CategoryFirst-passage times

So you want to do some path sampling…

Basic strategies, timescales, and limitations

Key biomolecular events – such as conformational changes, folding, and binding – that are challenging to study using straightforward simulation may be amenable to study using “path sampling” methods.  But there are a few things you should think about before getting started on path sampling.  There are fairly generic features and limitations that govern all the path sampling methods I’m aware of.

Path sampling refers to a large family of methods that, rather than having the goal of generating an ensemble of system configurations, attempt to generate an ensemble of dynamical trajectories.  Here we are talking about trajectory ensembles that are precisely defined in statistical mechanics.  As we have noted in another post, there are different kinds of trajectory ensembles – most importantly, the equilibrium ensemble, non-equilibrium steady states, and the initialized ensemble which will relax to steady state.  Typically, one wants to generate trajectories exhibiting events of interest – e.g., binding, folding, conformational change.

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More is better: The trajectory ensemble picture

The trajectory ensemble is everything you’ve always wanted, and more.  Really, it is.  Trajectory ensembles unlock fundamental ideas in statistical mechanics, including connections between equilibrium and non-equilibrium phenomena.  Simple sketches of these objects immediately yield important equations without a lot of math.  Give me the trajectory-ensemble pictures over fancy formalism any day.  It’s harder to make a mistake with a picture than a complicated equation.

A trajectory, speaking roughly, is a time-ordered sequence of system configurations.  Those configurations could be coordinates of atoms in a single molecule, the coordinates of many molecules, or whatever objects you like.  We assume the sequence was generated by some real physical process, so typically we’re considering finite-temperature dynamics (which are intrinsically stochastic due to “unknowable” collisions with the thermal bath).  The ‘time-ordered sequence’ of configurations really reflects continuous dynamics, so that the time-spacing between configurations is vanishingly small, but that won’t be important for this discussion.

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“Proof” of the Hill Relation Between Probability Flux and Mean First-Passage Time

The “Hill relation” is a key result for anyone interested in calculating rates from trajectories of any kind, whether molecular simulations or otherwise.  I am not aware of any really clear explanation, including Hill’s original presentation.  Hopefully this go-around will make sense.

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