7A Diffusion in domains with switching boundaries
In the analysis of narrow escape problems in Sec. 7.2, we assumed that the holes in
the boundary of the domain were always open. However, if we view each pore as a
stochastically gated ion channel, for example, then one has the problem of analyzing
the diffusion equation in a domain with a (partially) randomly switching boundary.
As a simple example, consider the diffusion equation on a finite interval with randomly switching boundary conditions at the two ends. This type of piecewise deterministic PDE has been analyzed by Lawley et al. [7] using the theory of random
iterative systems. These authors assumed that the left-hand boundary is Dirichlet,
and the right-hand boundary switches randomly between inhomogeneous Dirichlet
and either Neumann or Dirichlet. In both cases they showed that the solution of
the stochastic PDE converges in distribution to a random variable whose expectation satisfies a deterministic system of PDEs whose solution is a linear function of
x. They also found that the gradient of the solution is a much more complicated
function of parameters in the case of the Dirichlet-Neumann switching problem.
Mathematically speaking, we can view diffusion in a randomly switching environment as an example of a stochastic hybrid system involving a piecewise deterministic PDE. We will exploit this interpretation in order to present an alternative
approach to analyzing the diffusion equation with switching boundary conditions,
based on discretizing space and constructing the Chapman-Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system [5]. We show how
the CK equation can be used to determine the dynamics of the expectation of the
stochastic field, thus recovering the results of Lawley et al. [7]. We then construct a
hierarchy of equations for the rth moments, which take the form of r-dimensional
parabolic PDEs that couple to lower order moments at the boundaries. The rth moments of the stochastic PDE determine the statistics of r non-interacting Brownian
particles; although the particles are non-interacting, statistical correlations arise due
to the fact that they all move in the same randomly switching environment.
7A.1 Piecewise deterministic PDE
Consider a single Brownian particle drift-diffusing in the finite interval [0, L] with
a fixed absorbing boundary at x = 0 and a randomly switching gate at x = L; the
right-hand boundary randomly switches between absorbing and reflecting. In order
to keep track of the boundary state, we introduce the discrete random variable n(t) ∈
{0, 1} such that
√
(7A.1)
dX(t) = −Φ 0 (X)dt + 2DdW (t),
where Φ(x) is a potential energy function, with an inhomogeneous Dirichlet boundary condition at x = L if n(t) = 0 and a reflecting boundary at x = L if n(t) = 1.
Assume that transitions between the two states are given by the two-state Markov
process
1
β
0
1
(7A.2)
α
with fixed transition rates α, β . One way to analyze the given system, is to view
it as a piecewise SDE and to consider the differential Chapman-Kolmogorov (CK)
equation for the joint probability density pn (x,t) = E[p(x,t)1n(t)=n ]. This takes the
form
∂ pn (x,t)
∂
∂ 2 pn (x,t)
+ ∑ Anm pm (x,t),
=
[Φ 0 (x)pn (x,t)] + D
∂t
∂x
∂ x2
m=0,1
with A the matrix
A=
−β α
β −α
.
(7A.3)
(7A.4)
Equation (7A.3) is supplemented by the joint boundary conditions
p0 (0,t) = p1 (0,t) = 0,
−Φ 0 (L)p1 (L,t) − D
p0 (L,t) = η,
for some constant η, and the initial condition
∂ p1 (x,t) = 0,
∂ x x=L
pn (x, 0) = δ (x − y)ρn ,
where ρn is the stationary measure of the ergodic two-state Markov process generated by the matrix A,
∑
Anm ρm = 0,
ρ0 =
m=0,1
α
,
α +β
ρ1 =
β
.
