{"id":275,"date":"2019-03-30T20:16:08","date_gmt":"2019-03-30T20:16:08","guid":{"rendered":"http:\/\/statisticalbiophysicsblog.org\/?p=275"},"modified":"2022-11-02T21:05:48","modified_gmt":"2022-11-02T21:05:48","slug":"a-proof-of-the-discretized-hill-relation","status":"publish","type":"post","link":"https:\/\/statisticalbiophysicsblog.org\/?p=275","title":{"rendered":"A \u201cproof\u201d of the discretized Hill Relation"},"content":{"rendered":"

This is yet another one of those things where, after reading this, you\u2019re supposed to say, \u201cOh, that\u2019s obvious.\u201d And I admit it is kind of obvious \u2026 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.<\/p>\n

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 \u201cB\u201d, defined by us) is exactly<\/em> the inverse MFPT \u2026 so long as we measure that flow in the A-to-B steady state<\/a> based on initializing trajectories outside state B according to some distribution (state \u201cA\u201d, defined by us) and we remove trajectories reaching state B and re-initialize them in A according to our chosen distribution.<\/p>\n

<\/p>\n

\"C:\\Users\\zuckermd\\Box<\/p>\n

A demonstration of the Hill relation itself was previously given<\/a>. Here, we want to consider the discretized version of interest for Markov and related models. I emphasize at the outset that the discretized Hill relation is not exact <\/em>for Markov state models constructed in the usual way from an equilibrium ensemble of trajectories (or an approximation thereto). The relation is only exact for a special type of \u201chistory-augmented\u201d Markov model<\/a> constructed from trajectories harvested from the A-to-B steady state.<\/p>\n

So let\u2019s start our mental exercise by setting up and stating the discretized Hill relation. We\u2019ll consider a discrete set of regions (bins, cells, \u201cmicrostates\u201d) indexed by \"i\" or \"j\" which tile all of phase space. That is, every phase point (configuration and set of velocities if you wish) is in some cell but no phase point is in more than one cell. We\u2019ll also assume that the initial and target states, A and B, consist exactly of non-overlapping sets of these phase-space cells. If \"p^\alpha_i\" is the probability (fractional occupancy) of cell \"i\" in the \"\alpha\" = A-to-B steady state, with \"\sum_i p^\alpha_i = 1\" and \"T^\alpha_{ij}\" is the conditional probability to transition from \"i\" to \"j\" in \u201clag\u201d time \"\tau\" in the \"\alpha\" steady state, then<\/p>\n

\"k_{AB} \equiv \frac{1}{\mathrm{MFPT}(A \to B)}<\/p>\n

This looks messy at first but it\u2019s just a way of counting all the probability flow into state B in the A-to-B steady state, where the flow from \"i\" to \"j\" is \"p^\alpha_i \, T^\alpha_{ij}\" in time \"\tau\".<\/p>\n

We\u2019ll derive the discrete <\/em>relation by working backward from the steady state itself. Note that we\u2019re not proving the (continuous) Hill relation itself \u2013 that\u2019s already done. We\u2019re assuming the continuous form is true, and we will assume we have complete knowledge of the steady state. You can imagine we have a trillion copies of our system running independently and that together these systems constitute a steady state in which systems reaching B are re-initiated at A (according to our chosen distribution within A).<\/p>\n

So we\u2019ve got a lot of information in our hands. We just want to make sure it fits together the right way. And, like I said, I hope you\u2019ll agree all this is obvious \u2026 in the end.<\/p>\n

Let\u2019s first \u201cgenerate\u201d the transition probabilities \"T^\alpha_{ij}\" from the trillion copies of our system. We examine all our systems, and of those that are in cell \"i\" at a given time \"t\", we count the fraction that are found in cell \"j\" at time \"t+\tau\". We average this fraction over all the systems and over all \"t\" to obtain \"T^\alpha_{ij}\", which therefore depends on the chosen \u201clag\u201d time \"\tau\".<\/p>\n

