Validity conditions for stochastic chemical kinetics in diffusion-limited systems

Abstract

The chemical master equation (CME) and the mathematically equivalent stochastic simulation algorithm (SSA) assume that the reactant molecules in a chemically reacting system are “dilute” and “well-mixed” throughout the containing volume. Here we clarify what those two conditions mean, and we show why their satisfaction is necessary in order for bimolecular reactions to physically occur in the manner assumed by the CME and the SSA. We prove that these conditions are closely connected, in that a system will stay well-mixed if and only if it is dilute. We explore the implications of these validity conditions for the reaction-diffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily well-mixed, but can be partitioned into cubical subvolumes (voxels) that are. We show that the validity conditions, together with an additional condition that is needed to ensure the physical validity of the diffusion-induced jump probability rates of molecules between voxels, require the voxel edge length to have a strictly positive lower bound. We prove that if the voxel edge length is steadily decreased in a way that respects that lower bound, the average rate at which bimolecular reactions occur in the reaction-diffusion CME and SSA will remain constant, while the average rate of diffusive transfer reactions will increase as the inverse square of the voxel edge length. We conclude that even though the reaction-diffusion CME and SSA are inherently approximate, and cannot be made exact by shrinking the voxel size to zero, they should nevertheless be useful in many practical situations.

INTRODUCTION

Stochastic chemical kinetics is concerned with a system of molecules of N chemical species S₁, …, S_N undergoing M chemical reactions R₁, …, R_M inside some volume Ω at some temperature T. Its aim is to describe the behavior of X(t) ≡ (X₁(t), …, X_N(t)), where X_i(t) is the number of S_i molecules in Ω at time t. Its key premise is that there exists, for each reaction R_j, a propensity functiona_j(x) which satisfies

$$\begin{array}{ccc}\hfill a_j\left(\mathbf{x}\right)dt& \equiv & \mathrm{the}\mathrm{probability},\mathrm{if}\mathbf{X}\left(t\right)=\mathbf{x},\mathrm{that}{R}_j\mathrm{will}\mathrm{fire}\hfill \\ & & \mathrm{somewhere}\mathrm{inside}\Omega \mathrm{in}\mathrm{the}\mathrm{next}\mathrm{infinitesimal}\hfill \\ & & \mathrm{time}\mathrm{interval}[t,t+dt)(j=1,...,M).\hfill \end{array}$$ (1)

It is further assumed that when an R_j reaction does occur, it changes the molecular population of species S_i by ν_ij, thus changing the system's state from x to x + ν_j where ν_j ≡ (ν_1j, …, ν_Nj).

Under these assumptions, the laws of probability imply that the time evolution of X(t) can be described exactly in two equivalent ways: first via the chemical master equation (CME),

$$\begin{array}{ccc}\hfill \frac{\partial P(\mathbf{x},t|{\mathbf{x}}_0,t_0)}{\partial t}& =& \sum _{j=1}^M[a_j(\mathbf{x}-{\mathit{\nu }}_j)P(\mathbf{x}-{\mathit{\nu }}_j,t|{\mathbf{x}}_0,t_0)\hfill \\ & & -a_j\left(\mathbf{x}\right)P(\mathbf{x},t|{\mathbf{x}}_0,t_0)],\hfill \end{array}$$ (2)

where P(x, t | x₀, t₀) is the probability that X(t) will be equal to x given that X(t₀) = x₀ for t ⩾ t₀; and second via the stochastic simulation algorithm (SSA):

• 1° In state x at time t, generate two random numbers τ and j according to the joint probability density function $p(\tau ,j|\mathbf{x},t)={\mathrm{e}}^{-a_0\left(\mathbf{x}\right)\tau }{a}_j\left(\mathbf{x}\right)$, where $a_0\left(\mathbf{x}\right)\equiv {\sum }_{k=1}^M{a}_k\left(\mathbf{x}\right)$.

• 2° Actualize the next reaction by replacing t ← t + τ and x ← x + ν_j.

• 3° Record the new (x, t). Return to 1°, or else end the simulation.

Since the CME and the SSA can each be derived via mathematically rigorous reasoning from the above definitions of the propensity functions a_j and the state-change vectors ν_j,¹ it follows that the CME and the SSA are logically equivalent to each other. Anything that validates or invalidates or extends one will validate or invalidate or extend the other. In this paper, we are going to be concerned with validating and extending the CME and the SSA, so we will often refer to them jointly as the CME/SSA.

The only problematic step in establishing the physical validity of the CME/SSA is proving that chemical reactions physically occur in the manner prescribed by hypothesis 1. There is a close relationship between the propensity function defined in 1 and the reaction “rate” that appears in the ordinary differential equations of traditional deterministic chemical kinetics. Early work in stochastic chemical kinetics tended to view the propensity function as a kind of ad hoc stochastic extension of the classical reaction rate, with the latter having the more rigorous physical justification. But in fact, the reaction rates turn out to be approximations of the propensity functions in the thermodynamic (large system) limit. That being the case, we cannot rigorously derive propensity functions from the reaction rates. Propensity functions can be reliably inferred only by looking to molecular physics to see how molecules actually behave. Doing that inevitably requires adopting a specific, usually idealized model for bimolecular chemical reactions. The model we shall adopt here is a fairly common one which assumes that: (i) the reacting molecules are hard spheres which move about as solute molecules in a sea of very many, much smaller, chemically inert solvent molecules; (ii) two solute molecules chemically react by first colliding with each other, their velocities at the instant of collision being distributed according to the Maxwell-Boltzmann distribution; and (iii) two colliding molecules will immediately undergo a specific bimolecular reaction R_j with probability q_j (0 < q_j ⩽ 1), a parameter which in principle is determined by the physics of the two molecules.

In Secs. 2, 3, we review the derivations of the propensity function from molecular physics, with the aim of illuminating and clarifying the conditions that must be satisfied if those derivations are to be valid. In Secs. 4, 5, 6, we discuss the implications of these validity conditions for the reaction-diffusion extension of the CME/SSA to systems that are not completely homogeneous. In Sec. 7, we show that, provided its validity conditions are satisfied, the reaction-diffusion CME/SSA will give a plausible modeling of bimolecular chemical reactions in a solution. In Sec. 8, we summarize our conclusions, and offer some comments on the advantages and limitations of the reaction-diffusion CME/SSA.