α +β
(7A.5)
Equation (7A.3), and its higher-dimensional analog, is the starting point for analyzing the escape of a single particle from a bounded domain with switching gates
in the boundary [5, 6]. From this single-particle perspective, one could equally well
interpret the source of the switching to be changes in the conformational state of the
particle, rather than the gate(s), such that it can only pass through a gate when in
one of the two states, see Fig. 7A.1. (In this case, the diffusivity of the particle could
also depend on n.) However, this equivalence breaks down when there are multiple
particles. Even when the particles are non-interacting, statistical correlations arise
due to the fact that they all move in the same randomly switching environment; such
correlations would not occur if each particle independently switched between different conformational states with fixed gates. In order to explore the multi-particle
case, it is necessary to consider an alternative formulation of the problem, in which
equation (7A.3) is replaced by a piecewise deterministic diffusion equation:
∂u
∂ 2u
=D 2,
∂t
∂x
x ∈ [0, L],t > 0
with u satisfying the boundary conditions
2
(7A.6a)
n(t) = 0
n(t) = 1
(a)
β
α
(b)
β
α
Fig. 7A.1: (a) Switching gate and (b) switching conformational state.
u(0,t) = 0,
u(L,t) = η > 0 for n = 0,
∂x u(L,t) = 0 for n = 1.
(7A.6b)
For simplicity we have dropped the drift term V 0 (x). One can view a numerical
solution of equation (7A.6a) up to some time t as determining the probability density u(x,t) conditioned on a single realization {n(s), 0 ≤ s < t} of the stochastic
gate. Thus the conditional probability density u(x,t) can be interpreted as determining the density of multiple particles moving in the same random environment.
Each realization of the gate will typically generate a different solution u(x,t) so that
u(x,t) is a random field variable. The relationship between the single particle and
multi-particle perspectives is then established by averaging over realizations of the
stochastic gate:
pn (x,t)dx = E[u(x,t)1n(t)=n ].
(7A.7)
This immediately raises the question about the interpretation of higher-order moments of the random field variable u.
Following [4], we discretize space and construct the differential CK equation for
the resulting finite-dimensional stochastic hybrid system. The first step is to spatially
discretize the piecewise deterministic PDE (7A.6a) using a finite-difference scheme.
One of the nice features of this discretization is that we can incorporate the boundary
conditions into the resulting discrete Laplacian. Introduce the lattice spacing a such
that (N + 1)a = L for integer N, and let u j = u(a j), j = 0, . . . , N + 1. Then we obtain
the piecewise deterministic ODE
dui
=
dt
N
∑ ∆inj u j + ηa δi,N δn,0 ,
i = 1, . . . , N,
j=1
ηa =
ηD0
a2
(7A.8)
for n = 0, 1. Away from the boundaries (i 6= 1, N), ∆inj is given by the discrete Laplacian
D
∆inj = 2 [δi, j+1 + δi, j−1 − 2δi, j ].
(7A.9a)
a
3
On the left-hand absorbing boundary we have u0 = 0, whereas on the right-hand
boundary we have
uN+1 = η for n = 0,
uN+1 − uN−1 = 0 for n = 1.
These can be implemented by taking
∆10j =
D
[δ j,2 − 2δ j,1 ],
a2
∆N0 j =
D
[δN−1, j − 2δN, j ],
a2
∆11j =
D
[δ j,2 − 2δ j,1 ]
a2
(7A.9b)
and
2D
[δN−1, j − δN, j ].
a2
Let u(t) = (u1 (t), . . . , uN (t)) and introduce the probability density
∆N1 j =
Prob{u(t) ∈ (u, u + du), n(t) = n} = pn (u,t)du,
(7A.9c)
(7A.10)
where we have dropped the explicit dependence on initial conditions. The probability density evolves according to the following differential CK equation for the
stochastic hybrid system (7A.8), see also Sec. 7.4,
"
!
#
N
N
∂ pn
∂
=−∑
∑ ∆inj u j + ηa δi,N δn,0 pn (u,t) + ∑ Anm pm (u,t),
∂t
i=1 ∂ ui
j=1
m=0,1
(7A.11)
where A is the matrix
−β α
A=
.
