Reaction rates for a generalized reaction-diffusion master equation

Abstract

It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.

I. INTRODUCTION

Stochastic modeling has become a ubiquitous tool in the study of biochemical reaction networks [1–5], as the traditional approach of deterministic modeling has been shown to be unsuitable for some systems where species are present in low copy numbers, or systems with spatial inhomogeneities [3, 6]. Instead stochastic, spatially homogenous or inhomogeneous, models are employed.

Stochastic modeling can be carried out on multiple different scales. For processes occurring on the time scales typical of living cells we consider three different modeling scales: the spatially homogeneous well-mixed scale, the mesoscopic spatially heterogeneous scale, and the microscopic particle-tracking scale. In this paper the focus is on spatially heterogeneous modeling.

A prevalent model on the mesoscopic scale is the standard reaction-diffusion master equation, in which diffusion of individual molecules is modeled by discrete jumps between voxels, while reactions occur with a given intensity once molecules occupy the same voxel. The next subvolume method (NSM) [7] is an efficient algorithm for generating single trajectories of the system. The NSM has been implemented in several software packages, including URDME [8], PyURDME (www.pyurdme.org), STEPS [9], and MesoRD [10]. It is also available as a part of larger simulation frameworks such as StochSS (www.stochss.org) and E-Cell [11].

On the microscopic scale we model the molecules as hard spheres moving by normal diffusion. We track the continuous position of individual molecules, and molecules react with a probability upon collision. This model is commonly referred to as the Smoluchowski model [12], with the addition of a Robin boundary condition at the reaction radius of a pair of molecules. Algorithms aimed at accurately and efficiently simulating the Smoluchowski model for general systems have been implemented in E-Cell [11], Smoldyn [13], and MCell [14].

It has previously been shown that there is an inherent bound of several reaction radii on the spatial accuracy of the standard RDME compared to the Smoluchowski model [15, 16]. It was shown in [16] that by choosing correct mesoscopic reaction rates, the standard RDME could be made accurate all the way down to this lower bound. However, for mesh resolutions below this lower bound, the accuracy deteriorates.

In this paper we generalize the standard RDME by allowing molecules occupying neighboring voxels to react. Henceforth we refer to this generalization as the generalized RDME. The acronym RDME usually refers to the standard RDME, but to minimize the possibility of confusion as to which of the algorithms we are referring to, we will adopt the acronym sRDME for the standard RDME and gRDME for the generalized RDME. Similar generalizations have been considered previously in [17, 18]. In [17], Isaacson discretizes the Doi model [19] to obtain a convergent RDME. In [18], reaction rates are derived for a spherical model and applied to the RDME on a Cartesian mesh. In this paper we take a fundamentally different approach. By deriving reaction rates to match certain statistics of the Smoluchowski model, we arrive at analytical expressions for the reaction rates, and show that this approach yields accurate results down to a fundamental lower limit on the mesh size. This mesh size will be on the order of the reaction radius of two molecules.

Importantly, we derive reaction rates under specific assumptions on the dynamics of dissociating molecules, and we show with a simple example that not doing so may lead to reaction rates that are inaccurate for certain systems. We thus argue that it is crucial to take dissociations into account in the derivation of reaction rates for the gRDME.

The outline of the paper is as follows. In Section II we review the Smoluchowski model and the sRDME, and how they are connected through the mesoscopic reaction rates. In Section III we describe the generalized algorithm, and derive accurate mesoscopic reaction rates as well as the lower limit on the mesh size. Finally, in Section IV, we study two numerical examples, demonstrating the accuracy of the gRDME and how it can be used to simulate systems that are intractable with the sRDME.

II. BACKGROUND

A. Microscopic level

At this level of modeling we track the continuous position of individual molecules, modeled by hard spheres and moving by normal diffusion. Each species S_i has a diffusion constant D_i and a radius of σ_i, called the reaction radius. Consider two molecules, one of species S₁ and one of species S₂, with positions x_1n and x_2n at time t_n. The molecules can react according to $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$, where S₃ is some product. The probability distribution function (PDF) p(x₁, x₂, t|x_1n, x_2n, t_n) represents the probability that the positions of the molecules are given by x₁ and x₂ at time t; p then satisfies the Smoluchowski equation

$${\partial }_{t}p=D_1{\mathrm{\Delta }}_{{\mathbf{x}}_1}p+D_2{\mathrm{\Delta }}_{{\mathbf{x}}_2}p,$$ (1)

with a reactive Robin boundary condition

$${K\frac{\partial p_{{\mathbf{x}}_2-{\mathbf{x}}_1}}{\partial n}|}_{\Vert {\mathbf{x}}_2-{\mathbf{x}}_1\Vert =\sigma }=k_{a}p(\Vert {\mathbf{x}}_2-{\mathbf{x}}_1\Vert =\sigma ,t),$$ (2)

where D = D₁ + D₂, σ = σ₁ + σ₂ is the sum of the reaction radii, k_a is the microscopic reaction rate, and

$$K=\{\begin{array}{cc}2\pi \sigma D,\hfill & (2D)\hfill \\ 4\pi {\sigma }^{2}D,\hfill & (3D).\hfill \end{array}$$ (3)

The initial condition is given by

$$p({\mathbf{x}}_2-{\mathbf{x}}_1,t_n)=\delta (({\mathbf{x}}_2-{\mathbf{x}}_1)-({\mathbf{x}}_{2n}-{\mathbf{x}}_{1n}))$$ (4)

and, since we assume that there is no outer boundary, we enforce p(‖x₂ − x₁‖ → ∞, t) = 0.

It can be shown that with the change of variables

$$\mathbf{Y}=\sqrt{\frac{D_2}{D_1}}{\mathbf{x}}_1+\sqrt{\frac{D_1}{D_2}}{\mathbf{x}}_2$$ (5)

$$\mathbf{y}={\mathbf{x}}_2-{\mathbf{x}}_1,$$ (6)

we obtain two independent equations, where the equation for Y describes free diffusion, while the equation for y becomes

$${\partial }_{t}p(\mathbf{y},t)=D{\mathrm{\Delta }}_{\mathbf{y}}p(\mathbf{y},t),$$ (7)

with the boundary condition

$${K\frac{\partial p_{\mathbf{y}}}{\partial n}|}_{\Vert \mathbf{y}\Vert =\sigma }=k_a{p}_{\mathbf{y}}(\Vert \mathbf{y}\Vert =\sigma ,t).$$ (8)

The initial condition becomes p_y(y, t_n) = δ(y − y_n), and the outer boundary condition is now p_y(‖y‖ → ∞, t) = 0. This equation can be solved analytically in 3D [20], but the solution is difficult and expensive to evaluate numerically. Applying an operator split method to (7)–(8) can significantly simplify the process of sampling new positions from the PDF [21].

An S₃ molecule is assumed to dissociate according to an exponential distribution with mean k_d. Following a dissociation, the two products S₁ and S₂ are placed in contact a distance of σ apart.

A system of more than two molecules is not amenable to the direct approach of solving for the full PDF, due to the high dimensionality of the problem. A common approach is instead to approximate the full problem as a set of one- and two-body problems, by dividing the system into subsets of single and pairs of molecules according to the distances between them. We can obtain a good approximation of the full problem by updating each subset independently during short time steps Δt. This algorithm is commonly referred to as Green’s function reaction dynamics (GFRD) [22, 23]. All microscale computations in this paper are carried out using a variant of the GFRD algorithm described in [21].

B. Standard reaction-diffusion master equation

At the mesoscopic scale the simulation domain is discretized by N non-overlapping voxels, and diffusion is modeled as discrete jumps between the nodes of the voxels. The mesh may be either a Cartesian mesh, or an unstructured, tetrahedral (3D) or triangular (2D) mesh. A Cartesian mesh is suitable if the domain is simple, for instance a square or a cube, while an unstructured mesh has advantages for complicated domains. The jump coefficients between voxels are given by 2D/h² in the case of a Cartesian mesh, where h is the width of a voxel and D the diffusion rate of the molecule. For an unstructured mesh, the jump coefficients can be obtained from a finite element discretization of the diffusion equation [24]. Reactions occur with some intensity when molecules occupy the same voxel.

