The equilibrium trajectory ensemble is the best place to start.\u00a0 We\u2019ll \u201cmake\u201d this ensemble – in an old-fashioned thought-experiment way – by observing a large number of systems over a very long time.\u00a0 (You may imagine actual systems or computer simulations – it doesn\u2019t matter.)\u00a0 Regardless of how we started the systems, after enough time they will become fully decorrelated from one another, and we will have an equilibrium ensemble on our hands.<\/p>\n
![\"traj1\"](\"https:\/\/statisticalbiophysicsblog.org\/wp-content\/uploads\/2015\/08\/traj1-300x201.png\")
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In the figure, each arrow schematically represents one trajectory.\u00a0 Importantly, this is NOT a picture of atoms or particles interacting, but a schematic view of many fully independent systems projected onto a single copy of the configuration space.\u00a0 The arrow tip represents the present configuration and the (curved) arrow shaft shows some recent history of the trajectory.<\/p>\n
What can we learn from the equilibrium trajectory ensemble?\u00a0 In fact, anything we want to know about our system, but let\u2019s start with equilibrium quantities of interest, a.k.a. equilibrium \u201cobservables\u201d.\u00a0 Most prominently, if we take a fixed-time snapshot of the ensemble, such as all the configurations at the arrow tips, we have an equilibrium ensemble of configurations.\u00a0 That is, these configurations (x) will be distributed according to the Boltzmann factor, e^[-U(x)\/kT] of the potential energy U if the systems live at temperature T.\u00a0 The equilibrium distribution of configurations <\/em>is obtained from trajectories because dynamics are the ultimate source of equilibrium behavior: nature directly \u201cencodes\u201d only dynamics – we humans infer thermodynamics (e.g., equilibrium). Put another way, no molecule \u201cknows\u201d how it should participate in an ensemble or distribution; a molecule can only blindly behave according to the forces it feels.<\/p>\n With the equilibrium distribution of configurations in hand, we can calculate any equilibrium quantity we like.\u00a0 We can calculate the average of any observable \u2026 just by averaging the observable over the ensemble.\u00a0 We can obtain the potential of mean force along an arbitrary coordinate by histogramming our ensemble along that coordinate and taking a log.\u00a0 The free energy difference between any two configurational states (e.g., A and B in the figure below) can be obtained by taking the log of the ratio of counts of arrow tips in each state.<\/p>\n So what\u2019s different and special about the ensemble of trajectories, compared to the \u201cmere\u201d ensemble of configurations?\u00a0 Trajectories also tell us about dynamical processes – their rates and mechanisms.<\/p>\n From a trajectory ensemble, we can infer a \u201cmechanism\u201d for a process \u2026 but we cannot from a configurational ensemble.\u00a0 Assume we\u2019re interested in transitions from state A to B in the figure.\u00a0 Perhaps A is the closed state of an enzyme and B is the open state.\u00a0 The mechanism may then be defined as the weighted set of pathways taken from A to B – e.g., the fraction of trajectories taking the lower vs the upper pathway in the figure.\u00a0 Physically these might correspond to different orderings of events, such as which sub-domain of the enzyme opens first.\u00a0 (The enzyme adenylate kinase provides a good illustration of the potential for path heterogeneity.)<\/p>\n When we think about mechanism in the trajectory-ensemble picture, we see that some conventional conceptions of mechanism are inadequate for complex systems.\u00a0 For example, the notion of a single transition state or even a \u2018transition state ensemble\u2019 (e.g., of configurations as likely to reach A before B as the reverse) do not describe the sequence possible intermediates through which a complex system may pass, let alone the possible multiplicity of pathways.\u00a0 Our picture, in the literal sense, makes all this clear.<\/p>\n Can kinetic information – rate constants – be obtained from an equilibrium<\/em> ensemble of trajectories?\u00a0 Yes, because trajectories are intrinsically dynamical.\u00a0 The trick is to examine a suitable subset of trajectories.<\/p>\n In a previous post, we saw that the rate (expressed as the inverse of the mean first-passage time, MFPT) could be obtained from a steady-state ensemble of trajectories.\u00a0 This is because, according to the Hill relation<\/a>, the fraction of trajectories arriving to B from A per unit time in a steady state is exactly<\/em> the inverse MFPT.<\/p>\n To obtain a steady-state ensemble of trajectories transitioning from A to B – which in turn will yield the MFPT – we can use a labeling idea developed by vanden Eijnden and others (as re-formulated in Bhatt & Zuckerman, 2011) to select out a subset of the equilibrium trajectory ensemble that corresponds precisely to an A-to-B steady state.\u00a0 Quite simply, we label (or color red, in the figure) all trajectories which were more recently in A in than B.\u00a0 This labeling exactly dissects the equilibrium ensemble into the A-to-B and B-to-A steady-states.\u00a0 (Because our trajectory ensemble is in equilibrium, the same number of red trajectories will turn blue – on entering B – as will turn red per unit time, on average.)<\/p>\n![\"traj2\"](\"https:\/\/statisticalbiophysicsblog.org\/wp-content\/uploads\/2015\/08\/traj2-300x201.png\")
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