathcal {C}\end{array}\right. } \end{aligned}$$\end{document} y ˙ = 0 , for ( α , y ) ∈ A 2 γ 1 2 - y 2 γ 1 2 sign ( y ) , for ( α , y ) ∈ B 2 r γ 1 2 | ( r 2 - 1 ) sin α cos α | y , for ( α , y ) ∈ C 26b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \dot{\alpha } = {\left\{ \begin{array}{ll} 0, & \text {for } (\alpha ,y)\in \mathcal {A}\\ -8\frac{r^2-1}{r^2+1}\sin \alpha \cos \alpha \frac{\sqrt{\gamma _1^2 - y^2}}{\gamma _1^4}|y|, & \text {for } (\alpha ,y)\in \mathcal {B}\\ \frac{4r\textrm{sign}((r-1)\sin \alpha \cos \alpha )}{r^2+1}\\ \left[ y^2\left( \big (\frac{r^2-1}{r}\big )^2\frac{(\sin \alpha \cos \alpha )^2}{\gamma _1^4} - \big (\frac{r}{r^2-1}\big )^2\frac{(\gamma _2^4 - 1)}{(\sin \alpha \cos \alpha )^2}- \frac{1}{\gamma _1^4} \right) +\frac{\gamma _2^2-\gamma _1^2}{\gamma _1^2\gamma _2^2}\right] , & \text {for } (\alpha ,y)\in \mathcal {C}\end{array}\right. } \nonumber \\ \end{aligned}$$\end{document} α ˙ = 0 , for ( α , y ) ∈ A - 8 r 2 - 1 r 2 + 1 sin α cos α γ 1 2 - y 2 γ 1 4 | y | , for ( α , y ) ∈ B 4 r sign ( ( r - 1 ) sin α cos α ) r 2 + 1 y 2 ( r 2 - 1 r ) 2 ( sin α cos α ) 2 γ 1 4 - ( r r 2 - 1 ) 2 ( γ 2 4 - 1 ) ( sin α cos α ) 2 - 1 γ 1 4 + γ 2 2 - γ 1 2 γ 1 2 γ 2 2 , for ( α , y ) ∈ C Fig. 8 A Phase portrait in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , y)$$\end{document} ( α , y ) -space for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2$$\end{document} r = 2 . Example trajectories shown in blue. Boundaries between regions shown in red. Nullclines shown in green. B Snapshots of the cell configurations over time corresponding to the trajectory in A marked with (1) at initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0, y_0)$$\end{document} ( α 0 , y 0 )
|
39751668_p36
|
39751668
|
No Self-propulsion, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\nu = 0$$\end{document} ν = 0
| 3.997224 |
biomedical
|
Study
|
[
0.6672222018241882,
0.0008447192958556116,
0.33193305134773254
] |
[
0.9939056038856506,
0.005142126698046923,
0.0008437926298938692,
0.00010846286750165746
] |
en
| 0.999995 |
Summary of System Behaviour System ( 26 ) is symmetric around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} y = 0 . Together with the symmetry considerations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α , it therefore suffices to consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\le 0$$\end{document} y ≤ 0 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,\pi /2]$$\end{document} α ∈ [ 0 , π / 2 ] . We inspect the corresponding phase portrait in Fig. 8 . We find that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\dot{\alpha }},\dot{y})\equiv (0,0)$$\end{document} ( α ˙ , y ˙ ) ≡ ( 0 , 0 ) in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A , since in the absence of self-propulsion and shape relaxation, overlap avoidance is the only driver of change/movement. Throughout region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y}<0$$\end{document} y ˙ < 0 , indicating cells move apart as long as they interact. Further the boundary between region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B , given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=-\Gamma _{\mathcal{A}\mathcal{B}}=-\gamma _1$$\end{document} y = - Γ A B = - γ 1 is now a line of stationary points. Together, this suggests the following solution behaviour, which we will prove in the next paragraph. If we let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha (0),y(0))=(\alpha _0,y_0)$$\end{document} ( α ( 0 ) , y ( 0 ) ) = ( α 0 , y 0 ) be the initial conditions and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_0<0$$\end{document} y 0 < 0 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha _0<\pi /2$$\end{document} 0 < α 0 < π / 2 , we find that: i. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) lies in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A (no interaction), the solution will remain at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0, y_0)$$\end{document} ( α 0 , y 0 ) for all time. ii. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) lies in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C (four intersection points), it will move into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B in finite time and remain in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . iii. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) lies in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B (two intersection points) it will converge to a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _T,y_T)$$\end{document} ( α T , y T ) on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B boundary in finite time, see trajectory in Fig. 8 A, B. Proof of System Behaviour
|
39751668_p37
|
39751668
|
No Self-propulsion, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\nu = 0$$\end{document} ν = 0
| 3.751432 |
biomedical
|
Study
|
[
0.7012498378753662,
0.0007018090691417456,
0.2980484068393707
] |
[
0.9477418661117554,
0.049324944615364075,
0.0026860276702791452,
0.00024728142307139933
] |
en
| 0.999995 |
Part (i) follows trivially from the dynamical system ( 26 ). Part (ii): From ( 26a ) we see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y} <0$$\end{document} y ˙ < 0 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<0$$\end{document} y < 0 within region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C and on the boundary of region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C . We also have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\dot{y}| \ge 4 |y|/(r^2--1)$$\end{document} | y ˙ | ≥ 4 | y | / ( r 2 - - 1 ) which shows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\dot{y}|$$\end{document} | y ˙ | is bounded away from zero (note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>1$$\end{document} r > 1 ). Hence, we can conclude that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0, y_0) \in \mathcal {C}$$\end{document} ( α 0 , y 0 ) ∈ C then there exists a finite time, at which the trajectory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha (t), y(t))$$\end{document} ( α ( t ) , y ( t ) ) will cross the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C boundary and cross into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . Next we claim that region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B is invariant, i.e. that once a trajectory is in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B , it remains in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . To show that this is the case, we characterise the solution curves in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . We consider the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y)$$\end{document} ( α , y ) are in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and derive an equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\alpha )$$\end{document} y ( α ) . Dividing ( 26b ) by ( 26a ) we obtain 27 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}y}{\textrm{d}\alpha } = -\frac{1}{4r}\frac{r^2 + 1}{r^2 - 1}\frac{r^2\sin ^2\alpha + \cos ^2\alpha }{r \sin \alpha \cos \alpha }\frac{1}{y}. \end{aligned}$$\end{document} d y d α = - 1 4 r r 2 + 1 r 2 - 1 r 2 sin 2 α + cos 2 α r sin α cos α 1 y . Equation ( 27 ) can be solved explicitly to yield 28 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y(\alpha ) = -\sqrt{|y_0|^2 + \frac{1}{2r}\frac{r^2 + 1}{r^2-1}\log \left( \frac{(\cos \alpha )^{r^2}}{\sin \alpha }\frac{\sin \alpha _0}{(\cos \alpha _0)^{r^2}}\right) }. \end{aligned}$$\end{document} y ( α ) = - | y 0 | 2 + 1 2 r r 2 + 1 r 2 - 1 log ( cos α ) 2 sin α sin α 0 ( cos α 0 ) 2 . To show that region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B is invariant, we will show that once the trajectory moves from region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B , it is not possible for the trajectory to move back into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C again. To do this, we will show that if we start on the boundary of region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C , given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2 = \Gamma _{\mathcal {B}\mathcal {C}}^2$$\end{document} y 2 = Γ B C 2 as defined in ( 8 ), we will enter region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and will not intersect the line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2 = \Gamma _{\mathcal {B}\mathcal {C}}^2$$\end{document} y 2 = Γ B C 2 again, showing that it is not possible for the trajectory to cross this line a second time and enter back into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C . We substitute our initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_0^2 = \Gamma _{\mathcal {B}\mathcal {C}}^2(\alpha _0)$$\end{document} y 0 2 = Γ B C 2 ( α 0 ) into ( 28 ) and want to solve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2 = \Gamma _{\mathcal {B}\mathcal {C}}^2$$\end{document} y 2 = Γ B C 2 to see whether the trajectory y could intersect the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathcal {B}\mathcal {C}}$$\end{document} Γ B C again. If there are no additional solutions to this equation, then the trajectory does not cross the boundary between regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C again and hence does not re-enter region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C . From this procedure we obtain 29 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma _{\mathcal {B}\mathcal {C}}^2(\alpha ) = \Gamma _{\mathcal {B}\mathcal {C}}^2(\alpha _0) + \frac{1}{2r}\frac{r^2 + 1}{r^2-1}\log \left( \frac{(\cos \alpha )^{r^2}}{\sin \alpha }\frac{\sin \alpha _0}{(\cos \alpha _0)^{r^2}}\right) , \end{aligned}$$\end{document} Γ B C 2 ( α ) = Γ B C 2 ( α 0 ) + 1 2 r r 2 + 1 r 2 - 1 log ( cos α ) 2 sin α sin α 0 ( cos α 0 ) 2 , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathcal {B}\mathcal {C}}$$\end{document} Γ B C is defined in ( 8 ). Rearranging ( 29 ) and defining \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = \sin \alpha $$\end{document} s = sin α , we obtain 30 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{s}{(1-s^2)^{r^2/2}}\exp \left[ \frac{2(r^2-1)^3s^2(1-s^2)}{(r^2+1)(s^2 + r^2(1-s^2))}\right] \nonumber \\&\quad =\frac{\sin {\alpha _0}}{(\cos {\alpha _0})^{r^2}}\exp \left[ \frac{2(r^2-1)^3\sin ^2\alpha _0\cos ^2\alpha _0}{(r^2+1)(\sin ^2\alpha _0 + r^2\cos ^2\alpha _0)}\right] . \end{aligned}$$\end{document} s ( 1 - s 2 ) r 2 / 2 exp 2 ( r 2 - 1 ) 3 s 2 ( 1 - s 2 ) ( r 2 + 1 ) ( s 2 + r 2 ( 1 - s 2 ) ) = sin α 0 ( cos α 0 ) 2 exp 2 ( r 2 - 1 ) 3 sin 2 α 0 cos 2 α 0 ( r 2 + 1 ) ( sin 2 α 0 + r 2 cos 2 α 0 ) . We define the left-hand side of ( 30 ) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\mapsto f(s)$$\end{document} s ↦ f ( s ) with domain [0, 1). Differentiating f with respect to s shows that f is a strictly monotonically increasing function of s with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0) = 0$$\end{document} f ( 0 ) = 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(s) \rightarrow \infty $$\end{document} f ( s ) → ∞ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \rightarrow 1^{-}$$\end{document} s → 1 - . Since the right-hand side is always positive, there exists a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{sol}\in [0,1)$$\end{document} s sol ∈ [ 0 , 1 ) for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) and consequently a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{sol}=\arcsin {s_{sol}}\in [0,\pi /2]$$\end{document} α sol = arcsin s sol ∈ [ 0 , π / 2 ] such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\alpha _{sol})=-\Gamma _{\mathcal {B}\mathcal {C}}(\alpha _{sol})$$\end{document} y ( α sol ) = - Γ B C ( α sol ) . However, since we know that the initial condition satisfies these requirements, the unique solution is the initial condition i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{sol} = \alpha _0$$\end{document} α sol = α 0 . Therefore, after intersecting the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2 = \Gamma _{\mathcal {B}\mathcal {C}}^2$$\end{document} y 2 = Γ B C 2 and moving into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B , the trajectory will not intersect this boundary curve again. Hence, once the trajectory is in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B it will remain in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B .
|
39751668_p38
|
39751668
|
No Self-propulsion, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\nu = 0$$\end{document} ν = 0
| 3.375458 |
biomedical
|
Study
|
[
0.5336744785308838,
0.0010314880637452006,
0.4652939736843109
] |
[
0.9421981573104858,
0.05546269938349724,
0.001964287366718054,
0.00037481336039491
] |
en
| 0.999997 |
For part (iii), we start by noting that the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=-\Gamma _{AB}$$\end{document} y = - Γ AB describes a line of stationary points (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y} = {\dot{\alpha }} = 0$$\end{document} y ˙ = α ˙ = 0 ). Showing that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\alpha )$$\end{document} y ( α ) always intersects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Gamma _{\mathcal {A}\mathcal {B}}(\alpha )$$\end{document} - Γ A B ( α ) is equivalent to showing that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2(\alpha ) = \Gamma ^2_{\mathcal {A}\mathcal {B}}(\alpha )$$\end{document} y 2 ( α ) = Γ A B 2 ( α ) has a solution. Using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\alpha )$$\end{document} y ( α ) as defined in ( 28 ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathcal {A}\mathcal {B}}$$\end{document} Γ A B as defined in ( 8 ) this can be written as 31 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&y_0^2 + \frac{1}{2}\frac{r^2 + 1}{r^2-1}\log \left( \frac{(\cos \alpha )^{r^2}}{\sin \alpha }\frac{\sin \alpha _0}{(\cos \alpha _0)^{r^2}}\right) =r^2\sin {\alpha }^2+\cos {\alpha }^2. \end{aligned}$$\end{document} y 0 2 + 1 2 r 2 + 1 r 2 - 1 log ( cos α ) 2 sin α sin α 0 ( cos α 0 ) 2 = r 2 sin α 2 + cos α 2 . We need to show that ( 31 ) has a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,\pi /2]$$\end{document} α ∈ [ 0 , π / 2 ] . We define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\sin {\alpha }$$\end{document} s = sin α and use this to rewrite ( 31 ) as 32 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{s}{(1-s^2)^{r^2/2}}\exp \left[ 2\frac{(r^2-1)^2}{r^2+1}s^2\right] =\frac{\sin {\alpha _0}}{\cos {\alpha _0}^{r^2}}\exp \left[ 2\frac{r^2-1}{r^2+1}(y_0^2-1)\right] . \end{aligned}$$\end{document} s ( 1 - s 2 ) r 2 / 2 exp 2 ( r 2 - 1 ) 2 r 2 + 1 s 2 = sin α 0 cos α 0 2 exp 2 r 2 - 1 r 2 + 1 ( y 0 2 - 1 ) . We define the left-hand side as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\mapsto h(s)$$\end{document} s ↦ h ( s ) with domain [0, 1). Differentiating h ( s ) we find that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h'(s) > 0$$\end{document} h ′ ( s ) > 