Let p(x, t|x_n, t_n) be the probability that the system is found in state x at time t, given that it was in state x_n at time t_n. For brevity of notation, let p(x, t) = p(x, t|x_n, t_n). Let x_i· and x_·j denote the i-th row and the j-th column of the N × S state matrix x, respectively, where S is the number of species of the system. The sRDME is given by

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}p(\mathbf{x},t)=\sum _{i=1}^N\sum _{r=1}^M{a}_{ir}({\mathbf{x}}_{i·}-{\mathit{\mu }}_{ir})p({\mathbf{x}}_{1·},\dots ,{\mathbf{x}}_{i·}-{\mathit{\mu }}_{ir},\dots ,{\mathbf{x}}_{N·},t)-\sum _{i=1}^N\sum _{r=1}^M{a}_{ir}({\mathbf{x}}_{i·})p(\mathbf{x},t)+\sum _{j=1}^S\sum _{i=1}^N\sum _{k=1}^N{d}_{jik}({\mathbf{x}}_{·j}-{\mathit{\nu }}_{ijk})p({\mathbf{x}}_{·1},\dots ,{\mathbf{x}}_{·j}-{\mathit{\nu }}_{ijk},\dots ,{\mathbf{x}}_{·S},t)-\sum _{j=1}^S\sum _{i=1}^N\sum _{k=1}^N{d}_{ijk}({\mathbf{x}}_{·j})p(\mathbf{x},t),$$ (9)

where the propensity functions of the M chemical reactions are denoted by a_ir(x_i), μ_ir are the stoichiometry vectors associated with the reactions, d_ijk are the jump coefficients, and ν_ijk are stoichiometry vectors for diffusion events.

The sRDME is in general too high-dimensional to be solved by direct approaches. An alternative approach is to generate individual trajectories of the system with stochastic simulations. The NSM [7] is an efficient algorithm frequently used for that purpose.

C. Reaction rates for the standard reaction-diffusion master equation

Consider a system of two molecules, one of species S₁ and one of species S₂, that react according to $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$, where k_a and k_d are the microscopic reaction rates. Assume that the molecules diffuse in a square (2D) or cube (3D) with periodic boundary conditions. Without loss of generality, assume that the S₁ molecule is fixed at the origin, and that the S₂ molecule diffuses freely with a diffusion rate D. The S₂ molecule is initialized according to a uniform distribution.

Let ${\tau }_{\text{meso}}(k_a^{\text{meso}},h)$ be the mean association time of the two molecules on the mesoscopic scale, and let τ_micro(k_a) be the mean association time on the microscopic scale. Under the assumption that ${\tau }_{\text{meso}}(k_a^{\text{meso}},h)={\tau }_{\text{micro}}(k_a)$ holds, it is shown in [15, 16] that the mesoscopic association rate is given by

$$k_a^{\text{meso}}={\rho }^{(d)}(k_a,h)=\frac{k_a}{h^d}{[1+\frac{k_a}{D}{G}^{(d)}(h,\sigma )]}^{-1},$$ (10)

where d is the dimension,

$$G^{(d)}(h,\sigma )=\{\begin{array}{c}\frac{1}{2\pi }\text{ln }\left({\pi }^{-\frac{1}{2}}\frac{h}{\sigma }\right)-\frac{1}{4}(\frac{3}{2\pi }+C_2)\hspace{1em}(2\mathrm{D})\hfill \\ \frac{1}{4\pi \sigma }-\frac{C_3}{6h}\hspace{1em}(3\mathrm{D}),\hfill \end{array}$$ (11)

and

$$C_d\approx \{\begin{array}{cc}0.1951,\hfill & d=2\hfill \\ 1.5164,\hfill & d=3.\hfill \end{array}$$ (12)

The microscopic parameters are σ, the sum of the reaction radii of the molecules, D, the sum of the diffusion constants, and k_a, the microscopic reaction rate. To simplify the notation somewhat, we let ${\tau }_{\text{meso}}^{\rho }(k_a,h)≔{\tau }_{\text{meso}}({\rho }^{(d)}(k_a,h),h)$. For a reversible reaction we match the mean binding time for $h>h_{\infty }^{*}$, where

$$h_{\infty }^{*}\approx \{\begin{array}{c}\sqrt{\pi }\text{ exp }\left(\frac{3+2\pi C_2}{4}\right)\sigma \approx 5.1\sigma ,\hspace{1em}(2D)\hfill \\ \frac{2}{3}\pi C_3\sigma \approx 3.2\sigma ,\hspace{1em}(3D).\hfill & \hfill \end{array}$$ (13)

Let ${\tau }_{\text{meso}}^{\text{rebind}}(k_a^{\text{meso}},h)$ and ${\tau }_{\text{micro}}^{\text{rebind}}(k_a)$ denote the average rebinding times—that is, the average time until two molecules react, given that they have just dissociated—on the mesoscopic and microscopic scale, respectively. Again, to simplify notation, we let ${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)≔{\tau }_{\text{meso}}^{\text{rebind}}({\rho }^{(d)}(k_a,h),h)$. The rebinding times can be written in terms of the average binding times

$${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)={\tau }_{\text{meso}}^{\rho }(k_a,h)-{\tau }_{\text{meso}}^{\rho }(\infty ,h)$$ (14)

$${\tau }_{\text{micro}}^{\text{rebind}}(k_a)={\tau }_{\text{micro}}(k_a)-{\tau }_{\text{micro}}(\infty ),$$ (15)

where, for simplicty of notation, ${\tau }_{\text{meso}}^{\rho }(k_a\to \infty ,h)$ and τ_micro(k_a → ∞) are denoted by ${\tau }_{\text{meso}}^{\rho }(\infty ,h)$ and τ_micro(∞), respectively. That (14) and (15) hold can be realized by considering the following argument. Given a uniform initial distribution, τ_meso(∞) is the time until the molecules are in the same voxel for the first time. By subtracting that time from the total binding time, we obtain the rebinding time. A similar argument holds for the microscopic case. We immediately see that because ${\tau }_{\text{meso}}^{\rho }(k_a,h)={\tau }_{\text{micro}}(k_a)$ holds, the rebinding times will match if and only if ${\tau }_{\text{meso}}^{\rho }(\infty ,h)={\tau }_{\text{micro}}(\infty )$. This holds for $h=h_{\infty }^{*}$, and consequently

$${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)>{\tau }_{\text{micro}}^{\text{rebind}}(k_a)\text{ for }h>h_{\infty }^{*}$$ (16)

$${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)={\tau }_{\text{micro}}^{\text{rebind}}(k_a)\text{ for }h=h_{\infty }^{*}$$ (17)

$${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)<{\tau }_{\text{micro}}^{\text{rebind}}(k_a)\text{ for }h<h_{\infty }^{*}.$$ (18)

As a mesoscopic dissociation event is a combination of microscopic dissociation and the diffusion required to get well-mixed in a voxel, we require that ${\tau }_{\text{meso}}^{\text{rebind}}\ge {\tau }_{\text{micro}}^{\text{rebind}}$ should hold. For $h<h_{\infty }^{*}$ we cannot match the mean binding time while satisfying ${\tau }_{\text{meso}}^{\text{rebind}}\ge {\tau }_{\text{micro}}^{\text{rebind}}$, and the accuracy of the sRDME consequently deteriorates with decreasing h. Thus, $h_{\infty }^{*}$ is the finest spatial resolution attainable with the sRDME.

For a given $h>h_{\infty }^{*}$, we can compute the error in rebinding time as

$$|{\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)-{\tau }_{\text{micro}}^{\text{rebind}}(k_a)|=|{\tau }_{\text{meso}}(\infty ,h)-{\tau }_{\text{micro}}(\infty )|,$$ (19)

where the right-hand side thus is a measure of how well-resolved a system is. Details of the above theory can be found in [16].

III. THE GENERALIZED REACTION-DIFFUSION MASTER EQUATION

In the sRDME, molecules react only when they occupy the same voxel. In this section we extend this approach by allowing molecules occupying neighboring voxels to react. To connect the sRDME to the microscopic Smoluchowski model we determined the rate with which molecules react when occupying the same voxel. For the gRDME we need to obtain the rates for molecules occupying the same voxel, but also the rates for molecules occupying neighboring voxels. In [15, 16] we derive rates for the sRDME by matching the mean association times on the two scales. To uniquely determine both of the rates for the gRDME we need an additional constraint.

In Sec. IIIA we outline the algorithm, and in Sec. IIIB we derive mesoscopic parameters by trying to match certain statistics of the microscopic model to the corresponding statistics on the mesoscopic scale. In Sec. IIIC we determine the dissociation rate of a reversibly reacting pair of molecules, and in Sec. IIID we collect the results and summarize the algorithm.

