{"id":8,"date":"2015-04-21T12:32:07","date_gmt":"2015-04-21T12:32:07","guid":{"rendered":"http:\/\/statisticalbiophysicsblog.org\/?p=8"},"modified":"2015-07-01T18:17:52","modified_gmt":"2015-07-01T18:17:52","slug":"proof-of-the-hill-relation-between-probability-flux-and-mean-first-passage-time","status":"publish","type":"post","link":"https:\/\/statisticalbiophysicsblog.org\/?p=8","title":{"rendered":"\u201cProof\u201d of the Hill Relation Between Probability Flux and Mean First-Passage Time"},"content":{"rendered":"

The \u201cHill relation\u201d is a key result for anyone interested in calculating rates from trajectories of any kind, whether molecular simulations or otherwise.\u00a0 I am not aware of any really clear explanation, including Hill\u2019s original presentation.\u00a0 Hopefully this go-around will make sense.<\/p>\n

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The Hill relation applies to any \u201cfirst passage\u201d process – which could be diffusion from one region to another, chemical isomerization, protein folding, a cell-signaling process, or ….\u00a0 Quite simply, a first-passage process is one in which a system is initialized somehow (it doesn\u2019t matter how but the initialization should be well-specified or at least reproducible) and a hypothetical observer waits until a target state is reached or occurs (any target state will do).\u00a0 The observer could measure the time required for each occurrence, and the average time for such a process is called mean first passage time (MFPT).\u00a0 The inverse MFPT is one common definition for a rate constant, because the MFPT clearly describes the timescale for the process.<\/p>\n

The Hill relation tells us that in a steady state (SS) implementation of a first-passage process – where trajectories are initiated from some arbitrary state A and terminated upon reaching another, separate state B (at which time they are re-initiated at A) – that the inverse of the mean first-passage time (1\/MFPT) is given exactly<\/em> by the probability flux into B, i.e., the fraction of trajectories entering state B per second.<\/p>\n

\"math\"<\/a><\/p>\n

\u00a0\u00a0\u00a0 = Fraction of trajectories which started in A that arrive to B in steady state per unit time<\/strong><\/p>\n

We assume that trajectories are initiated according to some desired distribution from state A.\u00a0 (Different distributions of initial phase-space points will lead to different MFPTs, not surprisingly.\u00a0 We\u2019re just going to assume you\u2019re happy with some initial distribution, though it\u2019s worth noting that the choice of initial distribution requires a more subtle discussion: see the paper by Bhatt and Zuckerman.)<\/p>\n

\"img1\"<\/a><\/p>\n

To derive the relation, we will imagine running a very large number of fully independent simulations for a very long time.\u00a0 The schematic shows a set of red trajectories projected onto two coordinates.\u00a0 Importantly, we record the full history of each trajectory – the coordinates at all times – and we see a large number of A-to-B transitions.<\/p>\n

\"img2\"<\/a><\/p>\n

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Because we have all information about the trajectories in the steady state, we can go back and examine precisely where each was at all times in the past.\u00a0 This enables us to perform the simple transformation of \u201cstretching\u201d out each trajectory in such a way that we examine it in terms of its temporal progress toward state B.<\/em>\u00a0 This is like waiting at a bus stop with your smart phone and seeing the future arrival times of all the buses, regardless of the bus route.<\/p>\n

Hence we develop a picture like that shown below, with each red trajectory marching toward its arrival in state B.\u00a0 We know how long each trajectory will take to arrive because we (imagine we) have simulated each long into the past and future.<\/p>\n

\"img3\"<\/a><\/p>\n

We can calculate the probability flux into state B in two ways that must be equivalent.\u00a0 First, because our independent trajectories have been running for a very long time, we assume they have become fully decorrelated.\u00a0 Hence we expect that in any interval of time, the same fraction of trajectories will arrive to state B.\u00a0 In the figure, we see that 3 trajectories will arrive in the next two seconds.\u00a0 Equivalently, \u2153 of the trajectories will arrive in 2 seconds, so that the probability flux is \u2159 per second.<\/p>\n

We also expect that, because the trajectories are fully independent and de-correlated, in any time interval \u0394t, the fraction of trajectories arriving should be \u0394t\/MFPT.\u00a0 After all, the MFPT is the average time a trajectory should spend in transit, so our chance to observe it in time \u0394t should be \u0394t\/MFPT on average.\u00a0 (If you want to think about this more concretely, imagine restarting a single trajectory from A each time it arrives to B, creating a sequence of first-passage events.\u00a0 The average of the first-passage times will be the MFPT, by definition, and so the probability to observe an event will \u0394t\/MFPT.)\u00a0 In the example shown, we expect a fraction (2 sec)\/MFPT of the trajectories to arrive in the 2 sec time interval.\u00a0 When the MFPT is 6 seconds, then \u2153 of the trajectories should arrive in 2 sec.\u00a0 Conversely, if we see that \u2153 of the trajectories arrive in 2 sec, we know the MFPT = 6 sec.<\/p>\n

Thus we have it: Probability Flux = 1\/MFPT.\u00a0 Importantly, this is an exact result, not dependent on the type of dynamics or any assumptions about the states or the initial distribution (of starting points).<\/p>\n

The reason I consider the Hill relation a \u201cremarkable result,\u201d even if it gets to seem obvious when you think about it for too long, is that provides a means in principle<\/em> for calculating a very long timescale (the MFPT) from an arbitrarily short observation period (of a steady-state ensemble of trajectories).\u00a0 Although generating a steady-state ensemble in a naive way would require times of the same order as the MFPT itself, there are specialized methods that permit the indirect inference of steady-state information from short simulations.\u00a0 These are path-sampling methods including the weighted ensemble approach, milestoning, transition interface sampling, forward flux sampling, non-equilibrium umbrella sampling and others.<\/p>\n

The Hill relation also exemplifies the power of using the trajectory-ensemble picture.\u00a0 With a minimum of equations, the trajectory-ensemble view of statistical mechanics permits the development of powerful results and calculational methods.\u00a0 Stay tuned for more.<\/p>\n

References<\/p>\n