1 Introduction

Stochastic reaction–diffusion models have become a popular tool for modeling biological systems in which both noise in the chemical reaction process and the spatial diffusion of molecules play an important role. Such models have been used in a multitude of recent studies, examining the dynamics of synaptic transmission (Nadkarni et al. 2012); the MinCDE system in bacteria (Elf and Ehrenberg 2004); how proteins search for DNA binding sites (Isaacson et al. 2011); and studies of signaling in the cell membrane (Dushek et al. 2011). There are three primary stochastic reaction–diffusion models that these studies have made use of: the diffusion limited reaction model of Smoluchowski (Smoluchowski 1917; Keizer 1982), what we call the Doi model (Teramoto and Shigesada 1967; Doi 1976a, 1976b), and the reaction diffusion master equation (RDME) (Gardiner et al. 1976; Erban et al. 2007; Kang et al. 2012).

In the Doi and Smoluchowski models, the positions of molecules are represented as points undergoing Brownian motion. Bimolecular reactions between two molecules in the Doi model occur with a fixed probability per unit time when two reactants are separated by less than some specified “reaction radius.” The Smoluchowski model differs by representing bimolecular reactions in one of two ways; either occurring instantaneously, or with fixed probability per unit time, when two reactants’ separation is exactly the reaction-radius (Smoluchowski 1917; Keizer 1982). In this work, we focus on the former case (often called a pure absorption reaction). For both models, unimolecular reactions represent internal processes. They are assumed to occur with exponentially distributed times based on a specified reaction-rate constant. For general chemical systems, both the Doi and Smoluchowski models can be described by, possibly infinite, systems of partial integral differential equations (PIDEs) for the probability densities of having a given number of each chemical species and the corresponding locations of each molecule.

The RDME is spatially discrete, and given by a, possibly infinite, system of ODEs for the numbers of each chemical species located at each lattice site. It can be interpreted as an extension of the nonspatial chemical master equation (CME) (Gardiner et al. 1976; McQuarrie 1967; Gardiner 1996; Van Kampen 2001) model for stochastic chemical kinetics. Formally, the RDME has been shown to be an approximation to both the Doi model (Isaacson 2008) and the Smoluchowski model (Isaacson 2009; Isaacson and Isaacson 2009; Fange et al. 2010; Hellander et al. 2012) for appropriately chosen lattice spacings. These approximations break down for systems involving bimolecular reactions, in two or more dimensions, in the limit that the lattice spacing approaches zero (Isaacson 2009; Hellander et al. 2012). (Recently, we have suggested a new convergent RDME (CRDME) that converges to the Doi model as the lattice spacing is taken to zero (Isaacson 2012).)

There are a plethora of numerical methods and simulation packages that have been developed to study biological systems based on one of the Smoluchowski, Doi, or RDME models. These include the λ-ρ method (Erban and Chapman 2009; Lipkova et al. 2011), the CRDME (Isaacson 2012), the FPKMC (Donev et al. 2010; Takahashi et al. 2010), MCELL (Kerr et al. 2008), MesoRD (Fange et al. 2012), Smoldyn (Andrews and Bray 2004), STEPS (Hepburn et al. 2012), and URDME (Drawert et al. 2012). While these numerical methods and software packages have been used in many modeling efforts, it is still an open question how exactly the underlying mathematical models they approximate are rigorously related. Moreover, to compare the results of modeling studies, it would be helpful to understand how to choose parameters in the Doi (resp. Smoluchowski) model to accurately approximate the Smoluchowski (resp. Doi) model.

To address these questions, we investigate the rigorous relationship between the Doi and (pure absorption) Smoluchowski models. We begin in the next section by considering the special case of a single protein searching for a DNA binding site within the nucleus of a eukaryotic cell. The binding site is located at the origin, and the nucleus is modeled as a concentric sphere of radius R. With the further assumption that the initial distribution of the protein’s position is rotationally invariant, the Doi and Smoluchowski models can be simplified to spherically-symmetric diffusion equations. For the Doi model, the diffusing protein is allowed to bind with probability per unit time λ when within a reaction-radius, r b, of the binding site. This leads to a “reaction potential” in the PDE for the Doi model. In the Smoluchowski model, the protein reacts instantly upon reaching a separation r b from the binding site. This leads to the replacement of the reaction potential with a zero Dirichlet boundary condition on the sphere of radius r b around the origin.

Denote by p λ (r,t) the spherically symmetric probability density that the diffusing molecule is r from the origin at time t in the Doi model, and by ρ(r,t) the corresponding probability density in the Smoluchowski model. In Sect. 3, we numerically calculate the difference between p λ (r,t) and ρ(r,t) for biologically relevant parameter values. We find that p λ (r,t)→ρ(r,t) uniformly in r and t as λ→∞, with an empirical convergence rate that is \(O(\lambda^{-\frac{1}{2}})\). The same empirical convergence rate is observed for the corresponding binding time distributions and mean binding times of the two models. For sufficiently large, but fixed, values of λ, as r b is decreased the difference between the probability densities, binding time distributions, and mean binding times increase like \(r_{\textrm{b}}^{-1}\).

These results motivate our studies in Sect. 4, where we rigorously prove the convergence of p λ (r,t) to ρ(r,t) as λ→∞ with an \(O(\lambda^{-\frac{1}{2} + \epsilon})\) error bound (for all ϵ>0). Let μ n denote the nth eigenvalue of the generator of the Doi model (see (8)), with α n the nth eigenvalue of the generator of the Smoluchowski model (see (6)). Our approach is to first prove that for λ sufficiently large, if α n ≤M(λ) with M(λ) a specified increasing function of λ, then

$$\vert \alpha_n - \mu_n\vert \leq\frac{C}{\lambda^{\frac{1}{2} - \epsilon}} $$

for any fixed ϵ>0. The precise statement of this result is given in Theorem 4.1. This theorem is then used to show the corresponding eigenfunction convergence result in Lemma 4.1. Denote by ψ n (r) the nth eigenfunction of the generator of the Doi model (8), and ϕ n (r) the corresponding eigenfunction of the Smoluchowski model (6). We prove for λ sufficiently large and μ n <M(λ) that

$$\sup_{r \in [r_{\textrm{b}},R ]} \bigl \vert \phi_n(r) - \psi _n(r)\bigr \vert \leq\frac{C}{\lambda^{\frac{1}{2} - \epsilon}}. $$

Finally, the preceding two results are used to prove that

$$ \sup_{t \in (\delta,\infty )} \sup_{r \in (r_{\textrm{b}},R )} \bigl \vert p_{\lambda}(r,t) - \rho(r,t) \bigr \vert \leq \frac{C}{\lambda^{\frac{1}{2} - \epsilon}}, $$
(1)

for any ϵ>0 and δ>0 fixed. The precise statement of this result is given in Theorem 4.2.