(7A.12)
β −α
The left nullspace of the matrix A is spanned by the vector
1
ψ=
,
1
and the right nullspace is spanned by
1
α
ρ0
.
=
ρ≡
ρ1
α +β β
(7A.13)
(7A.14)
A simple application of the Perron-Frobenius theorem shows that the two state
Markov process with master equation
dPn (t)
= ∑ Anm Pm (t)
∂t
m=0,1
is ergodic with limt→∞ Pn (t) = ρn .
4
(7A.15)
7A.2 Moment equations and particle perspective
Since the drift terms in the CK equation (7A.11) are linear in the u j , it follows that
we can obtain a closed set of equations for the moment hierarchy.
First-order moments
Let
vn,k (t) = E[uk (t)1n(t)=n ] =
Z
pn (u,t)uk (t)du.
(7A.16)
Multiplying both sides of the CK equation (7A.11) by uk (t) and integrating with
respect to u gives (after integrating by parts and using that pn (u,t) → 0 as u → ∞
by the maximum principle)
dvn,k
=
dt
N
∑ ∆knj vn, j + ηa ρ0 δk,N δn,0 + ∑
j=1
Anm vm,k .
(7A.17)
m=0,1
We have assumed that the initial discrete state is distributed according to the stationary distribution ρn so that
Z
pn (u,t)du = ρn .
If we now retake the continuum limit a → 0 and set
Vn (x,t) = E[u(x,t)1n(t)=n ],
(7A.18)
then we obtain the system of equations
∂ 2V0
∂V0
= D 2 − βV0 + αV1
∂t
∂x
∂V1
∂ 2V1
= D 2 + βV0 − αV1
∂t
∂x
(7A.19a)
(7A.19b)
with
V0 (0,t) = V1 (0,t) = 0,
V0 (L,t) = ρ0 η > 0,
∂xV1 (L,t) = 0.
(7A.20)
To see why these are the correct boundary conditions, note that if n(t) = 0 and x = L,
then u(x,t) = η with probability one, and thus
V0 (L,t) = E[u(L,t)1n(t)=0 ] = ηP(n(t) = 0) = ηρ0 .
Deriving the other boundary conditions is similar.
It is now straightforward to recover the result of Lawley et al. [7] by determining
the steady-state solution of equations (7A.19a) and (7A.19b). First, note that
5
E[u(x,t)] = V0 (x,t) +V1 (x,t),
(7A.21)
Since equations equations (7A.19a) and (7A.19b) have a globally attracting steadystate, it follows that
lim E[u(x,t)] = V (x) ≡
t→∞
∑
Vn (x),
(7A.22)
n=0,1
where Vn (x) ≡ limt→∞ Vn (x,t). Adding equations (7A.19a) and (7A.19b) and using
the boundary conditions in equation (7A.20) gives
d 2V
= 0,
dx2
V (0) = 0,
and κ = V1 (L). Hence,
V (x) =
V (L) = ρ0 η + κ,
(7A.23)
x
[ρ0 η + κ],
L
with
d 2V1
β
− (α + β )V1 = − x(ρ0 η + κ)
2
dx
L
and V1 (0) = 0, ∂xV1 (L) = 0. It follows that
V1 (x) = ae−ξ x + beξ x +
with ξ =
p
(7A.24)
ρ1
(ρ0 η + κ)x,
L
α + β . The boundary conditions imply that
a = −b,
2ξ a cosh(ξ L) =
ρ1
(ρ0 η + κ),
L
which yields the solution
1 sinh(ξ x) x
V1 (x) = ρ1 (ρ0 η + κ) −
+
.
ξ L cosh(ξ L) L
(7A.25)
Finally, we obtain κ by setting x = L:
κ = ρ1 (ρ0 η + κ) 1 − (ξ L)−1 tanh(ξ L) ,
which can be rearranged to yield
κ = ρ1 ρ0 η
and thus [7]
V (x) =
1 − (ξ L)−1 tanh(ξ L)
ρ0 + ρ1 (ξ L)−1 tanh(ξ L)
x
η
.
