[Paper Review] Symmetry of forward and reverse path populations

Divesh Bhatt, Daniel M. Zuckerman|arXiv (Cornell University)|Feb 11, 2010

Insect Pheromone Research and Control3 citations

TL;DR

This paper formally establishes the symmetry of forward and reverse pathway populations in non-equilibrium steady states, showing exact symmetry holds when feedback mechanisms replicate equilibrium dynamics. For physically meaningful, deep basins, approximate symmetry is expected when intra-state relaxation is fast compared to inter-state transitions, providing a practical guideline for path sampling in molecular simulations.

ABSTRACT

In this note, we address formally the issue of symmetry for probabilities of different dynamical pathways in the forward and reverse directions of a conformational transition. Our discussion is based on a decomposition of equilibrium into opposing steady states, and makes clear the conditions necessary for symmetry to apply. From a practical point of view, we also discuss when approximate symmetry is to be expected.

Motivation & Objective

• To formally establish conditions under which forward and reverse pathway populations are symmetric in non-equilibrium steady states.
• To clarify when exact symmetry arises from feedback schemes that replicate equilibrium dynamics.
• To identify practical conditions under which approximate symmetry of pathway populations is expected in simulations.
• To provide a theoretical foundation for interpreting path populations in molecular dynamics and free energy calculations.

Proposed method

• Uses ensemble decomposition of trajectories in configuration space to analyze probability flows and pathway populations.
• Applies the detailed balance condition in equilibrium to derive exact symmetry of relative pathway probabilities in both forward and reverse directions.
• Analyzes special steady states where feedback of trajectories into source states exactly reproduces equilibrium distributions.
• Introduces the concept of 'deep' physical basins where internal relaxation is fast, enabling approximate symmetry.
• Compares exact symmetry under idealized feedback schemes with approximate symmetry under realistic simulation protocols.
• Builds on prior work by Crooks and vanden Eijnden to generalize symmetry conditions beyond equilibrium.

Experimental results

Research questions

• RQ1 Under what conditions does exact symmetry of forward and reverse pathway populations hold in non-equilibrium steady states?
• RQ2 How do feedback mechanisms that replicate equilibrium dynamics preserve symmetry in pathway populations?
• RQ3 When is approximate symmetry of pathway populations expected in practical simulations?
• RQ4 What defines a 'reasonably deep' physical basin in terms of timescales for symmetry to emerge?
• RQ5 How does the choice of state definitions affect the validity of forward–reverse path symmetry?

Key findings

• Exact symmetry of relative pathway populations between A and B is guaranteed if trajectories are fed back into the source state in a way that preserves the equilibrium probability distribution within that state.
• Symmetry holds for arbitrary state definitions when simulations are run in special steady states that exactly decompose equilibrium dynamics.
• Approximate symmetry of pathway populations is expected when intra-state relaxation times are much shorter than inter-state transition times, particularly for physically meaningful basins.
• The symmetry condition is robust when user-defined states correspond to deep physical basins of attraction, ensuring quasi-Markovian exit behavior.
• In the absence of exact feedback, the symmetry relation serves as a useful approximate guideline for path sampling and free energy calculations.