Nearly reducible finite Markov chains: Theory and algorithms

Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications.

Basis expansion leaping: a new method to solve the time-dependent Schrödinger equation for molecular quantum dynamics

A wide variety of molecular systems that have recently come into the reach of experimental and theoretical investigation is dominated by quantum phenomena. However, even state of the art quantum propagation techniques are either unsuitable for general application to molecular systems with strong interference and tunneling characteristics or are computationally prohibitive for systems with more than a few degrees of freedom. In this Letter, we introduce a novel quantum propagation technique with wide applicability, controllable accuracy, and efficient utilization of computational resources. Its performance is validated for tunneling and dissociating systems with 1, 2, and 3 degrees of freedom, and the scaling behavior with respect to system dimensionality and requested accuracy is discussed.

Interception of excited vibrational quantum states by O2 in atmospheric association reactions

Bimolecular reactions in Earth's atmosphere are generally assumed to proceed between reactants whose internal quantum states are fully thermally relaxed. Here, we highlight a dramatic role for vibrationally excited bimolecular reactants in the oxidation of acetylene. The reaction proceeds by preliminary adduct formation between the alkyne and OH radical, with subsequent O(2) addition. Using a detailed theoretical model, we show that the product-branching ratio is determined by the excited vibrational quantum-state distribution of the adduct at the moment it reacts with O(2). Experimentally, we found that under the simulated atmospheric conditions O(2) intercepts ~25% of the excited adducts before their vibrational quantum states have fully relaxed. Analogous interception of excited-state radicals by O(2) is likely common to a range of atmospheric reactions that proceed through peroxy complexes.

OH-Initiated Oxidation of Toluene. 2. Master Equation Simulation of Toluene Oxide Isomerization

In this work we continue our investigation of the toluene-OH-O2 system. We describe master equation modeling of the isomerization of toluene oxide, focusing on the formation of the cresols. A 15 isomer model is used. Simulations of both thermally activated processes and photolysis processes are described. In accord with experiment, it is found that photolysis with a high-energy light source should be expected to give a high yield of the thermal distribution of cresol products (dominated by the para isomer). Photolysis with a low-energy light source, on the other hand, favors formation of the thermally disfavored ortho isomer. Though the 15 isomer system is potentially an excellent test bed for the development of scalable master equation solution methods, existing scalable solution methods were found to fail on this system.

OH-Initiated Oxidation of Toluene. 1. Quantum Chemistry Investigation of the Reaction Path

In this paper we present the results of a detailed quantum chemistry investigation of the toluene-OH-O2 system, mostly at the B3LYP/6-311G(2df,2pd) level. We focus on OH addition followed by H abstraction to O2, a mechanism based on that proposed by Klotz et al. [Phys. Chem. Chem. Phys. 2000, 2, 227] to explain the experimentally observed photolysis products. A notable feature of the calculated minimum energy pathway is the formation of ketone intermediates during the isomerization from the toluene oxides to the cresols. The quantum chemistry results largely support the plausibility of the mechanism proposed by Klotz et al. The system provides a rich set of reactions with which to test statistical kinetic theories.

Model for Proton Transport Coupled to Protein Conformational Change: Application to Proton Pumping in the Bacteriorhodopsin Photocycle

A modeling method is presented for protein systems in which proton transport is coupled to conformational change, as in proton pumps and in motors driven by the proton-motive force. Previously developed methods for calculating pKa values in proteins using a macroscopic dielectric model are extended beyond the equilibrium case to a master-equation model for the time evolution of the system through states defined by ionization microstate and a discrete set of conformers. The macroscopic dielectric model supplies free energy changes for changes of protonation microstate, while the method for obtaining the energetics of conformational change and the relaxation rates, the other ingredients needed for the master equation, are system dependent. The method is applied to the photoactivated proton pump, bacteriorhodopsin, using conformational free energy differences from experiment and treating relaxation rates through three adjustable parameters. The model is found to pump protons with an efficiency relatively insensitive to parameter choice over a wide range of parameter values, and most of the main features of the known photocycle from very early M to the return to the resting state are reproduced. The boundaries of these parameter ranges are such that short-range proton transfers are faster than longer-range ones, which in turn are faster than conformational changes. No relaxation rates depend on conformation. The results suggest that an "accessibility switch", while not ruled out, is not required and that vectorial proton transport can be achieved through the coupling of the energetics of ionization and conformational states.