PHYSICAL JUSTIFICATION FOR THE PROPENSITY FUNCTION HYPOTHESIS

The implicit assumption in 1 that a chemical reaction is a physical event that occurs practically instantaneously dictates that every R_j in the CME/SSA will always be one of two types: either unimolecular or bimolecular. Reactions that are commonly called trimolecular or reversible practically always occur, at least in the cellular chemistry setting that we are primarily concerned with here, as a series of two or more unimolecular or bimolecular reactions. The rarity of true trimolecular reactions is a simple consequence of the rarity with which three molecules simultaneously collide with each other under well-mixed conditions; indeed, with hard-sphere molecules that virtually never happens.

Since quantum mechanics governs the way in which atoms arrange themselves into molecules, the dynamics of any unimolecular reaction S₁ → ⋅⋅⋅ is inherently stochastic. More specifically, quantum mechanics implies, at least on time scales of practical interest, that the probability that a particular S₁ molecule will undergo a unimolecular reaction in the next infinitesimal time dt will practically always be equal to some constant c_j multiplied by dt.² Summing the single-molecule reaction probability c_jdt over all x₁S₁ molecules in Ω gives, by the addition law of probability, Eq. 1 with a_j(x) = c_jx₁.

A propensity function for the bimolecular reaction S₁ + S₂ → ⋅⋅⋅ has been shown to be justified by molecular physics in two cases: when the reactant molecules are a dilute well-mixed gas, and when they are solute molecules in a dilute, well-mixed solution. Although the latter case is the only one of relevance for cellular chemistry, the simpler derivation in the former case illustrates more clearly the necessity of the dilute and well-mixed requirements. The propensity function for a dilute well-mixed gas is³

$$a_j(x_1,x_2)=\left(\pi {\sigma }_{12}^2{\overline{v}}_{12}{q}_j{\left|\Omega \right|}^{-1}\right)·x_1{x}_2\left(\mathrm{dilute}\mathrm{gas}\right).$$ (3)

Here, σ₁₂ is the average distance between the centers of a pair of reactant molecules at collision (the sum of their radii for hard sphere molecules); ${\overline{v}}_{12}=\sqrt{\left(8k_{\mathrm{B}}T\right)/\left(\pi m_{12}\right)}$ is their average relative speed, with m₁₂ their reduced mass, k_B Boltzmann's constant, and T the absolute temperature of the system; and q_j is the probability that an S₁−S₂ collision will result in an R_j reaction. The derivation of Eq. 3 goes as follows: $\pi {\sigma }_{12}^2·{\overline{v}}_{12}dt$ is the average “collision volume” that a randomly chosen S₂ molecule sweeps out relative to the center of a randomly chosen S₁ molecule in time dt. Dividing that collision volume by the system volume |Ω| gives, for reasons that will be elaborated below, the probability that the center of the S₁ molecule lies inside the collision volume, and hence the probability that the two molecules will collide in the next dt. That collision probability multiplied by the conditional probability q_j of a reaction given a collision yields, by the multiplication law of probability, the probability that the two molecules will react in the next dt. And finally, that single-pair reaction probability summed over all x₁x₂ distinct reactant pairs gives, by the addition law of probability, the probability defined in Eq. 1, thus establishing Eq. 3. For the bimolecular reaction 2S₁ → ⋅⋅⋅, the sum over all distinct reactant pairs would give a factor $\frac{1}{2}{x}_1(x_1-1)$ instead of x₁x₂. A “collision energy threshold” reaction model yields for q_j the well-known Arrhenius factor; however, we need not assume anything specific here about q_j other than it will have some value between 0 and 1.

The critical step in the foregoing derivation is the assertion that the ratio of {the average collision volume $\pi {\sigma }_{12}^2·{\overline{v}}_{12}dt$} to {the total system volume |Ω|} provides a valid estimate of the probability that the center of the S₁ molecule will lie inside the collision volume. This key assertion entails two assumptions: First, the probability that a reactant molecule will be found inside any small subvolume of Ω is independent of where inside Ω that subvolume is. That will be our definition of a “well-mixed” system. Note that this definition does not require that there be a perfectly regular placement of the reactant molecules inside Ω, nor that there be a large number of those molecules.

The second assumption implicit in the aforementioned assertion is that all but a negligibly small fraction of the containing volume Ω is physically accessible to the center of the randomly chosen S₁ molecule. That of course assumes Ω to be sufficiently convex that this would be true in the absence of any other molecules. But the presence of other reactant molecules will inevitably occlude some of Ω. Just how that occluded volume should be quantitatively taken into account is far from clear. But it is clear that the argument leading to Eq. 3 will be valid only if the volume occluded by the reactant molecules inside Ω is very small compared to |Ω|. That will be our definition of a “dilute” system. If we make the simplifying approximation that each S₁ molecule and S₂ molecule has a diameter that is the average of the two, namely, σ₁₂, then a rough estimate of the total volume that all those molecules occlude is

$$(x_1+x_2)·\frac{4}{3}\pi {\left(\frac{1}{2}{\sigma }_{12}\right)}^3=\frac{\pi }{6}(x_1+x_2){\sigma }_{12}^3.$$

Therefore, the diluteness assumption is essentially the order-of-magnitude requirement

$$(x_1+x_2){\sigma }_{12}^3\ll \left|\Omega \right|.$$ (4a)

Since |Ω| / (x₁ + x₂) is the average volume allotted to each reactant molecule inside Ω, then the cube root of that quantity estimates the average distance between the reactant molecules. Condition 4a evidently requires that average distance to satisfy

$${\left(\frac{\left|\Omega \right|}{x_1+x_2}\right)}^{1/3}\gg {\sigma }_{12}.$$ (4b)

So an equivalent way of stating the diluteness requirement is to say that the average distance between a pair of reactant molecules must be very large compared to their average diameter.

Since the minimum value of the factor (x₁ + x₂) in Eq. 4a for a reaction to occur is 2 (when x₁ = x₂ = 1), then a minimal practical requirement for diluteness would appear to be that the diameter |Ω|^1 / 3of the system must be at least an order of magnitude larger than the average diameter σ₁₂of a reactant molecule. It might be thought that this diluteness requirement is simply the requirement that the reactant molecules be “points.” But that would not be a correct assessment. As can be seen by putting σ₁₂ = 0 in the propensity function 3, two point molecules have zero probability of reacting with each other; therefore, satisfying the dilute conditions (4) by simply assuming the reactant molecules are points and setting σ₁₂ = 0 is not viable option.