It should be noted that the approximation of Dirichlet boundary conditions by reaction potentials is a well-studied problem in the context of the large-coupling limit in quantum physics (Glimm and Jaffe 1987). Our approach of proving convergence through successive eigenvalue, eigenfunction, and PDE solution estimates differs from the more standard resolvent and path integral estimates (Taylor 1996; Demuth 1980; BelHadjAli et al. 2011). A similar convergence rate of the Doi model solution to the Smoluchowski model solution was proven in \(L^{2}(\mathbb{R}^{3})\) in Demuth et al. (1993) (as opposed to the uniform convergence rate (1) we prove in a spherical domain). Denote by \(\mathbf {1}_{ [0,r_{\textrm{b}} ]}(r)\) the indicator function of the interval, [0,r b]. Convergence rates for eigenvalues of the general one-dimensional operator \(-\frac{d^{2}{}}{d{x}^{2}} + V(x) + \lambda W(x)\) as λ→∞, for \(x \in \mathbb{R}\), were proven in Gesztesy et al. (1988). In contrast, in Sect. 4, we study the spherically symmetric operator arising in the Doi model, \(-\frac{d^{2}{}}{d{r}^{2}} - \frac {2}{r} \frac{d}{dr} + \lambda \mathbf {1}_{ [0,r_{\textrm{b}} ]}(r)\) for r∈[0,R) with a Neumann boundary condition at R, directly.

2 Diffusion of a Protein to a Fixed DNA Binding Site

To study the rigorous relationship between the Doi and Smoluchowski models, we investigate the special case of the chemical reaction A+B→∅, with only one molecule of species A and one molecule of species B. We further assume the molecule of species B is located at the origin and stationary (D B=0). The molecule of species A is assumed to move within a sphere of radius R centered on the B molecule. We denote the diffusion constant of the A molecule by D and the reaction radius by r b. While idealized, this special case can be interpreted as a model for the diffusion of a DNA binding protein to a fixed DNA binding site (located at the center of a nucleus).

We now formulate the Doi and Smoluchowski models in this special case, with the additional assumption of spherical symmetry. This assumption will hold whenever the initial distribution of the A molecule is spherically-symmetric about the origin. Denote by ρ(r,t) the spherically-symmetric probability density that the A molecule is a distance r from the origin and has not reacted with the B molecule at time t. Then the Smoluchowski model reduces to

$$ \frac{\partial\rho}{\partial t} = D \varDelta_{r} \rho, \quad r_{\textrm{b}}< r < R, \, t > 0, $$
(2)

where Δ r denotes the spherically symmetric Laplacian in three-dimensions,

$$\varDelta_r \equiv\frac{1}{r^2}\frac{\partial}{\partial r} \biggl(r^2\frac{\partial}{\partial r} \biggr). $$

This equation is coupled with the reactive Dirichlet boundary condition, and zero Neumann boundary condition (so that the molecule remains in the “nucleus”)

$$\rho(r_{\textrm{b}},t) = 0, \qquad \frac{\partial\rho}{\partial r}(R,t) = 0, \quad t > 0. $$

Finally, we use a delta-function initial condition, ρ(r,0)=δ(r−r 0)/r 2, where r 0∈(r b,R) denotes the initial position of the diffusing molecule. With this initial condition ρ(r,0) has the normalization \(\int_{r_{\textrm {b}}}^{R} \rho(r,0) r^{2} \, dr = 1\).

Let p λ (r,t) label the corresponding spherically-symmetric probability density for the Doi model. In the special case we are considering, the PIDEs for the Doi model reduce to

$$ \frac{\partial p_{\lambda}}{\partial t} = D \varDelta_{r} p_{\lambda }- \lambda\, \mathbf {1}_{ [0,r_{\textrm{b}} ]}(r) \, p_{\lambda }(r,t), \quad0 \leq r < R, \, t > 0, $$
(3)

with the Neumann boundary condition,

and the same initial condition as the Smoluchowski model, p λ (r,0)=δ(r−r 0)/r 2. Note, probability is not conserved in either (2) or (3) due to the absorbing boundary condition in (2) and sink term in (3). If one desired a model that conserved probability, a second equation could be added to each model to keep track of the probability of being in the “reaction has occurred” state.

For simplicity, in what follows, we assume D=1. Equations (2) and (3) can be solved explicitly by separating variables. In solving (3), we impose continuity of the function and its derivative across the surface of discontinuity as justified by the results of Girsanov (1960) and Olenik (1961). The computations are standard so we give only the final results. We can write the solutions as the eigenfunction expansions

$$ \rho(r,t) = \sum_{n=1}^{\infty}{a_n \phi_n(r_0) \phi_{n}(r)e^{-\alpha_{n}t}} $$
(4)

and

$$ p_{\lambda}(r,t) = \sum _{n=1}^{\infty}{b_n \psi_n(r_0) \psi_{n}(r)e^{-\mu_{n}(\lambda )t}}. $$
(5)

