Stochastic diffusion processes on Cartesian meshes

Abstract

Diffusion of molecules is simulated stochastically by letting them jump between voxels in a Cartesian mesh. The jump coefficients are first derived using finite difference, finite element, and finite volume approximations of the Laplacian on the mesh. An alternative is to let the first exit time for a molecule in random walk in a voxel define the jump coefficient. Such coefficients have the advantage of always being non-negative. These four different ways of obtaining the diffusion propensities are compared theoretically and in numerical experiments. A finite difference and a finite volume approximation generate the most accurate coefficients.

1. Introduction

Small copy numbers of many molecular species in biological cells require stochastic models of the chemical reactions between the molecules and their diffusive motion. One example is gene expression where the number of molecules involved is small and only stochastic models can explain observations in experiments [1, 2]. Continuum models for the concentrations of the chemical species based on partial differential equations (PDEs) capture neither the randomness in the chemical reactions nor the fact that the number of molecules is integer.

In a well stirred system, there is no space dependence of the distribution of the species. Gillespie [3] invented an algorithm to simulate such chemical systems, the Stochastic Simulation Algorithm (SSA). The efficiency of the algorithm is improved in [4]. It is extended in [5, 6] to space-dependent systems where the diffusion of the molecules cannot be neglected.

The domain of interest is Ω with boundary ∂Ω. It is partitioned by a Cartesian mesh into compartments or voxels _i with volume | _i| and a node x_i in the center in [5]. The molecules jump between the voxels (or between the nodes in the lattice) with a certain probability. The time until a molecule jumps from _i to the adjacent _j is assumed to be exponentially distributed with parameter λ_ij. With n_i neighbours, the total jump propensity out of _i is

$${\lambda }_i=\sum _{j=1}^{n_i}{\lambda }_{ij}.$$ (1)

The diffusion propensity is λ_im_i with m_i molecules in _i. Then the SSA for the diffusion in the molecular system is:

1. Initialize the number of molecules m_k, k = 1, …, K, in the K voxels at t = 0.

2. Sample the exponentially distributed time Δt_k with rate λ_km_k to the first diffusion event in all K voxels and let t_k = Δt_k.

3. Determine the smallest t_k. Let t_i be the minimum of all t_k in voxel _i.

4. For the jump from _i, sample a jump to _j with probability θ_ij = λ_ij/λ_i.

5. Update t := t_i and molecule numbers m_i := m_i − 1 and m_j := m_j + 1.

6. Sample Δt_i and Δt_j with rates λ_im_i and λ_jm_j and let t_i = t + Δt_i and t_j = t + Δt_j and go to 3.

In the algorithm, the number of molecules m_i in each voxel is updated if it has changed after an event and a new time t_i is determined for the next event in the same voxel. The SSA generates one realization of a continuous time, discrete space Markov process. We will compare different ways of determining non-negative jump coefficients λ_ij for a Cartesian mesh where all voxels have equal size assuming that the rates to jump from _i to _j and back again are equal, λ_ij = λ_ji.

Let u_i(t) be the numerical approximation at x_i on the Cartesian mesh of the concentration u(x, t) satisfying the diffusion equation in the domain Ω

$$\frac{\partial u(\mathbf{x},t)}{\partial t}=\mathrm{\Delta }u(\mathbf{x},t),\mathbf{x}\in \mathrm{\Omega },t\ge 0,$$ (2)

with Neumann boundary condition n · ▽u = 0 at the boundary ∂Ω with the outward normal n. Choose the λ_ij coefficients such that they approximate the Laplacian in node i at x_i

$$\mathrm{\Delta }u({\mathbf{x}}_i,t)\approx \sum _{j=1}^{n_i}{\lambda }_{ji}{u}_j(t)-\sum _{j=1}^{n_i}{\lambda }_{ij}{u}_j(t)=\sum _{j=1}^{n_i}{\lambda }_{ij}(u_j(t)-u_i(t))=\sum _{j=1}^{n_i}{\lambda }_{ji}{u}_j(t)-{\lambda }_i{u}_i(t).$$ (3)

The node at x_i has n_i adjacent nodes at x_j used in the approximation. In the limit of a large total number of molecules $M={\sum }_{k=1}^K{m}_k$, the time dependent expected values of the concentrations ũ_i = m_i/(M| _i|) in the voxels in SSA with the jump coefficients in (3) converge to the deterministic concentrations u_i solving the discretized diffusion equation (2) using (3), see [7]. The distribution for the difference between ũ_i and u_i is given in [8].

In an equidistant Cartesian mesh, the Laplacian is discretized with a finite difference method (FDM) in [5, 6, 9]. Unstructured triangular meshes in 2D and tetrahedral meshes in 3D are better suited to represent complicated geometries inside the cell effectively. The coefficients for these unstructured meshes are derived from a finite element method (FEM) for the Laplacian in [10, 11] and with a finite volume method (FVM) in [12].