A convincing physics derivation of a bimolecular propensity function for the situation in which the reactant molecules are solute molecules in a solution eluded researchers for a long time. The fact that the standard diffusion equation implies that the average displacement of a solute molecule in time dt is proportional to $\sqrt{dt}$ seemed to suggest, at least on the basis of the foregoing derivation of the dilute gas result, that the probability of a reaction between two solute molecules in the next dt might be proportional to $\sqrt{dt}$ instead of dt. That would be totally inconsistent with hypothesis 1. But in what might be described as a refined, corrected, and stochastically extended version of the analysis of Collins and Kimball,⁴ it was recently shown that if the reactant molecules are dilute and well-mixed in the senses defined above, then a propensity function for S₁ + S₂ → ⋅⋅⋅ as defined in 2 does exist and is given by⁵

$$\begin{array}{ccc}& & a_j(x_1,x_2)=\left(\frac{4\pi {\sigma }_{12}^2{D}_{12}{\overline{v}}_{12}{q}_j{\left|\Omega \right|}^{-1}}{4D_{12}+{\sigma }_{12}{\overline{v}}_{12}{q}_j}\right)·x_1{x}_2\hfill \\ & & \left(\mathrm{dilute}\mathrm{solution}\right).\hfill \end{array}$$ (5)

Here, D₁₂ is the sum of the diffusion coefficients of the S₁ and S₂ molecules, and the other quantities are as previously defined. Note that the requirement for diluteness in a solution applies only to the reactant (solute) molecules, and not to the chemically inert solvent molecules. In the “fast-diffusion” regime $4D_{12}\gg {\sigma }_{12}{\overline{v}}_{12}{q}_j$, Eq. 5 reduces to the dilute gas result 3. At the opposite extreme $4D_{12}\ll {\sigma }_{12}{\overline{v}}_{12}{q}_j$, which is the “diffusion-limited” or “Smoluchowski” regime which typifies cellular systems, the factor in parentheses in Eq. 5 reduces to 4πσ₁₂D₁₂|Ω|⁻¹. This result corresponds to a well known deterministic rate result that can be obtained by adapting Smoluchowski's famous analysis of colloidal coagulation.⁶ The more rigorous derivation of Eq. 5 actually makes use of Smoluchowski's reasoning, but does so in a way that takes account of the often overlooked fact⁵ that the standard diffusion equation on which the Smoluchowski analysis is based is physically incorrect on small length scales (we will return to this point in Sec. 5). As in the case of the derivation of the dilute gas result 3, the derivation of Eq. 5 fails if the system is not dilute and well-mixed, for reasons that are basically the same as in the dilute gas case.

To summarize: Molecular physics provides the following justification for the CME/SSA when the reactant molecules are solute molecules in a solution with very many, much smaller, chemically inert solvent molecules:

1. A propensity function for the unimolecular reaction S₁ → ⋅⋅⋅ normally exists in the form a_j(x) = c_jx₁, where c_j is independent of both x and |Ω|.

2. A propensity function for the bimolecular reaction S₁ + S₂ → ⋅⋅⋅ normally exists provided the reactant (solute) molecules are dilute and well-mixed inside Ω. The propensity function then has the form a_j(x) = c_jx₁x₂, where c_j is independent of x and inversely proportional to |Ω|.

An examination of the derivation of Eq. 5⁵ reveals that the bimolecular reaction results in (ii) will not be true if the reactant molecules are not dilute and well-mixed. In those situations, there is at present no valid physics-based derivation of a propensity function, and hence no valid physics-based CME and SSA.

BEING DILUTE IS A NECESSARY AND SUFFICIENT CONDITION FOR STAYING WELL-MIXED

Diffusion not only serves to bring solute molecules together so that they can chemically react, it is also the main mechanism by which the system stirs itself. In order for the bimolecular reaction S₁ + S₂ → ⋅⋅⋅ to occur in the well-mixed setting assumed by the propensity function hypothesis 1, the S₁ and S₂ molecules need to diffuse around for a while before they react with each other. We will now show that will happen if and only if the system is dilute.⁷

Consider a randomly chosen S₁ molecule. According to the standard theory of diffusion, this molecule will move an average net distance $\sqrt{2D_1\tau }$ in a time τ. So since the diameter of Ω is on the order of |Ω|^1 / 3, the time needed for the S₁ molecule to randomly reposition itself inside Ω will be roughly the time τ_d defined by $\sqrt{2D_1{\tau }_{\mathrm{d}}}={\left|\Omega \right|}^{1/3}$:

$${\tau }_{\mathrm{d}}=\frac{{\left|\Omega \right|}^{2/3}}{2D_1}.$$ (6)

According to Eqs. 1, 5, the probability that this S₁ molecule will react with one of the x₂S₂ molecules inside Ω in the next infinitesimal dt is

$$\left(\frac{4\pi {\sigma }_{12}^2{D}_{12}{\overline{v}}_{12}{q}_j{\left|\Omega \right|}^{-1}{x}_2}{4D_{12}+{\sigma }_{12}{\overline{v}}_{12}{q}_j}\right)·dt.$$

But in the “diffusion-limited” regime assumed by Eq. 6, which as noted in connection with Eq. 5 is defined by $4D_{12}\ll {\sigma }_{12}{\overline{v}}_{12}{q}_j$, this probability reduces to

$$\left(4\pi {\sigma }_{12}{D}_{12}{\left|\Omega \right|}^{-1}{x}_2\right)·dt.$$

From this it is easy to show that the average time τ_r before the S₁ molecule will react with one of the x₂S₂ molecules inside Ω in the next infinitesimal dt is

$${\tau }_{\mathrm{r}}={\left(4\pi {\sigma }_{12}{D}_{12}{\left|\Omega \right|}^{-1}{x}_2\right)}^{-1}.$$ (7)

What is needed to secure the well-mixed condition for this reaction is that {the average time required for the S₁ molecule to become randomly repositioned inside Ω} be much less than {the average time before the S₁ molecule reacts with one of the x₂S₂ molecules inside Ω}, i.e., τ_d ≪ τ_r. Substituting into this condition from Eqs. 6, 7 yields the requirement

$$2\pi {\sigma }_{12}\left(D_{12}/D_1\right)x_2\ll {\left|\Omega \right|}^{1/3}.$$ (8)

Since D₁ and D₂ are typically the same order of magnitude, we can further approximate D₁₂ / D₁ ≈ 2, and so conclude that the well-mixed condition requires

$$4\pi x_2{\sigma }_{12}\ll {\left|\Omega \right|}^{1/3}.$$ (9)

In words, the sum of the diameters of x₂ (and also by symmetry x₁) “average” reactant molecules must be much less than the diameter of the system. From a practical point of view, this is basically the diluteness condition 4b. Thus we have shown that the reactant molecules in a diffusion-limited system will remain well-stirred if and only if they are dilute. We note in passing that essentially this same result can also be established in the ideal gas regime, $4D_{12}\gg {\sigma }_{12}{\overline{v}}_{12}{q}_j$.⁸

This is an intuitively plausible result that seems not to be widely appreciated: If the system is “dilute,” so that the average distance between reactant molecules is large compared to their average diameter, then a reactant molecule will usually have to wander around for a relatively long time before it chances to collide with another reactant molecule. That wandering around makes the system well-mixed. But if the reactant molecules crowd each other, then they will likely find reacting partners before they have wandered very far, and consequently they will react in a system that is not well-mixed.

