Simulation of deterministic turing machines
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.