The jump propensities λ_ij have to be non-negative to be meaningful in the SSA. The standard 5-point (2D) and 7-point (3D) approximations of the Laplace operator in (2) with FDM on a Cartesian mesh yield positive λ_ij for the neighbouring voxels but the FEM coefficients for an unstructured mesh may be negative for a poor mesh [11]. The numerical discretization of the diffusion equation (2) by a common FVM method may be inconsistent [13] and not converge to the analytical solution on a general unstructured mesh but the coefficients are non-negative.

If the jump coefficients are negative, the numerical solution of (2) will in general not be monotone and not satisfy the discrete maximum principle. For the discrete maximum principle to hold the coefficients must satisfy the following conditions, see [14]: (1) is valid for interior nodes; λ_ij ≥ 0; and λ_i is greater than the sum of the off-diagonal elements for at least one boundary node. Thus, the two first conditions apply for both the diffusion coefficients in the SSA and the spatial discretization of the diffusion equation.

Many papers are devoted to the derivation of consistent FEM and FVM approximations fulfilling the discrete maximum principle for triangular and quadrilateral meshes in 2D and tetrahedral meshes in 3D, e.g. [15, 16, 17, 18, 19]. They are nonlinear and the coefficients in (3) depend on the solution u_i making them less suitable as jump propensities in the SSA or rely on meshes with geometrical properties that may be difficult to achieve with a mesh generator. In [20, 21] it has been shown that it is impossible to construct a linear method, i.e. λ_ij is constant in (3), satisfying the discrete maximum principle for a linear elliptic equation on general quadrilateral meshes in 2D.

As an alternative, we solve the diffusion equation on a local domain and use the solution to calculate the mean first exit time (FET) from that domain for a molecule in Brownian motion. The molecule is released at x_i at t = 0 and after a random walk it leaves a subdomain defined by the convex hull of the adjacent nodes x_j, j = 1, …, n_i, for the first time at t = τ. The expected value of τ from this subdomain is the inverse of the rate λ_i. The mesh is Cartesian in 2D with different mesh sizes h_x and h_y in the x and y directions, respectively. Jumps are allowed in the coordinate directions and along the diagonals, see Fig. 1. The probability to exit from _i to _j depends on the distance between the nodes x_i and x_j i.e. h_x and h_y. The FET coefficients are always non-negative. They are compared with the methods above for approximations of the Laplacian. The coefficients derived with FDM, FEM, and FVM also depend only on h_x and h_y in the mesh. The jump coefficients obtained by the FET and the systematic comparison with the coefficients from numerical discretizations are the main contributions of this paper. The analysis is extended to the fully unstructured case in [22].

Figure 1.

The reference nine point stencil in 2D for the voxel ₀ with midpoint x₀ and its neighbours.

The expected FET can be utilized in a different way to solve (2), see [23] and the references therein. Stochastic simulations determine the FET in polygonal domains with general boundary and initial conditions in a Monte Carlo method suitable for high dimensions. Here we are interested in using FET to find the probabilities of the motion of molecules on a discrete lattice.

In the next section, we present the FDM, FEM, and FVM discretizations of the Laplacian and how to derive the jump coefficients from them. We compute expressions for the FET in Sect. 3 to derive new jump coefficients from the exit behavior of a diffusing molecule. In Sect. 4, we compare the four methods to obtain λ_i, λ_ij, and θ_ij in SSA. We perform numerical experiments in Sect. 5 and draw conclusions in the final section.

2. FDM, FEM, and FVM coefficients

The primal mesh is Cartesian in 2D and has nodes at x_i = (x_i1, x_i2) and the dual mesh consists of voxels _i with x_i in the center. The distance is h_x between the nodes in the x direction and h_y in the y direction. In the following, we consider the reference voxel ₀ with midpoint x₀ and area h_xh_y, see Fig. 1, and introduce the aspect ratio κ, such that h_x = κh and h_y = h. As we will only work on this reference voxel from now on, the notation is simplified by letting λ_0i → λ_i and θ_0i → θ_i in the remainder of the paper.

2.1. FDM

Discretize the Laplacian Δ with the FDM as in [24] to obtain the weights λ_i in the difference stencil for the mesh in Fig. 1. By symmetry, λ₁ = λ₅, λ₃ = λ₇, and λ₂ = λ₄ = λ₆ = λ₈ and by consistency, the weight in the center point is $-{\sum }_{j=1}^8{\lambda }_j$. The following coefficients obtained after Taylor expansion give a second order approximation of Δ

$${\lambda }_1=\frac{1-\kappa \alpha }{{\kappa }^2{h}^2},{\lambda }_2=\frac{\alpha }{2\kappa h^2},{\lambda }_3=\frac{{\kappa }^2-\kappa \alpha }{{\kappa }^2{h}^2},$$ (4)

where α is a free parameter. If α = 0 then we have the usual 5-point stencil. We wish all λ_i to be non-negative to fulfill the discrete maximum principle and to be useful as jump coefficients requiring

$$0\le \alpha \le min(\kappa ,\frac{1}{\kappa })\le 1.$$ (5)