THE REACTION-DIFFUSION CME/SSA

If the system is not well-mixed in Ω, but can be considered approximately well-mixed inside each subvolume or “voxel” Ω_k of some partitioning {Ω₁, …, Ω_K} of Ω, then one can try the following: Model the chemical reactions as events occurring wholly inside single voxels as is done in the standard CME/SSA, and model the diffusional movement of reactant molecules between adjacent voxels as concurrent “diffusive transfer reactions” in a way that is consistent with the standard diffusion equation. This is the strategy of what is commonly called the “reaction-diffusion master equation (RDME)” and the “spatial SSA” (or rather less commonly, the “spatial CME” and the “reaction-diffusion SSA”). We shall refer to these jointly as the “reaction-diffusion CME/SSA.” The original paper on this strategy was the 1976 paper of Gardiner et al.⁹ As was done in that paper, we will take the voxels to be cubes of edge length h, since that greatly simplifies the mathematics of the diffusive transfer reactions.

Parceling the chemical reactions out to the K voxels has the effect of replacing the M nominal reactions {R_j} by KM reactions {R_jk}, where R_jk is reaction R_j inside voxel Ω_k. The propensity function a_jk for R_jk is the propensity function a_j for R_j, but now referred to the voxel Ω_k and regarded as a function of x_k = (x_1k, …, x_Nk), where x_ik is the current number of S_i molecules in Ω_k. If R_j is a unimolecular reaction, the coefficient c_jk in the R_jk propensity function will be identical to the coefficient c_j for R_j. If R_j is a bimolecular reaction, the coefficient c_jk will be the factor in parentheses in Eq. 5, except the factor |Ω|⁻¹ there must be replaced by h⁻³, the reciprocal of the voxel volume. In all cases, the state-change vector ν_jk for R_jk is the ν_j for R_j, but confined to the space of x_k.

The reaction-diffusion CME/SSA aims to model the movement of an S_i molecule inside Ω in accordance with the standard Einstein diffusion equation

$$\frac{\partial p(\mathbf{r},t)}{\partial t}=D_i{\nabla }_{\mathbf{r}}^{2}p(\mathbf{r},t),$$ (10)

where p is the position PDF of a single S_i molecule, and D_i is the molecule's diffusion coefficient. But since in the reaction-diffusion CME/SSA we do not know the position r of any reactant molecule—we know only that the molecule is equally likely to be anywhere inside a particular voxel—we must be content to model only the transfer of some S_i molecule from its current voxel Ω_k to an adjacent voxel Ω_l. The reaction-diffusion CME/SSA accomplishes this by positing a “diffusive transfer reaction” $R_{ikl}^{\left(\mathrm{d}\right)}$, which has propensity function

$$a_{ikl}^{\left(\mathrm{d}\right)}\left({\mathbf{x}}_k\right)=\frac{D_i}{h^2}{x}_{ik}.$$ (11)

Thus, the reaction-diffusion CME/SSA assumes that $a_{ikl}^{\left(\mathrm{d}\right)}\left({\mathbf{x}}_k\right)dt$ is the probability that an S_i molecule in Ω_k will move to Ω_l in the next dt. We will examine in detail the justification for this assumption in Sec. 6. The state-change vector ${\mathit{\nu }}_{ikl}^{\left(\mathrm{d}\right)}$ for reaction $R_{ikl}^{\left(\mathrm{d}\right)}$ simply decreases x_ik by 1 and increases x_il by 1.

Since the state jumps induced by diffusional transfers of reactant molecules between voxels have the same mathematical character as the state jumps induced by chemical reactions (both are described by propensity functions and state-change vectors), then the reaction-diffusion CME/SSA is just the standard CME/SSA described in Sec. 1, but with the following re-interpretation of its symbols: the N-dimensional state vector x = {x_i} is now regarded as the KN-dimensional state vector {x_ik}; and the M propensity functions {a_j} and their associated state-change vectors {ν_j} are now regarded as those for the KM chemical reactions {R_jk} and the 2NB diffusive transfer reactions $\left\{R_{ikl}^{\mathrm{d}}\right\}$, where B is the total number of boundaries between adjacent voxels. The factor 2N here comes from the fact that a molecule of any of the N species can cross the boundary in either direction.

It is important to understand that the voxel strategy of the reaction-diffusion CME/SSA is inherently approximate. This can be seen in two ways. First is the artificiality of its assumption that the position PDFs of the reactant molecules are uniform inside each voxel and change discontinuously at the boundaries between voxels. Since the boundaries between voxels are imaginary constructs, no real chemical system will ever present itself in this way. Second is the artificiality of its assumption that each reactant molecule is always wholly inside a single voxel. Even if we were to stipulate (as we usually do) that a reactant molecule is “in” the voxel that contains its center of mass, the molecule's non-zero size can extend its physical presence to neighboring voxels. Since these artificialities are most severe near the boundaries of the voxels, and least severe near the centers of the voxels, it follows that the reaction-diffusion CME/SSA describes the system only to resolutionh. Increased accuracy can thus be obtained only by decreasing h.

But there is a price to be paid for making h smaller. For example, halvingh increases the total number of voxels by a factor of 2³ = 8. That causes factor-of-8 increases in the number of state variables {x_ik}, and in the number of chemical reaction channels {R_jk}, and (at least approximately) in the number of diffusive transfer reaction channels $\left\{R_{ikl}^{\mathrm{d}}\right\}$. Furthermore, as we will see later, halving h will also increase the average number of diffusive transfer events that will occur between successive chemical reaction events by a factor of ${(1/2)}^{‒2}=4$. In view of these substantial computational penalties for adopting a finer spatial resolution, it would appear that in making practical computations we should always choose h as large as possible. We will have more to say about choosing h later.

