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

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 pseudopotential approach we previously employed [Isaacson and Newby, Phys. Rev. E 88, 012820 (2013)] 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.

Figure 1

(a) Spherically symmetric first-passage-time density $\left(\varepsilon =0.01\right)$ and (b) max norm difference between the first-passage-time densities $\left(r_0=0.3\right)$. We set $D=1$ and $\widehat{\gamma }=\sqrt{\widehat{\mu }}/tanh\left(\sqrt{\widehat{\mu }}\right)-1$. With these choices, ${\widehat{k}}_{\mathrm{R}}={\widehat{k}}_{\mathrm{D}}$, so solutions to the Robin and Doi models have identical asymptotic expansions through $O\left({\varepsilon }^2\right)$.

Figure 2

First-passage-time density, with the target center fixed at ${\mathit{r}}_{\mathrm{b}}=({\theta }_b,{\varphi }_b,r_b)=(0,0,0.5)$. The initial position is ${\mathit{r}}_0=(0,0,r_0)$. The approximations (solid lines) are compared to normalized histograms obtained from Monte Carlo simulations, which are plotted as rectangles centered at the histogram value. The height of each rectangle represents a $95\%$ confidence interval. Thick white lines (gray rectangles) show the approximation of the pure-absorption problem. Thin white lines correspond to the approximation of the Robin (red rectangles) and Doi (black rectangles) partial reaction mechanisms. The parameter values are $\varepsilon =0.05,D=1,\widehat{\mu }=5$, and $\widehat{\gamma }=\sqrt{\widehat{\mu }}/tanh\left(\sqrt{\widehat{\mu }}\right)-1$.

Figure 3

Same as Fig. 2 with different choices of the target location ${\mathit{r}}_{\mathrm{b}}$ and initial position ${\mathit{r}}_0$ such that $\left|{\mathit{r}}_{\mathrm{b}}-{\mathit{r}}_0\right|=0.7$.

Figure 4

Approximation for the (a) first-passage-time density $f\left(t\right)$ and (b) joint distribution function $p(\mathit{r},t)$, where $\mathrm{Prob}[\mathbf{R}\left(t\right)\in {\mathcal{B}}_{d\mathit{r}}\left(\mathit{r}\right),t<T]=p(\mathit{r},t)d\mathit{r}$. The target radius is $\varepsilon =0.03$. The target and initial position are placed along the centerline $({\theta }_b=\pi ,{\theta }_0=0)$ at a distance of $r_b=r_0=0.35$.