The weight in the center is

$${\lambda }_0=2\frac{{\kappa }^2-\alpha \kappa +1}{{\kappa }^2{h}^2}>0$$ (6)

with a negative sign. If κ → ∞ then α → 0 in (5) for positive coefficients and ${\lambda }_0\to 2/h_y^2$. For a large κ, almost all jumps out of ₀ will be in the y direction with the same rate as in 1D. Accordingly, if κ → 0, then α → 0 by (5) and ${\lambda }_0\to 2/h_x^2$ and the diffusion behaves as one-dimensional in the x direction. The jump rate to node i is λ_i and the relative rates θ_i = λ_i/λ₀ between the nodes are

$${\theta }_1={\theta }_5=\frac{1-\alpha \kappa }{2({\kappa }^2-\alpha \kappa +1)},{\theta }_3={\theta }_7=\frac{{\kappa }^2-\alpha \kappa }{2({\kappa }^2-\alpha \kappa +1)},{\theta }_2={\theta }_4={\theta }_6={\theta }_8=\frac{\alpha \kappa }{4({\kappa }^2-\alpha \kappa +1)}.$$ (7)

With α = 0 and κ = 1, the non-zero relative rates are θ₁ = θ₃ = θ₅ = θ₇ = 1/4, the 5-point stencil for the Laplacian.

2.2. FEM

We approximate the Laplacian by bilinear basis functions on a rectangular element, see [14, 25]. The lumped mass matrix has h_xh_y on the diagonal. After division by h_xh_y, the coefficients corresponding to λ_i in (4) and (6) are

$${\lambda }_0=\frac{4({\kappa }^2+1)}{3{\kappa }^2{h}^2},{\lambda }_1=\frac{2-{\kappa }^2}{3{\kappa }^2{h}^2},{\lambda }_2=\frac{{\kappa }^2+1}{6{\kappa }^2{h}^2},{\lambda }_3=\frac{2{\kappa }^2-1}{3{\kappa }^2{h}^2}.$$ (8)

The free parameter α in (4) is here (κ²+1)/(3κ) with the following constraints on the mesh for non-negative coefficients

$$\frac{1}{\sqrt{2}}\le \kappa \le \sqrt{2}.$$ (9)

The relative rates are

$${\theta }_1={\theta }_5=\frac{2-{\kappa }^2}{4({\kappa }^2+1)},{\theta }_3={\theta }_7=\frac{2{\kappa }^2-1}{4({\kappa }^2+1)},{\theta }_2={\theta }_4={\theta }_6={\theta }_8=\frac{1}{8},$$ (10)

cf. (7).

2.3. FVM

The Laplacian in the FVM is approximated by (see [13, 26]):

$$\mathrm{\Delta }u({\mathbf{x}}_0)\approx \frac{1}{|{\mathcal{V}}_0|}{\int }_{{\mathcal{V}}_0}\mathrm{\Delta }ud\mathrm{\Omega }=\frac{1}{|{\mathcal{V}}_0|}{\int }_{v_0}\mathbf{n}·\nabla u\mathit{\text{ds}}=\frac{1}{|{\mathcal{V}}_0|}\sum _{j=1}^4{\int }_{v_{0_j}}\mathbf{n}·\nabla u\mathit{\text{ds}}$$ (11)

in voxel ₀ of area | ₀| = h_xh_y = κh² and with outward normal n(x) at the boundary $v_0={\cup }_{j=1}^4{v}_{0j}$. We now approximate the outward fluxes n · ∇u across the edges. Suppose that u_j is the solution in _j, j = 1, …, 8, surrounding ₀ in Fig. 1 and u₀ is the solution in ₀. The straightforward approximation of the outward flux on the edge ν₀₁ in (11) is (u₁ − u₀)/h_x. Similar approximations on the other edges yield the coefficients

$${\lambda }_0=\frac{2({\kappa }^2+1)}{{\kappa }^2{h}^2},{\lambda }_1=\frac{1}{{\kappa }^2{h}^2},{\lambda }_2=0,{\lambda }_3=\frac{1}{h^2},$$ (12)

corresponding to the case α = 0 in (4). There is no problem with negative jump propensities here. The relative rates in Step 4 in the SSA in Sect. 1 are

$${\theta }_1={\theta }_5=\frac{1}{2({\kappa }^2+1)},{\theta }_3={\theta }_7=\frac{{\kappa }^2}{2({\kappa }^2+1)},{\theta }_2={\theta }_4={\theta }_6={\theta }_8=0.$$ (13)