A. Generalized reactions

Consider a domain Ω discretized by a Cartesian mesh, and a single reversible reaction $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$. In the gRDME we allow reactions between molecules occupying neighboring voxels. Thus, if a molecule of species S₁ occupies the same voxel as a molecule of species S₂, they react with an intensity given by k₀. If the molecules instead occupy neighboring voxels they react with an intensity of k₁, where two voxels are neighbors if they share one side.

We can choose k₀ and k₁ freely, with the restriction that the total intensity should be constant. Call the total intensity $k_a^{\text{meso}}$. Let d be the dimension. Then, since each voxel has 2d neighbors, k₀ and k₁ must satisfy

$$k_0+2d{k}_1=k_a^{\text{meso}}.$$ (20)

Thus we can write

$$k_0=(1-2dr)k_a^{\text{meso}}$$ (21)

$$k_1=r{k}_a^{\text{meso}},$$ (22)

where 0 ≤ r ≤ 1/(2d).

Now assume that a molecule of species S₃ dissociates. We must determine where to place the two products S₁ and S₂. It may seem natural to place them in the same voxel with probability 1 − 2dr, and in neighboring voxels with probability 2dr. While this arguably would yield the most accurate results compared to microscopic simulations for a single reversible reaction, we show below that this approach is unsuitable in general.

First consider the single irreversible dissociation given by

$$P\stackrel{k_{\mathit{\text{deg}}}}{\to }{S}_1+S_2.$$ (23)

In this case the microscopic and mesoscopic rates will be the same; thus $k_d^{\text{meso}}=k_{\mathit{\text{deg}}}$, and the products are placed in the same voxel. Now consider that in addition to (23) we have the following reactions:

$$S_1\stackrel{k^{*}}{\to }{S}_1^{*}$$ (24)

$$S_1^{*}+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3.$$ (25)

Again, (24) is an irreversible unimolecular reaction and thus the mesoscopic and microscopic rates are the same. Now, if k* is large, the system (23)–(25) will be well approximated by

$$P\stackrel{k_{\mathit{\text{deg}}}}{\to }{S}_1^{*}+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3.$$ (26)

Had we derived rates for reaction (25) assuming that dissociating molecules are placed in neighboring voxels with some probability, we can see that the sequence (26) will be incorrectly simulated, as S₁ and S₂ are placed in the same voxel with probability 1 when P dissociates. Specifically, the rebinding dynamics of $S_1^{*}$ and S₂ will be incorrect, as the rebinding time will depend on whether they were produced from a dissociating S₃ or a dissociating P.

To summarize:

• Reactive molecules occupying the same voxel react with intensity $(1-2dr)k_a^{\text{meso}}$.

• Reactive molecules in neighboring voxels react with intensity $r{k}_a^{\text{meso}}$.

• When a molecule dissociates, the products are placed in the same voxel with probability 1.

The parameters r and $k_a^{\text{meso}}$ now have to be determined from the microscopic parameters k_a, σ and D.

B. Reaction rates

Consider the reversible reaction $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$. Assume that the initial state of the system is given by one molecule of species S₁ and one molecule of species S₂ in a square (2D) or a cubic (3D) domain Ω of width L with periodic boundary conditions. For simplicity, and without loss of generality, assume that the S₁ molecule is fixed at the origin while the S₂ molecule has a uniform initial distribution and a diffusion rate D = D₁ + D₂. On the microscopic scale the S₂ molecule moves by continuous Brownian motion. On the mesoscopic scale, Ω is subdivided into non-overlapping squares or cubes of width h. The S₂ molecule thus jumps between voxels with a total intensity of k_j = 2dD/h² in dimension d. Let ${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)$ denote the average time until the molecules react on the mesoscopic scale in the gRDME.

For the sRDME, it was shown in [15, 16] that by enforcing the constraint τ_meso = τ_micro we obtain mesoscopic reaction rates as given by (10). In addition, it was shown that ${\tau }_{\text{meso}}^{\text{rebind}}$ approaches ${\tau }_{\text{micro}}^{\text{rebind}}$ from above as $h\to h_{\infty }^{*}$. Therefore it seems reasonable to require that with the gRDME we obtain an approximation of ${\tau }_{\text{micro}}^{\text{rebind}}$ that is equal to or better than the approximation we obtain with the sRDME. The first constraint is therefore that the mean binding time agrees between the mesoscopic and the microscopic scales

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)={\tau }_{\text{micro}}(k_a),$$ (27)

and the second constraint will be that, given (27), $k_a^{\text{meso}}$ and r minimize the difference between the rebinding times at the mesoscopic and microscopic scales; that is, we want to minimize

$$|{\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)-{\tau }_{\text{micro}}^{\text{rebind}}(k_a)|,$$ (28)

under the assumption that (27) holds, where τ_meso,r is the average binding time (dependent on r and $k_a^{\text{meso}}$), and where ${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}$ is the average rebinding time in the gRDME. Note that with (27) satisfied we have ${\tau }_{\text{meso},0}(k_a^{\text{meso}},h)={\tau }_{\text{meso}}(k_a^{\text{meso}},h)$ and ${\tau }_{\text{meso},0}^{\text{rebind}}(k_a^{\text{meso}},h)={\tau }_{\text{meso}}^{\text{rebind}}(k_a^{\text{meso}},h)$.

1. Mean mesoscopic binding time

Again, assume that we have species S₁ and S₂, with one molecule of each, and that the S₁ molecule is fixed. The S₂ molecule is initialized according to a uniform distribution, and diffuses with diffusion rate D.

We start by deriving the mesoscopic mean binding time. To this end, let $M_s^i$ denote the average number of diffusive jumps required for the S₂ molecule to reach a voxel at distance i from the S₁ molecule, where the distance between two voxels is defined to be the smallest number of discrete jumps required to move from one voxel to the other. Let the set of all voxels at a distance i from the S₁ molecule be denoted by d_i (note that d₀ will then be a set of only one voxel; specifically the voxel occupied by the S₁ molecule), let t_j denote the average time for a diffusive jump, and let τ_i denote the average time for the S₂ and the S₁ molecule to react, given that the S₂ molecule is occupying a voxel at distance i from the S₁ molecule. Thus, t_j = h²/(2dD), and

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)=M_s^1{t}_j+{\tau }_1.$$ (29)

The first term, $M_s^1{t}_j$, represents the average time required for the S₂ molecule to reach d₁. The second term, τ₁, represents the remaining time until the molecules react, given that the S₂ molecule occupies a voxel in d₁.

a. Derivation of $M_s^1$

In this section we show that

$$M_s^1=\{\begin{array}{c}{\pi }^{-1}N\text{ ln }N+(C_2-1)N+O(1)\hspace{1em}(2\mathrm{D})\hfill \\ (C_3-1)N+O(\sqrt{N})\hspace{1em}(3\mathrm{D}),\hfill \end{array}$$ (30)

where C₂ and C₃ are defined in (12). Let $M_i^j$ denote the average number of steps required to diffuse from d_i to d_j. We also show that

$$M_2^1=\frac{N-2}{2d-1}.$$ (31)

To obtain (30) we first note that

$$M_s^1=M_s^0-M_1^0.$$ (32)

In [26] it is shown that

$$M_s^0=\{\begin{array}{c}{\pi }^{-1}N\text{ ln }N+C_{2}N+O(1)\hspace{1em}(2\mathrm{D})\hfill \\ C_{3}N+O(\sqrt{N})\hspace{1em}(3\mathrm{D}).\hfill \end{array}$$ (33)

Let $M_0^0$ be the average number of steps required to return to d₀, given that we start in d₀. The first jump of a molecule starting in d₀ always transfers the molecule to d₁, so we find that

$$M_1^0=M_0^0-1.$$ (34)

We know that $M_0^0=N$, shown in [25]. By combining (32), (33), and (34), we obtain (30).