0 and can therefore conclude that h is a strictly monotonically increasing function of s with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(0)=0$$\end{document} h ( 0 ) = 0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(s)\rightarrow \infty $$\end{document} h ( s ) → ∞ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\rightarrow 1^{-}$$\end{document} s → 1 - , i.e. its range is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document} [ 0 , ∞ ) . Since the right-hand side is always positive, there exists a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_T\in [0,1)$$\end{document} s T ∈ [ 0 , 1 ) for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _0,y_0)$$\end{document} ( α 0 , y 0 ) and consequently a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _T=\arcsin {s_T}\in [0,\pi /2]$$\end{document} α T = arcsin s T ∈ [ 0 , π / 2 ] such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\alpha _T)=-\Gamma _{\mathcal {A}\mathcal {B}}(\alpha _T)$$\end{document} y ( α T ) = - Γ A B ( α T ) . Finally, we need to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _T,y_T)$$\end{document} ( α T , y T ) is reached in finite time, where we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_T:=y(\alpha _T)$$\end{document} y T : = y ( α T ) . To show this, we substitute ( 28 ) into ( 26b ), the governing equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\alpha }}$$\end{document} α ˙ , separate variables and integrate over time from 0 to t , obtaining 33 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 8t\frac{r^2-1}{r^2+1} = \int _{\alpha _0}^{\alpha (t)}\frac{\gamma _1^4(\alpha )}{y(\alpha )\sin \alpha \cos \alpha \sqrt{\gamma _1^2(\alpha ) - y^2(\alpha )}}\,\textrm{d}\alpha =:I(\alpha (t)). \end{aligned}$$\end{document} 8 t r 2 - 1 r 2 + 1 = ∫ α 0 α ( t ) γ 1 4 ( α ) y ( α ) sin α cos α γ 1 2 ( α ) - y 2 ( α ) d α = : I ( α ( t ) ) . Our final task is to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\alpha \rightarrow \alpha _T^+}I(\alpha )=:I(\alpha _T)<\infty $$\end{document} lim α → α T + I ( α ) = : I ( α T ) < ∞ , which will imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _T,y_T)$$\end{document} ( α T , y T ) is reached in finite time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=I(\alpha _T)\frac{r^2+1}{r^2-1}\frac{1}{8r\sqrt{r}}$$\end{document} T = I ( α T ) r 2 + 1 r 2 - 1 1 8 r r . We use the substitution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\sin {\alpha }$$\end{document} s = sin α in ( 33 ) and obtain 34 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I(\alpha _T) = \lim _{s_t \rightarrow s_T^+}\int _{s_0}^{s_t}\frac{\gamma _1^4(s)}{y(s)s(1-s^2)\sqrt{\gamma _1^2(s) - y^2(s)}}\,\textrm{d}s. \end{aligned}$$\end{document} I ( α T ) = lim s t → s T + ∫ s 0 t γ 1 4 ( s ) y ( s ) s ( 1 - s 2 ) γ 1 2 ( s ) - y 2 ( s ) d s . To understand whether ( 34 ) converges, we need to understand the nature of the singularity at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = s_T$$\end{document} s = s T where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^2(s_T) - y^2(s_T) = 0$$\end{document} γ 1 2 ( s T ) - y 2 ( s T ) = 0 . Taylor expanding this term around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = s_T$$\end{document} s = s T gives 35 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma _1^2( s) - y^2(s)= (s-s_T)\left[ 2s_T\frac{r^2-1}{r} + \frac{(r^2+1)(s_T^2(r^2-1) + 1)}{2r(r^2-1)s_T (1-s_T^2)}\right] +\mathcal {O}(s-s_T)^2. \end{aligned}$$\end{document} γ 1 2 ( s ) - y 2 ( s ) = ( s - s T ) 2 s T r 2 - 1 r + ( r 2 + 1 ) ( s T 2 ( r 2 - 1 ) + 1 ) 2 r ( r 2 - 1 ) s T ( 1 - s T 2 ) + O ( s - s T ) 2 . Since the coefficient of the linear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s-s_T$$\end{document} s - s T term in ( 35 ) does not vanish, the singularity in the integrand is an inverse square root, and therefore is integrable. That is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(\alpha _T)<\infty $$\end{document} I ( α T ) < ∞ and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha (t),y(t)) =(\alpha _T,y_T)$$\end{document} ( α ( t ) , y ( t ) ) = ( α T , y T ) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = T<\infty $$\end{document} t = T < ∞ . This explains the computational results shown in Fig. 7 A, D, i.e. that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 perfect alignment is typically not achieved. Fig. 9 A , B Comparison of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t)$$\end{document} α ( t ) ( A ) and y ( t ) ( B ) for solutions of the full system (solid-back), solutions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 (dashed-red) and the small- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν approximation ( 43 ) (dotted-pink). Insets show dynamic for t small. C Zoomed in region of phase portrait showing corresponding trajectory for A and B. Parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0.1$$\end{document} ν = 0.1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2$$\end{document} r = 2
|
39751668_p39
|
39751668
|
No Self-propulsion, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\nu = 0$$\end{document} ν = 0
| 3.940811 |
biomedical
|
Study
|
[
0.8040411472320557,
0.0006584844668395817,
0.19530035555362701
] |
[
0.9888783097267151,
0.010337931104004383,
0.0006800013361498713,
0.00010383347398601472
] |
en
| 0.999997 |
General Effect of Small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν The case of a small but finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν is a singular pertubation of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 system. We can investigate its behaviour using asymptotic methods in the small- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν limit. To briefly summarise before going into details, the main differences between the small- and the zero- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν cases is that small- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν dynamics do not vanish in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A , nor do they terminate at the point at which they first hit the region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B boundary. Instead, in the small- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν case the trajectories travel within region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A and along the region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B boundary over slow timescales of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\nu )$$\end{document} O ( 1 / ν ) . So, in the small- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν case a trajectory starting in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} C will follow the zero- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν trajectory determined in Sect. 4.2 with an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\nu )$$\end{document} O ( ν ) correction, and move into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B and then to the region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B boundary in finite time. A trajectory starting in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A will also move into region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B over a slower time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\nu )$$\end{document} O ( 1 / ν ) . Given this, we focus our analysis on understanding what happens in region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B .
