CategoryFirst-passage times

A “proof” of the discretized Hill Relation

This is yet another one of those things where, after reading this, you’re supposed to say, “Oh, that’s obvious.” And I admit it is kind of obvious … after you think about it for a few minutes! So spend those few minutes now to learn one more cool thing about non-equilibrium trajectory physics.

In non-equilibrium calculations of transition processes, we often wish to estimate a rate constant, which can be quantified as the inverse of the mean first-passage time (MFPT). That is, one way to define a rate constant is just reciprocal of the average time it takes for a transition. The Hill relation tells us that probability flow per second into a target state of interest (state “B”, defined by us) is exactly the inverse MFPT … so long as we measure that flow in the A-to-B steady state based on initializing trajectories outside state B according to some distribution (state “A”, defined by us) and we remove trajectories reaching state B and re-initialize them in A according to our chosen distribution.

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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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