To obtain (31) we note that we can write $M_0^0$ as

$$M_0^0=1+\frac{1}{2d}+\frac{2d-1}{2d}(M_2^1+M_1^0).$$ (35)

To see that the above equality holds, start by considering a molecule in d₀. The first jump transfers the molecule to d₁; the second jump transfers it back to the origin with a probability of 1/(2d), or to d₂ with a probability of (2d − 1)/(2d). The average number of steps required to reach d₀ from d₂, is given by the average number of steps to reach d₁ plus the average number of steps to reach d₀, given that the molecule starts in d₁. Now, solving (35) for $M_2^1$ yields (31).

b. Derivation of τ₁

To obtain τ_meso,r for k_a < ∞, it remains to determine τ₁. To that end, assume that the S₂ molecule occupies a voxel in d₁, and that the intensity with which the molecules react in d₁ is given by $1/(r{k}_a^{\text{meso}})$. Then, to maintain a total intensity of $1/k_a^{\text{meso}}$, the molecules must react with an intensity of $1/[(1-2dr)k_a^{\text{meso}}]$ in d₀. We require that r ≥ 0, and that 0 ≤ 1 − 2dr ≤ 1. To simplify the notation we let $1/(r{k}_a^{\text{meso}})$ be denoted by p₁, and $1/[(1-2dr)k_a^{\text{meso}}]$ by p₀.

Let $t_e^0$ and $t_e^1$ denote the average time until the next event fires, given that the S₂ molecule occupies a voxel in d₀ or d₁, respectively. Then $t_e^0=1/(p_0^{-1}+t_j^{-1})$ and $t_e^1=1/(p_1^{-1}+t_j^{-1})$.

By assumption, the S₂ molecule initially occupies a voxel in d₁. The next event can either be: (1) a diffusive jump, with probability p₁/(p₁ + t_j), or (2) a reaction event with probability t_j/(p₁ + t_j).

Now assume that the next event is a diffusion event. Then: (1.1) the molecule jumps to d₂ with probability (2d − 1)/2d, or (1.2) the molecule jumps to d₀ with probability 1/(2d). Assume that the molecule jumps to d₀. Then the next event is: (1.2.1) a reaction with probability t_j/(p₀ + t_j), or (1.2.2) diffusion to d₁ with probability p₀/(p₀ + t_j). Thus, if the molecule is in the state (1.2), the time until the molecules react is given by

$${\tau }_0=\frac{t_j}{p_0+t_j}{t}_e^0+\frac{p_0}{p_0+t_j}(t_e^0+{\tau }_1)=t_e^0+\frac{p_0}{p_0+t_j}{\tau }_1.$$ (36)

Now instead assume that the molecule is in the state (1.1). The molecules cannot react until the S₂ molecule reaches d₁, and thus the average time until the molecules react is given by

$${\tau }_2=M_2^1{t}_j+{\tau }_1=\frac{N-2}{2d-1}{t}_j+{\tau }_1,$$ (37)

where $M_2^1$ is given by (31). To summarize:

The S₂ molecule initially occupies a voxel in d₁, and the S₁ molecule is fixed in d₀.

1. The S₂ molecule diffuses with probability p₁/(p₁ + t_j).

   • (1.1)
     The S₂ molecule jumps to d₂ with probability (2d − 1)/(2d). The average remaining time until the S₁ and S₂ molecules react is given by τ₂.
   • (1.2)
     The S₂ molecule jumps to d₀ with probability 1/(2d).

     • (1.2.1)
       The S₁ and S₂ molecules react with probability t_j/(p₀ + t_j).
     • (1.2.2)
       The S₂ molecule diffuses to d₁ with probability p₀/(p₀ + t_j). The average remaining time until the S₁ and S₂ molecules react is given by τ₁.

2. The S₁ and S₂ molecules react with probability t_j/(p₁ + t_j).

Putting it all together, we obtain

$${\tau }_1=\frac{t_j}{p_1+t_j}{t}_e^1+\frac{p_1}{p_1+t_j}(t_e^1+\frac{1}{2d}{\tau }_0+\frac{2d-1}{2d}{\tau }_2).$$ (38)

By inserting (36) and (37) into (38) and solving for τ₁ we obtain

$${\tau }_1=\frac{(N+2d-1)p_0+(N+2d-2)t_j}{2d(p_0+t_j)+p_1}{p}_1$$ (39)

$$\approx \frac{p_0+t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}},$$ (40)

after some cumbersome but straightforward algebra, where (40) follows by assuming N ≫ 1.

c. Analytical expression for τ_meso,r

Now, using (29), (30), and (40) we find that

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)\approx \{\begin{array}{c}[{\pi }^{-1}N\text{ ln }N+(C_2-1)N]t_j+\frac{p_0+t_j}{p_0+4r{t}_j}\frac{N}{k_a^{\text{meso}}}\hspace{1em}(2\mathrm{D})\hfill \\ (C_3-1)N{t}_j+\frac{p_0+t_j}{p_0+6r{t}_j}\frac{N}{k_a^{\text{meso}}}\hspace{1em}(3\mathrm{D}).\hfill \end{array}$$ (41)

d. Lower limit on the voxel size

It is of interest to know the smallest voxel size h for which we can match the mesoscopic mean binding time, τ_meso,r, with the microscopic mean binding time, τ_micro. In [15, 16] this problem was solved in the case of the sRDME for a general reversible reaction $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$. Similar results in the case of the gRDME can be obtained for the case of an irreversible reaction with k_a → ∞. As we will see in Eqs. (74)–(75), the lower bound for k_a → ∞ is in fact a fundamental lower bound for the gRDME.

We now let $h_{k_a,g}^{*}$ be the smallest voxel size for which we can choose reaction rates such that ${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)={\tau }_{\text{micro}}(k_a)$. It is easy to see that $h_{k_a,g}^{*}$ exists; the average time until two molecules on the mesoscopic scale occupy the same or neighboring voxels diverges in 2D and 3D, thus there must exist a smallest mesh size for which the mesoscopic mean binding time can match the mean microscopic binding time. We do not have analytical results for τ_micro on a square or a cube, but given that L ≫ σ is satisfied, an excellent approximation is provided by

$${\tau }_{\text{micro}}(k_a)=\{\begin{array}{c}\frac{1+\alpha F(\lambda )}{k_a}{L}^2\hspace{1em}(2\mathrm{D})\hfill \\ \frac{L^3}{k_{\mathrm{C}\mathrm{K}}}\hspace{1em}(3\mathrm{D}),\hfill \end{array}$$ (42)

where

$$\lambda ={\pi }^{\frac{1}{2}}\frac{\sigma }{L}$$

$$\alpha =\frac{k_a}{2\pi D}$$ (43)

$$F(\lambda )=\frac{\text{ln}(1/\lambda )}{{(1-{\lambda }^2)}^2}-\frac{3-{\lambda }^2}{4(1-{\lambda }^2)},$$

and where k_CK = 4πσDk_a/(4πσD + k_a) is the classical mesoscopic reaction rate, valid for large volumes, derived by Collins and Kimball in [27]. The expression in 2D was derived in [28], following the approach devised in [29].

Since molecules are allowed to react with molecules occupying neighboring voxels, we obtain

$${\tau }_1\to 0\text{ for }{k}_a\to \infty ,r>0$$ (44)

and thus

$${\tau }_{\text{meso},\mathrm{r}}\to M_s^1{t}_j\text{ for }{k}_a\to \infty ,r>0.$$ (45)

We know $M_s^1$ from (30), and we have t_j = h²/(2dD) by definition. We now obtain $h_{k_a,g}^{*}$ by solving

$$M_s^1{t}_j={\tau }_{\text{micro}}(\infty )$$ (46)

for h.

In 3D, (46) becomes

$$\frac{(C_3-1)L^3}{6Dh}=\frac{L^3}{4\pi \sigma D},$$ (47)

since $M_s^1~(C_3-1)N$ for N ≫ 1, and k_CK → 4πσD as k_a → ∞. Solving (47) for h yields

$$h=\frac{2}{3}(C_3-1)\pi \sigma \approx 1.0815\sigma .$$ (48)

In 2D, (46) becomes

$$\frac{h^2}{4D}\left[{\pi }^{-1}\frac{L^2}{h^2}\text{ln }\right(\frac{L^2}{h^2})+(C_2-1)\frac{L^2}{h^2}]=\frac{L^2}{k_a}+\frac{F(\lambda )}{2\pi D}{L}^2,$$ (49)

and for k_a → ∞, we have

$$\frac{L^2}{k_a}+\frac{F(\lambda )}{2\pi D}{L}^2\to \frac{\text{ln }\left({\pi }^{-\frac{1}{2}}\frac{L}{\sigma }\right)-\frac{3}{4}}{2\pi D}{L}^2.$$ (50)

In (50) we used that λ ≈ 0 for L ≫ σ. Now (49) reduces to

$${\pi }^{-1}\text{ ln }\left(\frac{L}{h}\right)+\frac{C_2-1}{2}={\pi }^{-1}\text{ ln }\left({\pi }^{-\frac{1}{2}}\frac{L}{\sigma }\right)\frac{3}{4\pi }.$$ (51)

