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WIREs Comput Mol Sci
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Optimal reaction coordinates

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The dynamic behavior of complex systems with many degrees of freedom is often analyzed by projection onto one or a few reaction coordinates. The dynamics is then described in a simple and intuitive way as diffusion on the associated free‐energy profile. In order to use such a picture for a quantitative description of the dynamics one needs to select the coordinate in an optimal way so as to minimize non‐Markovian effects due to the projection. For equilibrium dynamics between two boundary states (e.g., a reaction), the optimal coordinate is known as the committor or the pfold coordinate in protein folding studies. While the dynamics projected on the committor is not Markovian, many important quantities of the original multidimensional dynamics on an arbitrarily complex landscape can be computed exactly. In this study, we summarize the derivation of this result, discuss different approaches to determine and validate the committor coordinate, and present three illustrative applications: protein folding, the game of chess, and patient recovery dynamics after kidney transplant. WIREs Comput Mol Sci 2016, 6:748–763. doi: 10.1002/wcms.1276 This article is categorized under: Structure and Mechanism > Computational Biochemistry and Biophysics Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics
(a) Free energy as a function of optimal reaction coordinate for the game of chess; x was rascaled so that D(x) = 1. The boards show representative positions for the regions on the landscape. (b) Probability to win, estimated from the free‐energy profile (red line), using MSM (blue line), and directly from the games (crosses). The inset shows the left part of the plot on a logarithmic scale. (Reprinted with permission from Ref . Copyright 2011)
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Free‐energy landscapes as functions of optimal reaction coordinates. The coordinates are rescaled so that the diffusion coefficient is D(x) = 1. (a) FIP35 protein‐folding trajectory (200 µs) with 15 folding–unfolding events. (b) HP35 Nle/Nle mutant protein‐folding trajectory (301 µs) with 160 folding–unfolding events. The representative structures for the regions of the landscape show a trajectory snapshot closest to the average structure of the region. Colors code the root‐mean‐square (rms) fluctuations of atomic positions around the average structure. (Reprinted with permission from: Ref Copyright 2011; Ref . Copyright 2013)
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p(TP|q) criterion applied to a model system (symbols) using a suboptimal coordinate (a) and putative optimal coordinate (b). The line shows the theoretical maximum of 2q(1 − q). The plots were prepared using the fep1d.py script.
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FC,1 criterion applied to a model system. (a) FC,1 increases with increasing Δt indicating that x coordinate is suboptimal. (b) FC,1 is approximately constant, indicating that the putative coordinate Ropt closely approximates the committor. The plots were prepared with the fep1d.py script. (Reprinted with permission from Ref . Copyright 2015)
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(a) Time evolution of kidney transplant patient trajectories projected on the optimal biomarker x (optimal coordinate); x was rescaled so that D(x) = 1. The color indicates the final clinical classification of the patient: primary function (black), delayed graft function (red), and acute rejection (blue). (b) Likelihood of a positive outcome as a function of position along the optimal biomarker: estimated directly from trajectories (blue) and from diffusion on the free‐energy profile (red). The black line shows the free‐energy profile. (Reprinted with permission from Ref . Copyright 2014)
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Theoretical and Physical Chemistry > Statistical Mechanics
Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics
Structure and Mechanism > Computational Biochemistry and Biophysics

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