Kinetic theory of bimolecular reactions in liquid. III. Reversible association–dissociation: A+B⇄C

Multisite reversible association in membranes and solutions: From non-Markovian to Markovian kinetics

The role of diffusion on the kinetics of reversible association to a macromolecule with two inequivalent sites is studied. Previously, we found that, in the simplest possible description, it is not sufficient to just renormalize the rate constants of chemical kinetics, but one must introduce direct transitions between the bound states in the kinetic scheme. The physical reason for this is that a molecule that just dissociated from one site can directly rebind to the other rather than diffuse away into the bulk. Such a simple description is not valid in two dimensions because reactants can never diffuse away into the bulk. In this work, we consider a variety of more sophisticated implementations of our recent general theory that are valid in both two and three dimensions. We compare the predicted time dependence of the concentrations for a wide range of parameters and establish the range of validity of various levels of the general theory.

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Reversible Stochastically Gated Diffusion-Influenced Reactions

An approximate but accurate theory is developed for the kinetics of reversible binding of a ligand to a macromolecule when either can stochastically fluctuate between reactive and unreactive conformations. The theory is based on a set of reaction-diffusion equations for the deviations of the pair distributions from their bulk values. The concentrations are shown to satisfy non-Markovian rate equations with memory kernels that are obtained by solving an irreversible geminate (i.e., two-particle) problem. The relaxation to equilibrium is not exponential but rather a power law. In the Markovian limit, the theory reduces to a set of ordinary rate equations with renormalized rate constants.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

Time-dependent rate coefficients for diffusion-influenced reactions with centrosymmetric potentials

Simple closed-form expressions are presented for the time-dependent rate coefficients of diffusion-influenced reactions in the presence of spherically symmetric potentials. For diffusion-controlled contact reactions, our expression reproduces the first two terms in both the short- and long-time expansions of the rate coefficient. At intermediate times, agreement with numerical results for the Debye-Hückel potential is found to be within a few percent for a wide range of parameters. For diffusion-influenced contact reactions (described by the radiation boundary condition), the agreement is even better. When the reactivity depends on the distance between the reactants (e.g., exponentially), our analytic result is less accurate, because it reproduces the two terms in the long-time expansion only to the linear order of the reciprocal of the diffusion coefficient. Our results should prove useful in the analysis of experimental data for diffusion-influenced reactions with centrosymmetric interaction potentials.

Rate kernel theory for pseudo-first-order kinetics of diffusion-influenced reactions and application to fluorescence quenching kinetics

Theoretical foundation of rate kernel equation approaches for diffusion-influenced chemical reactions is presented and applied to explain the kinetics of fluorescence quenching reactions. A many-body master equation is constructed by introducing stochastic terms, which characterize the rates of chemical reactions, into the many-body Smoluchowski equation. A Langevin-type of memory equation for the density fields of reactants evolving under the influence of time-independent perturbation is derived. This equation should be useful in predicting the time evolution of reactant concentrations approaching the steady state attained by the perturbation as well as the steady-state concentrations. The dynamics of fluctuation occurring in equilibrium state can be predicted by the memory equation by turning the perturbation off and consequently may be useful in obtaining the linear response to a time-dependent perturbation. It is found that unimolecular decay processes including the time-independent perturbation can be incorporated into bimolecular reaction kinetics as a Laplace transform variable. As a result, a theory for bimolecular reactions along with the unimolecular process turned off is sufficient to predict overall reaction kinetics including the effects of unimolecular reactions and perturbation. As the present formulation is applied to steady-state kinetics of fluorescence quenching reactions, the exact relation between fluorophore concentrations and the intensity of excitation light is derived.

Corrections to the law of mass action and properties of the asymptotic t=∞ state for reversible diffusion-limited reactions

For diffusion-limited reversible A+A<==>B reactions we reexamine two fundamental concepts of classical chemical kinetics-the notion of "chemical equilibrium" and the "law of mass action." We consider a general model with distance-dependent reaction rates, such that any pair of A particles, performing standard random walks on sites of a d-dimensional lattice and being at a distance mu apart of each other at time moment t, may associate forming a B particle at the rate k+(mu). In turn, any randomly moving B particle may spontaneously dissociate at the rate k-(lambda) into a geminate pair of As "born" at a distance lambda apart of each other. Within a formally exact approach based on Gardiner's Poisson representation method we show that the asymptotic t=infinity state attained by such diffusion-limited reactions is generally not a true thermodynamic equilibrium, but rather a nonequilibrium steady state, and that the law of mass action is invalid. The classical concepts hold only in case when the ratio k+(mu)k-(mu) does not depend on mu for any mu.

The integral encounter theory of multistage reactions containing association-dissociation reaction stages. III. Taking account of quantum states of reactants

The formalism developed in Part I [K. L. Ivanov, N. N. Lukzen, A. A. Kipriyanov, and A. B. Doktorov, Phys. Chem. Chem. Phys. 6, 1706 (2004)] of the present contribution is extended to treat the reacting particles with internal quantum states. Initial spatial correlations of reactants are considered in the framework of this formalism as well.

Influence of diffusion on the kinetics of excited-state association–dissociation reactions: Comparison of theory and simulation

Several recent theories of the kinetics of diffusion influenced excited-state association--dissociation reactions are tested against accurate Brownian dynamics simulation results for a wide range of parameters. The theories include the relaxation time approximation (RTA), multiparticle kernel decoupling approximations and the so-called kinetic theory. In the irreversible limit, none of these theories reduce to the Smoluchowski result. For the pseudo-first-order target problem, we show how the RTA can be modified so that the resulting formalism does reduce correctly in the irreversible limit. We call this the unified Smoluchowski approximation, because it unites modern theories of reversible reactions with Smoluchowski's theory of irreversible reactions.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Rigorous derivation of the long-time asymptotics for reversible binding

Using an iterative solution in Laplace-Fourier space, we supply a rigorous mathematical proof for the long-time asymptotics of reversible binding in one dimension. The asymptotic power law and its concentration dependent prefactor result from diffusional and many-body effects which, unlike for the corresponding irreversible reaction and in classical chemical kinetics, play a dominant role in shaping the approach to equilibrium.