L 1 + (ρ1 /ρ0 )(ξ L)−1 tanh(ξ L)
In the limit ξ → ∞ (fast switching),
6
(7A.26)
V (x) =
x
η.
L
Second-order moments
Let
vn,kl (t) = E[uk (t)ul (t)1n(t)=n ] =
Z
pn (u,t)uk (t)ul (t)du.
(7A.27)
Multiplying both sides of the CK equation (7A.11) by uk (t)ul (t) and integrating
with respect to u gives (after integration by parts)
dvn,kl
=
dt
h
i
n
n
∆
v
+
∆
v
∑ k j n, jl l j n, jk + ηa δn,0 vn,k δl,N + vn,l δk,N +
N
j=1
∑
Anm vm,kl .
m=0,1
(7A.28)
If we now retake the continuum limit a → 0, we obtain a system of equations for the
equal-time two-point correlations
Cn (x, y,t) = E[u(x,t)u(y,t)1n(t)=n ],
(7A.29)
given by
∂ 2C0
∂ 2C0
∂C0
= D 2 + D0
− βC0 + αC1
∂t
∂x
∂ y2
∂C1
∂ 2C1
∂ 2C1
= D 2 + D1
+ βC0 − αC1 .
∂t
∂x
∂ y2
(7A.30a)
(7A.30b)
The two-point correlations couple to the first-order moments via the boundary conditions:
C0 (0, y,t) = C0 (x, 0,t) = C1 (x, 0,t) = C1 (0, y,t) = 0
(7A.31a)
and
C0 (L, y,t) = ηV0 (y,t), C0 (x, L,t) = ηV0 (x,t), ∂xC1 (L, y,t) = ∂yC1 (x, L,t) = 0.
(7A.31b)
As in the case of the first-moment equations, we can solve for the steady-state
correlations explicitly. Define
lim E[u(x,t)u(y,t)] = C(x, y) ≡
t→∞
∑
Cn (x, y),
n=0,1
where Cn (x, y) = limt→∞ Cn (x, y,t). Adding equations (7A.30a) and (7A.30b) then
gives
∂ 2C ∂ 2C
+ 2 = 0,
(7A.32)
∂ x2
∂y
7
with boundary conditions
C(0, y) = C(x, 0) = 0
(7A.33a)
and
C(L, y) = ηV0 (y) +C1 (L, y),
C(x, L) = ηV0 (x) +C1 (x, L).
(7A.33b)
Using separation of variables, we have C(x, y) = f (x)g(y) with
f 00 (x)
g00 (y)
=−
= ±µ 2
f (x)
g(y)
for a constant µ. The general solution is
A0
xy + ∑ An [sinh(nπx/L) sin(nπy/L) + sin(nπx/L) sinh(nπy/L)].
L2
n>0
(7A.34)
The 2nd-order moment C1 satisfies the equation
C(x, y) =
∂ 2C1 ∂ 2C1
+
− (α + β )C1 (x, y) = −βC(x, y)
∂ x2
∂ y2
(7A.35)
with
C1 (x, 0) = C1 (0, y) = 0,
∂xC1 (L, y) = ∂yC1 (x, L) = 0.
The general solution of C1 is
C1 (x, y)
= ρ1C(x, y) + B0 [y sinh(ξ x) + x sinh(ξ y)]
p
+ ∑n>0 Bn sinh( (nπ/L)2 + ξ 2 x) sin(nπy/L)
p
+ ∑n>0 Bn sin(nπx/L) sinh( (nπ/L)2 + ξ 2 y),
(7A.36)
From the boundary conditions (7A.33b),
ρ0C(L, y)
= ηV0 (y) + B0 [y sinh(ξ L) + L sinh(ξ y)]
p
+ ∑n>0 Bn sinh( (nπ/L)2 + ξ 2 L) sin(nπy/L).
