Article pubs.acs.org/JPCB
Diffusing Diffusivity: Survival in a Crowded Rearranging and Bounded Domain Rohit Jain and Kizhakeyil L. Sebastian* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India ABSTRACT: We consider a particle diffusing in a bounded, crowded, rearranging medium. The rearrangement happens on a time scale longer than the typical time scale of diffusion of the particle; as a result, effectively, the diffusion coefficient of the particle varies as a stochastic function of time. What is the probability that the particle will survive within the bounded region, given that it is absorbed the first time it hits the boundary of the region in which it diffuses? This question is of great interest in a variety of chemical and biological problems. If the diffusion coefficient is a constant, then analytical solutions for a variety of cases are available in the literature. However, there is no solution available for the case in which the diffusion coefficient is a random function of time. We discuss a class of models for which it is possible to find analytical solutions to the problem. We illustrate the method for a circular, two-dimensional region, but our methods are easy to apply to diffusion in arbitrary dimensions and spherical/rectangular regions. Our solution shows that if the dimension of the region is large, then only the average value of the diffusion coefficient determines the survival probability. However, for smaller-sized regions, one would be able to see the effects of the stochasticity of the diffusion coefficient. We also give generalizations of the results to N dimensions.
■
INTRODUCTION It is a well-known fact that many chemical reactions are diffusion-limited;1,2 that is, the rate of reaction is governed by the rate of diffusion of the reacting particles. This becomes even more important when one has reactions in confined geometries involving a small number of particles. In many biological systems, such as cells or cellular compartments, many important processes occur at the boundaries, viz., gene transcription,3,4 interneuron translation,5,6 regulation of calcium ions inside cells,7 membrane protein diffusion,8 and so forth. The first step in determining the kinetics of such processes is to calculate the probability distribution at the time at which the particle arrives at the boundary for the first time. Known as the first-passage time, this can be calculated directly from the survival probability of a particle starting within the region and being absorbed at the boundary as soon as it arrives there. Firstpassage times and survival probabilities have been calculated for a variety of cases and have found applications in different areas, which have been discussed in the books by Redner9 and Risken.10 See also the book by Bagchi,11 which stresses on the chemical applications. The same kind of mathematical problem is discussed by Carlslaw and Jaeger in the context of heat conduction.12 In the cases in which analytical results are available, one has always considered a homogeneous medium in which diffusion takes place, as a result of which diffusion coefficient D is a constant, characteristic of the medium and particle. However, biological systems are far from uniform, and the heterogenous nature of these systems produces a wide scatter in the values of diffusion coefficient D.13,14 In the fluorescence photobleach recovery and single particle tracking (SPT) experiments performed on biological membranes, the © 2016 American Chemical Society
observed scatter in diffusivity is found to be much larger than the expected experimental error. Moreover, with a larger heterogeneity, the distribution of D values becomes broader. Particle diffusion in confined heterogeneous systems has been studied in SPT experiments on cell membranes,8,15 and the effects of macromolecular crowding and confinment13,14,16,17 have been analyzed. Benichou et al.18 argued that geometry plays a significant role in diffusion-controlled kinetics. In particular, the distance between the reacting species becomes a key parameter in crowded and confined systems. Computer simulations, performed for diffusion of a tracer particle in crowded environments, also suggest a deviation from an otherwise normal diffusion.8,19,20 In a recent article, Metzler et al. have analyzed a spatially varying diffusivity model. By simulating a two-dimensional (2D) Langevin equation, they found that diffusion along the radial direction is anomalous.20 These results, in spite of being interesting and giving a better insight into the problem, do not provide a simple analytical expression for survival probability. Most of the work done on this topic is computational,8,19,20 and there does not seem to be any model that provides an analytical solution for the firstpassage time/survival probability, although diffusion in a confined, heterogeneous medium has been a topic of considerable experimental interest in the recent past. In the present work, we consider the “diffusing diffusivity model”21−23 (the term was first coined by Slater and Chubynsky22) and apply it to the problem of diffusion in a Received: June 16, 2016 Revised: July 27, 2016 Published: August 1, 2016 9215
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B ∞
heterogenous confined system. For a class of diffusing diffusivity models, we are able to calculate the survival probability analytically and analyze the effects of heterogeneity on the survival probability in a bounded region. In the next section, we outline very briefly the results for the survival probability for diffusion in a 2D circular homogeneous region. These results are quite well known8 and are useful as a reference as well as introduce our notation. In the next section, we introduce our model and give an exact answer for the survival probability, and the results are presented in the Result section. Our approach, although specifically illustrated for diffusion in two dimensions, is easily extended to diffusion in arbitrary dimensions. For details of particle diffusion in Ndimensional regions with a spherical symmetry, see Appendix B.