|
39751668_p40
|
39751668
|
Slow-Moving Cells, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 < \nu \ll 1$$\end{document} 0 < ν ≪ 1
| 3.927611 |
biomedical
|
Study
|
[
0.6566849946975708,
0.0006968300440348685,
0.3426181375980377
] |
[
0.9814505577087402,
0.017045028507709503,
0.0013596131466329098,
0.00014481303514912724
] |
en
| 0.999995 |
Asymptotic expansion To understand the system behaviour in the limit of small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν , we first write both y and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α as asymptotic series in powers of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν . That is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(t) \sim y_0(t) + \nu y_1(t)$$\end{document} y ( t ) ∼ y 0 ( t ) + ν y 1 ( t ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t) \sim \alpha _0(t) + \nu \alpha _1(t)$$\end{document} α ( t ) ∼ α 0 ( t ) + ν α 1 ( t ) . Substituting these into the governing equations ( 10 ) and considering the leading-order equations, we obtain exactly the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 that we analysed in Sect. 4.2 . In this case, we know that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_0(t) \rightarrow \bar{y}$$\end{document} y 0 ( t ) → y ¯ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _0(t) \rightarrow \bar{\alpha }$$\end{document} α 0 ( t ) → α ¯ in some finite time T , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^2(\bar{\alpha }) = \bar{y}^2$$\end{document} γ 1 2 ( α ¯ ) = y ¯ 2 . In the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 , the analysis ends here. However, in the singular case of small but finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν , there is an additional slow timescale after this finite time. To analyse this slow timescale behaviour, we transform into the slow timescale regime defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = T + \tau /\nu $$\end{document} t = T + τ / ν , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = O(1)$$\end{document} τ = O ( 1 ) , and now expand \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(t) = Y(\tau ) \sim Y_0(\tau ) + \nu Y_1(\tau )$$\end{document} y ( t ) = Y ( τ ) ∼ Y 0 ( τ ) + ν Y 1 ( τ ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (t) = A(\tau ) \sim A_0(\tau ) + \nu A_1(\tau )$$\end{document} α ( t ) = A ( τ ) ∼ A 0 ( τ ) + ν A 1 ( τ ) . On substituting the slow timescale and these new expansions into the governing equations ( 10 ), we obtain 36a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \frac{\textrm{d}Y}{\textrm{d}\tau }&= \nu \sin A - f(Y,A), \end{aligned}$$\end{document} ν d Y d τ = ν sin A - f ( Y , A ) , 36b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \frac{\textrm{d}A}{\textrm{d}\tau }&= g(Y,A)f(Y,A), \end{aligned}$$\end{document} ν d A d τ = g ( Y , A ) f ( Y , A ) , where we introduce the following shorthand functions 37a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(Y,A) = 2\frac{\sqrt{\gamma _1^2(A) - Y^2}}{\gamma _1^2(A)}, \end{aligned}$$\end{document} f ( Y , A ) = 2 γ 1 2 ( A ) - Y 2 γ 1 2 ( A ) , 37b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&g(Y,A) = 4\frac{r^2-1}{r^2+1}\frac{Y\sin A\cos A}{\gamma _1^2(A)}. \end{aligned}$$\end{document} g ( Y , A ) = 4 r 2 - 1 r 2 + 1 Y sin A cos A γ 1 2 ( A ) .
|
39751668_p41
|
39751668
|
Slow-Moving Cells, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 < \nu \ll 1$$\end{document} 0 < ν ≪ 1
| 4.087844 |
biomedical
|
Study
|
[
0.6685146689414978,
0.0010230281623080373,
0.3304623067378998
] |
[
0.9811059832572937,
0.01611395925283432,
0.0026143044233322144,
0.0001657626562518999
] |
en
| 0.999996 |
The O (1) terms in equations ( 36 ) generate a duplication of information, with both giving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(Y_0,A_0) = 0$$\end{document} f ( Y 0 , A 0 ) = 0 , or equivalently 38 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma _1^2(A_0(\tau )) = Y_0^2(\tau ). \end{aligned}$$\end{document} γ 1 2 ( A 0 ( τ ) ) = Y 0 2 ( τ ) . This tells us that the leading-order slow-time dynamics are confined to the line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^2(\alpha ) = y^2$$\end{document} γ 1 2 ( α ) = y 2 , the boundary between regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B . The remaining goal of our analysis is to determine the precise dynamics of this slow motion. There are several ways to proceed at this point. One method involves continuing to the next asymptotic order and deriving an appropriate solvability condition. Another way involves combining the full equations ( 36 ) in a way that removes the duplication of information, then taking the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \rightarrow 0$$\end{document} ν → 0 to obtain an independent evolution equation. We proceed via the latter, since this involves significantly less algebra. By substituting ( 36a ) into ( 36b ) and dividing through by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν , we obtain 39 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{\textrm{d}A}{\textrm{d}\tau } = g(Y,A)\left( \sin A - \frac{\textrm{d}Y}{\textrm{d}\tau }\right) . \end{aligned}$$\end{document} d A d τ = g ( Y , A ) sin A - d Y d τ . The equation ( 39 ) has leading-order form 40 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}A_0}{\textrm{d}\tau } = 4\frac{r^2-1}{r^2+1}\frac{Y_0 \sin A_0\cos A_0}{\gamma _1^2(A_0)}\left( \sin A_0 - \frac{\textrm{d}Y_0}{\textrm{d}\tau }\right) . \end{aligned}$$\end{document} d A 0 d τ = 4 r 2 - 1 r 2 + 1 Y 0 sin A 0 cos A 0 γ 1 2 ( A 0 ) sin A 0 - d Y 0 d τ . The equations ( 38 ) and ( 40 ) represent enough information to determine the slow evolution along the region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} A - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} B boundary. It is more straightforward to see this if we use the direct time derivative