Equating terms on the two sides of this equation and using equations (7A.25) and
(7A.26) to give
ρ1 sinh(ξ y)
y
V0 (y) = (ρ0 η + κ)
+ ρ0 ,
ξ L cosh(ξ L)
L
we find that
ρ0 A0 = ηρ0 (ρ0 η + κ) + B0 L sinh(ξ L),
ρ1
η(ρ0 η + κ)
+ B0 L = 0,
ξ L cosh(ξ L)
and
8
(7A.37a)
(7A.37b)
q
ρ0 An sinh(nπ) = Bn sinh( (nπ/L)2 + ξ 2 L),
n > 0.
(7A.37c)
The first two equations determine A0 , B0 and the remaining equations determine Bn
in terms of An .
The final step is to determine the coefficients An , n > 0 using the other boundary condition ∂xC1 (L, y) = 0. (By symmetry the boundary conditions at y = L are
automatically satisfied.) After some algebra, one finds that [4]
L
γm Am + ∑ Γmn An = −Λm ,
2
n>0
m > 0,
(7A.38)
where
Γmn
nmπ 2
sinh(nπ)
sinh(nπ)
n+m+1
=
(−1)
ρ1
+ ρ0
,
L2
(nπ/L)2 + (mπ/L)2
(nπ/L)2 + ξ 2 + (mπ/L)2
and
Λm = (B0 ξ cosh(ξ L) + ρ1 A0 /L2 )
L2
mπ/L
(−1)m+1 + (−1)m+1 B0 2
sinh(ξ L).
mπ
ξ + (mπ/L)2
Introducing the change of coefficients (for n > 0)
bn = sinh(nπ)An ,
A
Fig. 7A.1: Plot of C(x, x) for L = η = 1, ξ = 10, and various values ρ0 = 0.75, 0.5, or 0.25. The
Fourier series representation of C(x, x) is truncated after 200 terms. A 50,000-dimensional version
of the infinite dimensional system found in equation (7A.39) is solved to estimate the Fourier
coefficients.
9
equation (7A.38) can be rewritten as
bm + ∑ (−1)n+m+1
γbm A
n>0
where
γbm =
1
2
i
h ρ n
ρ0 n
1
bn = − Λm ,
+
A
n2 + m2 n2 + (ξ L/π)2 + m2
m
(7A.39)
h
i
p
p
ρ1 πcoth(mπ) + ρ0 π 2 + (ξ L/m)2 cotanh( (mπ)2 + (ξ L)2 ) .
If we assume that the infinite-dimensional matrix equation (7A.39) has a unique
bm ∼ 1/m2 for large m and thus
solution, then taking the limit m → ∞ shows that A
Am ∼ e−mπ /m2 for large m. In Figure 7A.1 we plot estimates of C(x, x) by truncating its Fourier series expansion in equation (7A.34), where the coefficients are
estimated by solving a truncated version of equation (7A.39). We find that the numerical solution converges to a unique solution, except for a small boundary layer
around x = L, which shrinks as more terms in our numerical approximation scheme
are included.
Higher-order moments
Equations for rth order moments r > 2 can be obtained in a similar fashion. Let
(r)
vn,k1 ...kr (t) = E[uk1 (t) . . . ukr (t)1n(t)=n ] =
Z
pn (u,t)uk1 (t) . . . ukr (t)du.
(7A.40)
Multiplying both sides of the CK equation (7A.11) by uk1 (t) . . . ukr (t) and integrating with respect to u gives (after integration by parts)
(r)
dvn,k1 ...kr
dt
(r)
(r−1)
= ∑rl=1 ∑Nj=1 ∆knl j vn,k1 ...kl−1 jkl+1 ...kr + ηa δn,0 ∑rl=1 vn,k1 ...kl−1 kl+1 ...kr δkl ,N
(r)
+ ∑m=0,1 Anm vm,k1 ...kr .