An often voiced concern with the reaction-diffusion CME/SSA arises in connection with its rule that bimolecular chemical reactions can occur only between molecules that are in the same voxel. That would seem to allow a reaction between two molecules that are relatively far apart in the same voxel, yet not allow reactions between two molecules that are very close together but separated by a voxel boundary. However, there is an answer to this concern: the hypothesis that reactant molecules are “well-mixed” inside any voxel precludes us from making positional distinctions between molecules in the same voxel. In particular, a reactant molecule that is “near a boundary of its voxel” is not a reactant molecule that is “uniformly distributed inside its voxel”, and the latter is the only kind of reactant molecule that the reaction-diffusion CME/SSA is able to say anything about.

DIFFUSIVE TRANSFER REACTIONS

The reaction-diffusion CME/SSA transfers an S_i molecule from its present voxel Ω_k to an adjacent voxel Ω_l in accordance with the propensity function 11. In the Appendix, we give a concise proof of the fact that in the limith → 0 this molecule transfer strategy exactly replicates the dynamics prescribed by the standard diffusion equation 10. Given that result, one might be tempted to conclude that, at least in the absence of chemical reactions, the reaction-diffusion CME/RRA becomes exact in the limit h → 0. But that is not true.

To see why, consider a single S_i molecule in some interior voxel. Equation 11 stipulates that the probability that this molecule will jump to a particular one of its adjacent voxels in the next infinitesimal time dt is (D_i / h²)dt. Then by the addition law of probability, the probability that the molecule will, in the next dt, jump to either of the two adjacent voxels along a particular Cartesian axis, say the x-axis, is 2(D_i / h²)dt. That implies that the time it takes the molecule to leave its present voxel for either of those adjacent voxels is an exponential random variable with mean h² / (2D_i). Thus, the average time it takes the molecule to move to a distance h along the x-axis is h² / (2D_i). Therefore, the average speed of the molecule along the x-axis while making that voxel-to-voxel transition is

$${\overline{s}}_{\mathrm{vv}}=\frac{\mathrm{avg}\mathrm{dist}\mathrm{moved}}{\mathrm{avg}\mathrm{time}}=\frac{h}{h^2/\left(2D_i\right)}=\frac{2D_i}{h}.$$ (12)

Note that this voxel transition speed will be less than the molecule's average instantaneous speed$\overline{s}$ along the x-axis,

$${\overline{s}}_{\mathrm{vv}}<\overline{s},$$ (13)

since, owing to the irregular back and forth motion of the diffusing molecule, the total distance it travels along the x-axis during that voxel transition will be larger than h.

Equations 12, 13 imply the disturbing result that the molecule's average instantaneous speed $\overline{s}$ along any Cartesian axis, as well as the average speed ${\overline{s}}_{\mathrm{vv}}$ with which it moves from voxel to voxel in a given Cartesian direction, both go to infinity as h → 0. It is important to understand that this result is not a harmless reflection of the fact that the Maxwell-Boltzmann velocity distribution allows molecular speeds to be unbounded. The Maxwell-Boltzmann distribution says that the velocity along any Cartesian axis of a particle of mass m at absolute temperature T will be a normal random variable with mean zero and variance k_BT / m. That implies that the root-mean-square velocity of our S_i molecule along any Cartesian axis is $\sqrt{k_{\mathrm{B}}T/m_i}$, and to a factor of order unity that is the “average speed” of the S_i molecule along the x-axis. So even though the Maxwell-Boltzmann distribution does allow molecular speeds that are arbitrarily large, its average speed is finite. In contrast, the average speed $\overline{s}$ predicted by the diffusional jumping rule 11grows without bound as h → 0. There are at least two other ways of demonstrating this unphysical prediction of the standard diffusion equation.¹⁰

Since, as we have just seen, the average instantaneous x-axis speed of an S_i molecule moving according to the voxel-hopping rule of the reaction-diffusion CME/SSA is always greater than 2D_i / h, and since statistical thermodynamics stipulates that the average instantaneous x-axis speed of a molecule of mass m_i at temperature T is approximately $\sqrt{k_{\mathrm{B}}T/m_i}$, then a minimal condition for the reaction-diffusion CME/SSA to conform to the requirements of statistical thermodynamics is $\sqrt{k_{\mathrm{B}}T/m_i}>2D_i/h$. Solving this for h, we conclude that a minimal condition for the reaction-diffusion CME/SSA to be physically acceptable is

$$h>2D_i\sqrt{\frac{m_i}{k_{\mathrm{B}}T}}.$$

In fact, an analysis based on Langevin's more accurate theory of Brownian molecular diffusion reveals that the voxel-hopping rule 11 will be accurate only if this inequality is strongly satisfied, i.e., only if¹¹

$$h\gg D_i\sqrt{\frac{m_i}{k_{\mathrm{B}}T}}.$$ (14)

To summarize: In the limit h → 0, the voxel-hopping rule 11 converges exactly to the behavior predicted by the diffusion equation. But the diffusion equation, despite its name, does not accurately describe the physical motion of a molecule undergoing simple Brownian diffusion on all length scales; more specifically, it fails on length scales smaller than that prescribed by condition 14, where the molecule actually moves ballistically. If the voxel size h is steadily decreased below the value in 14, the voxel-hopping rule's modeling of a diffusing molecule's physical motion becomes more and more inaccurate, and eventually catastrophically inaccurate. For example, if h is reduced below 2D_i / c where c is the speed of light, then Eq. 12 implies that S_i molecules will be moving from voxel to voxel at an average speed greater than c. Even aside from the strictures of special relativity, it should be obvious that there is no physically plausible mechanism by which the velocity of a solute molecule could be continually and rapidly switched between “very fast in one direction” to “very fast in the opposite direction” by collisions with the surrounding much smaller and much less massive solvent molecules. We conclude that if the diffusional motion of molecules in the reaction-diffusion CME/SSA is to remain correct from the perspective of physics, then h must obey the lower bound condition 14.

LOWER BOUNDS ON h

Since smaller values of h (i) improve the accuracy of the key assumption that the PDFs of the reactant molecules do not vary appreciably over any one voxel and (ii) cause the diffusive transfer propensity function 11 to more accurately model the Einstein diffusion equation, it is tempting to conclude that the accuracy of the reaction-diffusion CME/SSA can be made as great as desired simply by taking h sufficiently small. However, in Sec. 5 we saw that there is a caveat to (ii), in that if h is smaller than allowed by condition 14 then the molecular motion prescribed by the diffusive transfer propensity function 11 will be physically incorrect. This means that we are not free to take h arbitrarily small, and we certainly cannot take the limit h → 0.