of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^2 (A_0) = Y_0^2$$\end{document} γ 1 2 ( A 0 ) = Y 0 2 , which is 41 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sin A_0 \cos A_0 (r^2 - 1) \frac{\textrm{d}A_0}{\textrm{d}\tau } = r Y_0 \frac{\textrm{d}Y_0}{\textrm{d}\tau }. \end{aligned}$$\end{document} sin A 0 cos A 0 ( r 2 - 1 ) d A 0 d τ = r Y 0 d Y 0 d τ . Combining this with ( 40 ) we can deduce that 42a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left( 4 \frac{(r^2 - 1)^2}{r^2 + 1} \sin ^2 A_0 \cos ^2 A_0 + r Y_0^2\right) \frac{\textrm{d}A_0}{\textrm{d}\tau } = 4 r \frac{(r^2 - 1)}{r^2 + 1} Y_0 \sin ^2 A_0 \cos A_0,\qquad \end{aligned}$$\end{document} 4 ( r 2 - 1 ) 2 r 2 + 1 sin 2 A 0 cos 2 A 0 + r Y 0 2 d A 0 d τ = 4 r ( r 2 - 1 ) r 2 + 1 Y 0 sin 2 A 0 cos A 0 , 42b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left( 4 \frac{(r^2 - 1)^2}{r^2 + 1} \sin ^2 A_0 \cos ^2 A_0 + r Y_0^2\right) \frac{\textrm{d}Y_0}{\textrm{d}\tau } = 4 \frac{(r^2 - 1)^2}{r^2 + 1} \sin ^3 A_0 \cos ^2 A_0. \end{aligned}$$\end{document} 4 ( r 2 - 1 ) 2 r 2 + 1 sin 2 A 0 cos 2 A 0 + r Y 0 2 d Y 0 d τ = 4 ( r 2 - 1 ) 2 r 2 + 1 sin 3 A 0 cos 2 A 0 . Then, since we are considering \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0,\pi /2]$$\end{document} α ∈ [ 0 , π / 2 ] and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0 \in [0,\pi /2]$$\end{document} A 0 ∈ [ 0 , π / 2 ] , we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0(\tau ) = \sin (A_0(\tau ))$$\end{document} s 0 ( τ ) = sin ( A 0 ( τ ) ) , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \in $$\end{document} s 0 ∈ . Under this substitution, ( 42 ) can be rewritten as 43a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{\textrm{d}s_0}{\textrm{d}\tau } = \frac{4r(r^2 - 1) Y_0 s_0^2 (1 - s_0^2)}{4(r^2 - 1)^2 s_0^2 (1 - s_0^2) + r (r^2 + 1)Y_0^2}, \end{aligned}$$\end{document} d s 0 d τ = 4 r ( r 2 - 1 ) Y 0 s 0 2 ( 1 - s 0 2 ) 4 ( r 2 - 1 ) 2 s 0 2 ( 1 - s 0 2 ) + r ( r 2 + 1 ) Y 0 2 , 43b \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{\textrm{d}Y_0}{\textrm{d}\tau } = \frac{4(r^2 - 1)^2 s_0^3 (1 - s_0^2)}{4(r^2 - 1)^2 s_0^2 (1 - s_0^2) + r (r^2 + 1)Y_0^2}. \end{aligned}$$\end{document} d Y 0 d τ = 4 ( r 2 - 1 ) 2 s 0 3 ( 1 - s 0 2 ) 4 ( r 2 - 1 ) 2 s 0 2 ( 1 - s 0 2 ) + r ( r 2 + 1 ) Y 0 2 . Solving ( 43 ) numerically, we can compare our numerical solution of the reduced problem to a numerical solution of the full problem. In Fig. 9 , we see that the asymptotic solution gives good agreement with the full numerical solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0.1$$\end{document} ν = 0.1 , solved using ode15s in MATLAB. We also see that for all initial conditions, the trajectory will move to the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -\frac{1}{\sqrt{r}})$$\end{document} ( α , y ) = ( 0 , - 1 r ) , corresponding to perfect alignment.
|
39751668_p42
|
39751668
|
Slow-Moving Cells, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 < \nu \ll 1$$\end{document} 0 < ν ≪ 1
| 4.253384 |
biomedical
|
Study
|
[
0.935209333896637,
0.0006490109954029322,
0.06414168328046799
] |
[
0.995955228805542,
0.003427483607083559,
0.0005499905673786998,
0.00006733970803907141
] |
en
| 0.999995 |
If we substitute the solution of the algebraic constraint ( 38 ), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_0 = -\frac{1}{\sqrt{r}}\sqrt{(r^2--1)s_0^2 + 1}$$\end{document} Y 0 = - 1 r ( r 2 - - 1 ) s 0 2 + 1 into ( 43a ) we can reduce the dynamics to a single ODE 44 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}s_0}{\textrm{d}\tau } = -\frac{4\sqrt{r}(r^2 - 1) \sqrt{(r^2-1)s_0^2 + 1} s_0^2 (1 - s_0^2)}{4(r^2 - 1)^2 s_0^2 (1 - s_0^2) + (r^2 + 1)(r^2-1)s_0^2 + r^2 + 1}. \end{aligned}$$\end{document} d s 0 d τ = - 4 r ( r 2 - 1 ) ( r 2 - 1 ) s 0 2 + 1 s 0 2 ( 1 - s 0 2 ) 4 ( r 2 - 1 ) 2 s 0 2 ( 1 - s 0 2 ) + ( r 2 + 1 ) ( r 2 - 1 ) s 0 2 + r 2 + 1 . On separating variables and integrating, we find that 45 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 4\sqrt{r}(r^2-1)\tau&= \int _{s_0(0)}^{s_0(\tau )} \frac{4(r^2 - 1)^2 s_0^2 (1 - s_0^2) + (r^2 + 1)(r^2-1)s_0^2 + r^2 + 1}{s_0^2 (1 - s_0^2) \sqrt{(r^2-1)s_0^2 + 1}} \, \textrm{d}s_0\nonumber \\&=: I(s_0(\tau )). \end{aligned}$$\end{document} 4 r ( r 2 - 1 ) τ = ∫ s 0 ( 0 ) s 0 ( τ ) 4 ( r 2 - 1 ) 2 s 0 2 ( 1 - s 0 2 ) + ( r 2 + 1 ) ( r 2 - 1 ) s 0 2 + r 2 + 1 s 0 2 ( 1 - s 0 2 ) ( r 2 - 1 ) s 0 2 + 1 d s 0 = : I ( s 0 ( τ ) ) . We note that we have shown in Sect. 4 that the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -\frac{1}{\sqrt{r}})$$\end{document} ( α , y ) = ( 0 , - 1 r ) is a stable steady state for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} α > 0 . We also know that all trajectories solving ( 43 ) move towards this point. To understand this long-term behaviour in more detail, we must understand what happens as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 0^{+}$$\end{document} α → 0 + or equivalently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{s_0 \rightarrow 0^+} I(s_0(\tau ))$$\end{document} lim s 0 → 0 + I ( s 0 ( τ ) ) . If this integral converges, then perfect alignment in reached in finite time. However, if this integral diverges, then it takes infinite time to reach perfect alignment where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s_0(\tau ), Y_0(\tau )) = (0,-\frac{1}{\sqrt{r}})$$\end{document} ( s 0 ( τ ) , Y 0 ( τ ) ) = ( 0 , - 1 r ) . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \rightarrow 0$$\end{document} s 0 → 0 , the integrand in ( 45 ) behaves like \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{r^2+1}{s_0^2}$$\end{document} r 2 + 1 s 0 2 and hence the integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(s_0(\tau ))$$\end{document} I ( s 0 ( τ ) ) diverges. Therefore, the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -\frac{1}{\sqrt{r}})$$\end{document} ( α , y ) = ( 0 , - 1 r ) of perfect alignment is only reached in infinite time.