If we now retake the continuum limit a → 0, we obtain a system of equations for the
equal-time r-point correlations
(r)
Cn (x, y) = E[u(x1 ,t)u(x2 ,t) . . . u(xr ,t)1n(t)=n ],
(7A.41)
given by
(r)
∂C0
∂t
(r)
∂C1
∂t
(r)
∂ 2C0
(r)
(r)
− βC0 + αC1
2
∂
x
l=1
l
r
= D0 ∑
(r)
∂ 2C0
(r)
(r)
+ βC0 − αC1
2
l=1 ∂ xl
r
(7A.42a)
= D1 ∑
10
(7A.42b)
The r-point correlations couple to the (r − 1)-order moments via the boundary conditions:
(r)
(r)
(r)
= ∂xl C1 (x1 , . . . , xr ,t)
= 0,
= C1 (x1 , . . . , xr ,t)
C0 (x1 , . . . , xr ,t)
xl =0
xl =0
xl =L
and
(r)
C0 (x1 , . . . , xr ,t)
(r−1)
xl =L
= ηC0
(x1 , . . . , xl−1 , xl+1 . . . , xr ,t),
(7A.43a)
for l = 1, . . . , r.
Particle perspective
We now relate the rth moments of the piecewise deterministic PDE considered
above to statistics of multiple Brownian particles diffusing in a randomly switching
environment. The first observation is that the first-moment equations (7A.19a) and
(7A.19b) are identical in form to the CK equation (7A.3) describing a single Brownian particle diffusing under the same switching boundary conditions with Φ(x) = 0.
That is, pn (x,t) = Vn (x,t). One remaining issue is the interpretation of the inhomogeneous term η from the single particle perspective. This can be addressed by
noting that if we set
πn (x) =
1
Vn (x),
ρn η
where Vn (x) is the steady-state solution of equations (7A.19a) and (7A.19b), then
∂ 2 π0
− β [π0 − π1 ],
∂ y2
∂ 2 π1
0 = D 2 + α[π0 − π1 ]
∂y
0=D
(7A.44a)
(7A.44b)
with boundary conditions
π0 (0) = π1 (0) = 0,
π0 (L) = 1,
∂y π1 (L) = 0.
We recognize equations (7A.44a) and (7A.44b) as steady-state backward equations
for the CK equation (7A.3) when Φ(x) = 0 and η = 0, with πn (x) corresponding to
the hitting probability that the particle exits at the end x = L and n(0) = n, see also
Sec. 7.6.
Furthermore, let πnr (x1 , . . . , xr ) be the probability that r Brownian particles all
exit at x = L given that the initial positions of the Brownian particles are x1 , . . . , xr
and n(0) = n. Then
11
πnr (x1 , . . . , xr ) =
1
(r)
lim Cn (x1 , . . . , xr ,t),
ρn η r t→∞
(7A.45)
(r)
where Cn is the rth moment defined in equation (7A.41). Though the particles are
non-interacting, the probability that all r particles exit at x = L is not the product
of the probabilities of each particle exiting at x = L because the particles are all
diffusing in the same randomly switching environment. Equation (7A.45) follows
from writing down the backward equation for the joint probability density for r
particles. The crucial step is determining the appropriate inhomogeneous boundary
condition for the resulting r-dimensional time-independent PDE that determines the
splitting probability. The boundary condition takes the form
(r−1)
(r)
= π0
(x1 , . . . , xl−1 , xl+1 . . . , xr ),
(7A.46)
π0 (x1 , . . . , xr )
xl =L
for l = 1, . . . , r. This ensures that if one of the particles starts on the right-hand
boundary when the latter is in the state n = 0, then the particle is immediately removed and thus one just has to determine the splitting probability that the r − 1
remaining particles also exit at the right-hand boundary. Finally, performing a similar scaling to the first-moments yields the desired result.