Critically, this transition probability \"T^\alpha_{ij}\" will depend on the intra-cell distribution. For example, in the \"\alpha\" steady state, perhaps more trajectories are closer to one edge of the cell than another.<\/p>\n

<\/p>\n

As the simplest possible instance, consider the case of simple diffusion in one dimension, as sketched above. In the \"\alpha\" steady-state, the density of trajectories decreases linearly as B is approached. Therefore the probability to transition from any cell \"i\" is lower<\/em> to the subsequent cell \"j\" closer to B than to the preceding cell! Importantly, this distribution is different from the equilibrium distribution, which would be used in a standard Markov state model, and which is simply constant in \"x\".<\/p>\n

Moving on. Now that we have the \"T^\alpha_{ij}\", we must obtain the \"p^\alpha_i\" values. We can do this in two ways. Most simply, we can simply calculate the fractional occupancies from our ensemble of systems in the \"\alpha\" steady state. Alternatively, we could use the set of \"T^\alpha_{ij}\" values within a linear algebra (matrix) formulation to calculate the \"p^\alpha_i\", so long as we appropriately set up the source and sink boundary conditions. In any case, we can certainly obtain the steady-state probabilities.<\/p>\n

Now let\u2019s recall our actual goal \u2013 to calculate the probability flow per second into target state B, which is equivalent to the rate (inverse MFPT) according to the continuous Hill relation. Our reference value, which is correct by definition, is just the fraction of all trajectories which newly arrive to B when we examine the ensemble every \"\tau\".<\/p>\n

We want to obtain this same fraction of newly arriving trajectories from the set of \"T^\alpha_{ij}\" and \"p^\alpha_i\" values. Let\u2019s consider just one pair of cells \"i\" and \"j\", where \"j\" is part of B and \"i\" is not. \"T^\alpha_{ij}\" gives the conditional<\/em> probability to arrive in B after a \"\tau\" interval, but we\u2019re interested in the actual probability which will arrive. Well, we know that \"i\" contains probability \"p^\alpha_i\" at all times in steady state. Thus, the actual probability arriving from cell \"i\" to cell \"j\" will be \"p^\alpha_i \, T^\alpha_{ij}\" because \"T^\alpha_{ij}\" gives the fraction of the cell \"i\" probability which will make the transition.<\/p>\n

Now the discrete Hill relation is simply a sum of the \"p \, T\" terms over all cells \"i\" which transition to cells within B. We don\u2019t have to consider any cells which only lead to indirect transitions to B, perhaps over multiple \"\tau\" intervals: after all, everything which makes an indirect transition ultimately makes a direct transition – and we\u2019re including all the direct transitions at every time.<\/p>\n

To summarize, the discrete Hill relation is correct whenever \"T^\alpha_{ij}\" values are correct, and those in turn will be unbiased when calculated based on the steady-state<\/em> distribution of trajectories within every cell \"i\". That is, the fundamental transition probabilities must be estimated from a specific initial distribution within each cell that corresponds to the steady-state of interest. The equilibrium distribution is wrong in this case, and that shouldn’t be surprising given that we’re calculating a non-equilibrium quantity.<\/p>\n

That\u2019s it! I hope in the end all this seems obvious \u2026 but hopefully it wasn’t a total waste of your time!!<\/p>\n","protected":false},"excerpt":{"rendered":"

This is yet another one of those things where, after reading this, you\u2019re supposed to say, \u201cOh, that\u2019s obvious.\u201d And I admit it is kind of obvious \u2026 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 […]<\/p>\n","protected":false},"author":6,"featured_media":277,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,8],"tags":[],"class_list":["post-275","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-first-passage-times","category-trajectory-physics"],"_links":{"self":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/275","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=275"}],"version-history":[{"count":10,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/275\/revisions"}],"predecessor-version":[{"id":288,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/275\/revisions\/288"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/media\/277"}],"wp:attachment":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=275"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=275"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}