In another gradient approximation involving the diagonal voxels, let ∂u/∂x ≈ (u₁+u₂-(u₀+u₃))/2h_x in the corner to ₂, see Fig. 1. With the same kind of approximation for ∂u/∂y and in the other corners of ₀ and a linear variation of the gradient along v_0j, the coefficients are

$$\begin{array}{cc}{\lambda }_1={\lambda }_5=\frac{1-{\kappa }^2}{2{\kappa }^2{h}^2},& {\lambda }_2={\lambda }_4={\lambda }_6={\lambda }_8=\frac{1+{\kappa }^2}{4{\kappa }^2{h}^2},\\ {\lambda }_3={\lambda }_7=\frac{{\kappa }^2-1}{2{\kappa }^2{h}^2},& {\lambda }_0=4{\lambda }_2.\end{array}$$ (14)

These are non-negative only if κ = 1. Then they are the same as for the FDM with α= 1 in Sect. 2.1.

The jump coefficients derived here are defined by FDM (4), FEM with bilinear basis functions (8) as in [14], and a standard FVM (12). Additional constraints for non-negativity are necessary for FDM (5) and FEM (9).

3. First exit time

Let x₀ be a node inside a simply connected domain ω with boundary ∂ω. The FET of a molecule from ω initially at x₀ at t = 0 is denoted by the random variable τ. The mean value E_τ = E[τ] and the location on ∂ω where the molecule left ω can be computed from the probability density function (PDF) p(x, t) for the position x of the molecule at time t. Then p(x, t) satisfies the following PDE with diffusion coefficient D in ω:

$$\frac{\partial p(\mathbf{x},t)}{\partial t}=D\mathrm{\Delta }p(\mathbf{x},t),\mathbf{x}\in \omega ,p(\mathbf{x},t)=0,\mathbf{x}\in \partial \omega ,p(\mathbf{x},0)=\delta (\mathbf{x}-{\mathbf{x}}_0).$$ (15)

Then p(x, t) is the probability density for the position x of a diffusing molecule provided that it has not left ω until time t [27, 28, 29, 30]. The probability that the molecule is still inside ω at t is the survival probability

$$S(t)={\int }_{\omega }p(\mathbf{x},t)d\omega =P(\tau \ge t).$$ (16)

Here dω denotes integration over the domain ω. Let further n(x) be the outward normal of ∂ω. The PDF p_ω(t) for a molecule to exit at time t is then derived from (15), (16), and Gauss' formula

$$p_{\omega }(t)=-\frac{\partial S(t)}{\partial t}=-{\int }_{\omega }D\mathrm{\Delta }pd\omega =-D{\int }_{\partial \omega }\mathbf{n}·\nabla p\mathit{\text{ds}}\ge 0.$$ (17)

Here ds denotes integration over the boundary ∂ω. The expected value of the exit time is obtained from p_ω(t)

$$E_{\tau }={\int }_0^{\infty }t{p}_{\omega }(t)\mathit{\text{dt}}={\int }_0^{\infty }S(t)\mathit{dt}.$$ (18)

For the Markov property to hold in the SSA simulations, the exit time distribution has to be exponential. Since S(0) = 1, lim_t→∞ S(t) = 0, and S(t) decays monotonically it can be approximated by an exponential S(t) ≈ S̃(t) = exp(–λt) and we can e.g. choose λ such that the two distributions have the same mean value in (18). With this S̃, by (18) E_τ = 1/λ and by (17) the approximate PDF of the exit time τ is

$${\stackrel{\sim }{p}}_{\omega }(t)=\lambda exp(-\lambda t),$$ (19)

i.e. τ is exponentially distributed. The diffusion propensity λ₀ in the SSA in Sect. 1 to jump out of ₀ is then 1/E_τ with the corresponding domain ω₀ embedding ₀, see Fig. 2a.

Figure 2.

(a) The voxel ₀ embedded in the rectangle ω₀. (b) Boundary treatment for a corner.

The flux of molecules out of ω₀ at a point on ∂ω₀ is −D▽p · n. Let ∂ω₀ consist of non-overlapping sections ∂ω_j such that ∂ω₀ = ∪_j∂ω_j. The probability that the molecule leaves ω₀ through ∂ω_j of the boundary at t is

$$p_{\partial \omega j}(t)=-D{\int }_{\partial {\omega }_j}\mathbf{n}·\nabla p(\mathbf{x},t)\mathit{ds}.$$ (20)

Given that the exit time out of ω₀ is t, the conditional probability for the molecule to exit along the edge ∂ω_j is

$${\theta }_j(t)=\frac{p_{\partial {\omega }_j}(t)}{p_{\partial {\omega }_0}(t)}=\frac{{\int }_{\partial {\omega }_j}\mathbf{n}·\nabla p(\mathbf{x},t)\mathit{\text{ds}}}{{\int }_{\partial {\omega }_0}\mathbf{n}·\nabla p(\mathbf{x},t)\mathit{\text{ds}}}.$$ (21)

The probability in the SSA to leave ω₀ through ∂ω_j at time τ is then the expected value

$${\theta }_j=E[{\theta }_j(\tau )]={\int }_0^{\infty }{\theta }_j(t)p_{\omega }(t)\mathit{\text{dt}}$$ (22)

and by (21) Σ_jθ_j = 1. No matter how we choose ω₀, the θ_j are always non-negative.