m =−∞ n = 1
e
2 −TDkmn
⟨m , n|r0 , ϕ0⟩
This may be written in the usual notation as ∞
J|m|(k |m| nr )J|m|(k |m| nr0) πR2J|m2 |+ 1(k |m| nR )
m =−∞ n = 1 2
e−TDk|m|n e im(ϕ − ϕ0)
(3)
Survival probability :free(r0 , ϕ0 , T ), given that the particle was initially at (r0, ϕ0), is given by
∫0
:free(r0 , ϕ0 , T ) =
DIFFUSION IN A CIRCULAR REGION Free Diffusion in a Homogeneous Medium. Consider a free particle undergoing Brownian motion in two dimensions. The probability, P(r, ϕ, t), of finding the particle at the point having polar co-ordinates, (r, ϕ), at time t obeys the diffusion equation
2π
∫0
R
r dr P(r , ϕ , T |r0 , ϕ0)
dϕ
(4)
Integrations over r and ϕ are easily done to get ⎛ 2 ⎞ J0 (k 0nr0) −k 2 DT e n0 ⎟ k R J (k R ) n = 1 ⎝ 0 n ⎠ 1 0n ∞
∑⎜
:free(r0 , ϕ0 , T ) =
(5)
The survival probability for a particle, starting with an initial distribution uniformly spread in the disk r ≤ R, would then be
⎛1 ∂ ⎛ ∂ ⎞ ∂P(r , ϕ , t ) 1 ∂2 ⎞ ⎜r ⎟ + = D⎜ ⎟P(r , ϕ , t ) ∂t ⎝ r ∂r ⎝ ∂r ⎠ r 2 ∂ϕ2 ⎠
1 πR2
Sfree(R , T ) = (1)
∫0
2π
dϕ0
∫0
R
r0 dr0 :free(r0 , ϕ0 , T )
This gives ∞
In the following, it is useful to use the operator notations of quantum mechanics and write the above also as
Sfree(R , T ) =
∑ n=1
4 −γ02nDT / R2 e γ02n
(6)
2
We note that DT/R is a dimensionless quantity; therefore, defining the reduced variable, τ = DT/R2, one can rewrite the survival probability as function of τ as
(2)
where the ket vector, |P(t)⟩, is defined by ⟨r, ϕ|P(t)⟩ = P(r, ϕ, t). Operator 3̂ is defined by eqs 1 and 2. At the initial time, t = 0, we assume the particle to be at (r0, ϕ0). |r, ϕ⟩ is the position ket vector, with the particle at position (r, ϕ), and is normalized 1 according to ⟨r , ϕ|r0 , ϕ0⟩ = r δ(r − r0)δ(ϕ − ϕ0). Obviously,
∞
̃ (τ ) = Sfree
∑ n=1
4 −γ02nτ e γ02n
(7)
Note that we use “tilde” for quantities expressed in dimensionless variables. The first few values of γ0n are 2.40283, 5.52008, 8.65373, 11.7915, .... Thus, in general, S(τ) has a multiexponential decay, and in the asymptotic limit of a large τ, the first eigenmode is the most important one. Hence, in the large τ limit (i.e., τ ≫ 1), we have
0
2π ∫∞ 0 r dr∫ 0 dϕ ⟨r, ϕ|r0, ϕ0⟩ = 1. To find the survival probability, we need to solve eq 1, subject to the absorbing boundary condition P(R, ϕ, t) = 0. For this, one can use the eigenfunctions of operator 3̂ , satisfying the same boundary condition. These eigenfunctions are
⟨r , ϕ|m , n⟩ =
∞
∑ ∑
P(r , ϕ , T |r0 , ϕ0) =
■
∂|P(t )⟩ = −D 3̂ |P(t )⟩ ∂t
∞
∑ ∑ ⟨r , ϕ|m , n⟩
P(r , ϕ , T |r0 , ϕ0) =
̃ (τ ) ∼ Sfree
e imϕ RJ|m|+ 1(k |m| nR ) π J|m|(k |m| nr )
2 2 4 −γ012τ 4 e = 2 e−γ01DT / R 2 γ01 γ01
(8)
For short times (i.e., τ ≪ 1), the sum can be evaluated (see Appendix A) to get
where m = 0, ±1, ±2, ... and J|m| is the Bessel function of the first kind, of order |m|. k|m|n = γ|m|n/R, where γ|m|n denotes the nth (=1, 2, 3, ...) root of the Bessel function, satisfying J|m|(γ|m|n) = 0. The eigenvalue of 3̂ corresponding to this eigenfunction is k2mn. |m, n⟩ are normalized and form a complete set. Formally, the probability of finding the particle at (r, ϕ) at time t = T, given that it started at (r0, ϕ0) at time t = 0, is given by
̃ (τ ) ≈ 1 − Sfree
4 1/2 1 3/2 1 2 τ +τ+ τ + τ + ··· π 3 π 8 (9)
This expression is in good agreement with the exact result given in eq 7 at small values of τ (see Figure 2a). Diffusing Diffusivity. Now, we consider exactly the same problem as earlier, but the diffusion happens in a crowded but rearranging medium. This makes the diffusion coefficient change in a random manner as time passes. A cartoon representation of the situation is given in Figure 1. A drunken (random) walker is walking on top of a steep cylinderical hill and will fall to his death if he hits the periphery of the circle.