7A.3 Stochastically gated diffusion-limited reactions
Another example of a PDE with switching boundaries arises when extending the
classical analysis of diffusion-limited reaction rates presented in Sec. 2.4.1 to the
case that a target protein can switch between two conformational states n = 0, 1,
and is only reactive in the open state n = 0. That is, the boundary of the target can
be treated as randomly switching. For unbounded domains exterior to the target,
this problem was first studied by Szabo et al [10], who assumed that it is irrelevant
whether it is the target protein or the diffusing ligands that switch between conformational states. Although this symmetry holds for a pair of reacting particles, it
breaks down when a single protein is surrounded by many ligands [11, 2, 9, 3, 8, 1].
For one of the basic simplifying assumptions of Smoluchowski theory is that one
can ignore many-particle effects, namely, correlations in the dynamics of the ligands. However, correlations arise when the protein switches between reactive states,
since this is simultaneously experienced by all of the ligands. In other words, all of
the ligands diffuse in the same randomly switching environment. (A similar issue
arises in the case of proteins escaping through a randomly switching membrane
[5, 6]). On the other hand, such correlations don’t arise if the ligands are assumed
to independently switch between conformational states. We will ignore the effects
of correlations here.
As in the case of stochastic ion channels (see Chap. 3), the switching dynamics
is described by a two-state Markov process with switching rates α, β :
12
α
(closed) (open).
(7A.47)
β
We will treat the target as a fixed sphere Ω of radius a at the origin in R3 and denote
the concentration of background diffusing molecules exterior to Ω by c(x,t). The
associated diffusion equation takes the form
∂ c(x,t)
= D∇2 c(x,t),
∂t
c(x, 0) = 1,
subject to the far–field boundary condition c(x,t) = 1 for x → ∞. The boundary
conditions are
∂ c ∂ c =
k
c(a,t),
if
n(t)
=
0,
= 0, if n(t) = 1
4πa2 D
on
∂ r r=a
∂ r r=a
Introducing the conditional densities
cn (x,t) = E[c(x,t)In(t)=n ],
(7A.48)
we obtain the differential Chapman-Kolmogorov (CK) equation
∂ c0
= D∇2 c0 − β c0 + αc1 ,
∂t
∂ c1
= D∇2 c1 + β c0 − αc1 ,
∂t
(7A.49a)
(7A.49b)
with boundary conditions
4πa2 D
∂ c0 = kon c0 (a,t),
∂ r r=a
The far-field conditions become
c0 (∞) = ρ0 ≡
α
,
α +β
∂ c1 = 0.
∂ r r=a
c1 (∞) = ρ1 ≡
β
.
α +β
(7A.50)
and the initial conditions are cn (x, 0) = ρn . Given the steady-state solution (c0 (x), c1 (x)),
which exists in 3D, the stochastically-gated reaction rate is given by
∂ c0 (x) kSG = 4πa2 D
.
(7A.51)
∂ r r=a
The calculation of kSG was originally carried out be Szabo et al. [10], and we review
their analysis here.
Exploiting the spherical symmetry of the problem, we set un (r) = rcn (r) such
that the steady-state version of (7A.49) becomes
13
d 2 u0
− β u0 + αu1 ,
dr2
d 2 u1
0 = D 2 + β u0 − αu1 ,
dr
0=D
(7A.52a)
(7A.52b)
with un (r) → ρn r as r → ∞. Adding equations (7A.52a) and (7A.52b) with u(r) =
u0 (r) + u1 (r) yields d 2 u/dr2 = 0 with solution u(r) = A + r for the unknown constant A. Equation (7A.52a) can then be rewritten as
D
d 2 u0
− (α + β )u0 = −α(A + r),
dr2
which has the solution
−λ r
u0 (r) = Γ e
α
+
(r + A),
α +β
λ=
r
α +β
.