There is, furthermore, another lower bound on h which in practice is usually more restrictive than 14. It arises from the presumption that the CME/SSA correctly describes bimolecular reactions occurring inside individual voxels. In order for that to be true, the reactant molecules for all bimolecular reactions must be dilute and well-mixed inside each voxel. As shown in Secs. 2, 3 for the bimolecular reaction S₁ + S₂ → ⋅⋅⋅ inside a volume Ω, being dilute and staying well-mixed requires satisfaction of conditions (4) and 9, conditions which in all practical circumstances amount to the requirement that |Ω|^1 / 3 be very much larger than σ₁₂. Thus, the reaction-diffusion CME/SSA requires that the voxel edge length must be much larger than σ₁₂:

$$h\gg {\sigma }_{12}.$$ (15)

Calculations using parameter values roughly typical of protein molecules in water at room temperature suggest that the h-bound in 15 will be several thousand times larger than the h-bound in 14. In any case, both conditions 14, 15 must be satisfied if the reaction-diffusion CME/SSA is to be physically valid.

BEHAVIOR OF THE AVERAGE REACTION RATES FOR SMALL VOXELS

It has been argued elsewhere¹² that, in the limit h → 0, bimolecular reactions stop occurring in the reaction-diffusion CME/SSA. In view of the positive lower bounds on h in Eqs. 14, 15, it might be objected that the limit h → 0 is not allowed. But let us suppose that h is decreased in a restricted way that does not violate conditions 14, 15. In that case it is fair to ask, does the average rate at which bimolecular reactions occur inside Ω decrease along with h? In this section, we will show that it does not.

For this demonstration, we will consider a system that is well-mixed throughout its entire volume Ω. We begin by examining how chemical and diffusive transfer reactions in a single voxel behave as h is made smaller and smaller, but not smaller than allowed by conditions 14, 15. Since the decrease in h is not accompanied by any change in either Ω or the total number x_i of S_i molecules inside Ω, the average density of the S_i molecules in Ω will remain constant at the value x_i / |Ω|. The average number of S_i molecules inside any one voxel will therefore be $\left(x_i/\left|\Omega \right|\right)·h^3\equiv {\overline{x}}_i$, and thus will decrease with h like h³. Notice that if h is taken small enough, we could have ${\overline{x}}_i\ll 1$; however, satisfaction of the general diluteness condition ensures that that will not be due to the voxel being too small to accommodate more than one reactant molecule, but simply to there being many more voxels than reactant molecules in Ω.

The average propensity function for the unimolecular chemical reaction S₁ → ⋅⋅⋅ inside a single voxel has the form $c{\overline{x}}_1$, where c is independent of h. Since ${\overline{x}}_1\propto h^3$, the average propensity function for a unimolecular chemical reaction in a single voxel will therefore decrease with h like h³.

The average propensity function for the bimolecular chemical reaction S₁ + S₂ → ⋅⋅⋅ inside a single voxel has the form $c{\overline{x}}_1{\overline{x}}_2$ where c ∝ h⁻³. Here we have invoked the proviso that h be large enough to satisfy condition 15—see point (ii) at the end of Sec. 2. Since ${\overline{x}}_i\propto h^3$, the average propensity function for a bimolecular chemical reaction in a single voxel will therefore decrease with h like h⁻³h³h³ = h³. Note that this is the same small-h behavior as for a unimolecular chemical reaction.

By Eq. 11, the average propensity function for the diffusive transfer reaction in which an S_i molecule leaves a given voxel for a particular adjacent voxel is $\left(D_i/h^2\right){\overline{x}}_i$, where D_i is independent of h. Since ${\overline{x}}_i\propto h^3$, the average propensity function for a diffusive transfer of an S_i molecule from a particular voxel to a particular adjacent voxel will therefore decrease with h like h⁻²h³ = h.

To summarize, the average propensity function for any chemical reaction inside a particular voxel will decrease with h like h³, while the average propensity function for any diffusive transfer reaction from a particular voxel to a particular adjacent voxel will decrease with h like h. However, our goal is to assess the behavior of these reactions over the entire volume Ω. To do that, we will use the fact that propensity functions are additive; because, the probability for an R₁ reaction or an R₂ reaction to occur in the next infinitesimal time dt is, by the addition law of probability,¹³

$$a_{1}dt+a_{2}dt=(a_1+a_2)dt.$$

As a consequence of propensities being additive, the average propensity function for a chemical reaction anywhere inside Ω will be the sum of all the single voxel chemical reaction propensities. We can estimate that sum as the product of {the average single-voxel chemical reaction propensity}, which we have just seen is ∝h³, times {M × the total number of voxels}. Since the total number of voxels is |Ω| / h³, we conclude that the h-dependence of the average propensity function for a chemical reaction anywhere inside Ω will be ∝h³ × h⁻³ = h⁰, i.e., the average propensity function for a chemical reaction anywhere inside Ω will be independent ofh.

What about diffusive transfer reactions? The average propensity function for those will be the product of {the average single-molecule diffusive transfer propensity}, which we found above is ∝h, times {2 × N × the total number of interfaces between adjacent voxels}; the factor 2 × N here accounts for the fact that at each such interface, a molecule of any of the N species can cross in either direction. An interior voxel will have 6 such interface boundaries, each of which is shared with one other voxel. Since the ratio of the number of voxels on the boundary of Ω to the number of interior voxels will approach zero with h, the total number of interfaces between adjacent voxels will be approximately $6\times \frac{1}{2}\times$ the total number of voxels |Ω| / h³. The h-dependence of the total number of interfaces between adjacent voxels is therefore approximately ∝h⁻³. Thus, the h-dependence of the average propensity function for a diffusive transfer reaction anywhere inside Ω will be approximately ∝h × h⁻³ = h⁻².

Since the reciprocal of the propensity function of a reaction gives the average time between those reactions, we have thus proved the following: If h is decreased in a way that respects the lower bounds 14, 15, then the average time between chemical reactions (unimolecular or bimolecular) inside Ω does not change, while the average time between diffusive transfer reactions decreases approximately like h².

