Practical approximations for diffusion-influenced reactions on two-dimensional surfaces

Theory of bi-molecular association dynamics in 2D for accurate model and experimental parameterization of binding rates

The dynamics of association between diffusing and reacting molecular species are routinely quantified using simple rate-equation kinetics that assume both well-mixed concentrations of species and a single rate constant for parameterizing the binding rate. In two-dimensions (2D), however, even when systems are well-mixed, the assumption of a single characteristic rate constant for describing association is not generally accurate, due to the properties of diffusional searching in dimensions d ≤ 2. Establishing rigorous bounds for discriminating between 2D reactive systems that will be accurately described by rate equations with a single rate constant, and those that will not, is critical for both modeling and experimentally parameterizing binding reactions restricted to surfaces such as cellular membranes. We show here that in regimes of intrinsic reaction rate (ka) and diffusion (D) parameters ka/D > 0.05, a single rate constant cannot be fit to the dynamics of concentrations of associating species independently of the initial conditions. Instead, a more sophisticated multi-parametric description than rate-equations is necessary to robustly characterize bimolecular reactions from experiment. Our quantitative bounds derive from our new analysis of 2D rate-behavior predicted from Smoluchowski theory. Using a recently developed single particle reaction-diffusion algorithm we extend here to 2D, we are able to test and validate the predictions of Smoluchowski theory and several other theories of reversible reaction dynamics in 2D for the first time. Finally, our results also mean that simulations of reactive systems in 2D using rate equations must be undertaken with caution when reactions have ka/D > 0.05, regardless of the simulation volume. We introduce here a simple formula for an adaptive concentration dependent rate constant for these chemical kinetics simulations which improves on existing formulas to better capture non-equilibrium reaction dynamics from dilute to dense systems.

Green's function for reversible geminate reaction with volume reactivity

The kinetics of a diffusing particle near a reversible trap may be described by an extension of the Feynman-Kac equation to the case of reversible binding, which can occur within a finite reaction sphere. We obtain the Green's function solution for the Laplace transform of this equation when the particle is initially either bound or unbound. We study the solution in the time-domain by either inverting the Laplace transform numerically or propagating the partial differential equation in the time-domain. We show that integrals of this solution over the reaction sphere agree with previously obtained solutions.

Exact Green's function of the reversible diffusion-influenced reaction for an isolated pair in two dimensions

We derive an exact Green's function of the diffusion equation for a pair of disk-shaped interacting particles in two dimensions subject to a backreaction boundary condition. Furthermore, we use the obtained function to calculate exact expressions for the survival probability and the time-dependent rate coefficient for the initially unbound pair and the survival probability of the bound state. The derived expressions will be of particular utility for the description of reversible membrane-bound reactions in cell biology.