Here, ϕ n (r) are the eigenfunctions of the Smoluchowski model and satisfy the equation

$$ -\varDelta_r \phi_n(r) = \alpha _n \phi_n(r), \quad r_{\textrm{b}}< r < R, $$
(6)

with the boundary conditions ϕ n (r b)=0 and \(\frac {\partial\phi_{n}}{\partial r}(R) = 0\). We extend these functions to [0,R] by defining ϕ n (r)=0 for r∈[0,r b]. α n denotes the eigenvalue of the Smoluchowski model corresponding to ϕ n (r), and satisfies the equation f(α n )=0, where

$$ f(\mu) = \frac{R\sqrt{\mu}-\tan(\sqrt{\mu }(R-r_{\textrm{b}}))}{R\mu\tan(\sqrt{\mu}(R-r_{\textrm {b}}))+\sqrt{\mu}}. $$
(7)

Solving (6) we find that

$$ \phi_n(r) =\frac{1}{r} \biggl[\frac{\sin(\sqrt {\alpha_n}(R-r))}{R\sqrt{\alpha_n}}-\cos\bigl( \sqrt{\alpha _n}(R-r)\bigr) \biggr],\quad r_{\textrm{b}}<r<R. $$

ψ n (r) denote the eigenfunctions of the Doi model (3), satisfying the equation

$$ -\varDelta_r \psi_n(r) + \lambda \mathbf {1}_{ [0,r_{\textrm{b}} ]} \psi_n(r) = \mu_n \psi _n(r), \quad0 \leq r < R, $$
(8)

with the Neumann boundary condition \(\frac{\partial\psi_{n}}{\partial r}(R) = 0\). Here, μ n labels the eigenvalue of the Doi model corresponding to ψ n (r), solving the equation f(μ n )=A(μ n ,λ) with

$$ A(\mu,\lambda)= \begin{cases} \sqrt{\frac{1}{\lambda-\mu}}\tanh (\sqrt{\lambda-\mu}r_{\textrm{b}} ), & \mu<\lambda,\\[5pt] \sqrt{\frac{1}{\mu- \lambda}}\tan (\sqrt{\mu -\lambda}r_{\textrm{b}} ), & \mu>\lambda. \end{cases} $$
(9)

We will sometimes use the notation μ n (λ) to make explicit the dependence of the eigenvalues on λ. Solving (8), we find that

$$ \psi_n(r) = \begin{cases} \psi^{\mathrm{in}}_{n}(r), & 0<r<r_{\textrm{b}},\\ \psi ^{\mathrm{out}}_{n}(r), & r_{\textrm{b}}\leq r<R, \end{cases} $$

where

$$ \psi^{\mathrm{in}}_{n}(r) = \begin{cases} \frac{\tfrac{1}{R\sqrt{\mu_n}}\sin (\sqrt{\mu_n}(R-r_{\textrm{b}}))-\cos(\sqrt{\mu_n}(R-r_{\textrm {b}}))}{\sinh(r_{\textrm{b}}\sqrt{\lambda-\mu_n})} (\frac {\sinh(r\sqrt{\lambda-\mu_n})}{r} ), &\mu_n<\lambda,\\[5pt] \frac{\tfrac{1}{R\sqrt{\mu_n}}\sin(\sqrt{\mu _n}(R-r_{\textrm{b}}))-\cos(\sqrt{\mu_n}(R-r_{\textrm{b}}))}{\sin (r_{\textrm{b}}\sqrt{\mu_n-\lambda})} (\frac{\sin(r\sqrt {\mu_n-\lambda})}{r} ), &\mu_n>\lambda, \end{cases} $$
(10)

and

$$ \psi^{\mathrm{out}}_{n}(r) =\frac{1}{r} \biggl[ \frac{\sin (\sqrt{\mu_n}(R-r))}{R\sqrt{\mu_n}}-\cos\bigl(\sqrt{\mu _n}(R-r)\bigr) \biggr]. $$
(11)

Finally, let \((u(r),v(r))=\int_{0}^{R} {u(r)v(r)r^{2}} \,dr\) denote the usual L 2 inner product. The normalization constants a n and b n are then given by

$$a_n =\frac{1}{(\phi_n(r),\phi_n(r))}, \qquad b_n =\frac{1}{(\psi_n(r),\psi_n(r))}. $$

3 Difference Between Smoluchowski and Doi Models for Biologically Relevant Parameters

If we interpret (2) and (3) as models for the diffusion of a protein that has just entered the nucleus (r 0=R) to a DNA binding site, then typically r b would be between .1 and 10 nm (Kühner et al. 2004; Dushek et al. 2011; Andrews and Bray 2004), R between 1 and 10 μm, and D between 1 and 20 μm2 s−1. In the following, we assume that all spatial units are in micrometers and time is in seconds, with R=r 0=1 μm, D=10 μm2 s−1, and λ having units of s−1. We also assume that p λ (r,0)=ρ(r,0)=δ(r−r 0)/4πr 2. p λ and ρ are then the same as in the previous section, but rescaled by (4π)−1.

We numerically evaluated p λ (r,t) and ρ(r,t) in MATLAB using the eigenfunction expansions (5) and (4). The series were truncated at the first term with magnitude smaller than 10−10. In evaluating these series numerically, it is necessary to calculate a number of the eigenvalues μ n and α n . For each term of the eigenfunction expansions, the transcendental equations for the corresponding Doi eigenvalue, f(μ n )=A(μ n ,λ), and the Smoluchowski eigenvalue, f(α n )=0, were solved to 25 digits of precision using the Mathematica Reduce function. In Fig. 1(a), we show the solution to the Smoluchowski model (2), ρ(r,t), when r b=10−3 μm. For short times, the solution is localized near R, while a boundary layer develops near r=r b as t increases. Figure 1(b) shows the maximum absolute difference between p λ and ρ for a discrete set of points,

$$ \bigl\Vert p_{\lambda}(r,t) - \rho(r,t)\bigr\Vert_{\infty} \equiv\max _{i} \max_{j} \bigl \vert p_{\lambda}(r_i,t_j) - \rho(r_i,t_j) \bigr \vert , $$

where r i and t j are given in Appendix C by Listings 1 and 2. For each fixed value of r b, we see that as λ→∞ the difference between p λ and ρ converges to zero like λ −1/2. As r b is decreased, the absolute difference increases by approximately an order of magnitude for each fixed value of λ (for λ sufficiently large).