|
39751668_p43
|
39751668
|
Slow-Moving Cells, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 < \nu \ll 1$$\end{document} 0 < ν ≪ 1
| 4.329947 |
biomedical
|
Study
|
[
0.9939228892326355,
0.00030739023350179195,
0.005769721232354641
] |
[
0.9983153343200684,
0.001220448175445199,
0.0004200459225103259,
0.0000441512129327748
] |
en
| 0.999996 |
The Role of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν The analysis of this section allows us to obtain a full understanding of the effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν on the alignment mechanism between two cells. As a reminder, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν was defined in ( 4 ) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =\frac{v\eta }{\sigma }\sqrt{\pi A}$$\end{document} ν = v η σ π A and can be interpreted as the ratio of self-propulsion and overlap avoidance strength in the presence of friction. We find that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu = 0$$\end{document} ν = 0 trajectories do not move towards the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,y) = (0, -1/\sqrt{r})$$\end{document} ( α , y ) = ( 0 , - 1 / r ) , and hence perfect alignment is not achieved when there is no self-propulsion. This highlights the importance of self-propulsion in the alignment mechanism and shows that both overlap avoidance and self-propulsion are needed for full alignment. The analysis of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\nu \ll 1$$\end{document} 0 < ν ≪ 1 represents a singular perturbation of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 case. We find that there is a set of initial conditions that lead to perfect alignment (for small, but positive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν ), however, achieving perfect alignment takes infinite time. Quantifying computationally how favourable different model parameters are to alignment, we find a surprisingly good qualitative agreement with the results for large collective of cells found in Leech et al. : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 leads to very little alignment, however, for larger values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν alignment strength decreases as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} ν increases. The fact that the analysis for a symmetric two-cell situation recapitulates the results for large cell collectives indicates that it is a property of the alignment mechanism itself, not a result of the many-cell situation. Based on our analysis we can conclude that, while it is the overlap avoidance that causes interacting cells to align, (smaller amounts of) self-propulsion ensure that these interactions do not stop too quickly. However, if self-propulsion is too strong (compared to overlap avoidance), cells can “escape” the interaction by being pushed past each other.
|
39751668_p44
|
39751668
|
Lessons for (Collective) Cell Alignment
| 4.223381 |
biomedical
|
Study
|
[
0.990368664264679,
0.00029365374939516187,
0.009337710216641426
] |
[
0.9984862804412842,
0.0009584539802744985,
0.0005131973884999752,
0.000042001272959169
] |
en
| 0.999997 |
The Role of the Aspect Ratio r . For the dependence of alignment strength on the aspect ratio r , we found that for the symmetric two-cell system, there is an optimal aspect ratio for alignment: Interactions between close to circular cells will not lead to much velocity alignment. On the other hand, interactions between very elongated cells can more easily lead to cells moving past each other at which point the interaction stops. Note that this is different to the results for collectives found in Leech et al. : There larger aspect ratios lead to more collective alignment. This could be, because in a collective, cell movement past each other might be hindered by other cells. Another possible reason could be, that alignment in Leech et al. referred to “nematic alignment”, which also captures situations where cells are side-by-side, but move in opposite directions. This is not something we can describe well with the symmetry condition imposed in this work, where alignment only describes velocity alignment, i.e. cells move in the same directions. A different framework would be necessary for further analytical investigations.
|
39751668_p45
|
39751668
|
Lessons for (Collective) Cell Alignment
| 4.125735 |
biomedical
|
Study
|
[
0.9987326264381409,
0.00023195573885459453,
0.001035436987876892
] |
[
0.9992571473121643,
0.0002657764416653663,
0.00043669258593581617,
0.0000403204612666741
] |
en
| 0.999998 |
Model Summary In this work, we have presented a number of results to help us understand the alignment mechanism between two interacting cells in the model presented in Leech et al. . These results have led to a more in-depth understanding of how model ingredients lead to alignment. Specifically we investigated the interplay between self-propulsion and overlap avoidance and the role of the aspect ratio of the cell. The analysed model describes self-propulsion and overlap avoidance, in reaction to which cells change their position, orientation and aspect ratio. We derived an analytical framework which led to a dynamical system, for which there are many mathematical tools available.
|
39751668_p46
|
39751668
|
Summary of the Results
| 4.085853 |
biomedical
|
Study
|
[
0.9992423057556152,
0.0001764488115441054,
0.0005813198513351381
] |
[
0.9984214305877686,
0.0003551491245161742,
0.0011662140022963285,
0.000057217726862290874
] |
en
| 0.999995 |
Deformable Cells In the full 3D (in variable space) system, which allows for shape deformations, a classic linear stability analysis is degenerate. Performing a dynamical asymptotic analysis, we find that the situation where both cells have their preferred shape and are perfectly aligned side-by-side, is a half-stable steady state of the system for elongated cells, and an unstable steady state for wide cells, indicating the fleeting nature of perfect alignment. This stability analysis also gives us insight into the timescales at play during the cell interactions. We find that the relative sizes of the two non-dimensional parameters determines the ordering of the change in aspect ratio and position, and that the change in orientation always occurs last, over a much slower timescale. We find that this change in orientation is unaffected by the shape change parameter, suggesting that deformability is not essential for alignment to occur. In the computational analysis done in Leech et al. , it is found that allowing for changes in aspect ratio leads to greater alignment. The fact that we do not see this in the two-cell analysis, suggests that this result is due to the collective behaviour of the system, as opposed to the underlying alignment mechanism. We also find that if the non-dimensional shape change parameter is small, cells will quickly restore their preferred aspect ratio, after which the system behaves as though cell shape is rigid. This is reflected in the computational results in Leech et al. by noting that a strong shape-restoring force leads to little change in behaviour from the rigid cell case.
|
39751668_p47
|
39751668
|
Summary of the Results
| 4.261957 |
biomedical
|
Study
|
[
0.999355137348175,
0.00028775373357348144,
0.0003571614797692746
] |
[
0.99906986951828,
0.0002160335861844942,
0.0006517847650684416,
0.00006230971484910697
] |
en
| 0.999995 |
Non-deformable Cells We then consider a 2D (in variable space) limit of the full system in more detail, specifically, the limit in which the cells are rigid and keep their preferred aspect ratio. We computationally quantify the effect of model parameters on the overall alignment properties of the system. We find that both no self-propulsion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =0$$\end{document} ν = 0 and too much self-propulsion lead to less alignment, in striking qualitative agreement with the computation results in Leech et al. for cell collectives. This underlines the difference between active matter, that self-propels, and passive matter, which doesn’t. We are also able to make a significant amount of analytic progress in understanding the singular nature of the small self-propulsion limit. We find that cells are quickly pushed to a single point of overlap over a fast timescale, before the small self-propulsion causes cells to align while maintaining a single point of overlap over a much slower timescale. Together with the quantification of alignment, this leads to a greater understanding of how overlap avoidance combined with self-propulsion leads to alignment, see the discussion in Sect. 4.4 . Hence, even though many of the complex details of the collective behaviour in Leech et al. are not captured in the analysis for two cells here, our analysis gives a plausible explanations for the observed behaviour. When analysing how alignment properties depend on the aspect ratio, we find that for our symmetric two-cell system there is an optimal aspect ratio for alignment. This is in contrast to the collective results in Leech et al. , where larger aspect ratios lead to more alignment. This difference could be due to limitations of this work due to the imposed symmetry condition, which limit the notion of alignment to “velocity alignment”.