̂
P(r , ϕ , T |r0 , ϕ0) = ⟨r , ϕ|e−3T |r0 , ϕ0⟩
Using the completeness relation Σm,n|m, n⟩⟨m, n| = 1, we can write 9216
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B
realizations of it. We denote this averaging by ⟨. . . ⟩+ . This leads to ⟨P(r , ϕ , T )⟩+ =
T dt 0
∑ ⟨r , ϕ|m , n⟩⟨e− ∫
2 D(t )kmn
⟩+
n,m
⟨m , n|P(0)⟩
D(t), being the diffusion coefficient, can have only positive T
2
values. Calculation of the average, ⟨e− ∫0 dt D(t )kmn⟩+ , is easily performed for the model that we introduced in the earlier paper.23 Following that paper, we put D(t ) = ξ12(t ) + ξ22(t )
where ξ1(t) and ξ2(t) are independent, identically distributed random functions of t. We assume (ξ1(t), ξ2(t)) to be the position vector of a 2D Brownian oscillator having frequency ω and “diffusion coefficient” F. Therefore, ξi(t) obeys the Langevin equation
Figure 1. The walker will fall to his death if he hits the boundary of the top region. What is the probability of him surviving for time T?
ξi(̇ t ) + ωξi(t ) = ηi(t )
The top of the hill has obstacles, which keep on rearranging as time passes. What is the probability of his survival for time T? One of the models that has been suggested for this is the diffusing diffusivity model.21,22,24 We have recently given an analytic solution to one such model, in which we calculated the spatial probability distribution for a particle diffusing into a heterogeneous rearranging environment.23 In the model, the diffusion coefficient is taken to be a random function of time, changing on a slow time scale in comparison to the time scale of observation of the diffusional motion of the particle. In such a case, the diffusion equation (eq 2) becomes
where ηi(t) is Gaussian white noise, having correlation function ⟨ηi(t) ηj(t′)⟩= 2Fδijδ(t − t′). This means that the diffusion coefficient is a random function, whose probability distribution function, π(D, t), obeys the Fokker−Planck equation23 ∂π (D , t ) ∂ ∂ = 4FD + 2ωD π (D , t ) ∂t ∂D ∂D
{
with D0 =
dt D(t )3̂
|P(0)⟩
∑ ⟨r , ϕ|m , n⟩ e
2F . ω
1 D0
e−D / D0 ,
Assuming the initial distribution to be the T
Using the complete set of states, |m, n⟩, we get P(r , ϕ , T ) =
(11)
2
equilibrium one, the average, ⟨e− ∫0 dt D(t )kmn⟩+ , can be calculated using phase space path integration.23,25,26 Alternately, one can evaluate it on the basis of the fact that the average may be written as
Formally, the solution of this is T
}
This equation has the equilibrium solution πe(D) =
∂|P(t )⟩ = −D(t )3̂ |P(t )⟩ ∂t
|P(T )⟩ = e− ∫0
(10)
T
⟨e− ∫0
T
2 − ∫0 dt D(t )kmn
⟨m , n|P(0)⟩
2 dt D(t )kmn
⟩+ =
∫ dDf ∫ dDiπa(Df , T |Di , 0)πe(Di) (12)
n,m
where πa(D, T|Di, 0) is the probability of having diffusion coefficient D at time T, given that the diffusion coefficient was Di at time t = 0. It obeys the reaction diffusion equation
Here, kmn is a root of the Bessel equation, Jm(kmnR) = 0. As D(t) is a random function of time, one has to average over all
Figure 2. Comparisons of the exact vs approximate expressions for survival probability at short times (see main text) (a) for the constant diffusivity case, exact (eq 7, solid line) and approximate (eq 9, dashed line), and (b) for the diffusing diffusivity case, exact (eq 23, solid line) and approximate (eq 24, dashed line). 9217
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B
Figure 3. Comparisons of the survival probability of normal diffusion (eq 7) with that in the case of diffusing diffusivity, obtained using eq 17: (a) linear plots suggesting that with increasing ρ the survival probability for diffusing diffusivity approaches that for normal diffusion, with the average diffusion coefficient and (b) log-linear plots indicating that the decay of survival probability remains exponential in τ for large values of τ; however, the rate of decay depends on parameter ρ.