Fig. 1
figure 1

(a) Solution to the Smoluchowski model (2) for r b=10−3 μm. Note that the t-axis uses a logarithmic scale. (b) Absolute difference in p λ (r,t) and ρ(r,t) as λ and r b are varied

Full size image

For many biological models, the statistics of the random variable for the time at which the diffusing molecule first binds to the binding site are of interest. For example, in Isaacson et al. (2011), we studied how this time was influenced by volume exclusion due to the spatially varying density of chromatin inside the nucleus of mammalian cells. We subsequently denote these random variables by T Doi and T Smol. Statistics of these random variables can be calculated from the cumulative distribution functions

$$ \operatorname{Prob}[T_{\textrm{Doi}}< t ] = 1 - 4 \pi\int_0^R p_{\lambda}(r,t) r^2 \, dr, $$

and

$$ \operatorname{Prob}[T_{\textrm{Smol}}< t ] = 1 - 4 \pi\int_{r_{\textrm{b}}}^R \rho(r,t) r^2 \, dr. $$

We evaluated \(\operatorname{Prob}[T_{\textrm{Doi}}< t ]\) and \(\operatorname{Prob}[T_{\textrm{Smol}}< t ]\) by analytically integrating the eigenfunction expansions (5) and (4), see Appendix A, and evaluating the truncated series in MATLAB. (Using the same method as described above for evaluating p λ (r,t) and ρ(r,t).)

Figure 2(a) shows \(\operatorname{Prob}[T_{\textrm {Smol}}< t ]\) for varying r b and demonstrates a constant increase in the binding time as r b is decreased (on a logarithmic scale). Figure 2(b) shows the absolute difference in binding time distributions,

$$ \bigl\Vert \operatorname{Prob}[T_{\textrm{Doi}}< t ] - \operatorname{Prob}[T_{\textrm{Smol}}< t ] \bigr\Vert_{\infty} \equiv\max_{t_j} \bigl \vert \operatorname{Prob}[T_{\textrm{Doi}}< t_j ] - \operatorname{Prob}[T_{\textrm{Smol}}< t_j ] \bigr \vert , $$

where t j is given by Listing 2 in Appendix C. We again observe an empirical \(\lambda^{-\frac{1}{2}}\) convergence rate of \(\operatorname{Prob}[T_{\textrm {Doi}}< t ]\) to \(\operatorname{Prob}[T_{\textrm{Smol}}< t ]\) as λ→∞. For a biologically relevant binding radius of 10−3 μm, when λ=1011 s−1 the absolute difference between the two distributions is on the order of 10−3.

Fig. 2
figure 2

(a) Smoluchowski binding time distributions for varying r b. Note the t-axis is logarithmic. (b) Absolute difference in binding time distributions from the Smoluchowski and Doi models as λ is varied

Full size image

In addition to the probability densities and binding time distributions, we also examined the mean time for the diffusing protein to find the binding site (when starting a distance r 0 from the origin). The mean binding time for the Doi model can be found in a simple closed form by solving the corresponding mean first passage time problem (24), see Appendix B, and is given by

$$ \mathbb {E}[T_{\textrm{Doi}} ](r_0) = \int _0^{\infty} \operatorname{Prob}[T_{\textrm{Doi}}< t ] \, dt = \begin{cases} u^{-}(r_0), \quad r_0 < r_{\textrm{b}}, \\ u^{+}(r_0), \quad r_0 > r_{\textrm{b}}. \end{cases} $$
(12)

Here, for \(\hat{\lambda} = \lambda/ D\), we have that

Similarly, the mean binding time in the Smoluchowski model can be found by solving the mean first passage time problem (25), and is given by

$$ \mathbb {E}[T_{\textrm{Smol}} ](r_0) = \frac{r_{\textrm {b}}^2-r_0^2}{6D}+ \frac{R^3}{3D} \biggl[\frac{1}{r_{\textrm {b}}}- \frac{1}{r_0} \biggr], \quad r_0 > r_{\textrm{b}}. $$
(13)

Figure 3(a) shows the mean binding time in the Doi model as λ is varied for r 0=R.

Fig. 3
figure 3

(a) Mean binding time of Doi model vs λ. (b) Relative difference in mean binding times (15)

Full size image

The difference between the two mean binding times, for r 0>r b, is then given by

(14)

This difference is \(O(\lambda^{-\frac{1}{2}})\), consistent with the results we prove in the next section on the uniform convergence of p λ to ρ as λ→∞. (14) also demonstrates that for large, but fixed values of λ the absolute difference in mean binding times for biologically relevant parameter choices will increase like \(r_{\textrm{b}}^{-1}\) as r b is decreased. In Fig. 3(b), the relative difference,

$$ \frac{\vert \mathbb {E}[T_{\textrm{Smol}} ] - \mathbb {E}[T_{\textrm{Doi}} ]\vert }{\mathbb {E}[T_{\textrm{Smol}} ]}, $$
(15)

is graphed as λ is varied for r 0=R, illustrating the λ −1/2 convergence of \(\mathbb {E}[T_{\textrm{Doi}} ]\) to \(\mathbb {E}[T_{\textrm{Smol}} ]\). For r b=10−3 μm the mean binding times differ by less than 1 % when λ=1011 s−1.

Each of Figs. 1(b), 2(b), and 3(b) illustrate a λ −1/2 empirical convergence rate, consistent with the bound (1) we prove in the next section (Theorem 4.2). Moreover, they demonstrate that as r b is decreased larger values of λ are required to ensure the absolute difference between the two models remains below a fixed tolerance. In all three cases, once λ is sufficiently large that the λ −1/2 convergence rate can be observed, for biologically relevant parameter values the absolute difference appears to scale like \(r_{\textrm {b}}^{-1}\) as r b is decreased.