D
Hence, the steady-state solution takes the form
α
Aα
Γ −λ r
e
+
+
,
r
α +β
r
β
Aβ
Γ
+
.
c1 (r) = − e−λ r +
r
α +β
r
c0 (r) =
(7A.53a)
(7A.53b)
with two unknown constants A,Γ . The latter are determined by imposing the boundary conditions:
Γ
Aα
Γ −λ a
α
Aα
4πa2 D − 2 [1 + λ a]e−λ a − 2 = kon
e
+
+
a
a
a
α +β
a
and
∂ c1 Γ
Aβ
0=
= 2 [1 + λ a]e−λ r − 2 .
∂ r r=a a
a
The second equation relates Γ and A
Γ [1 + λ a]e−λ a = Aβ ,
so that the first equation can be rewritten as
kon
Aβ
α
α +β
−A 2 =
+ Aα +
a .
a
4πDa3 1 + λ a
α +β
Rearranging, we obtain the following expression for the constant A:
kon
α + β + αλ a
kon αa
A 1+
=−
,
kD (α + β )
1+λa
kD (α + β )2
with kD = 4πDa. Combining these various results, the reaction rate is
14
Γ
Aα
kD (α + β )
kSG = akD − 2 [1 + λ a]e−λ a − 2 = −
A.
a
a
a
Finally, substituting for A yields
kSG =
kon
kon α
α + β + αλ a −1
1+
,
α +β
kD (α + β )
1+λa
kSG =
kon kD α[1 + λ a]
.
kD (α + β )(1 + λ a) + kon [β + (1 + λ a)α]
that is,
(7A.54)
Note that in the limit β → 0 (gate always open), we recover the classical result
k∞ =
kD kon
.
kD + kon
Supplementary references
1. Benichou, O., Moreau, M., Oshanin, G.: Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories. Phys. Rev. E 61, 3388-3406 (2000).
2. Berezhkovskii, A. M., Yang, D.-Y., Sheu, S.-Y., Lin, S. H.: Stochastic gating in diffusioninfluenced ligand binding to proteins: Gated protein versus gated ligands. Phys. Rev. E 54,
4462-4464 (1996).
3. Berezhkovskii, A. M., Yang, D.-Y., Lin, S. H., Makhnovskii, Yu. A., Sheu, S.-Y.:
Smoluchowski-type theory of stochastically gated diffusion-influenced reactions. J. Chem.
Phys. 106, 6985 (1997).
4. Bressloff, P. C., Lawley, S. D.: Moment equations for a piecewise deterministic PDE J. Phys. A
48 105001 (2015).
5. Bressloff, P. C., Lawley, S. D.: Escape from a potential well with a randomly switching boundary. J. Phys. A 48 225001 (2015).
6. Bressloff, P. C., Lawley, S. D.: Escape from subcellular domains with randomly switching
boundaries. Multi-scale Model. Simul. 13 1420-1445 (2015)
7. Lawley, S. D., Mattingly, J. C., Reed, M. C. Stochastic switching in infinite dimensions with
applications to random parabolic PDEs. (2015).
8. Makhnovskii, Yu. A., Berezhkovskii, A. M., Sheu, S.-Y., Yang, D.-Y., Kuo, J., Lin, S. H.:
Stochastic gating influence on the kinetics of diffusion-limited reactions. J. Chem. Phys. 108,
971-983 (1998).
9. Spouge, J. L., Szabo, A., Weiss, G. H.: Single-particle survival in gated trapping. Phys. Rev. E
54, 2248-2255 (1996).
10. Szabo, A., Shoup, D., Northrup, S. H., McCammon, J. A.: Stochastically gated diffusion?influenced reactions. J. Chem. Phys. 77, 4484-4493 (1982).
11. Zhou, H.-X., Szabo, A.: Theory and Simulation of Stochastically-Gated Diffusion-Influenced
Reactions. J. Phys. Chem. 100, 2597-2604 (1996).
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