|
39751668_p48
|
39751668
|
Summary of the Results
| 4.291695 |
biomedical
|
Study
|
[
0.998978853225708,
0.00035099455271847546,
0.000670219655148685
] |
[
0.9989892840385437,
0.00016580014198552817,
0.0007863003411330283,
0.00005861989120603539
] |
en
| 0.999997 |
The modelling framework presented in Leech et al. is not specific to ellipse-shaped cells and could be applied to a number of different cell shapes. Provided the points of intersection could be found analytically, a similar approach to analyse and understand the system in greater depth could be taken. It would be interesting to apply the same analytical framework to different cell shapes to see how the results differ. This would be of particular interest if we wanted to apply the modelling framework to bacteria, for example, which are often modelled as spherocylinders instead of ellipses . In this work we did not deeply analyse the behaviour of the system for wide cells with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{r}<1$$\end{document} r ¯ < 1 , such as keratocytes. In this case a different symmetry condition might be appropriate to capture alignment: The two cells could be assumed to move one behind the other. In Leech et al. we also considered cell-cell adhesions. The analysis presented here could be extended to include such cell-cell contacts. In a first approximation, one could assume such contacts are permanent to then minimise a combined 2-cell energy. This will be subject of future work. Our approach could also be used to understand the dynamics of more than two cells, e.g. three cells placed in such a way that their centres of mass form a triangle. Finally, looking to other methods of mathematically analysing agent-based models, it would be beneficial to see whether any progress could be made on coarse-graining the model to obtain an equivalent continuum model such that further results on the collective behaviour of the system for many cells could be obtained. In the recent work of Merino-Aceituno et al. this is done for a similar model, which uses a smooth overlap potential.
|
39751668_p49
|
39751668
|
Further Work
| 4.176845 |
biomedical
|
Study
|
[
0.9988828301429749,
0.00022705766605213284,
0.0008900673710741103
] |
[
0.996877908706665,
0.000505537202116102,
0.0025516056921333075,
0.00006504785415017977
] |
en
| 0.999996 |
Brainstem interneuronal excitability can be investigated by recording the recovery cycle of the blink reflex (BRrc). When two electrical stimuli of equal intensity (first stimulus or Conditioning, second stimulus or Test) are delivered to the supraorbital nerve, the second R2 response or R2 test amplitude is influenced by the interstimulus interval (ISI). Particularly, when the ISI is short (shorter than 200 ms), the R2 test is inhibited and gradually recovers with longer ISIs (longer than 500 ms) . Brainstem reflexes can be functionally abnormal in some neurodegenerative diseases, underlying dysfunction of cortico-thalamic, basal ganglia, and brainstem loops . In disorders characterized by dopaminergic lack such as Parkinson's disease (PD) and cranio-cervical dystonias, there is evidence of increased excitability of blink reflex measured by the recovery cycle [ 1 , 3 – 6 ]. In blepharospasm where the hyperexcitability of BRrc is considered as one the most consistent finding , it has been suggested that an altered influence of the sensorimotor cortices on the basal ganglia and brainstem could play a role in the pathophysiology of dystonia in addition to dopaminergic dysfunction . Taken together, these evidences lead to the hypothesis of a direct influence of basal ganglia on BRrc excitability . Crucially, in line with this data, enhancement of blink reflex excitability has been already demonstrated in patients with juvenile myoclonic epilepsy, prompting disinhibition of cortico-thalamic pathways involved in the excitability of brainstem circuits . Moreover, in animal models, the beta-band (16 Hz) stimulation of the subthalamic nucleus enhanced blink reflex excitability in normal rats as well as in 6-hydroxydopamine induced model of PD and human patients with PD .
|
39096396_p0
|
39096396
|
Introduction
| 4.611111 |
biomedical
|
Study
|
[
0.9992437362670898,
0.0005052807973697782,
0.00025097368052229285
] |
[
0.9886007905006409,
0.000534197548404336,
0.010581153444945812,
0.0002838005602825433
] |
en
| 0.999997 |
Several studies have reported a modulating role of different non-invasive brain stimulation (NIBS) techniques on brainstem excitability in healthy subjects and patients. In 2009, De Vito et al. showed that subthreshold low frequency (1-Hz) repetitive transcranial magnetic stimulation (rTMS) reduced blink reflex excitability in a group of 10 healthy volunteers: long-lasting reduction of blink reflex recovery cycle was interpreted as the consequence of reduced cortical excitability and therefore reduced cortico-nuclear facilitation of brainstem interneuronal circuitry. In another study by Cabib et al. , transcranial direct current stimulation (tDCS) was used as NIBS technique able to modify membrane polarization and modulate the probability to generate action potentials . Anodal (excitatory) tDCS over central cortices induced a persistent increase of BR excitability, also evident 10 min after stimulation, with a larger ipsilateral than contralateral effect. Authors hypothesized that these effects could underlie a modulatory effect of tDCS on descending cortico-nuclear pathways, as indirectly suggested by increased facilitation of R1 in case of unilateral hemispheric damage . However, the same authors also reported the ability of constant electrical currents to sensitize trigeminal neurons, as proved by the mild cutaneous sensation induced by the stimulation . Low-frequency rTMS (inhibitory) over the anterior cingulate cortex has been demonstrated to reduce the blink reflex hyperexcitability and was associated with a clinical improvement in patients with blepharospasm .
|
39096396_p1
|
39096396
|
Introduction
| 4.301261 |
biomedical
|
Study
|
[
0.9987720847129822,
0.0006087920046411455,
0.0006190959247760475
] |
[
0.7682984471321106,
0.000855900114402175,
0.2303721159696579,
0.0004736155387945473
] |
en
| 0.999997 |
Transcranial alternating current stimulation (tACS) is a relatively new NIBS device, that uses two electrodes placed on the scalp, with the electrodes alternating as the anode and cathode and creating an alternating direction of current flowing through the target region; unlike TMS or tDCS, tACS can modulate brain oscillations. Brain oscillations are the rhythmic patterns of electrophysiological activity in the neural tissue, naturally occurring in the brain, that can be revealed by EEG analysis. Different frequency bands, like theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz) and gamma (≥ 30 Hz) have been related to different functions of the human brain involved in specific tasks .
