Reactive Flux Formalism Research Articles

Statistical Mechanical Theories of Membrane Permeability.

A popular theoretical framework to compute the permeability coefficient of a molecule is provided by the classic Smoluchowski-Kramers treatment of the steady-state diffusive flux across a free-energy barrier. Within this framework, commonly termed "inhomogeneous solubility-diffusion" (ISD), the permeability, P, is expressed in closed form in terms of the potential of mean force and position-dependent diffusivity of the molecule of interest along the membrane normal. In principle, both quantities can be calculated from all-atom MD simulations. Although several methods exist for calculating the position-dependent diffusivity, each of these is at best an estimate. In addition, the ISD model does not account for memory effects along the chosen reaction coordinate. For these reasons, it is important to seek alternative theoretical formulations to determine the permeability coefficient that are able to account for the factors ignored by the ISD approximation. Using Green-Kubo linear response theory, we establish the familiar constitutive relation between the flux density across the membrane and the difference in the concentration of a permeant molecule, j = PΔC. On this basis, we derive a time-correlation function expression for the nonequilibrium flux across a membrane that is reminiscent of the transmission coefficient in the reactive flux formalism treatment of transition rates. An analysis based on the transition path theory framework is exploited to derive alternative expressions for the permeability coefficient. The different strategies are illustrated with stochastic simulations based on the generalized Langevin equation in addition to unbiased molecular dynamics simulations of water permeation of a lipid bilayer.

Transition rate theory, spectral analysis, and reactive paths.

The kinetics of a dynamical system dominated by two metastable states is examined from the perspective of the activated-dynamics reactive flux formalism, Markov state eigenvalue spectral decomposition, and committor-based transition path theory. Analysis shows that the different theoretical formulations are consistent, clarifying the significance of the inherent microscopic lag-times that are implicated, and that the most meaningful one-dimensional reaction coordinate in the region of the transition state is along the gradient of the committor in the multidimensional subspace of collective variables. It is shown that the familiar reactive flux activated dynamics formalism provides an effective route to calculate the transition rate in the case of a narrow sharp barrier but much less so in the case of a broad flat barrier. In this case, the standard reactive flux correlation function decays very slowly to the plateau value that corresponds to the transmission coefficient. Treating the committor function as a reaction coordinate does not alleviate all issues caused by the slow relaxation of the reactive flux correlation function. A more efficient activated dynamics simulation algorithm may be achieved from a modified reactive flux weighted by the committor. Simulation results on simple systems are used to illustrate the various conceptual points.

Rate theory of ion pairing at the water liquid-vapor interface: A case of sodium iodide.

Studies on ion pairing at interfaces have been intensified recently because of their importance in many chemical reactive phenomena, such as ion-ion interactions that are affected by interfaces and their influence on kinetic processes. In this study, we performed simulations to examine the thermodynamics and kinetics of small polarizable sodium iodide ions in the bulk and near the water liquid-vapor interface. Using classical transition state theory, we calculated the dissociation rates and corrected them with transmission coefficients obtained from the reactive flux formalism and Grote-Hynes theory. Our results show that in addition to affecting the free energy of ions in solution, the interfacial environments significantly influence the kinetics of ion pairing. The results on the relaxation time obtained using the reactive flux formalism and Grote-Hynes theory present an unequivocal picture that the interface suppresses ion dissociation. The effects of the use of molecular models on the ion interactions as well as the ion-pair configurations at the interface are also quantified and discussed.

Effect of wall-mediated hydrodynamic fluctuations on the kinetics of a Brownian nanoparticle.

The reactive flux formalism (Chandler 1978 J. Chem. Phys.68, 2959-2970. (doi:10.1063/1.436049)) and the subsequent development of methods such as transition path sampling have laid the foundation for explicitly quantifying the rate process in terms of microscopic simulations. However, explicit methods to account for how the hydrodynamic correlations impact the transient reaction rate are missing in the colloidal literature. We show that the composite generalized Langevin equation (Yu et al. 2015 Phys. Rev. E91, 052303. (doi:10.1103/PhysRevE.91.052303)) makes a significant step towards solving the coupled processes of molecular reactions and hydrodynamic relaxation by examining how the wall-mediated hydrodynamic memory impacts the two-stage temporal relaxation of the reaction rate for a nanoparticle transition between two bound states in the bulk, near-wall and lubrication regimes.