4 Rigorous Convergence Results

In this section, we study the rigorous relationship between the Doi model (3) and Smoluchowski model (2). In Sect. 4.1, we derive a rigorous error bound on the rate of convergence of the Doi eigenvalues, μ n (λ), to the Smoluchowski eigenvalues, α n , as λ→∞. In Sect. 4.2, we obtain similar estimates for the convergence of the Doi eigenfunctions, \(\psi^{\mathrm{out}}_{n}\), to the Smoluchowski eigenfunctions, ϕ n . We obtain our main result in Sect. 4.3 where we use these eigenvalue and eigenfunction estimates to show the uniform convergence in space and time of the solution to the Doi model (3), p λ (r,t), to the solution of the Smoluchowski model (2), ρ(r,t), as λ→∞. (With the error bound (1).)

4.1 Eigenvalue Estimates

In this subsection, we derive estimates for the difference between the Doi, μ n (λ), and Smoluchowski, α n , eigenvalues. We start by proving some properties of the functions A(μ,λ) and f(μ) that we will find useful.

Proposition 4.1

A(μ,λ) is monotone increasing in μ. Furthermore, for 0<μ<λ, A is positive.

Proof

Positivity is trivial since tanh is positive. Note also that \(A(0,\lambda) =\tanh(\sqrt{\lambda}r_{\textrm{b}}) /\allowbreak \sqrt{\lambda}\). A simple computation shows that for μ<λ

and for μ>λ

The result follows since for u≥0, sinh(u)≥u and sin(u)≤u. □

Let the vertical asymptotes of f(μ) be denoted by β n . They satisfy the equation

$$ R\beta_n \tan\bigl(\sqrt{\beta_n}(R-r_{\textrm{b}}) \bigr)+\sqrt{\beta _n}=0;\quad\beta_n >0. $$

Recalling that α n denote the eigenvalues of the Smoluchowski model, satisfying f(α n )=0, then

Proposition 4.2

We have the following:

  1. 1.

    0<α 1<β 1<α 2<⋯<β n <α n+1<⋯.

  2. 2.

    f′(μ)<0 and f(μ)>0 on [0,α 1) and (β n ,α n+1) for n≥1.

Proof

1. Let κ=1−r b/R and \(x=R\sqrt{\mu}\). We make the change of variable in f(μ) and obtain

$$ f(\mu)\equiv\tilde{f}(x) =\frac{R (x-\tan( \kappa x) )}{x^2\tan(\kappa x)+x}=\frac{R (1-\frac{\tan(\kappa x)}{x} )}{x\tan(\kappa x)+1}\equiv \frac{N(x)}{D(x)}. $$

Let d n be such that N(d n )=0 and η n be such that D(η n )=0. In terms of the old variables, we have \(d_{n} =R\sqrt{\alpha_{n}}\) and \(\eta_{n}=R\sqrt{\beta_{n}}\). N(x)=0 implies that tan(κx)=x and D(x)=0 implies that \(\tan(\kappa x) =-\frac{1}{x}\). Note that the functions tan(κx), x and \(-\frac{1}{x}\) are all monotone increasing. Finally, if we let \(\theta_{n} =\frac{\pi}{2\kappa}(2n-1)\) be the vertical asymptotes of tan(κx) one easily checks that we have

$$ 0<d_1<\theta_1<\eta_1< \cdots<d_n<\theta_n<\eta_n\cdots. $$

This proves 1.

2. N>0 on ⋃ n≥1(θ n ,d n+1)∪(0,d 1), and N<0 on ⋃ n≥1(d n ,θ n ). Similarly, D>0 on ⋃ n≥1(η n ,θ n+1)∪(0,θ 1) and D<0 on ⋃ n≥1(θ n ,η n ). Thus, it follows that \(\tilde{f}(x) >0\) if and only if x∈⋃ n≥1(η n ,d n+1)∪(0,d 1). Next, we show that f′<0. Note

and

Since |sin(θ)|≤|θ|, it follows that N′(x)≤0 and that D′(x)≥0, with equality in both only when x=0. From what we have shown above, \(\tilde{f}(x)> 0\) if and only if both N(x)>0 and D(x)>0 so that

 □

Proposition 4.3

The Doi eigenvalues μ n (λ) satisfy for n such that μ n (λ)≤λ

$$ 0<\mu_1(\lambda)<\alpha_1 < \beta_1< \mu_2(\lambda)<\alpha _2<\cdots<\beta_{n-1}< \mu_{n}(\lambda)<\alpha_{n}. $$

Proof

Proposition 4.2 implies that f(μ) is positive and decreases monotonically from infinity to zero on (β n ,α n+1). Since A(μ,λ) is positive and increasing in μ for μ≤λ, we find f(μ)=A(μ,λ) once on each interval (β n ,α n+1). The proof of Proposition 4.2 also shows that f≤0 on (α n ,β n ), and hence there are no other possible roots for α 1<μ≤λ. As f(0)=r b and \(A(0,\lambda) = \lambda^{-\frac{1}{2}} \tanh(\sqrt{\lambda} r_{\textrm{b}}) < r_{\textrm{b}}\), a similar argument implies 0<μ 1<α 1. □

We will also have need for the following.

Proposition 4.4

Let {γ n } denote the eigenvalues for the (positive) radially symmetric Laplacian on [0,R), with zero Neumann boundary conditions on the ball of radius R. Let α n and μ n be as above. Then the following hold:

  1. 1.

    The μ n (λ) are continuous and monotone increasing in λ for all n≥1

  2. 2.

    For all n≥1 and any fixed λ, we have that

    $$ \mu_n(0) = \gamma_n = \biggl(\frac{(n-1)\pi}{R} \biggr)^2\leq\mu _n(\lambda) $$

Remark 4.1

The proof is a straightforward application of the variational minimax principle of Poincaré. We do not show it here.