|
39096396_p2
|
39096396
|
Introduction
| 4.080316 |
biomedical
|
Study
|
[
0.999110996723175,
0.00028232071781530976,
0.000606758170761168
] |
[
0.5897328853607178,
0.2590954005718231,
0.1500757485628128,
0.0010960109066218138
] |
en
| 0.999998 |
Two main mechanisms have been suggested to explain tACS effects, entrainment and spike-timing dependent plasticity (STDP). Entrainment is the synchronization of two oscillatory systems occurring when a driving external oscillatory force coordinates another oscillating system . The STDP proposed by Zaehle et al. refers to the ability of different frequencies of tACS to induce long-term potentiation (LTP) or depression (LTD) : in particular, if a neuron is stimulated at the same or lower frequency of its endogenous frequency, the alternating current would lead to potentiation; conversely, if higher stimulation frequencies than the endogenous ones are used, a post-synaptic spike delivered from external stimulation will arrive before the pre-synaptic spike, weakening of the synapse, by means of an LTD-like mechanism.
|
39096396_p3
|
39096396
|
Introduction
| 4.214082 |
biomedical
|
Study
|
[
0.9993122816085815,
0.00018192344577983022,
0.0005058306851424277
] |
[
0.9691531658172607,
0.0022446971852332354,
0.028459720313549042,
0.0001424454094376415
] |
en
| 0.999995 |
In this study, we evaluate the effect of alternating currents at different frequency ranges (10-Hz or alfa band and 20-Hz, beta band) on the blink reflex excitability as tested by BRrc in a group of healthy subjects. Given that it is not known whether tACS-induced modulation depends on local activity variations or involves broader networks, the primary aim of this study is to understand if alternating currents can influence subcortical structures. Secondarily, our objective is to verify if a beta-band frequency (20-Hz) stimulation over the sensori-motor cortex can increase blink reflex excitability, similarly to patients with PD or blepharospasm and in animal models undergoing a beta-band deep stimulation of the subthalamic nucleus.
|
39096396_p4
|
39096396
|
Introduction
| 4.125861 |
biomedical
|
Study
|
[
0.9993594288825989,
0.0004477954935282469,
0.00019279970729257911
] |
[
0.9993237257003784,
0.0002511975762899965,
0.0003351893974468112,
0.0000898952639545314
] |
en
| 0.999997 |
We initially recruited 17 healthy volunteers; 2 of them were excluded: 1 refused to complete the entire experimental procedure and 1 started a steroid treatment for medical issues some days after the first stimulation session. We finally enrolled a group of 15 healthy subjects (mean age ± SD: 27.4 ± 2.7, 11 females), all right-handed, as assessed by Edinburgh Inventory . None of the participants suffered from any systemic or neurological disorders, as assessed by a clinical neurologist, female subjects were not examined during the menstrual phase (from 5 days before to 5 days after menstruation); none of them was taking any drug known to alter neuromuscular excitability or any medical therapy for three months before the inclusion. All subjects gave written informed consent before enrollment and the study was approved by the Ethical Committee of the University of Palermo and conducted in accordance with the Declaration of Helsinki. All subjects had to fill out a specific form to detect any adverse reaction after the stimulation .
|
39096396_p5
|
39096396
|
Subjects
| 4.013616 |
biomedical
|
Study
|
[
0.9926173090934753,
0.007084470707923174,
0.00029818457551300526
] |
[
0.9963927865028381,
0.002442662138491869,
0.00048798249918036163,
0.0006766223232261837
] |
en
| 0.999995 |
During the study subjects laid down supine, with their eyes gently closed, on a comfortable examination bed, in a quiet and dimly lit room. Ag–AgCl surface recording electrodes were placed over the orbicularis oculi muscle of both sides (mid-lower eyelid and temple). In all volunteers, the cathode of the stimulating electrode was placed over the right supraorbital notch and the anode 3 cm away, over the skin of the frontal bone. Skin impedance was lower than 5 kΩ. The ground electrode was placed over the nasion. Blink reflex was recorded with a KeyPoint Electromyographic System and was obtained from stimulation of the right supraorbital nerve. The duration of the stimulus was 0.2 ms and the stimulus intensity was set to three times the intensity (mean ± SD: 11 ± 5 mA) needed to obtain a reproducible ipsilateral R2 with an amplitude of at least 50 μV in five consecutive trials. This intensity was maintained constant for all the experimental procedure.
|
39096396_p6
|
39096396
|
Blink reflex and blink reflex recovery cycle
| 4.169207 |
biomedical
|
Study
|
[
0.9977589845657349,
0.002036964986473322,
0.00020401425717864186
] |
[
0.9983069896697998,
0.0010603712871670723,
0.0003927855577785522,
0.00023974735813681036
] |
en
| 0.999997 |
To obtain BR recovery curves, we used the original technique described by Kimura . Briefly filter settings were 20 Hz-10 kHz. We only considered the R2 ipsilateral to the side of stimulation (right supraorbital nerve). We delivered paired stimuli at a constant current at interstimulus intervals (ISI) of 100, 150, 200, 300, 400, 500 e 750 ms. At each ISI, recordings were repeated 4 times at random intervals of at least 20–40 s to avoid habituation. Data were analyzed offline. The amplitude (μV) of R2 responses was measured after the first (conditioning) and the second (test) stimulus. The outcome measure was the R2 amplitude ratio (R2AR), calculated as follows: R2AR = (R2 test amplitude)/(R2 conditioning amplitude) × 100.
|
39096396_p7
|
39096396
|
Blink reflex and blink reflex recovery cycle
| 4.089289 |
biomedical
|
Study
|
[
0.9994915723800659,
0.0002882115950342268,
0.00022022402845323086
] |
[
0.9994159936904907,
0.00025793013628572226,
0.000267861905740574,
0.000058293462643632665
] |
en
| 0.999998 |
Participants were seated on a comfortable chair in a dimly lit room. tACS was applied at a fixed intensity of 1 mA delivered by a DC stimulator (Brainstim, EMS, Bologna, Italy). We used saline-soaked sponge electrodes (5 × 7 cm) and flexible elastics to fixate the electrode on the head. The center of the active electrode was placed over C4 (with the long axis anterior–posterior) and the reference electrode over Pz according to the International 10–20 EEG System, this placement was associated with a lower risk of flickering sensations . During real or sham stimulation, impedance was kept below 10 kΩ. Every subject underwent three types of stimulation for 10 min: a. alpha-band stimulation at a fixed frequency of 10 Hz with no direct current offset; b. beta-band stimulation at a fixed frequency of 20 Hz with no direct current offset; c. sham stimulation (20 Hz) with the current turned on for 30 s, with 5 s of fade-in and fade-out, and then turned off. The order of conditions was randomized across participants and all sessions were separated by ≥ 3 days. Participants were blinded to the condition.
|
39096396_p8
|
39096396
|
Transcranial alternating current stimulation (tACS)
| 4.155472 |
biomedical
|
Study
|
[
0.9946568012237549,
0.005011909641325474,
0.00033133287797681987
] |
[
0.995618462562561,
0.003483572043478489,
0.0004813926643691957,
0.00041661440627649426
] |
en
| 0.999996 |
Subsets and Splits
SQL Console for rntc/test-pp-aa
The query retrieves a sample of documents that are clinical cases with an educational score above 3, providing limited analytical value.
Clinical Cases Sample
Returns a sample of 100 clinical case documents, providing a basic overview of the document type's content.