Computation of Rate Constants for Diffusion of Small Ligands to and from Buried Protein Active Sites.

The diffusion of ligands to actives sites of proteins is essential to enzyme catalysis and many cellular signaling processes. In this contribution we review our recently developed methodology for calculation of rate constants for diffusion and binding of small molecules to buried protein active sites. The diffusive dynamics of the ligand obtained from molecular dynamics simulation is coarse grained and described by a Markov state model. Diffusion and binding rate constants are then obtained either from the reactive flux formalism or by fitting the time-dependent population of the Markov state model to a phenomenological rate law. The method is illustrated by applications to diffusion of substrate and inhibitors in [NiFe] hydrogenase, CO-dehydrogenase, and myoglobin. We also discuss a recently developed sensitivity analysis that allows one to identify hot spots in proteins, where mutations are expected to have the strongest effects on ligand diffusion rates.

Composite generalized Langevin equation for Brownian motion in different hydrodynamic and adhesion regimes.

We present a composite generalized Langevin equation as a unified framework for bridging the hydrodynamic, Brownian, and adhesive spring forces associated with a nanoparticle at different positions from a wall, namely, a bulklike regime, a near-wall regime, and a lubrication regime. The particle velocity autocorrelation function dictates the dynamical interplay between the aforementioned forces, and our proposed methodology successfully captures the well-known hydrodynamic long-time tail with context-dependent scaling exponents and oscillatory behavior due to the binding interaction. Employing the reactive flux formalism, we analyze the effect of hydrodynamic variables on the particle trajectory and characterize the transient kinetics of a particle crossing a predefined milestone. The results suggest that both wall-hydrodynamic interactions and adhesion strength impact the particle kinetics.

The dynamics of single enzyme reactions: A reconsideration of Kramers' model for colored noise processes

The utility of an approximate heuristic version of Kramers' theory of reaction rates that was earlier used [Chaudhury and Cherayil, J. Chem. Phys. 125, 024904 (2006)] to successfully describe the nonexponential waiting time distributions of the enzyme beta-galactosidase is reassessed. The original model, based on the Smoluchowski equation, is reformulated in terms of the phase space variables of the reaction coordinate, without neglecting inertial contributions. A new derivation of the Fokker-Planck equation (FPE) that describes the dynamics of this coordinate is presented. This derivation, based on functional methods, provides a more direct alternative to the existing distribution function approach used by Hanggi and Mojtabai [Phys. Rev. A 26, 1168 (1982)]. The time-dependent coefficients in the FPE, when incorporated into the exact expression for the transmission coefficient obtained from a reactive-flux formalism [Kohen and Tannor, J. Chem. Phys. 103, 6013 (1995)], are found to yield virtually the same results as the earlier heuristic model.

Inverse Temperature Transition of a Biomimetic Elastin Model: Reactive Flux Analysis of Folding/Unfolding and Its Coupling to Solvent Dielectric Relaxation

The inverse temperature transition (ITT) of a biomimetic model for elastin, capped GVG(VPGVG) in liquid water, is investigated by a comprehensive classical molecular dynamics study. The temperature dependence of the solvation structure and dynamics of the octapeptide are compared using three common force fields, CHARMM, GROMOS, and OPLS. While these force fields differ in quantitative detail, they all predict this octapeptide to undergo a "folding transition" to closed conformations upon heating and a subsequent "unfolding transition" to open conformations at still higher temperatures, thus reproducing the ITT scenario. The peptide kinetics is analyzed within the reactive flux formalism applied to the largest-amplitude mode extracted from principal component analysis, and the solvent's dielectric fluctuations are obtained from the total water dipole autocorrelations. Most importantly, preliminary evidence for an intimate coupling of peptide folding/unfolding dynamics, and thus the ITT, and dielectric relaxation of bulk water is given, possibly being consistent with a "slave mode" picture.