Proposition 4.5

For any \(L \in \mathbb{R}_{+}\), define the index set \(\mathcal{A}(L) = \{n| \alpha_{n}\leq L\}\). If we let \(\vert \mathcal{A}(L)\vert \equiv \operatorname{card}(\mathcal{A}(L))\) then given δ>0 there exists constants \(C_{1}^{*}(\delta)\) and \(C_{2}^{*}(\delta)\) such that for L≥δ

$$ C_1^{*}(\delta)\sqrt{L} \leq\bigl \vert \mathcal{A}(L) \bigr \vert \leq C_2^{*}(\delta)\sqrt{L}. $$
(16)

Proof

Write \(\widetilde{L} =R\sqrt{L}\) and again define κ=1−r b/R. Then \(\vert \mathcal{A}(L)\vert \) is just the number of solutions to tan(κx)=x which lie in the interval \([0,\widetilde{L}]\). This number is well approximated by the number of vertical asymptotes of tan(κx). It then follows that

$$ \frac{\kappa\widetilde{L}}{\pi}-1 \leq\bigl \vert \mathcal{A}(L)\bigr \vert \leq \frac{\kappa\widetilde{L}}{\pi}+1 $$

so that

$$ \sqrt{L} \biggl(\frac{R-r_{\textrm{b}}}{\pi} -\frac{1}{\sqrt {L}} \biggr) \leq\bigl \vert \mathcal{A}(L)\bigr \vert \leq\sqrt{L} \biggl(\frac{R-r_{\textrm{b}}}{\pi} + \frac{1}{\sqrt{L}} \biggr). $$

If L≥δ the choice \(C_{1}^{*}(\delta) = \frac {R-r_{\textrm{b}}}{\pi} -\frac{1}{\sqrt{\delta}}\) and \(C_{2}^{*}(\delta) = \frac{R-r_{\textrm{b}}}{\pi} +\frac{1}{\sqrt{\delta}}\) gives the proposition. □

Remark 4.2

In the remainder we assume δ=4π 2/(R−r b)2 so that \(C_{1}^{*}=(R-r_{\textrm{b}})/2\pi\).

We now give our main convergence estimate for the eigenvalues of the Doi model. The following theorem can be regarded as the heart of the subsequent computations.

Theorem 4.1

Let \(0<\sigma_{0} < \tfrac{1}{4}\) and define M(λ)≡K 0 λ σ for K 0>1. For any fixed σ∈(0,σ 0], there exists λ 0>0 such that for λ≥λ 0, \(\frac{M(\lambda)}{\lambda}\leq \frac{1}{2}\). Then for α n ≤M(λ), μ n (λ)→α n , \(\mathcal{O}(\lambda^{-(\frac{1}{2}-2\sigma)})\).

Remark 4.3

Note that in the remainder C will denote an arbitrary constant that may depend on R, r b, and λ 0. We will also subsequently assume λ 0>1.

Proof

Recall that the Doi eigenvalues μ n (λ) satisfy

$$ f\bigl(\mu_{n}(\lambda)\bigr) = A\bigl(\mu_{n}(\lambda), \lambda\bigr). $$

As before, let κ=1−r b/R and let \(x=R\sqrt{\mu }\). Recalling the definitions of D(x) and N(x) from Proposition 4.2, define

$$ B(x;\lambda) = D(x)\tilde{A}(x,\lambda)\equiv \frac{ [x\tan (\kappa x)+1 ]\tanh (r_{\textrm{b}}\sqrt{1-\frac {x^2}{\lambda R^2}} )}{\sqrt{\lambda}\sqrt{1-\frac {x^2}{\lambda R^2}}}. $$

It follows that the rescaled Doi eigenvalues x n (λ) satisfy

$$ N\bigl(x_n(\lambda)\bigr) = B\bigl(x_n(\lambda); \lambda\bigr). $$

For K 0>1, we choose \(\lambda\geq(2K_{0})^{\frac{1}{1-\sigma_{0}}}\) so that \(\frac{M(\lambda)}{\lambda} \leq\frac{1}{2}\). Recall \(f(\mu)\equiv\tilde{f}(x)\). We restrict to \(\{x : \frac {x^{2}}{R^{2}}\leq M(\lambda), \tilde{f}(x) \geq0\}\), and let h(u)=(1−u)−1/2. Since h is monotone, it follows that

$$ \frac{1}{\sqrt{\lambda}\sqrt{1-\frac{x^2}{\lambda R^2}}} \equiv \frac{h(\tfrac{x^2}{R^2\lambda})}{\sqrt{\lambda}}\leq\frac {h(\tfrac{1}{2})}{\sqrt{\lambda}} = \frac{\sqrt{2}}{\sqrt{\lambda}}. $$

As shown in Proposition 4.2, \(\tilde{f}(x) \geq0\) implies \(-\frac{1}{x} \leq\tan(\kappa x)\leq x\) so that

Write x n (λ)=d n −ϵ n . (Recall \(d_{n} = R\sqrt{\alpha_{n}}\).) Then

$$ N(d_n - \epsilon_n) = B\bigl(x_n( \lambda);\lambda\bigr). $$

Applying the mean value theorem, we get, for some e n ∈(x n (λ),d n ) that

$$ N(d_n) -N'(e_n)\epsilon_n = B\bigl(x_n(\lambda);\lambda\bigr). $$

As N(d n )=0 and N′(e n )<0,

$$ \bigl \vert N'(e_n)\bigr \vert \epsilon_n \leq\frac{ [1 + x_n^2 ]\sqrt{2}}{\sqrt{\lambda}}, $$

which gives that

$$ \epsilon_n \leq \frac{\sqrt{2} [1+x^2_n ] (2e_n^2\cos^2(\kappa e_n) )}{R\sqrt{\lambda} (2\kappa e_n -\sin(2\kappa e_n) )} \leq\frac{\sqrt{2} [1+d^2_n ]e_n}{\kappa R\sqrt{\lambda } (1 -\tfrac{\sin(2\kappa e_n)}{2\kappa e_n } )} \leq\frac{C [1+d^2_n ]d_n}{\sqrt{\lambda} (1 -\tfrac{\sin(2\kappa e_n)}{2\kappa e_n } )}. $$
(17)

For any fixed c∈(0,κ  x 1(λ 0)), monotonicity of the eigenvalues implies \(0<\frac{c}{2\kappa}\leq x_{1}(\lambda_{0})\leq x_{n}(\lambda) <e_{n} <d_{n}\) and, therefore, c<2κe n . Define \(l(\theta) = 1-\frac{\sin(\theta)}{\theta}\). It follows easily that for θ≥c there exists 0<m<1 such that

$$ m \leq l(\theta)\leq1. $$

Using this bound in (17), in the original unscaled variables we find that

$$ R(\sqrt{\alpha_n} -\sqrt{ \mu_n})\leq\frac{CR\sqrt{\alpha _n} [1+R^2\alpha_n ]}{\sqrt{\lambda}}, $$
(18)

which implies

For α n ≤M(λ)≡K 0 λ σ, M(λ)>1 implies

 □

Remark 4.4

Of interest is the possibility of tighter estimates here. In fact, one can show that if f″(μ)>0 wherever f(μ)>0 we actually have

$$ \alpha_n-\mu_n \leq\frac{C\alpha_n}{\lambda^{\frac{1}{2}}}. $$

Theorem 4.1 and Proposition 4.4 immediately implies

Corollary 4.1

For any fixed n, we have that μ n (λ) converges monotonically to α n as λ→∞.