■
∂πa(D , t |Di , 0) ⎧ ∂ ⎛ ⎞ ∂ = ⎨ ⎜4FD + 2ωD⎟ ⎝ ⎠ ⎩ ∂D ∂t ∂D 2 − kmn D}πa(D , t |Di , 0)
RESULTS Survival probability :diff (R , T , r0 , ϕ0), given that the particle started at (r0, ϕ0) at t = 0, is given by
(13)
:diff (R , T , r0 , ϕ0) =
On solving eq 13 and using it in eq 12, we get (see the previous paper23 for more details) ⟨e
:diff (R , T , r0 , ϕ0) =
with
=
⎛ ⎜ 1+ ⎝
ω2R2
e
− ωT ( 1 +
⎞ ⎛ + 1⎟ − ⎜ 1 + ⎠ ⎝ 2
2 4Fγmn
ω2R2
2 4Fγmn
ω2R2
2 4Fγmn ω2R2
− 1)
∞
⎞ −2ωT − 1⎟ e ⎠ 2
Sdiff (R , T ) =
4Fγ 2 1 + 2mn2 ω R
(n /2 − 1)
( )( )
D0 Γ
n 2
D D0
e−D / D0 , which is a generalized
gamma distribution that is easy to analyze. The results are somewhat different. For example, if n = 4, the short-time distribution would be
e−|x| /
D0T (|x|
+ D0T )
4D0T
dϕ r⟨P(r , ϕ , T )⟩+
⎛ 2 ⎞ J0 (k 0nr0) f (R , T ) ⎟ k R J ( k R ) 0n n = 1 ⎝ n ⎠ 1 0n
, which is different from
∞
̃ (ρ , τ ) = Sdiff
24
the experimentally observed exponential distribution. Hence, we shall not discuss this case further. Other forms of π(D, t) can be taken care of by generalizing eqs 11 and 12. If one had a π(D, t) that obeys ∂π (D , t ) = -̂ π (D , t ), where -̂ is any Fokker−Planck operator, ∂t which describes the diffusive motion of D in the region D > 0, ∂π (D , t ) then eq 11 needs to be modified to a = -̂ π (D , t ), ∂t
∑⎜
(15)
4 f (R , T ) γ02n 0n
(16)
We now introduce the dimensionless quantities defined as 2F T ωR τ = ω 2 and ρ = 2F . Notice that the average equilibrium R diffusivity value is D0 = 2F/ω and hence variable τ is similar to the one defined in the normal diffusion case. Physically, τ is the ratio of the mean-square displacement of the particle (=D0T) to the square of the size (=R2) of the region, whereas ρ is the ratio of the size of the region (=R) to the average displacement of the particle during the time of relaxation of the medium (=ω−1). In terms of these new variables, the survival probability is given by
for m = 0, 1, 2, .... It may be noted that taking D(t) = ∑in= 1ξ2i (t), which is a generalization of eq 10, one would obtain equilibrium distribution function 1
∑ n=1
(14)
πe(D) =
2π
As before, we consider the initial distribution to be the one in which the particle is distributed uniformly in the region r ≤ R. For this, we get
fmn (R , T ) 2 4Fγmn
∫0
dr
∞
⟩+ = fmn (R , T )
4 1+
R
In this, only the m = 0 mode survives. Hence
T
2 − ∫0 dt D(t )kmn
∫0
∑ n=1
4 ̃ f (ρ , τ ) γ02n 0n
(17)
with f0̃ n (ρ , τ ) 4 1+ =
a a
2 where -̂a = -̂ − kmn D . For any diffusional process that attains a steady final state, the eigenvalues of -̂ and hence those of -̂a are generally discrete. Hence, πa(Df, T|Di, 0) can be written as a sum of exponentials in t, leading to the conclusion that time decay will generally be exponential.
