A Comparison of Bimolecular Reaction Models for Stochastic Reaction–Diffusion Systems

Abstract

Stochastic reaction–diffusion models have become an important tool in studying how both noise in the chemical reaction process and the spatial movement of molecules influences the behavior of biological systems. There are two primary spatially-continuous models that have been used in recent studies: the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching a fixed separation (called the reaction-radius). The Doi model uses reaction potentials, whereby two molecules react with a fixed probability per unit time, λ, when separated by less than the reaction radius. In this work, we study the rigorous relationship between the two models. For the special case of a protein diffusing to a fixed DNA binding site, we prove that the solution to the Doi model converges to the solution of the Smoluchowski model as λ→∞, with a rigorous $O(\lambda^{-\frac{1}{2} + \epsilon})$ error bound (for any fixed ϵ>0). We investigate by numerical simulation, for biologically relevant parameter values, the difference between the solutions and associated reaction time statistics of the two models. As the reaction-radius is decreased, for sufficiently large but fixed values of λ, these differences are found to increase like the inverse of the binding radius.

Appendices

Appendix A: Cumulative Binding Time Distributions

The cumulative binding time distributions we evaluated in Sect. 3, $\operatorname{Prob}[T_{\textrm{Smol}} < t ]$ and $\operatorname{Prob}[T_{\textrm{Doi}}< t ]$ are given by the series expansions

and

where for

$$ h(\alpha) = \frac{1}{\alpha^{\frac{3}{2}} R} \bigl[ \bigl( (R-r_{\textrm{b}}) \sqrt{\alpha} \bigr) \cos \bigl((R-r_{\textrm{b}}) \sqrt{\alpha} \bigr) - (1 + R r_{\textrm {b}}\alpha ) \sin \bigl( (R-r_{\textrm{b}}) \sqrt{\alpha} \bigr) \bigr], $$

we have

and

with

$$ \ell_n = \frac{\tfrac{1}{R\sqrt{\mu_n}}\sin (\sqrt{\mu_n}(R-r_{\textrm{b}}))-\cos(\sqrt{\mu_n}(R-r_{\textrm {b}}))}{\lambda- \mu_n}. $$

Appendix B: Mean Binding Time

Let $\mathbb {E}[T_{\textrm{Doi}} ]$ denote the mean time at which the two molecules in the Doi model (3) first react when initially separated by r ₀. $\mathbb {E}[T_{\textrm{Doi}} ]$ can be shown to satisfy (Gardiner 1996; Van Kampen 2001)

$$ \varDelta_{r_0} \mathbb {E}[T_{\textrm{Doi}} ](r_0) - \hat{\lambda } \, \mathbf {1}_{\{r < r_{\textrm{b}}\}}(r) \mathbb {E}[T_{\textrm {Doi}} ](r_0) =-\frac{1}{D}, \quad0 \leq r_0 < R, $$

(24)

with the boundary condition,

(Here, $\hat{\lambda} = \lambda/ D$.) The solution to (24) is given by (12).

The mean time, $\mathbb {E}[T_{\textrm{Smol}} ](r_{0})$, at which the two molecules in the Smoluchowski model (2) first react when initially separated by r ₀ can be shown to satisfy (Gardiner 1996; Van Kampen 2001)

$$ \varDelta_{r_0} \mathbb {E}[T_{\textrm{Smol}} ](r_0) =-\frac{1}{D}, \quad r_{\textrm{b}}< r_0 < R, $$

(25)

with the boundary conditions,

$$\mathbb {E}[T_{\textrm{Smol}} ](r_{\textrm{b}}) = 0, \qquad \frac {\partial \mathbb {E}[T_{\textrm{Smol}} ]}{\partial r_0}(R) = 0. $$

The solution to (25) is given by (13).

Appendix C: Discrete Space and Time Points

The spatial evaluation points, r _i , are generated in MATLAB by

Listing 1

r _i points

The time evaluation points, t _j , are generated in MATLAB by

Listing 2

t _j points