The time to leave ₀ and enter a neighbouring voxel _i, i = 1, …, n₀, is exponentially distributed in (19). The rate at which a molecule reaches _i is λ_i = θ_iλ and the time for the jump to occur is exponentially distributed with rate λ_i. The SSA algorithm in Sect. 1 defines a continuous time, discrete Markov process [27] with these jump coefficients.

The coefficients for the voxel in Fig. 1 are calculated by choosing ω₀ to be the rectangle defined by the surrounding nodes x₁, …, x₈, see Fig. 2a. By this choice, we arrive at jump coefficients that are comparable to the FDM, FEM, and FVM coefficients. We compute the jump rate λ₀ by (16) and (18). The boundary ∂ω₀ is partitioned into ∂ω_j, j = 1, …,8, centered around the nodes x_j. We choose two parameters 0 ≤ β_x,β_y ≤ 1 such that |∂ω₁| = |∂ω₅| = 2β_yh_y and |∂ω₃| = |∂ω₇| = 2β_xh_x, where | · | is the length of an edge. Consequently, the length of ∂ω₂, ∂ω₄, ∂ω₆, and ∂ω₈ is (1 − β_x)h_x + (1 − β_y)h_y. Then θ_j follows from (21) and (22).

The analytical solution of (15) with x₀ = (h_x,h_y) in the rectangular domain ω = [0, 2h_x] × [0, 2h_y] is obtained by separation of variables and an expansion in the eigenfunctions

$$p(x,y,t)=\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }\frac{1}{h_x{h}_y}sin\left(\frac{k\pi }{2}\right)sin\left(\frac{j\pi }{2}\right)sin\left(\frac{k\pi x}{2h_x}\right)sin\left(\frac{j\pi y}{2h_y}\right)·exp(-(\frac{k^2}{4h_x^2}+\frac{j^2}{4h_y^2}\left){\pi }^2\mathit{\text{Dt}}\right)=\frac{1}{\kappa h^2}\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{(-1)}^{j+k}sin\left(\frac{(2k-1)\pi x}{2\kappa h}\right)sin\left(\frac{(2j-1)\pi y}{2h}\right)·exp(-\frac{{\pi }^2\mathit{\text{Dt}}}{4{\kappa }^2{h}^2}({(2j-1)}^2+{\kappa }^2{(2\kappa -1)}^2)).$$ (23)

This formula converges rapidly when D·t is not too small. A different formula is available when D · t is small, cf. [23, 31].

Choosing β_x = β_y = 1 corresponds to α = 0 in the FDM in Sect. 2.1. In that case, jumps only happen along the coordinate axes of the mesh and coordinate splitting allows us to simulate two one dimensional diffusions instead. The choice β_x = β_y = 0 corresponds to α = 1 in Sect. 2.1 with only the nodes in the corners involved.

By (16), (18) and (23) the jump rate from x₀ out of ω₀ is

$$\frac{1}{{\lambda }_0}=\frac{64{\kappa }^2{h}^2}{{\pi }^{4}D}\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\frac{{(-1)}^{j+k}}{(2j-1)(2k-1)({(2j-1)}^2+{\kappa }^2{(2k-1)}^2)}.$$ (24)

There are two parameters in the definition of the FET coefficients, β_x and β_y. They determine ∂ω_j and θ_j in (21) and (22). At the edge where x = 2κh, p_∂ω1 is given by (20)

$$p_{{\partial }_{\omega 1}}(t,{\beta }_y,\kappa )={\int }_{(1-{\beta }_y)h}^{(1+{\beta }_y)h}\frac{\partial p(x,y,t)}{\partial x}\mathit{\text{dy}}=\frac{2}{{\kappa }^2{h}^2}\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{(-1)}^{k+1}\frac{2k-1}{2j-1}sin\left(\frac{(2j-1)\pi {\beta }_y}{2}\right)·exp(-\frac{{\pi }^2\mathit{\text{Dt}}}{4{\kappa }^2{h}^2}((2j-1)+{\kappa }^2{(2k-1)}^2)).$$ (25)

Hence, θ₁ in (22) depends nonlinearly on β_y and κ, and accordingly for θ₃ and β_x.