4.2 Eigenfunction Estimates

In this subsection, we carry over the estimates for the eigenvalues obtained in the last subsection to obtain the uniform convergence in r of the eigenfunctions as λ→∞. Though unstated, in the remainder all theorems and lemmas include the assumptions of Theorem 4.1.

Lemma 4.1

The (unnormalized) Doi and Smoluchowski eigenfunctions satisfy

$$ \sup_{r\in[r_{\textrm{b}},R]}{\bigl \vert \phi_n(r)-\psi ^{\mathrm{out}}_n(r)\bigr \vert } = \mathcal{O} \bigl(\lambda^{-(\frac{1}{2} -\frac{3\sigma}{2})} \bigr) $$

for μ n <M(λ)=K 0 λ σ as λ→∞.

Proof

Note that

We find

Similarly, we have

Combining these with (18)

For II, we have

It follows that

$$ \bigl \vert \phi_n(r)- \psi^{\mathrm{out}}_n(r)\bigr \vert \leq \frac{CK_0}{\lambda^{\frac{1}{2} -\frac{3\sigma}{2}}} \biggl[\frac{1}{\lambda^{\sigma/2}}+\sqrt{K_0} \biggr] \leq\frac{C_1(r_{\textrm{b}},R,K_0)}{\lambda^{\frac{1}{2} -\frac {3\sigma}{2}}}. $$
(19)

This concludes the proof. □

We now prove several uniform properties of the eigenfunctions we will use in the next subsection.

Lemma 4.2

  1. 1.

    There exist a λ 0, C 2=C 2(r b,R), such that for all λ≥λ 0 and \(n \in \mathbb {Z}^{+}\)

    $$ \max \Bigl(\sup _{r\in[r_{\textrm{b}},R]}\bigl \vert \psi _n^{\mathrm{out}}(r)\bigr \vert , \sup_{r\in[r_{\textrm{b}},R]}\bigl \vert \phi _n(r)\bigr \vert \Bigr) \leq C_2. $$
    (20)
  2. 2.

    Let \(b_{n} = \Vert\psi_{n}\Vert^{-2}_{2}\) and \(a_{n} =\Vert\phi _{n}\Vert^{-2}_{2}\). Then there exists C 3 such that for all λ≥λ 0

    (21)

Proof

  1. 1.

    We start by defining for z≥0, and r b≤r≤R the auxiliary function

    $$ H(z,r) := \frac{1}{r} \biggl[\frac{\sin(\sqrt{z}(R-r))}{R\sqrt {z}}-\cos\bigl(\sqrt{z}(R-r) \bigr) \biggr]. $$

    Note ϕ n (r)≡H(α n ,r) and \(\psi_{n}^{\mathrm{out}}(r)\equiv H(\mu_{n},r)\). Now for z≥z 0>0 we have that

    By Corollary 4.1, there exists λ 0 such that for λ≥λ 0

    $$ \alpha_1\geq\mu_1(\lambda)\geq\mu_1( \lambda_0)\geq\frac{\alpha _1}{2}>0. $$

    Note that we are using the fact that both the Doi and Smoluchowski eigenvalues can be written in non-decreasing order. Choosing z 0=α 1/2, proves the first part of the lemma.

  2. 2.

    To prove the second part start by defining

    $$ h(z):= \int_{r_{\textrm{b}}}^{R}{ \bigl(H(z,r) \bigr)^2 r^2\,dr}. $$

    Once again, we have that \(\Vert\psi^{\mathrm{out}}_{n}\Vert^{2}_{2}\equiv h(\mu _{n})\) and \(\Vert\phi_{n}\Vert^{2}_{2} \equiv h(\alpha_{n})\). A priori we have that h(z)>0 for all z≥0. An explicit computation shows that for z>0, h(z) is continuous, lim z→∞ h(z)=(R−r b)/2>0, and \(\lim_{z\to0} h(z) = (R^{3} - r_{\textrm{b}}^{3}) / 3 R^{2}\). With the positivity of h(z) on [0,∞), these results imply that A:=infh(z)>0. It then follows that

    $$ a_n =\frac{1}{\Vert\phi_n\Vert^{2}_2} \equiv\frac{1}{h(\alpha _n)}\leq \frac{1}{A}=:C_3 $$

    and

    $$ b_n =\frac{1}{\Vert\psi_n\Vert^{2}_2}\leq\frac{1}{\Vert\psi ^{\mathrm{out}}_n\Vert^{2}_2}\equiv \frac{1}{h(\mu_n)}\leq\frac{1}{A}=:C_3. $$

    This concludes the proof of the lemma.

 □

These results imply that

Lemma 4.3

There exists C 4 such that for n with μ n (λ)≤α n ≤M(λ)

$$ \vert b_n - a_n\vert \leq \frac{C_4}{\lambda^{\frac{1}{2} -\frac {3\sigma}{2}}}. $$

Proof

We start by noting that

To get the last line, we have used Lemmas 4.1 and 4.2. A direct computation shows that

(Here, we have used that \(\sqrt{\lambda- \mu_{n}} \geq\sqrt{\lambda- M(\lambda)} \geq\sqrt{\frac{\lambda}{2}}\).) Using the preceding bounds, we find that

The choice \(C_{4} = CC_{3}^{2} \lambda_{0}^{- \frac{3 \sigma}{2}} + 2C_{1}C_{2}C_{3}^{2}(R^{3}-r_{\textrm{b}}^{3}) / 3\) gives the lemma. □

4.3 An Error Estimate for the Convergence of the Doi to Smoluchowski Model

We now show the uniform convergence of the Green’s function of the radially symmetric Doi PDE (3) to the Green’s function of the radially symmetric Smoluchowski PDE (2) model. The error bound we give shows that the convergence of the Doi model to the Smoluchowski model cannot be expected to be faster than O(λ −1/2) as λ→∞.

