Reaction coordinates and rates from transition paths

Abstract

The molecular mechanism of a reaction in solution is reflected in its transition-state ensemble and transition paths. We use a Bayesian formula relating the equilibrium and transition-path ensembles to identify transition states, rank reaction coordinates, and estimate rate coefficients. We also introduce a variational procedure to optimize reaction coordinates. The theory is illustrated with applications to protein folding and the dipole reorientation of an ordered water chain inside a carbon nanotube. To describe the folding of a simple model of a three-helix bundle protein, we variationally optimize the weights of a projection onto the matrix of native and nonnative amino acid contacts. The resulting onedimensional reaction coordinate captures the folding transition state, with formation and packing of helix 2 and 3 constituting the bottleneck for folding.carbon nanotubes ͉ chemical kinetics ͉ protein folding ͉ transition-state theory ͉ Grotthuss mechanism I dentifying the molecular mechanism of a reaction in solution, such as protein folding or enzyme-catalyzed chemistry, poses serious challenges because of the large number of coupled degrees of freedom (1-5). The identification and characterization of populated intermediate states along reaction paths is only a first step. Understanding the mechanism at a molecular level requires in addition the characterization of the transitions between those populated intermediates. The goal then is (i) to identify what is common to the transitions in a rare molecular reaction (or significant subsets thereof), and (ii) to find coordinates that not only measure the progress of the reaction but also are useful to characterize the reaction dynamics. The former leads to the concept of a transition state, the latter to that of a reaction coordinate.Transition states can be thought of as configurations ''intermediate'' between reactants and products. In one widely used definition (6-11), the ensemble of transition states comprises those configurations that have an equal probability of reaching reactant and product regions. The chance of proceeding to reactants or products first can be quantified by the splitting (or commitment) probability introduced by Onsager for ion-pair recombination (12). The splitting probability is defined as the fraction of trajectories reaching the reactant region first when initiated from a given configuration with random MaxwellBoltzmann velocities, and possibly averaged over noise for stochastic dynamics. We note that one of the difficulties arising from the above definition of transition states is that splitting probabilities cannot be measured experimentally, not even in a single-molecule measurement with atomic resolution. The reason is that multiple initializations with precise atomic positions, including those of solvent molecules, are required. Here, we will show how this difficulty can be circumvented by calculating average splitting probabilities (13) from transitionpath and equilibrium ensembles that can also be measured experimentally.Fro...

Role of the Prestructured Surface Cloud in Crystal Nucleation

“…Given two metastable states, A (e.g., the liquid phase) and B (e.g., the solid phase), separated by a FE barrier, the perfect RC is the committor function p B ðxÞ, the probability that dynamical trajectories initiated at configuration x, consisting of the positions of all particles, actually reach the crystalline phase B [4,5]. This function provides a quantitative measure for the progress of a reaction in the sense that it tells us what is likely to happen next [4,6]. However, computing the committor directly requires sampling a large number of trajectories and moreover does not yield any new insight into the nucleation mechanism.…”

Diffusion maps, clustering and fuzzy Markov modeling in peptide folding transitions

“…In poor representations we expect large hysteresis effects because distant, slowly equilibrating regions of conformation space are erroneously not separated in projection. 8 By contrast, in a good representation the projection provides us with suitable "reaction coordinates" 9 along which the motion has minimal memory effects; 10 even driven motion in such coordinates is expected to stay close to equilibrium, thus minimizing dissipation effects because of coupling to unrelaxed motions transverse to the chosen reaction coordinate. Examples of work motivated by these ideas and making use of biased sampling in reduced coordinates include accelerating stochastic simulations 11,12 and systematically identifying low dimensional parametrizations of a biomolecule's free energy surface.…”

A benchmark for reaction coordinates in the transition path ensemble

“…23 Methods that do not rely on evaluating committor are available as well. [24][25][26][27][28] Best and Hummer proposed to examine the distribution of the probability of a trajectory being a transition path, given that the trajectory passes through configurations with a fixed value of a candidate coordinate-the distribution would have the sharpest peak if the candidate is the reaction coordinate. 24 This method was later shown to be equivalent to optimizing the separatrix by Peters.…”

“…[24][25][26][27][28] Best and Hummer proposed to examine the distribution of the probability of a trajectory being a transition path, given that the trajectory passes through configurations with a fixed value of a candidate coordinate-the distribution would have the sharpest peak if the candidate is the reaction coordinate. 24 This method was later shown to be equivalent to optimizing the separatrix by Peters. 29 Recently, Peters et al reported that reaction coordinate can be identified from an ensemble of short trajectory swarms 25 if dynamics of the system along the reaction coordinate follows a one-dimensional Smoluchowski equation.…”