We can rewrite the equation above to get

$${\pi }^{-1}\text{ ln }\left({\pi }^{\frac{1}{2}}\frac{\sigma }{h}\right)=\frac{1-C_2}{2}-\frac{3}{4\pi }.$$ (52)

Solving for h yields

$$h=\sqrt{\pi }\text{ exp }\left(\frac{3+2\pi (C_2-1)}{4}\right)\sigma \approx 1.0599\sigma$$ (53)

To summarize, we find that

$$h_{\infty ,g}^{*}=\{\begin{array}{c}\sqrt{\pi }\text{ exp }\left(\frac{3+2\pi (C_2-1)}{4}\right)\sigma \approx 1.0599\sigma \hspace{1em}(2\mathrm{D})\hfill \\ \frac{2}{3}(C_3-1)\pi \sigma \approx 1.0815\sigma \hspace{1em}(3\mathrm{D}).\hfill \end{array}$$ (54)

2. Mean mesoscopic rebinding time

To satisfy the second constraint (28) we need both the microscopic and the mesoscopic mean rebinding times. The microscopic rebinding time is derived in [16], as

$${\tau }_{\text{micro}}^{\text{rebind}}=\frac{L^d}{k_a}.$$ (55)

The mesoscopic rebinding time is simply given by

$${\tau }_{\text{meso}}^{\text{rebind}}={\tau }_0,$$ (56)

as τ₀ by definition is the time until an S₁ and an S₁ molecule react, given that they start in the same voxel. We have already derived τ₀ in terms of τ₁ in (36), and we thus obtain

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}\approx t_e^0+\frac{p_0}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}.$$ (57)

immediately from (39)–(40).

3. Solving for r and $k_a^{\text{meso}}$

We now want r and $k_a^{\text{meso}}$ to satisfy the constraints (27) and (28). It will prove useful to divide the problem into two cases:

$$\text{Case }1:h\ge h_{\infty }^{*}$$ (58)

$$\text{Case }2:h_{\infty }^{*}>h\ge h_{\infty ,g}^{*}.$$ (59)

It turns out that in case 1 we get r = 0, effectively reducing the generalized algorithm to the standard algorithm.

a. Case 1

We assume that r and $k_a^{\text{meso}}$ have been chosen to satisfy the first constraint (27) and then show that for $h\ge h_{\infty }^{*}$ we have

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)⪆{\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)$$ (60)

Since ${\tau }_{\text{meso}}^{\text{rebind},\rho }(k_a,h)\ge {\tau }_{\text{micro}}^{\text{rebind}}(k_a)$, it immediately follows that for $h\ge h_{\infty }^{*}$, the gRDME and the sRDME agree.

We first note that we already know that

$${\tau }_{\text{meso}}=M_s^0{t}_j+{\tau }_0$$ (61)

$${\tau }_{\text{meso},\mathrm{r}}=M_s^1{t}_j+{\tau }_1^g$$ (62)

where τ_i, as previously defined, is the average time until the molecules react, given that the S₂ molecule is in d_i. The superscript g indicates that it is the average time in the case of the gRDME, and omission of the superscript indicates that it is the average time in the case of the sRDME.

We have assumed that (27) is satisfied and consequently

$$0={\tau }_{\text{meso}}-{\tau }_{\text{meso},\mathrm{r}}=(M_s^0-M_s^1)t_j+({\tau }_0-{\tau }_1^g)=N{t}_j+({\tau }_0-{\tau }_1^g),$$ (63)

where the second equality follows from (30) and (33). We know that ${\tau }_{\text{meso}}^{\text{rebind}}={\tau }_0$, so we get

$${\tau }_{\text{meso}}^{\text{rebind}}={\tau }_1^g-N{t}_j.$$ (64)

Thus

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}\ge {\tau }_{\text{meso}}^{\text{rebind}}$$ (65)

$$\iff {\tau }_0^g\ge {\tau }_1^g-N{t}_j$$ (66)

$$\iff {\tau }_1^g-{\tau }_0^g\le N{t}_j.$$ (67)

We have already shown that

$${\tau }_0^g=t_e^0+\frac{p_0}{p_0+t_j}{\tau }_1^g>\frac{p_0}{p_0+t_j}{\tau }_1^g$$ (68)

$${\tau }_1^g\approx \frac{p_0+t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}.$$ (69)

Now (67) becomes

$$\frac{p_0+t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}-\frac{p_0}{p_0+t_j}\frac{p_0+t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}\le N{t}_j$$ (70)

$$\iff \frac{1}{p_0+2dr{t}_j}\frac{1}{k_a^{\text{meso}}}\le 1.$$ (71)

By definition, $p_0=1/(1-2dr)k_a^{\text{meso}}$, so (71) becomes

$$\frac{1}{\frac{1}{(1-2dr)k_a^{\text{meso}}}+2dr{t}_j}\frac{1}{k_a^{\text{meso}}}\le 1$$ (72)

$$\iff 1\le {(1-2dr)}^{-1}+2dr{t}_j{k}_a^{\text{meso}}.$$ (73)

Since 1 − 2dr ≤ 1, we have (1 − 2dr)⁻¹ ≥ 1, and $2dr{t}_j{k}_a^{\text{meso}}\ge 0$ so (73) is satisfied for all r and $k_a^{\text{meso}}$. Thus (65) holds for all r and $k_a^{\text{meso}}$.

What remains is to determine r and $k_a^{\text{meso}}$ for $h<h_{\infty }^{*}$.

b. Case 2

We will proceed in two steps. First we show that for $h=h_{\infty ,g}^{*}$ with ${\tau }_1\gg t_e^0$ we have

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)\approx {\tau }_{\text{micro}}(k_a)$$ (74)

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)\approx {\tau }_{\text{micro}}^{\text{rebind}}(k_a),$$ (75)

for $k_a^{\text{meso}}=k_a/h^d$ and 1 − 2dr = 0, and for $h<h_{\infty ,g}^{*}$

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)⪅{\tau }_{\text{micro}}^{\text{rebind}}(k_a).$$ (76)

Note that we have already shown that we can satisfy (27) at least down to $h=h_{\infty ,g}^{*}$. The assumption ${\tau }_1\gg t_e^0$ means, in words, that the average time until two molecules react, given that they are one voxel apart, is much longer than the average time until the first event, given that they occupy the same voxel. Unless the microscopic reaction rate is very high, this should be a reasonable assumption for most systems. The necessity of this assumption is realized by considering two molecules in the same voxel. Now, if the average microscopic rebinding time is smaller than the average time until the first diffusion event on the mesoscopic scale, we could not hope to find mesoscopic rates that will yield a match between the mesoscopic rebinding time and the microscopic rebinding time.

To show that (74) and (75) hold, we first note that we already know that

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)=M_s^1{t}_j+{\tau }_1,$$ (77)

and from assuming $h=h_{\infty ,g}^{*}$, it follows that

$$M_s^1{t}_j={\tau }_{\text{micro}}(\infty ),$$ (78)

and thus

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h_{\infty ,g}^{*})={\tau }_{\text{micro}}(\infty )+{\tau }_1.$$ (79)

Eqs. (39) and (40) yield, for 1 − 2dr = 0 and $k_a^{\text{meso}}=k_a/h^d$,

$${\tau }_1=\frac{N}{k_a^{\text{meso}}}=\frac{N{h}^d}{k_a}=\frac{L^d}{k_a}={\tau }_{\text{micro}}^{\text{rebind}}(k_a).$$ (80)

We now have

$${\tau }_{\text{meso},\mathrm{r}}(\frac{k_a}{h^d},h_{\infty ,g}^{*})={\tau }_{\text{micro}}(\infty )+{\tau }_{\text{micro}}^{\text{rebind}}(k_a)={\tau }_{\text{micro}}(k_a),$$ (81)

and thus (74) holds. Since we have assumed ${\tau }_1\gg t_e^0$ and 1 − 2dr = 0, we get

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(\frac{k_a}{h^d},h_{\infty ,g}^{*})={\tau }_0\approx {\tau }_1={\tau }_{\text{micro}}^{\text{rebind}}(k_a),$$ (82)

and we have shown that (75) holds.