SUMMARY AND CONCLUSIONS

The physical validity of the CME/SSA model of chemical kinetics hinges solely on there being a sound basis in molecular physics for the propensity function hypothesis 1. We began by showing that if the reactant molecules are hard-sphere solute molecules in solution with very many much smaller chemically inert solvent molecules, then the propensity function hypothesis can be justified from molecular physics only if the reactant molecules of all bimolecular reactions are dilute and well-mixed inside the containing volume Ω. Here, “dilute” means that the total volume occluded by the reactant molecules (but not the solvent molecules) is only a very small fraction of |Ω|, or equivalently that the average distance between two reactant molecules is very large compared to their diameters. “Well-mixed” means that the probability of finding the center of a reactant molecule inside any small subvolume of Ω is independent of the location of that subvolume. We presented an argument showing that ordinary diffusion will suffice to maintain the well-mixed condition if and only if the diluteness condition is satisfied, i.e., the system will stay well-mixed if and only if it is dilute.

We next examined the reaction-diffusion extension of the CME/SSA. It assumes that even if the reactant molecules are not well-mixed inside Ω, we can partition Ω into a set of cubic voxels Ω₁, …, Ω_K, each of edge length h, such that the reactant molecules are approximately well-mixed inside each voxel. Chemical reactions are then viewed as occurring wholly inside single voxels in accordance with the usual propensity functions of the CME/SSA, while the diffusion of reactant molecules between adjacent voxels is modeled as diffusive transfer reactions with propensity functions of the form 11. The obvious physical artificiality of the two assumptions (i) that the distributions of the reactant molecules are perfectly uniform inside each voxel and change discontinuously at voxel boundaries, and (ii) that each reactant molecule always lies wholly inside a single voxel, makes it clear that this is an inherently approximate, coarse grained description, its spatial resolution being the voxel edge length h.

Although a smaller value of h will yield a more finely resolved description, we showed that physics considerations impose two lower bounds on h which prevent us from taking h arbitrarily small. First is the lower bound stipulated by condition 14. It ensures that the propensity function 11 for diffusive transfer reactions is physically accurate; taking h smaller than allowed by condition 14 will result in movement of the reactant molecules between voxels that is unphysical, despite being consistent with the predictions of the standard diffusion equation. Second is the lower bound on h stipulated by condition 15. It ensures that bimolecular reactions occurring inside voxels can be described using propensity functions of the form 5; taking h smaller than allowed by condition 15 will make it impossible for two reactant molecules to be “dilute” inside a single voxel, as is required to derive the bimolecular propensity function in a voxel. The lower bound 15, namely, h ≫ σ₁₂, will usually be the controlling one, since it will typically be several orders of magnitude larger than the lower bound 14.

We finally showed that if a chemical system is well-mixed inside its full volume Ω and we decrease hin a way that respects the lower bounds14, 15, then contrary to what might be inferred from a recent analysis,¹² the average rate at which both unimolecular and bimolecular chemical reactions occur inside Ω does not change. That, of course, is exactly what we should expect: the rate at which chemical reactions occur inside Ω should not be affected by how finely we subdivide Ω into imaginary voxels. We also showed that the average rate at which diffusive transfer reactions occur inside Ω increases with decreasing h approximately like h⁻². So, if we imagine a time-line on which all reactions are recorded by placing a dot at the instant they occur, then as h gets smaller, the “chemical reaction dots” will on average stay as they are, but the “diffusive transfer dots” will become more numerous, filling in the spaces between the chemical reaction dots.

At least three earlier works^{14, 15, 16} have suggested modifications to the rate constant in the bimolecular propensity function 5 in an attempt to allow the reaction-diffusion CME/SSA to be used with very small values of h. These three analyses differ in the specific reactions they consider, the basic assumptions they make, the inference logic they use, and the specific results they obtain; thus it is very difficult to compare them with each other or with our work here. Fange et al.¹⁴ appear to push the lower bound on h all the way to zero. That would contradict the lower bound 14, which we claim arises whenever the diffusional hopping rule 11 is used. But the results of both Erban and Chapman¹⁵ and Hellander et al.¹⁶ do imply positive lower bounds on h. Hellander et al.¹⁶ have suggested that those two lower bounds more or less agree, and they concluded that no h-dependent modification of the bimolecular rate constant in Eq. 5 will allow the reaction-diffusion CME/SSA to be correct unless h > πσ₁₂. If that is true, then it would appear that such modifications cannot yield much in the way additional latitude for h beyond our restriction of h ≫ σ₁₂ on the conventional reaction-diffusion CME/SSA. We have made no attempt in our paper to find a generalization of the bimolecular propensity function 5 that would make it correct when the volume occluded by all the reactant molecules in a single voxel is not negligibly small compared to the voxel volume |Ω| = h³. But since that occluded volume will depend on the numbers x_1k, …, x_Nk of reactant molecules that are currently in voxel k, then it would seem that any physics-based correction to the propensity function 5 will inevitably also change its dependence on the voxel population variables x_1k, …, x_Nk to something considerably more complicated than the simple factor x_1kx_2k.

In the quest for a lower bound on h, we should bear in mind that a practical lower bound might exist that is larger than the lower bound h ≫ σ₁₂, owing to the fact that the computational complexity of the reaction-diffusion CME/SSA increases so rapidly with decreasing h. For example, merely halving h increases by a factor of 8 both the number of state variables and the number of reaction channels, and also increases by a factor of 4 the average number of diffusive transfer events that occur between successive chemical reaction events. So even if decreasing h is desirable, it might not be feasible. From a practical standpoint, the goal in using the reaction-diffusion CME/SSA should be to use a value for h that is just small enough to capture the spatial non-uniformities in the system. Anything smaller will only make the calculation more difficult.

Our conclusion that the reaction-diffusion CME/SSA is inherently approximate, and cannot be made arbitrarily accurate by taking h arbitrarily small, immediately prompts the following question: Are there finer-scale models that can be used in those regions of space where one requires greater resolution than can be provided by the reaction-diffusion CME/SSA? There is of course molecular dynamics, which meticulously tracks the movement of not only all the reactant (solute) molecules but also all the solvent (usually water) molecules; however, that is such a computationally intensive enterprise that it is rarely seen as a practical option. A less detailed but more feasible approach would be to simulate the movement of only the reactant (solute) molecules. But there are serious difficulties in doing that too: It is true that we can simulate the unimpeded x-displacement of a molecule, as prescribed by the Einstein diffusion equation A3, from time t to a later time t + Δt with the formula

$$x_{t+\Delta t}=x_t+n\sqrt{2D\Delta t},$$ (16)

where n is a sample of the normal random variable with mean 0 and variance 1; and similarly for the y- and z-displacements. The problem is that we have no way of knowing where the molecule went between times t and t + Δt, and therefore no way of knowing whether it might have reacted with another solute molecule during that Δt interval. Efforts to find out by taking Δt progressively smaller will initially be hampered by the inherently fractal nature of the trajectory described by Eq. 16, and will ultimately be thwarted by the fact that Eq. 16 will be physically incorrect unless Δt ≫ τ, where τ = mD / (k_BT) and m is the molecule's mass.¹⁰

One way around this problem might be to use, instead of the stepping algorithm 16 which is prescribed by the Einstein model of diffusion, the stepping algorithm that is prescribed by the more accurate Langevin model of diffusion.¹⁷ The latter agrees with the former for Δt ≫ τ, but unlike the former it properly segues to ballistic motion as Δt is reduced below τ. At least for sufficiently small Δt, the Langevin stepping algorithm would solve the problem of not knowing where the molecules went between times t and t + Δt; because, for Δt ≪ τ, the predicted trajectory will be a nearly straight line, and a smooth interpolation will therefore be warranted. But using such a small Δt will be extremely time consuming.

