Theory of Diffusion-Influenced Reaction Networks

Abstract

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.

Diffusion-induced competitive two-site binding

“…This kinetics is identical to that where site 2 does not exist [Eqs. (2) and (3)]. Thus, in this model, the kinetics of binding to the first site is unaffected by the presence of the second site.…”

“…II A. By applying the theory of diffusion-influenced reaction networks, 2 we obtain a set of the diffusion-modified rate equations, in which the chemical reaction rates become coupled (Sec. II B).…”

“…. .L (2). By using the steady-state approximation for these encounter complexes, the kinetic scheme in Fig.…”

“…Since this ligand is already close to the macromolecule, rebinding will occur faster than binding a ligand from the bulk. In this paper, we will bring our general theory of coupled diffusion-influenced reactions 2 to bear on this problem. We will see that even the simplest version of our formalism leads to the modification of the standard chemical kinetic scheme by introducing a new reaction channel that directly connects the two states that are singly occupied.…”

Reaction-Diffusion Systems: Self-Balancing Diffusion and the Use of the Extent of Reaction as a Descriptor of Reaction Kinetics

“…In this text we are concerned with the latter type. We only mention a recent work on the statistical theory of kinetics altered by diffusion which operates with stoichiometry (Gopich and Szabo, 2018) but not in the sense of constraints as described in this work. Several reviews have recently been published overviewing the basics of a thorough mathematical description of reaction-diffusion systems at the macroscopic level.…”