It remains to show (76). To this end, we simply note that

$${\tau }_{\text{micro}}(k_a)={\tau }_{\text{meso},\mathrm{r}}=M_s^1{t}_j+{\tau }_1>{\tau }_{\text{micro}}(\infty )+{\tau }_1,$$ (83)

since $M_s^1{t}_j>{\tau }_{\text{micro}}(\infty )$ for $h<h_{\infty ,g}^{*}$. Thus

$${\tau }_{\text{micro}}^{\text{rebind}}={\tau }_{\text{micro}}(k_a)-{\tau }_{\text{micro}}(\infty )>{\tau }_1\approx {\tau }_0={\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}.$$ (84)

and (76) follows.

The second step is to show that for $h_{\infty ,g}^{*}<h<h_{\infty }^{*}$, still assuming ${\tau }_1\gg t_e^0$, we have

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)\approx {\tau }_{\text{micro}}(k_a)$$ (85)

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)\approx {\tau }_{\text{micro}}^{\text{rebind}}(k_a),$$ (86)

for

$$k_a^{\text{meso}}=\left(\frac{t_j{Q}^2+k_a/h^d}{t_j{Q}^2+Q{k}_a/h^d}\right)\frac{k_a}{h^d}$$ (87)

$$r=\frac{DQ(Q-1)}{2dD{Q}^2+k_a/h^{d-2}},$$ (88)

where

$$Q=\frac{N{t}_j}{{\tau }_{\text{micro}}(\infty )-{\tau }_{\text{meso},\mathrm{r}}(\infty )}=\{\begin{array}{c}{\left[\frac{2}{\pi }\text{ln}\right(\frac{h}{h_{\infty ,g}^{*}}\left)\right]}^{-1}\hspace{1em}(2D)\hfill \\ {[(C_3-1)(\frac{h}{h_{\infty ,g}^{*}}-1\left)\right]}^{-1}\hspace{1em}(3D).\hfill \end{array}$$ (89)

We show this by first noting that we already know that

$${\tau }_{\text{meso},\mathrm{r}}(k_a^{\text{meso}},h)=M_s^1{t}_j+{\tau }_1$$ (90)

$${\tau }_{\text{micro}}^{\text{rebind}}(k_a)={\tau }_{\text{micro}}(k_a)-{\tau }_{\text{micro}}(\infty )=\frac{N{h}^d}{k_a}$$ (91)

$${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)={\tau }_0\approx \frac{p_0}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}.$$ (92)

Consequently we satisfy (85) if

$$M_s^1{t}_j+{\tau }_1={\tau }_{\text{micro}}(k_a),$$ (93)

which, by (39) and (40), approximately holds if

$$\frac{p_0+t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}={\tau }_{\text{micro}}(k_a)-M_s^1{t}_j.$$ (94)

To satisfy (86), we must, by (91) and (92), satisfy

$$\frac{p_0}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}={\tau }_{\text{micro}}^{\text{rebind}}(k_a).$$ (95)

Subtracting both the right-hand and left-hand side of (95) from (94), we obtain

$$\frac{t_j}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}={\tau }_{\text{micro}}(k_a)-M_s^1{t}_j-{\tau }_{\text{micro}}^{\text{rebind}}(k_a).$$ (96)

By definition, $p_0=1/(1-2dr)k_a^{\text{meso}}$ and t_j = h²/(2dD), so (96) yields, after some straight-forward algebra,

$$k_a^{\text{meso}}=(\frac{(1-2dr)N{t}_j}{{\tau }_{\text{micro}}(k_a)-M_s^1{t}_j-{\tau }_{\text{micro}}^{\text{rebind}}(k_a)}-1)\frac{D}{r{h}^2(1-2dr)}.$$ (97)

Since ${\tau }_{\text{meso},\mathrm{r}}(\infty )=M_s^1{t}_j$ and ${\tau }_{\text{micro}}^{\text{rebind}}(k_a)={\tau }_{\text{micro}}(k_a)-{\tau }_{\text{micro}}(\infty )$, (97) becomes

$$k_a^{\text{meso}}=\frac{D}{r{h}^2}(\frac{N{t}_j}{{\tau }_{\text{micro}}(\infty )-{\tau }_{\text{meso},\mathrm{r}}(\infty )}-\frac{1}{1-2dr})$$ (98)

$$=\frac{D}{r{h}^2}(Q-\frac{1}{1-2dr}).$$ (99)

With $k_a^{\text{meso}}$ as in (99), we want to find r such that (95) is satisfied. Since ${\tau }_{\text{micro}}^{\text{rebind}}=L^d/k_a=N{h}^d/k_a$, we obtain

$$\frac{p_0}{p_0+2dr{t}_j}\frac{N}{k_a^{\text{meso}}}=\frac{N{h}^d}{k_a}$$ (100)

$$\iff \frac{1}{1_{2}dr{t}_j{p}_0^{-1}}\frac{1}{k_a^{\text{meso}}}=\frac{h^d}{k_a}$$ (101)

$$\iff (1+2dr{t}_j{p}_0^{-1})k_a^{\text{meso}}=\frac{k_a}{h^d}.$$ (102)

Since t_j = h²/2dD and p₀ = 1/(1 − 2dr)k_a, (102) becomes

$$\frac{r{h}^2}{D}(1-2dr){(k_a^{\text{meso}})}^2+k_a^{\text{meso}}=\frac{k_a}{h^d}.$$ (103)

We expand the first term of the left-hand side to get

$$\frac{r{h}^2}{D}(1-2dr){(k_a^{\text{meso}})}^2=\frac{D}{r{h}^2}[(1-2dr)Q^2-2Q+\frac{1}{1-2dr}].$$ (104)

Thus

$$\frac{r{h}^2}{D}(1-2dr){(k_a^{\text{meso}})}^2+k_a^{\text{meso}}=\frac{D}{r{h}^2}[(1-2dr)Q^2-Q],$$ (105)

and (103) becomes

$$\frac{D}{r{h}^2}[(1-2dr)Q^2-Q]=\frac{k_a}{h^d},$$ (106)

yielding

$$r=\frac{DQ(Q-1)}{2dD{Q}^2+\frac{k_a}{h^d}}.$$ (107)

Inserting r above into (99) yields (87).

It remains to show that $k_a^{\text{meso}}>0$ and 0 < r < 1/(2d) hold for $k_a^{\text{meso}}$ and r given by (87) and (88). We first show that Q > 1, from which r > 0 follows. Thus we should show that

$$Q=\frac{N{t}_j}{{\tau }_{\text{micro}}(\infty )-{\tau }_{\text{meso},\mathrm{r}}(\infty )}>1$$ (108)

holds. We start by showing that (108) holds in 3D. By (30),

$${\tau }_{\text{meso},\mathrm{r}}(\infty ,h)\approx (C_3-1)N{t}_j=(C_3-1)\frac{L^3}{h^3}\frac{h^2}{2dD},$$ (109)

for N ≫ 1. We have already shown that

$${\tau }_{\text{micro}}(\infty )={\tau }_{\text{meso},\mathrm{r}}(\infty ,h_{\infty ,g}^{*})=(C_3-1)\frac{L^3}{{(h_{\infty ,g}^{*})}^3}{t}_j,$$ (110)

so (108) becomes

$$Q=\frac{\frac{L^3}{h^3}\frac{h^2}{2dD}}{(C_3-1)\frac{L^3}{{(h_{\infty ,g}^{*})}^3}\frac{{(h_{\infty ,g}^{*})}^2}{2dD}-(C_3-1)\frac{L^3}{h^3}\frac{h^2}{2dD}}=\frac{\frac{1}{h}}{(C_3-1)(\frac{1}{h_{\infty ,g}^{*}}-\frac{1}{h})}>1$$ (111)

$$\iff (C_3-1)(\frac{h}{h_{\infty ,g}^{*}}-1)<1.$$ (112)

Since $h_{\infty }^{*}/h_{\infty ,g}^{*}=C_3/(C_3-1)$, and, by assumption, $h<h_{\infty }^{*}$, we obtain

$$(C_3-1)(\frac{h}{h_{\infty ,g}^{*}}-1)<(C_3-1)(\frac{C_3}{C_3-1}-1)=1.$$ (113)

Thus Q > 1, and, as a consequence, r > 0. In 2D we have

$${\tau }_{\text{meso},\mathrm{r}}(\infty ,h)=[{\pi }^{-1}N\text{ ln }N+(C_2-1)N]t_j$$ (114)

$$=[{\pi }^{-1}\frac{L^2}{h^2}\text{ln}\frac{L^2}{h^2}+(C_2-1)\frac{L^2}{h^2}]\frac{h^2}{2dD}$$ (115)

$$={\pi }^{-1}\frac{L^2}{2dD}\text{ ln }\frac{L^2}{h^2}+(C_2-1)\frac{L^2}{2dD}.$$ (116)

