Abstract
Modern methods for sampling rugged landscapes in state space mainly rely on knowledge of the relative probabilities of microstates, which is given by the Boltzmann factor for equilibrium systems. In principle, trajectory reweighting provides an elegant way to extend these algorithms to non-equilibrium systems, by numerically calculating the relative weights that can be directly substituted for the Boltzmann factor. We show that trajectory reweighting has many commonalities with Rosenbluth sampling for chain macromolecules, including practical problems which stem from the fact that both are iterated importance sampling schemes: for long trajectories the distribution of trajectory weights becomes very broad and trajectories carrying high weights are infrequently sampled, yet long trajectories are unavoidable in rugged landscapes. For probing the probability landscapes of genetic switches and similar systems, these issues preclude the straightforward use of trajectory reweighting. The analogy to Rosenbluth sampling suggests though that path-ensemble methods such as PERM (pruned-enriched Rosenbluth method) could provide a way forward.
Highlights
The Boltzmann factor, which describes exactly the relative probability of microstates at equilibrium in systems whose dynamics obeys detailed balance, forms the cornerstone of a plethora of simulation methods in the physical sciences
The reference system is typically easier to sample than the target system
Trajectory reweighting provides an apparently elegant way to extend biased sampling methods developed for equilibrium steady states, to non-equilibrium systems whose dynamics do not obey detailed balance
Summary
The Boltzmann factor, which describes exactly the relative probability of microstates at equilibrium in systems whose dynamics obeys detailed balance, forms the cornerstone of a plethora of simulation methods in the physical sciences. Trajectory reweighting provides a way to generalise a plethora of biased sampling methods to non-equilibrium systems [23,24,25] It has been used in the context of Onsager–Machlup path probabilities to reweight Brownian dynamics trajectories [26,27], while in kinetic Monte-Carlo schemes trajectory weights [28] and reweighting [29] have been successfully exploited for first-passage time problems [30,31,32,33] and steady-state parametric sensitivity analysis [19,23,34]. We illustrate these issues using the simple case of a birth–death process, before suggesting ways in which the limitations of the method might be overcome
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