Henceforth, the diffusion coefficient D is taken to be 1. Examples of time dependent solutions to (16), (17) and (21) using (23) are found in Fig. 3 for κ = 1.4 and h = 1. We approximate the cumulative survival time distribution S(t) by an exponential distribution with the same mean value to preserve the Markov property in the SSA. In this way, the average waiting time for a jump sampled from e^–λ₀t is the same as from the accurate S(t). The higher order moments $\mathbb{E}[t^n]={\int }_0^{\infty }{t}^n{p}_{\omega }(t)\mathit{\text{dt}}$ are $n!/{\lambda }_0^n$ for the exponential distribution and can be computed from (16) and (17). In Table 1, we see that the second moments agree well between the approximation by an exponential and the exact S(t). The products of the infinite sums in (23) to compute S(t) and (24) to compute λ₀ are rewritten as Cauchy products and truncated after 10³ terms.

Figure 3.

FET quantities calculated for κ = 1.4, h = 1, D = 1 and β_x = β_y = 0.5: (a) S(t) and approximation by an exponential with λ₀ = 2.5841 as in (24) and Table 2. (b) Exit time distributions p_ω(t) = −∂S(t)/∂t. (c) θ_j(t) calculated by (21) and (25).

Table 1.

Higher order moments for the exact exit time distribtuion S(t) and the approximating exponential exp(−λ₀t).

	S(t)	exp(−λ0t)
[t]	0.3870	0.3870
[t2]	0.2267	0.2995
[t3]	0.1866	0.3477

The jump coefficients θ_j are computed for a t > 0 in Fig. 3c. For small t and κ > 1 jumps are more likely to occur in the y direction, i.e. to x₃ and x₇ since θ₃ > θ₁, θ₂. When t is large, it follows from (25) that the sums are well approximated by their first term, c_j exp(−γt), where γ = π²D(1 + κ²)/4κ²h², resulting in

$${\theta }_j\approx \frac{c_{j}exp(-\gamma t)}{{\sum }_{k=1}^8{c}_{k}exp(-\gamma t)}=\frac{c_j}{{\sum }_{k=1}^8{c}_k},$$ (26)

the constant behaviour for large times we see in Fig. 3b.

4. Comparison of the coefficients

The FEM, FVM, and FET define the coefficient λ₀ without any parameter. We can then choose the free parameter α such that the FDM coefficient corresponds to either of them:

$${\alpha }_{\mathit{\text{FEM}}}=\frac{1+{\kappa }^2}{3\kappa },{\alpha }_{\mathit{\text{FVM}}}=0,{\alpha }_{\mathit{\text{FET}}}=\frac{{\kappa }^2(1-\frac{1}{2}{\lambda }_0{h}^2)+1}{\kappa }.$$ (27)

The values of λ₀ and α for k = 1 and k = 1.4 are found in Table 2.

Table 2.

The jump coefficients for different methods when κ = 1 and κ = 1.4 and α is such that the FDM has the same λ₀.

		FEM	FVM	FET
κ = 1	λ0·h2	2.6667	4	3.3934
	α	0.6667	0	0.3033
κ = 1.4	λ0·h2	2.0136	3.0204	2.5841
	α	0.7048	0	0.3054

By choosing α as in (27) we obtain the same relative jump coefficients θ_j as the FEM (10) and the FVM (13). The relative jump coefficients in FET depend on β_x and β_y in Sect. 3. By solving

$${\theta }_{1,\mathit{\text{FET}}}({\beta }_y)={\theta }_{1,\mathit{\text{FDM}}},{\theta }_{3,\mathit{\text{FET}}}({\beta }_x)={\theta }_{3,\mathit{\text{FDM}}},$$ (28)

numerically for β_x and β_y and using that by symmetry

$$\begin{array}{cc}{\theta }_{5,\mathit{\text{FET}}}={\theta }_{1,\mathit{\text{FET}}},& {\theta }_{3,\mathit{\text{FET}}}={\theta }_{7,\mathit{\text{FET}}},\\ {\theta }_{j,\mathit{\text{FET}}}=(1-2{\theta }_{1,\mathit{\text{FET}}}-2{\theta }_{3,\mathit{\text{FET}}})/4,& j=2,4,6,8,\end{array}$$ (29)

the FET jump coefficients agree with those from FDM. With κ = 1.4, the values of β_x and β_y are 0.511 and 0.738, respectively. With this choice of β_x and β_y, the FET coefficients define a second order approximation of the Laplacian. In Fig. 4, we see the behaviour of β_x, β_y, λ₀ and the appropriate α for different κ. as κ increases, λ₀h² → 2 converging again to the one-dimensional case, cf. (24). In Fig. 4b, we find that an increase in κ affects β_y - regulating the contribution from the shorter edge - more than β_x, which only slightly depends on κ.

Figure 4.

(a) The α parameter (right) in FDM to achieve the same h²λ₀ as in FET in (27) and the corresponding λ₀h². (b) The β coefficients in FET in (28) to achieve the same relative jump coefficients θ_j for the same λ₀ as in FDM.