In 2D, similarly as in the 3D case, we have ${\tau }_{\text{micro}}(\infty )={\tau }_{\text{meso},\mathrm{r}}(\infty ,h_{\infty ,g}^{*})$. Thus

$$Q=\frac{\frac{L^2}{h^2}\frac{h^2}{2dD}}{[{\pi }^{-1}\frac{L^2}{2dD}\text{ln}\frac{L^2}{{(h_{\infty ,g}^{*})}^2}+(C_2-1)\frac{L^2}{2dD}]-[{\pi }^{-1}\frac{L^2}{2dD}\text{ln}\frac{L^2}{h^2}+(C_2-1)\frac{L^2}{2dD}]}$$ (117)

$$=\frac{1}{2{\pi }^{-1}(\text{ln}\frac{L}{h_{\infty ,g}^{*}}-\text{ln}\frac{L}{h})}=\frac{1}{2{\pi }^{-1}\text{ ln}\frac{h}{h_{\infty ,g}^{*}}}.$$ (118)

Since, by assumption,

$$1<\frac{h}{h_{\infty ,g}^{*}}<\frac{h_{\infty }^{*}}{h_{\infty ,g}^{*}},$$ (119)

and

$$\frac{h_{\infty }^{*}}{h_{\infty ,g}^{*}}=\text{exp }[\frac{3+2\pi C_2}{4}-\frac{3+2\pi (C_2-1)}{4}]=\text{exp }\left(\frac{\pi }{2}\right),$$ (120)

we obtain

$$Q=\frac{1}{2{\pi }^{-1}\text{ ln}\frac{h}{h_{\infty ,g}^{*}}}>\frac{1}{2{\pi }^{-1}\text{ ln }\left(\text{exp}\frac{\pi }{2}\right)}=1.$$ (121)

Thus Q > 1 holds in both 2D and 3D, and we have r > 0. Note that with Q > 1, $k_a^{\text{meso}}>0$ follows immediately. It remains to show r < 1/(2d). To this end, we simply note that

$$r=\frac{D{Q}^2-DQ}{2dD{Q}^2+\frac{k_a}{h^{d-2}}}=\left(\frac{D{Q}^2-DQ}{D{Q}^2+\frac{k_a}{2d{h}^{d-2}}}\right)\frac{1}{2d},$$ (122)

where

$$\frac{D{Q}^2-DQ}{D{Q}^2+\frac{k_a}{2d{h}^{d-2}}}<1$$ (123)

holds, since Q > 1.

Thus 0 < r < 1/(2d) and $k_a^{\text{meso}}>0$ for $h_{\infty ,g}^{*}<h<h_{\infty }^{*}$.

C. Dissociation rates

Consider the same setup as before, with one S₁ molecule and one S₂ molecule reacting reversibly according to $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$. Above we have determined how to choose the mesoscopic association rates, so what remains is to determine the dissociation rate. This can be done completely analogously to the case of the sRDME. We thus follow the approach of [16], and conclude that we must have

$$\frac{{(k_d^{\text{meso}})}^{-1}}{{\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}+{(k_d^{\text{meso}})}^{-1}}=\frac{k_d^{-1}}{{\tau }_{\text{micro}}^{\text{rebind}}+k_d^{-1}},$$ (124)

to obtain a steady state on the mesoscopic scale that matches the steady state of the microscopic scale. Thus it follows immediately that for $h_{\infty ,g}^{*}\le h\le h_{\infty }^{*}$, we should have

$$k_d^{\text{meso}}=k_d,$$ (125)

because ${\tau }_{\text{meso},\mathrm{r}}^{\text{rebind}}(k_a^{\text{meso}},h)={\tau }_{\text{micro}}^{\text{rebind}}(k_a)$ holds.

D. Summary of the algorithm

Assume that we have a cubic (3D) or square (2D) domain of width L, discretized by a Cartesian mesh with voxels of width h. Consider a reversible reaction $S_1+S_2\underset{k_d}{\overset{k_a}{\rightleftarrows }}{S}_3$, where k_a and k_d are the microscopic reaction rates. Let D = D₁ + D₂, where D₁ and D₂ are the diffusion rates of species S₁ and S₂, respectively. Let σ = σ₁ + σ₂ be the reaction radius of an S₁ and an S₂ molecule.

The critical mesh sizes are given by

$$h_{\infty }^{*}\approx \{\begin{array}{cc}5.1\sigma \hfill & (2D)\hfill \\ 3.2\sigma \hfill & (3D),\hfill \end{array}$$ (126)

for the sRDME, and the critical mesh sizes for the gRDME are given by

$$h_{\infty ,g}^{*}\approx \{\begin{array}{cc}1.06\sigma \hfill & (2D)\hfill \\ 1.08\sigma \hfill & (3D),\hfill \end{array}$$ (127)

We now wish to simulate this system on the mesoscopic scale with the gRDME. The results of this section can be summarized as follows:

• For $h\ge h_{\infty }^{*}$: The gRDME reduces to the sRDME. Thus r = 0 and $k_a^{\text{meso}}={\rho }^{(d)}(k_a,h)$. Molecules react only when occupying the same voxel. The dissociation rate is given by $k_d^{\text{meso}}=h^d{k}_d{k}_a^{\text{meso}}/k_a$, as shown in [16].

• For $h_{\infty ,g}^{*}<h<h_{\infty }^{*}$: We match both the mean binding time, and the mean rebinding time, of the S₁ and S₂ molecules by choosing r and $k_a^{\text{meso}}$ as in (87) and (88). Now molecules react with an intensity of $r{k}_a^{\text{meso}}$ when occupying neighboring voxels, and with an intensity of $(1-2dr)k_a^{\text{meso}}$ when occupying the same voxel. The dissociation rate is simply given by $k_d^{\text{meso}}=k_d$.

• For $h<h_{\infty ,g}^{*}$ we can no longer match the mean rebinding time, and the accuracy deteriorates with decreasing h.

IV. NUMERICAL RESULTS

A. Rebinding-time distributions

Consider a system of two species S₁ and S₂, with one molecule of each. The S₁ molecule is fixed at the origin, while the S₂ molecule diffuses freely in space. In [16] it was shown that the sRDME matched the microscopic rebinding-time distribution for a reversibly reacting pair down to $t^{*}~{(h_{\infty }^{*})}^2/(2D)$. For t < t*, the behavior is inevitably going to be different, as the accuracy of the sRDME is inherently limited by the spatial resolution.

With the gRDME, we can match both the average binding times as well as the average rebinding times for $h_{\infty }^{*}\ge h\ge h_{\infty ,g}^{*}$, and thus we could hope that also the error in distribution will be small at timescales of ${(h_{\infty ,g}^{*})}^2/(2D)<t<{(h_{\infty }^{*})}^2/(2D)$.

In Fig. 1 we compare the microscopic rebinding-time distribution (obtained using the microscopic algorithm described in [21]) to the rebinding-time distribution for the gRDME. As we can see, there is a good match down to a spatial resolution of approximately σ. For the finest meshes, the behavior at really short time scales is incorrect due to dissociating particles starting in the same voxel, but not reacting until they are in neighboring voxels. This introduces an error on the order of the voxel size, which will be on the order of the size of the molecules.

FIG. 1.

In (a) and (b) we have plotted the rebinding-time distributions in 3D. For the sRDME we have a good match between the microscopic and mesoscopic simulations for $h\approx h_{\infty }^{*}$, while the average rebinding time is underestimated for finer meshes. For the gRDME we see that the microscopic and mesoscopic distributions agree well for $h_{\infty ,g}^{*}<h<h_{\infty }^{*}$ down to spatial resolution almost on the order of the size of the molecules, or a temporal resolution of approximately ${(h_{\infty ,g}^{*})}^2/(2D)$. In (c) and (d) we have plotted the rebinding-time distributions in 2D. The conclusions are the same as for the 3D case. The parameters in (a) and (b) are given by σ = 2 × 10⁻⁹ m, D = 2 × 10⁻¹² m² s⁻¹, L = 5.145 × 10⁻⁷ m, and k_a = 10⁻¹⁸ m³ s⁻¹. The parameters in (c) and (d) are given by σ = 2 × 10⁻⁹ m, D = 2 × 10⁻¹⁴ m² s⁻¹, L = 5.2 × 10⁻⁷ m, and k_a = 10⁻¹² m² s⁻¹.

