Uniform asymptotic approximation of diffusion to a small target: Generalized reaction models

Abstract

The diffusion of a reactant to a binding target plays a key role in many biological processes. The reaction-radius at which the reactant and target may interact is often a small parameter relative to the diameter of the domain in which the reactant diffuses. We develop uniform in time asymptotic expansions in the reaction-radius of the full solution to the corresponding diffusion equations for two separate reactant-target interaction mechanisms: the Doi or volume reactivity model, and the Smoluchowski-Collins-Kimball partial absorption surface reactivity model. In the former, the reactant and target react with a fixed probability per unit time when within a specified separation. In the latter, upon reaching a fixed separation, they probabilistically react or the reactant reflects away from the target. Expansions of the solution to each model are constructed by projecting out the contribution of the first eigenvalue and eigenfunction to the solution of the diffusion equation, and then developing matched asymptotic expansions in Laplace-transform space. Our approach offers an equivalent, but alternative, method to the pseudo-potential approach we previously employed in [1] for the simpler Smoluchowski pure absorption reaction mechanism. We find that the resulting asymptotic expansions of the diffusion equation solutions are identical with the exception of one parameter: the diffusion limited reaction rates of the Doi and partial absorption models. This demonstrates that for biological systems in which the reaction-radius is a small parameter, properly calibrated Doi and partial absorption models may be functionally equivalent.

Interlacing relaxation and first-passage phenomena in reversible discrete and continuous space Markovian dynamics

“…(26), which guarantees f (s, k) = F k * (k) (s) to be negative at s = −μ k . The truncated Newton series (27) involving the first N terms is then given by…”

“…In a broader context, first passage concepts were invoked in studies of kinetics in complex media, such as reactions in fractal-like [12,13] and planar domains [14,15], in inhomogeneous cellular environments [16][17][18][19], in the study of neural networks [20,21], ultra cold atoms [22], as well as in diverse narrow escape problems [23][24][25][26][27][28] and so-called intermittent search strategies involving searching agents with internal dynamics [29,30] (see also [31] for a review).…”

Reaction Kinetics in the Few-Encounter Limit

“…In the case of a diffusive exploration of (rugged) energy landscapes the short-time behavior is dominated by free diffusion, rendering the shape of the potential essentially irrelevant. 34,35 It will be interesting to extend the applications of the theory outlined in Appendix A.1 and to explore the physical consequences of fewencounter kinetics also in narrow escape problems [48][49][50][51][52][53] and diffusion on higher-dimensional graphs. Extending the work to irreversible dynamics will be challenging, whereas long-time asymptotics are still accessible.…”

Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus

“…In the case that there is no inactivation, so that the protein simply diffuses until reaching the nucleus, this is an example of a classical diffusion-limited first passage time problem [10]. First passage time problems are widely used in the study of chemical reactions [11,12], with a variety of asymptotic results and exact solution techniques when the target site is small or a basic geometrical shape such as a sphere [13][14][15][16].…”