5. Numerical results

We will now examine the performance of the coefficients in SSA simulations of the diffusion of molecules. A square Ω = [0, 20] × [0, 20] is discretized in the experiments into n_x nodes or grid points in the x-direction and n_y = κ(n_x – 1) + 1 in the y-direction. The boundary is defined by the first and the last nodes in the x and y directions. We choose κ = 1 and κ = 1.4. The molecules are reflected as they reach the boundary.

The methods are compared for high copy numbers where there are convergence results due to Kurtz [7, 8]. This approach is similar to the comparison of methods for the numerical solution of PDEs. Their convergence to the analytical solution is usually measured when the step length vanishes. To validate a stochastic method one can examine the moments of the solution. Here we focus on the first moment. According to Kurtz [7, 8] the mean value of the concentrations of molecules computed by the SSA converges to the deterministic solution of the diffusion equation in the limit of large molecule numbers. In an example with low copy numbers, the variance is compared for different approximations at the steady state.

The distribution of molecules is compared to the analytical solution of the diffusion equation with Neumann boundary conditions in (2) with u(x, 0) = δ(x – x₀). To implement the reflecting boundary condition, we add the contributions θ_j from the standard voxel at the boundary, which are outside the domain, to θ_j in the interior, see Fig. 2b.

Let M be the number of molecules and the final time T = 5. The molecules are released at t = 0 at the center x₀ = (10, 10) of the domain. The error e in the simulations is defined by

$$e^2=\sum _{i=1}^{n_x}\sum _{j=1}^{n_y}|{\mathcal{V}}_{ij}|{|\frac{m_{ij}}{M|{\mathcal{V}}_{ij}|}-u(x_{ij},T)|}^2,$$ (30)

where m_ij is the copy number in voxel (i, j), | _ij| = κh² is the volume of voxel _ij and x_ij is the node coordinate. As the FDM, the FEM, and the FVM are second order accurate, their errors will behave as $\mathcal{O}(h^2+M^{-\frac{1}{2}})$ because of the space discretization and the statistical error.

In the case κ = 1, we have β_x = β_y = β. The β value is chosen such that θ_i,FET = θ_i,FDM with α = 0.3033. The relative jump coefficients vary with β in Fig. 5a. At β ≈ 0.56 they agree with the FDM coefficients in (7), see also Fig. 4b. We calculate the error on a grid with n_x = n_y = 21 grid points and M = 5 · 10⁶ and see that the accuracy varies with β in Fig. 5b. The minimum is at β ≈ 0.56 where the FET is equal to an FDM in Fig. 5a and is second order accurate in h.

Figure 5.

Comparison between FDM with α = α_FET and FET at κ = 1: (a) Relative jump propensities depending on parameter β in FET and the constant FDM coefficients. (b) Error e for different values of β.

In Fig. 5b, the error increases monotonically when β decreases from 0.56 to 0, which corresponds to α = 1 in the FDM. Then the molecules only jump along the diagonals, which means - if they are released in one voxel - they do not reach all voxels in the domain but only half of them in a checkerboard pattern. If the molecule starts in the voxel above, then it will reach the other half of the voxels. This particular distribution of the diffusing molecules explains the high error.

The FET jump coefficients for different β = β_x =β_y are compared to FDM coefficients for κ = 1.4 in Fig. 6a. The absolute error for the same parameters as before is displayed in Fig. 6b with a minimum at β ≈ 0.5. The behaviour is similar in both Fig. 5b (κ = 1) and Fig. 6b (κ =1.4). The isolines of the error for combinations of β_x and β_y are found in Fig. 6c.

Figure 6.

Comparison between FDM with α = α_FET and FET at κ = 1.4: (a) Relative jump propensities for FET depending on parameter β and the constant FDM coefficients. (b) Absolute error for different values of β = β_x = β_y. (c) Level curves of the error for different values of β_x and β_y with the red dot indicating the minimum.

We observe in Fig. 6a that the match with the FDM coefficients occurs for β_x = 0.511 and β_y = 0.738, which coincides with the minimum - indicated by the red dot - in Fig. 6c. The FET viewed as an approximation to the Laplacian is second order accurate there. The simplification β = β_x = β_y has its minimum at β ≈ 0.5 in Fig. 6b. This value corresponds to integration in (25) and in Fig. 2a between the boundaries of the adjacent voxels ₁, ₂, and ₃ to obtain θ₁, θ₂, and θ₃, respectively.

In Figs. 6b, c, the error increases again when β_x and β_y approach 0 as in Fig. 5, which corresponds to the particular diffusion explained earlier where the molecules only reach half of the domain.

We measure the relative error e/‖u‖ in Fig. 7a for the four different methods where e is calculated as in (30) and

Figure 7.