B. Convergence of the generalized RDME

The dynamics of some systems is resolved only at a fine spatial resolution. In particular, it has been shown that fast rebinding events can affect e.g. the response time of a MAPK pathway [6]. We consider the system

$$\{\begin{array}{cc}\hfill S_1\hfill & \hfill \stackrel{k_d}{\to }{S}_{11}+S_{12}\stackrel{k_a}{\to }{S}_2\hfill \\ \hfill S_2\hfill & \hfill \stackrel{k_d}{\to }{S}_{21}+S_{22}\stackrel{k_a}{\to }{S}_3,\hfill \end{array}$$ (128)

which has a behavior similar to the MAPK pathway of [6]. Due to the possibility of fast rebinding events, the long-term dynamics of the system is affected by spatial correlations between newly produced molecules.

We start with an initial population of 100 S₁ molecules, with none of the other species present. The system is simulated for 2s, during which we sample the state of the system at 201 evenly distributed points between t = 0 and t = 2. We simulate the system with both the sRDME, as well as with the gRDME, for different voxel sizes. Let 𝒮 = {S₁, S₁₁, S₁₂, S₂, S₂₁, S₂₂, S₃}. We define the error, E(h), to be

$$E(h)=\frac{1}{201}\sum _{i=1}^{201}\sum _{S\in 𝒮}|{[S]}_{h,i}^{\mathit{\text{meso}}}-{[S]}_i^{\mathit{\text{micro}}}|,$$ (129)

where ${[S]}_i^{\mathit{\text{micro}}}$ is the average population of S at time t_i, obtained with the microscopic algorithm from [21], and where ${[S]}_{h,i}^{\mathit{\text{meso}}}$ denotes the average population of S at time t_i obtained at the mesoscopic scale with voxel size h.

After a dissociation of either an S₁ or S₂ molecule from (128), the products can rebind quickly to produce an S₂ or S₃ molecule, respectively. On the microscopic scale, the products are in contact after a dissociation event, and thus the spatial correlation will be significant. At the mesoscopic scale, the products are placed in the same voxel after a dissociation. If the voxel size is large compared to the size of the molecules, the spatial correlation will be less than on the microscale. Thus, to simulate (128) accurately, we would expect a fine mesh resolution to be required.

Let σ_i be the reaction radius of molecule S_i, and σ_ij the reaction radius of molecule S_ij. The parameters of the model are given by

$$\begin{array}{cc}\{\begin{array}{c}{k}_d=10{\mathrm{s}}^{-1}\hfill \\ k_a={10}^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}\hfill \end{array}\hfill & \{\begin{array}{c}{\sigma }_1={10}^{-9}\mathrm{m}\hfill \\ {\sigma }_{11}={\sigma }_{12}=0.8\times {10}^{-9}\mathrm{m}\hfill \\ {\sigma }_2=2\times {10}^{-9}\mathrm{m}\hfill \\ {\sigma }_{21}={\sigma }_{22}=1.8\times {10}^{-9}\mathrm{m}\hfill \\ {\sigma }_3=2.5\times {10}^{-9}\mathrm{m}.\hfill \end{array}\hfill \end{array}$$ (130)

For simplicity, we let all species have the same diffusion rate, D = 10⁻¹² m² s⁻¹. The S₁ molecules are initialized uniformly in a cube of volume 10⁻¹⁸ m³.

There is a critical lower bound on the mesh size associated with each of the system’s bimolecular reaction events

$$\{\begin{array}{c}{h}_1≔h_{\infty }^{*}({\sigma }_{11}+{\sigma }_{12})\approx 5.0815·{10}^{-9}\hfill \\ h_2≔h_{\infty }^{*}({\sigma }_{21}+{\sigma }_{22})\approx 1.1433·{10}^{-8}.\hfill \end{array}$$ (131)

We know that for $h>h_{\infty }^{*}$, we are unable to match both the mesoscopic mean association time and the mesoscopic mean rebinding time to the corresponding microscopic quantities. Thus, for h > max{h₁, h₂}, we will overestimate the rebinding time for both reactions, and consequently underestimate the average S₃ concentration.

For h₂ > h > h₁ the dynamics is less obvious; we are underestimating the average rebinding time for the first reaction, but overestimating the average rebinding time for the second. As we can see in Fig. 2 (b), the positive and negative errors partly cancel out in this regime. At first the error decreases with decreasing h, but as we approach h₁, it starts to increase again. The behavior of the sRDME is hard to predict, and a priori we cannot be sure that a particular choice of h is suitable.

FIG. 2.

In (a) we plot the average number of S₃ molecules as a function of time. As we can see, for a larger value of the voxel size h, we underestimate the number of S₃ molecules. For a very fine mesh, the number of S₃ molecules is overestimated with the sRDME. Somewhere in between we may obtain a good approximation compared to the microscopic results also with the sRDME, but then the concentration of other species in the system will be incorrect. For simulations with the generalized algorithm, the average number of S₃ molecules is underestimated for coarse meshes, but as we refine the mesh, the dynamics approach that of the microscopic simulations. In (b) we see that the total error, as defined in (129), decreases down to a mesh size of h₁ for the gRDME, while the error for the sRDME first decreases slightly but then increases as we refine the mesh further. We obtain an almost perfect match between the microscopic simulations and the gRDME as h approaches h₁, and all the way down to h_2,g. The average on the microscopic scale is based on 2000 trajectories, giving 95% confidence intervals of width less than 0.2% of the mean for most time points, and the mesoscopic results are based on approximately 500 trajectories, giving 95% confidence intervals of width less than 1.0% of the mean for most time points.

In contrast, we see that the gRDME has a more predictable behavior, converging with decreasing h, and yielding an almost perfect match for h < h₁. The difference in behavior is due to the gRDME matching the average rebinding time also for $h<h_{\infty }^{*}$, all the way down to $h_{2,g}≔h_{\infty ,g}^{*}({\sigma }_{21}+{\sigma }_{22})\approx 3.8934·{10}^{-9}$.

At the finest mesh sizes a diffusion event will move a particle a distance on the order of the radius of the molecules. Thus, for a dilute system such as the above, it is worth noting that an efficient microscopic method taking advantage of the large distances between molecules will be significantly faster than the gRDME. For the problem above, the algorithm in [21] is more than an order of magnitude faster than the gRDME at the finest mesh sizes (one realization on the microscopic scale took around 8s, while a simulation with the gRDME at the finest mesh size took around 150s; note that both implementations were crude and not optimized). However, in the case of a less dilute system, we expect the gRDME to be more competitive. The execution time on the microscopic scale increases almost quadratically with the number of molecules, but linearly with the number of molecules on the mesoscopic scale. A possible application of the gRDME is in hybrid methods, where part of the system can be simulated at the coarser sRDME scale, and only some parts of the system at the most fine-grained and expensive gRDME scale.

V. SUMMARY

For the sRDME there is a lower bound on the mesh size, $h_{\infty }^{*}$, below which the accuracy deteriorates. For $h>h_{\infty }^{*}$ we match the mean binding time of two molecules with the mesoscopic reaction rate given by ρ^(d)(k_a, h). For $h=h_{\infty }^{*}$ we match both the mean binding time and the mean rebinding time of the two molecules.

Some systems display fine-grained dynamics, requiring a fine spatial resolution to be simulated at the mesoscopic scale. By generalizing the sRDME to allow reactions between molecules in neighboring voxels, we obtain a lower bound on the mesh size given by $h_{\infty ,g}^{*}$, where $h_{\infty ,g}^{*}$ is on the order of the reaction radius of a pair of molecules. We derived analytical expressions for the reaction rates, and showed that we match both the mean binding time and the mean rebinding time for $h_{\infty ,g}^{*}\le h\le h_{\infty }^{*}$. For $h>h_{\infty }^{*}$, the gRDME and the sRDME agree.

We studied the accuracy of the gRDME in two numerical examples. In the first example we showed that we not only match the mean rebinding time for $h_{\infty ,g}^{*}\le h\le h_{\infty }^{*}$, but that we also obtain a good match between the rebinding-time distributions at the two scales. In the second example we considered a system that cannot be accurately simulated with the sRDME, as the mesh resolution required is below the fundamental lower limit $h_{\infty }^{*}$. We showed that with the gRDME we are able to simulate the system to high accuracy, and we showed that we obtain convergence to the microscopic simulations with decreasing mesh size h.