Transition Path Theory
An overview of the theory for analyzing the statistical properties of the reactive trajectories by which transitions occur between a set of initial and final states in a Markov model, and thereby obtain e.g. the highest flux pathways between these states.
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Transition Path Theory Eric Vanden-Eijnden
7.1
Introduction
Markov State Models (MSMs) are meant to be a way to analyze complex time-series data from molecular dynamics (MD) simulations. But MSMs can be complicated themselves and require analysis tools that go beyond simple “lookand-see” techniques. In this chapter we describe a set of such tools based on the framework of Transition Path Theory (TPT) originally introduced in [4] and further developed in [1, 5, 7, 8, 12]. In a nutshell, these tools are aimed at understanding the mechanism and rate of specific reactions in the system, that is, transitions between any particular states or group of states of interest in the MSM. Beside the theory recalled in Sect. 7.2, the main upshots of this chapter are two algorithms presented in Sect. 7.4 that permit to generate directly reactive trajectories (i.e. trajectories by which a specific transition of interest occurs) and loop-erased reactive trajectories (i.e. reactive trajectories from which we have extracted the productive pieces when they progress from the reactant to the product state). We begin by setting up notation and recalling a few basic concepts of the theory of discrete-time Markov chains that will prove useful in the sequel [10]. Let Tij ≡ Tij (τ ), i, j = 1, . . . , N denote the entries of the probability transition maE. Vanden-Eijnden (B) Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA e-mail: [email protected]
trix over the N states of the MSM and let us assume that this matrix satisfies a detailed balance condition (time-reversibility) with respect to the equilibrium probability distribution πi , i.e. πi Tij = πj Tj i ,
∀i, j = 1, . . . , N.
(7.1)
This condition implies that the transition matrix Tij admits the spectral decomposition Tij =
N
(k)
(k)
λk ψi ψj πj
(7.2)
k=1
where the eigenvalue/eigenvector pairs (λk , ψ (k) ) satisfy N
(k)
(k)
Tij ψj = λk ψi ,
k = 1, . . . , N
(7.3)
j =1
with the normalization condition N
ψi(k) ψi(l) πi = δk,l ,
k, l = 1, . . . , N. (7.4)
i=1
Assuming ergodicity, the detailed balance condition (7.1) also implies that all the eigenvectors (1) but the first λ1 = 1 (associated with ψi = 1) are in the interval (−1, 1) and can be ordered as 1 = λ1 > |λ2 | ≥ · · · ≥ |λN | ≥ 0. If we denote by μi (n) the probability to observe the system in state i after n steps (i.e. at time nτ ), this proba-
G.R. Bowman et al. (eds.), An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, Advances in Experimental Medicine and Biology 797, DOI 10.1007/978-94-007-7606-7_7, © Springer Science+Business Media Dordrecht 2014
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bility also admits a spectral decomposition μi (n) =
N
(k)
ck λnk ψi πi ,
ck =
N
(k)
ψj μj (0)
j =1
k=1
(7.5) and so does the probability current Fij (n) entering the forward Kolmogorov equation for μi (n): μi (n + 1) − μi (n) =
N
Fij (n),
j =1
Fij (n) = μj (n)Tj i − μi (n)Tij . (7.6) Explicitly: Fij (n) =
N
(k)
ck λ
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