Performance on a square domain with κ = 1.4 (a) Convergence for the FDM with α = 0.3054, the FVM, the FEM, and the FET with β_x = β_y= 0.5 and β_x = 0.511 and β_y = 0.738. The reference curve decays as h² (b) Absolute error for the FDM method for different parameters α. (c) Relative error versus the number of jumps needed in each simulation. (d) Variance around steady state.

$${\Vert u\Vert }^2=\sum _{i=1}^{n_x}\sum _{j=1}^{n_y}|{\mathcal{V}}_{ij}|{|u(x_{ij},T)|}^2.$$ (31)

We discretize the square into n_x = 11, 21, 31 and 41 nodes and choose κ = 1.4, such that n_y = 15,29,43 and 57 and κh = 2,1,0.67 and 0.5. We set $M=10n_x^4$ in order to keep the statistical error smaller than the spatial error. In Fig. 7b, we display the error in the FDM for α ≤ 1/κ, where κ = 1.4, M = 5 · 10⁶ and n = 21. The error increases slightly with increasing α and the fluctuations indicate the size of the statistical error. Furthermore, we count the number of jumps for each simulation. This number is proportional to the number of generated random numbers and the administration of an event in the system - and therefore the computational work - and we plot the error against it in Fig. 7c. To examine the behaviour for low copy numbers we distribute M = 2240 molecules evenly across all voxels for n_x = 21 and n_y = 29. We compare the fluctuating molecule numbers m_ij to the spatially constant steady state concentration μ = 0.0025 in three realizations in the time interval [0, 5]. In Fig. 7d the sample variances

$${\sigma }^2=\frac{1}{n_x{n}_y-1}\sum _{i=1}^{n_x}\sum _{j=1}^{n_y}{|\frac{m_{ij}}{M|{\mathcal{V}}_{ij}|}-\mu |}^2,$$ (32)

for FEM, FVM and FET with β_x = 0.511 and β_y = 0.738 agree well.

The three methods FDM, FVM, and FEM converge to the analytical solution of the diffusion equation with rate 2 in Fig. 7a, as expected. There is a unique choice of β_x and β_y such that the FET corresponds to the FDM and becomes second order, too. With β_x = β_y = 0.5, the FET is not a consistent approximation of Δ and no reduction in e/‖u‖ is observed when h is lowered from 1 to 0.5. The relative error stagnates at 0.02 − 0.03 which is sufficiently low for most problems. The minimum error for the FDM is obtained for α = 0 in Fig. 7b. This method is the standard five point approximation and is equal to the FVM in (12) and (13). In Table 2, the FEM has the smallest λ₀. As the exit time is distributed exponentially, that means that the FEM has the longest expected time between events, corresponding to fewer jumps being simulated in [0, T]. For a given error, the number of jumps should be as low as possible for the best efficiency. As the FEM corresponds to a mixed FDM with α = 0.7048 it generates a higher error than the standard five point FDM with α = 0, see Fig. 7b. The FET or the FDM with α = 0.3054 on the other hand results in a smaller error while still jumping less than the FVM or the FDM with α = 0. In the comparison in Fig. 7c, the difference between the methods is small. In Fig. 7d, the three methods show similar fluctuations around steady state in low copy number simulations.

6. Conclusions and discussion

We evaluate four different ways of computing jump coefficients between voxels for stochastic simulation of diffusion in a Cartesian mesh; as a distinctive feature we include jumps along the diagonals in the mesh. The method based on the first exit time (FET) is equivalent to a finite difference method (FDM) of second order accuracy if the parameters are chosen appropriately. The most accurate and efficient method compared to the solution of the diffusion equation on a square is to use the coefficients of the finite volume method (FVM) or equivalently a FDM with jumps in the coordinate directions without including the diagonals. If we count the number of jumps to obtain a certain relative error, the differences between the methods are minor. The FVM and FET coefficients are always non-negative. There are restrictions on the mesh for the finite element method (FEM) to generate non-negative coefficients and for a given mesh there are restrictions on the FDM for non-negativity.

A generalization of these methods to an unstructured mesh consisting of triangles and tetrahedra as in [11] is desirable for improved geometrical flexibility. The straightforward FVM in Sect. 2.3 is easily adapted to a triangulated domain. A FEM with linear basis and test functions will generate jump coefficients as in [11]. A FDM consistent with the Laplacian on such a mesh is not easily derived analytically. However, the FVM may not be consistent and the jump coefficients of the FEM may be negative. On a triangular mesh, FVM is consistent if the mesh is of Delaunay type derived from a Voronoi tessellation [32]. A Cartesian mesh has the properties of a Delaunay mesh making FVM consistent there. The FET is easily transformed to an unstructured mesh. The analytical solution of (15) is no longer possible but p(x, t), λ₀, and θ_j can be computed numerically. The jump coefficients are non-negative but there is no FDM to help us determine β. Taking β = 0.5 works fairly well in Sect. 5 for reasonable error levels. The inconsistency with the Laplacian plays a role only for very small errors in the numerical experiments. Hence, this would be the natural choice for an unstructured mesh. The generalization to triangular and tetrahedral meshes of the calculation of the diffusion jump coefficients is the subject of [22].