Numerical simulation of transmission coefficient using c-number Langevin equation

We numerically implement the reactive flux formalism on the basis of a recently proposed c-number Langevin equation [Barik et al., J. Chem. Phys. 119, 680 (2003); Banerjee et al., Phys. Rev. E 65, 021109 (2002)] to calculate transmission coefficient. The Kramers’ turnover, the T2 enhancement of the rate at low temperatures and other related features of temporal behavior of the transmission coefficient over a range of temperature down to absolute zero, noise correlation, and friction are examined for a double well potential and compared with other known results. This simple method is based on canonical quantization and Wigner quasiclassical phase space function and takes care of quantum effects due to the system order by order.

Quantum mechanical canonical rate theory: A new approach based on the reactive flux and numerical analytic continuation methods

We present the reactive flux analytic continuation (RFAC) method, based on the quantum reactive flux formalism combined with a numerical analytic continuation approach to calculate quantum canonical rates in condensed phase systems. We express the imaginary time reactive-flux correlation function in terms of a frequency dependent rate constant, and use path integral formalism to derive a working expression suitable for Monte Carlo simulation techniques. The imaginary time data obtained by simulation is analytically continued to the real time using the maximum entropy method to obtain the reaction rate. Motivated by the success of the method to predict the rates for a simple one dimensional parabolic barrier model, we assess its accuracy for a condensed phase reaction modeled by a double-well coupled to a harmonic bath. We note that the method is applicable to a more general Hamiltonian as long as the reaction coordinate can be identified. The reaction rates computed in this fashion are in very good agreement with analytic and numerically exact results. We demonstrate the applicability of the method for a wide range of model parameters and temperatures.

Classical Approximation to Nonradiative Electronic Relaxation in Condensed Phase Systems

We present a study of the classical limit of nonradiative electronic relaxation in condensed phase systems. The discrete Hamiltonian representing an impurity in a condensed phase environment is mapped onto a continuous form using the Meyer−Miller approach. The classical electronic relaxation rate is obtained within the framework of the reactive flux formalism and is compared to the fully quantum mechanical result, and to a mixed quantum−classical approximation. Similar to the case of vibrational relaxation, we find that the fully classical treatment is closer to the fully quantum mechanical rate than the mixed quantum−classical treatment. We provide a time domain analysis of the results.

Derivation of Kramers’ formula for condensed phase reaction rates using the method of reactive flux

Kramers’ formula for the rate of barrier crossing as a function of solvent friction is here rederived using the method of reactive flux. In the reactive flux formalism trajectories are started at the top of the barrier and propagated forward for a short time, to determine whether they are reactive or not. In isolated molecules it is customary to associate with each set of initial conditions a reactivity index (traditionally known as the characteristic function), which is 1 for a reactive trajectory and 0 for a nonreactive trajectory. In this paper we suggest that if the solvent interaction with the system is treated stochastically, it is appropriate to generalize the reactivity index to fractional values between 0 and 1, to take into account an ensemble average over different stochastic histories. We show how this fractional reactivity index can be calculated analytically, by using an analytic solution of the phase space Fokker–Planck equation. Starting with the distribution δ(x)δ(u−u0) that originates at the top of a parabolic barrier (x=0) at t=0, the fraction of the distribution function that is to the right of x=0, in the limit that t→∞, is the fractional reactivity index. The analytical expression for the fractional reactivity index leads immediately to Kramers’ expression for the rate constant. The derivation shows explicitly that the dynamical origin of Kramers’ prefactor is trajectories that recross the barrier. The evolution of the phase space distribution that originates at the top of the barrier highlights an interesting underlying phase space structure of this system, which may be considered as a paradigm for dissipative systems whose underlying dynamics is unstable.