Theorem 4.2

Fix any \(0< \sigma< \frac{1}{4}\) (see Theorem 4.1). For t≥δ>0, there exists a function u(t) and λ 0, such that for all λ≥λ 0 we have

$$ \sup_{r\in[r_{\textrm{b}},R]}{\bigl \vert \rho(r,t)-p_{\lambda }(r,t)\bigr \vert }\leq\frac{u(t)}{\lambda^{\frac{1}{2}-2\sigma}}. $$
(22)

Moreover, for t∈[δ,∞), u(t) is uniformly bounded.

Proof

The main idea is to use the series representation of the solutions to both models to estimate the error. There will be a proliferation of constants which we shall repeatedly and unceremoniously denote by C. For r∈(r b,R), we begin by writing:

We deal with the finitely indexed sum, I, first. Define the index set \(\mathcal{A}_{\lambda}= \{n \mid\alpha_{n}<M(\lambda)\}\). From now on, for simplicity of presentation, we write ψ n for \(\psi_{n}^{\mathrm{out}}\). Let I=I a +I b +I c +I d , where

Recalling that γ n denotes the nth eigenvalue of the radically symmetric Laplacian on [0,R) with a zero Neumann boundary condition at R (see Proposition 4.4), we find

Here, we have applied Proposition 4.4, Lemma 4.1, and Lemma 4.2 (in particular (19), (20), and (21)). The same argument shows

and using Lemma 4.3, also, we find

Finally, we have that

For s≥0, 1−e −s≤|s| so that, using the same lemmas as before and Theorem 4.1,

We now bound the tail of the series, II. First define the index set \(\mathcal{B}_{\lambda}= \{n \mid\alpha_{n}\geq M(\lambda)\}\). We now specify the choice K 0>4π 2/(R−r b)2 which guarantees \(C_{1}^{*} \sqrt{K_{0}} > 1\) (see Remark 4.2 after the proof of Proposition 4.5). Using the uniform bounds on ψ n , ϕ n , a n , and b n and Proposition 4.5 we find

We thus obtain the error estimate that

$$ \bigl \vert \rho(r,t)-p_{\lambda}(r,t) \bigr \vert \leq C \Biggl[\frac {1}{\lambda^{\frac{1}{2} -\frac{3\sigma}{2}}}\sum_{n=1}^{\infty }{e^{-\gamma_{n}t}} + \frac{1}{\lambda^{\frac{1}{2} -2\sigma}}\sum_{n=1}^{\infty}{t \, e^{-\mu_{n}t}} + \sum_{n\geq\lambda^{\frac {\sigma}{2}}}{e^{-\gamma_{n}t}} \Biggr]. $$
(23)

We estimate the terms in (23) one at a time. First

while

Finally, we bound the third term in (23):

Combining the preceding estimates, we have

Here, we absorbed the maximum of the many constants into C and

$$ C(\sigma) \equiv \biggl(\frac{1}{\widetilde{C}e} \biggl[{\frac {1}{2\sigma}-2} \biggr] \biggr)^{{\frac{1}{2\sigma}-2}}. $$

Let

$$ u(t):= C \biggl[\frac{1}{\sqrt{t}} \biggl(1+\frac{C(\sigma)}{t^{\frac {1}{2\sigma}-2}} \biggr)+1 \biggr]. $$

For t≥δ>0, we see that u(t) is uniformly bounded, concluding the proof. For any ϵ sufficiently small, the choice \(\sigma= \frac{\epsilon}{2}\) then gives the bound (1) stated in the Introduction. □

Remark 4.5

Note that even for t≥δ>0, u(t)→∞ as σ→0 because u(t) blows up like \((\frac{1}{\sqrt{\delta\sigma}} )^{\frac{1}{\sigma}}\). That said, we expect this divergence is an artifact of our eigenfunction expansion-based method of proof.

5 Conclusion

We have shown for the special case of two molecules that may undergo the annihilation reaction, A+B→∅, with one of the two molecules stationary, the solution to the Doi model (24) converges to the solution of the Smoluchowski model (25) as λ→∞. A rigorous asymptotic convergence rate that is \(O(\lambda^{-\frac {1}{2} + \epsilon})\), for all fixed ϵ>0, was proven for the maximum difference between the two models over all r∈(r b,R) and t∈(δ,∞) (for any fixed δ>0). Numerical evaluation of the exact eigenfunction expansions, binding time distributions, and mean binding times illustrated this convergence rate, and demonstrated that for sufficiently large fixed values of λ the difference between the two models for realistic parameter regimes scaled like \(r_{\textrm{b}}^{-1}\) as r b was decreased. For biologically relevant values of r b, such as the reaction-radius for a protein diffusing to a fixed DNA binding site, it was found that λ should be chosen at least as large as 1011 s−1 for the mean binding time in the two models to differ by less than 1 %.

There are a number of extensions of the current work that would aid in clarifying the rigorous relationship between the Smoluchowski and Doi models. Foremost would be the study of more detailed, biologically realistic, models in which multiple diffusing and reacting chemical species are present. Such models require the introduction of unbinding reactions, C→A+B, which in the Smoluchowski model requires the use of unbinding radii (a separation, greater than r b, at which to place the newly created molecules to avoid their immediate rebinding). As numerical methods to solve the Doi and Smoluchowski models have been used to study biological systems, understanding how parameters in the Doi model should be chosen so as to accurately approximate the Smoluchowski model would greatly aid in comparing predictions by these different approaches.