Simulation of deterministic turing machines What are the best known upper and lower bounds for simulating t steps of certain models of deterministic turing machines (1 tape, 1 tape with read only input tape, 2 tape, multi tape, with/without random access to memory)?

Edit: I am mainly interested in efficient simulations with the same or with weaker models. Simulations with stronger models are quite efficient in most of the cases.
For what regards (apparently) "weaker" models, one of the recent new results that I saw is that a deterministic Turing machine can be efficiently simulated by a 2-tag system:
Damien Woods and Turlough Neary, On the time complexity of 2-tag systems and
small universal Turing machines (FOCS 2006)

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Theorem 1: Given a single tape deterministic Turing machine $M$ that computes in time $t$ then there is a 2-tag system TM that simulates the computation
of $M$ and computes in polynomial time $O(t^4(\log t)^2)$.

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Corollary 1: The small UTMs of Minsky, Rogozhin and others are polynomial time, $O(t^8(\log  t)^4)$, simulators of Turing machines.