A promising alternative molecule-tracking approach is the extended Green's Function Reaction Dynamics (eGFRD) method of Takahashi, Tănase-Nicola, and ten Wolde.¹⁸ It corrals each pair of solute molecules by surrounding the pair with an imaginary absorbing surface, so that absorption of either molecule on that surface will signal that the corral has been breached. eGFRD then solves the two-molecule Einstein diffusion equation to obtain the earliest of (i) the times to absorption of each molecule, and (ii) the time to a bimolecular reaction between the molecules. It avoids difficulties arising from the small-scale deficiencies of the Einstein diffusion equation by imposing appropriate boundary conditions on the solution of that equation, the most critical of which is a special “radiation” boundary condition⁵ at the collision surface of the two molecules if they are able to react with each other. The overall procedure allows one to advance the system from one reaction to the next without skipping over any reaction. A caveat is that the present eGFRD method appears to be computationally efficient only if the system is fairly dilute.¹⁸ As we have shown in this paper, dilute systems are typically amenable to the even more computationally efficient CME/SSA. The development of efficient, fine-scale simulation strategies is an important on-going effort, but a more detailed discussion of that topic is beyond the scope of this paper.

APPENDIX: PROOF THAT THE D/h² VOXEL-JUMP PROBABILITY RATE AGREES WITH THE EINSTEIN DIFFUSION EQUATION IN THE LIMIT h → 0

Consider a system composed of a single molecule inside a right cylindrical volume of length L and constant cross sectional area A. Let the cylinder's axis coincide with the x-axis, and let the cylinder's end faces be at x = 0 and x = L. Subdivide this volume into K = L / h right cylindrical voxels, each of length h and cross sectional area A, by means of planes through the points x_k ≡ k · h (k = 1, …, K). Number the voxels so that voxel k is the one occupying the x-axis interval [x_{k − 1}, x_k) ≡ [x_k − h, x_k), where x₀ = 0. Assume that the molecule, in any voxel, will jump to a particular adjacent voxel in the next infinitesimal time interval [t, t + dt) with probability (D / h²)dt.

Let q(k, t) be the probability that the solute molecule is in voxel k at time t. Then it follows from our assumption and the addition and multiplication laws of probability that, for any interior voxel k ∈ [2, K − 1],

$$\begin{array}{ccc}\hfill q(k,t+dt)& =& q(k+1,t)·\left(\frac{D}{h^2}\right)dt\hfill \\ & & +q(k-1,t)·\left(\frac{D}{h^2}\right)dt\hfill \\ & & +q(k,t)·\left[1-2\left(\frac{D}{h^2}\right)dt\right]+o\left(dt\right).\hfill \end{array}$$ (A1)

Here, the first term on the right is the probability that the solute molecule is in voxel k + 1 at time t and then jumps to voxel k in the next dt; the second term is the probability that the molecule is in voxel k − 1 at time t and then jumps to voxel k in the next dt; the third term is the probability that the molecule is in voxel k at time t and does not jump across either boundary of that voxel in the next dt; and the last term, which satisfies o(dt) / dt → 0 as dt → 0, recognizes that the probabilities for all routes to voxel k at time t + dt via other voxels at time t are of higher order than 1 in dt. If we subtract q(k, t) from both sides of Eq. A1, divide through by dt, and then take the limit dt → 0, we obtain the following exact time evolution equation for q(k, t) for any interior voxel k:

$$\begin{array}{c}\hfill \frac{\partial q(k,t)}{\partial t}=\left(\frac{D}{h^2}\right)\left[q(k-1,t)-2q(k,t)+q(k+1,t)\right].\end{array}$$ (A2)

The standard (Einstein) diffusion equation for the probability density function p(x, t) for the x-coordinate of a solute molecule with diffusion coefficient D is

$$\frac{\partial p(x,t)}{\partial t}=D\frac{{\partial }^{2}p(x,t)}{\partial x^2}.$$ (A3)

Since by definition p(x, t) · Adx gives the probability that the x-coordinate of the solute molecule will at time t be in the infinitesimal interval dx at x, it follows by the addition law of probability that the two functions p and q are related by

$$q(k,t)={\int }_{x_k-h}^{x_k}p(x,t)Adx(k=1,...,K).$$ (A4)

For sufficiently small h, Eq. A4 can be approximated

$$q(k,t)\genfrac{}{}{0pt}{}{.}{=}p(x_k,t)Ah(\mathrm{if}h\mathrm{is}\mathrm{small}),$$ (A5)

an approximation that becomes exact in the limit h → 0. Substituting Eq. A5 into Eq. A2, dividing through by Ah, and then using the fact that x_{k ± 1} = x_k ± h, we convert equation A2 for the function q into an equation for the function p:

$$\begin{array}{c}\hfill \frac{\partial p(x_k,t)}{\partial t}\genfrac{}{}{0pt}{}{.}{=}\left(\frac{D}{h^2}\right)\left[p(x_k-h,t)-2p(x_k,t)+p(x_k+h,t)\right].\end{array}$$ (A6)

This equation has been derived from Eqs. A2 and Eq. A5, and since the former is exact while the latter becomes exact in the limit h → 0, then Eq. A6 becomes exact in the limit h → 0. Since further

$$\underset{h\to 0}{lim}\frac{p(x_k-h,t)-2p(x_k,t)+p(x_k+h,t)}{h^2}=\frac{{\partial }^{2}p(x_k,t)}{\partial x_k^2},$$

we conclude that the h → 0 limit of Eq. A6 is exactly the Einstein diffusion equation A3. Thus we have proved that the molecular motion produced by the voxel-hopping probability rate D / h² converges, in the limit h → 0, to the motion predicted by the standard diffusion equation A3.