{"id":431,"date":"2020-04-26T00:21:10","date_gmt":"2020-04-26T00:21:10","guid":{"rendered":"http:\/\/statisticalbiophysicsblog.org\/?p=431"},"modified":"2020-04-26T00:21:14","modified_gmt":"2020-04-26T00:21:14","slug":"from-continuous-to-discrete-time-exercise-7","status":"publish","type":"post","link":"https:\/\/statisticalbiophysicsblog.org\/?p=431","title":{"rendered":"From continuous to discrete time (Exercise 7)"},"content":{"rendered":"

Give yourself a pat on the back if you\u2019ve come this far. You have used simple exact solutions to differential equations to grasp the essentials of non-equilibrium processes. But there\u2019s the physical process on the one hand, and the mathematical description on the other. We\u2019ve used continuous-time math thus far. We now move to discrete time and get a taste for \u201cMarkov state models,\u201d which implicitly employ time discretization in the field of biomolecular simulation.<\/p>\n

<\/p>\n

Let\u2019s first review the questions from last time, based on the following pictures.<\/p>\n

\"C:\\Users\\zuckermd\\Box\\blog\\figs\\dbl-well.png\"<\/p>\n

<\/p>\n

    \n
  1. \n Draw a discrete<\/em> free<\/em> energy-level diagram corresponding to the continuous two state (A,B) system sketched above. Your diagram should consist of three free energy levels (A, B, barrier). Write down the Arrhenius expressions<\/a> (in which the rate constant is proportional to the Boltzmann factor of the free-energy barrier height) in both directions. If the conformational free energy \"F_i\" is defined according to \"P^{eq}_i \propto \exp(-F_i/k_B T)\", show that using Arrhenius-like rate constants causes detailed-balance to be satisfied.<\/p>\n
      \n
    1. \n In the Arrhenius picture, the rate constant for a transition is given by the product of a prefactor \"k_0\", which can be considered an attempt frequency, and a Boltzmann factor of the barrier height, which can be considered the probability of success. One ambiguity is whether the Boltzmann factor should be given in terms of the potential or free energy. We shall see that the free energy makes the estimate more reasonable and convenient, but note that the whole formulation is somewhat ad hoc and non-rigorous. We have <\/p>\n

        <\/span>   <\/span>\"\[k_{AB} = k_0 \, e^{-(F^\ddagger - F_A)/k_B T} \hspace{1cm} k_{BA} = k_0 \, e^{-(F^\ddagger - F_B)/k_B T}.\]\"<\/p>\n

      Having used the same<\/em> prefactor \"k_0\" for both rates leads to the satisfaction of detailed balance based on the equilibrium Boltzmann factor given above, \"P^{eq}_i = C \exp(-F_i/k_B T)\", where the unknown constant \"C\" is the same<\/em> for both A and B. Detailed balance is confirmed by simple multiplication: <\/p>\n

        <\/span>   <\/span>\"\[P^{eq}_A k_{AB} = C k_0 e^{-F^\ddagger/k_B T} = P^{eq}_B k_{BA}\]\"<\/p>\n<\/li>\n<\/ol>\n<\/li>\n

    2. \n What reasonable explanation can be given for the Arrhenius rate expression in terms of continuous-space equilibrium-ish statistical mechanics? And why should entropy<\/em> differences enter the expression based on a one-dimensional picture?<\/p>\n
        \n
      1. \n The dimensionless Boltzmann factor of the free energy difference (between barrier top and initial state) represents a relative equilibrium<\/em> probability. In the context of a non-equilibrium transition, a hand-waving explanation is that this dimensionless probability represents the chances to surmount the barrier given a constant attempt frequency \"k_0\". Entropy enters the free energy of the initial state in the standard way, so that larger entropy means lower free energy and hence lower success probability. The intuition is that entropy for a single state characterizes the effective<\/a> width of that state, so a wider state means a trajectory will reach the edge less often and effectively have a lower attempt frequency. Of course, we have assumed the attempt frequency is constant, so the entropy of the initial state can be said to correct this frequency.<\/li>\n<\/ol>\n<\/li>\n
      2. \n Describe a procedure by which you could computationally estimate the rate constants of a continuous two-state system by running trajectories.<\/p>\n
          \n
        1. \n We could simply start many trajectories in state A and run molecular dynamics until each reaches B. From that data, we could calculate the mean first-passage time (MFPT) and estimate the rate as 1\/MFPT. Or we could calculate correlation functions from the simulated trajectories and use them<\/a> to estimate rates.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

          It\u2019s finally time to connect our knowledge of continuous-time<\/em> systems to discrete-time<\/em> dynamical descriptions of discrete states, known as \u201cMarkov state models.\u201d That is, how do Markov state models arise from continuous descriptions? We have just made (above) a crude connection between continuous potentials and discrete states, and this will suffice for now. So we will implicitly assume our discrete states are single deep energy basins that indeed behave in a Markovian fashion \u2013 i.e., the probability of transitions out of the state do not depend on the path taken to get to the state. Note that this is very <\/em>special property<\/a> and one you should not <\/em>generally expect to be satisfied in a tractable way for an atomically accurate description of a complex system \u2013 though this is a topic for another day.<\/p>\n

          The following questions build directly on notes<\/a> you have already been studying. We will adopt the somewhat annoying notation that \"T_{ji} = T_{ji}(\Delta t)\" refers to the \"i \to j\" transition probability. There\u2019s a technical reason for this if you want to deal with the matrices, but at least I want to be consistent with my prior notes.<\/p>\n

            \n
          1. \n Give the full definition of \"T_{ji}(\Delta t)\" and explain why it\u2019s dimensionless.<\/li>\n
          2. \n If \"P_i(t)\" is the time-dependent probability of state \"i\", show that \"P_i(t+\Delta t) = \sum_j T_{ij} P_j(t)\".<\/li>\n
          3. \n Using the exact solutions to the continuous-time<\/em> two-state probabilities, \"P_A(t), P_B(t)\", which we derived previously<\/a>, calculate the discrete-time<\/em> transition probabilities \"T_{AB}(\Delta t)\" and \"T_{AA}(\Delta t)\" for transitions into<\/em> state A. This problem is trickier than it sounds (though the math is easy), but don\u2019t worry the derivation is there for you in the notes<\/a>. The key trick is to write \"P_A(t+\Delta t)\" in terms of \"P_A(t)\" and \"P_B(t)\" using the fact that \"P_A + P_B = 1\" for any\/all \"t\".<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

            Give yourself a pat on the back if you\u2019ve come this far. You have used simple exact solutions to differential equations to grasp the essentials of non-equilibrium processes. But there\u2019s the physical process on the one hand, and the mathematical description on the other. We\u2019ve used continuous-time math thus far. We now move to discrete […]<\/p>\n","protected":false},"author":6,"featured_media":440,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,19],"tags":[],"class_list":["post-431","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-general-biophysics","category-non-equilibrium-physics"],"_links":{"self":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/431","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=431"}],"version-history":[{"count":5,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/431\/revisions"}],"predecessor-version":[{"id":442,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/posts\/431\/revisions\/442"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=\/wp\/v2\/media\/440"}],"wp:attachment":[{"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=431"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=431"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statisticalbiophysicsblog.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=431"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}