⎛ ⎜ 1+ ⎝
2γ02n ρ2
2γ02n ρ2
e
−ρ 2 τ ( 1 +
⎞2 ⎛ + 1⎟ − ⎜ 1 + ⎠ ⎝
2γ02n ρ2
2γ02n ρ2
− 1)
⎞ 2 − 2ρ 2 τ − 1⎟ e ⎠
1+
2γ02n ρ2
(18)
We now consider the limiting cases, for which simple approximations can be obtained. 9218
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B Radius of the Region Is Large. In the limit in which both ρ2 (i.e., 2γ20n/ρ2 ≪ 1) and ρ2τ are very large, or in terms of 2F ω2
dimensioned variables R ≫
(=
D0 ω
) and T ≫ ω
−1
be noticed that this expression is different from the similar expression in eq 9. The above results show that by carefully analyzing the shortand long-term dependences of survival probabilities, one can, at least in principle, say whether the diffusivity is diffusing. The results obtained here can easily be extended to the case in which a particle is diffusing in N-dimensional space with a spherical symmetry. For details, refer to Appendix B.
, we
have 2
2
2
f0̃ n (ρ , τ ) ≃ e−γ0nτ = e−γ0nD0T / R
(19)
■
Therefore, for large values of ρ (i.e., 2γ20n/ρ2 ≪ 1), one expects that S̃diff(ρ, τ) will approach S̃free(τ), that is, ̃ (ρ , τ ) = Sfree ̃ (τ ) lim Sdiff
(20)
ρ→∞
Further, as the size of the region increases, the values of τ at which the agreement begins will become smaller. In Figure 3a, we have plotted survival probability as a function of variable τ for different values of ρ. We see that as we increase the value of ̃ (ρ, τ) approaches that for S̃free(τ). ρ the curve for Sdiff We also observe that for large values of τ, the n = 1 term is the most significant. In particular, in the long time limit, the decay of survival probability is ̃ (ρ , τ ) ≈ Sdiff
2 2 4 −γ012τ 4 e = 2 e−γ01D0T / R 2 γ01 γ01
ω2R2
( ρ2 = 2F ), which also implies a larger compartment size and faster relaxation of the surroundings, the process of diffusion in a crowded, rearranging environment becomes similar to the normal diffusion process. This behavior is expected, as with larger regions to explore the diffusing particle would have sampled different diffusive regimes, thereby attaining an equilibrium value for the diffusion coefficient before eventually being absorbed at the periphery. On the other hand, for a smaller value of ρ, the particle may not have sampled regions with different diffusivities and therefore the problem of diffusing diffusivity is qualitatively different from that of the normal diffusional process.
(21)
This is exactly the same as that for normal diffusion (see eq 8) but with diffusion coefficient D being replaced by the average diffusion coefficient, D0 = 2F/ω. Radius of the Region Is Small, That Is, ρ ≪ 1. In the long time limit, τ ≫ 1, we get ̃ (ρ , τ ) ≈ Sdiff
8 2ρ − e γ013
2 γ01ρτ
=
8ωR −2 e F γ013
■
APPENDIX A Here, we consider the problem of calculating the sum in eq 9. We consider the sum, U(τ), defined by U(τ) = ∑n(4/(γ20n)) 2 e−γ0nτ. Its Laplace transform
F / Rγ01T
(22)
Thus, the survival probability, although decaying exponentially in T at large values of τ, has different R dependences in the small (eq 21) and large ρ (eq 22) limits. This fact becomes more clear in the log-linear plots of Sdiff(ρ, τ) plotted against τ for different values of ρ (see Figure 3b). With ρ ≪ 1 and τ ≪ 1, we have lim f0̃ n (ρ , τ ) =
ρ→ 0
U̅ (p) =
∞
∑ n=1
∑ n
for τ ≪ 1
d τ e − pτ U ( τ ) =
∑ n
4 γ02n(γ02n + p)
(25)
(γ02n
J1(i p ) I1( p ) 1 = = J ( i p )2 i p I ( p )2 p + p) 0 0
(26)
where Iν(z) is the modified Bessel function of order m. Taking the p → 0 limit of eq 26, we get
(23)
This sum can be evaluated exactly for small values of τ, that is, τ ≪ 1 (for the evaluation, one may use eq 27 of Appendix A) to give ̃ (ρ → 0, τ ) ∼ 1 − 2τ1/2 + τ + Sdiff
∞
Using eq 3 on page 61 of volume II of the book by Erdelyi et al.27 and putting z = i√p in that equation, we get
1 1 + τγ02n
4 1 γ02n 1 + τγ02n
∫0
⎛ 1 ⎞ 4 1 ⎟⎟ = ∑ ⎜⎜ 2 − 2 p n ⎝ γ0n (γ0n + p) ⎠
The survival probability in this limit becomes ̃ (ρ , τ ) = Sdiff
CONCLUSIONS
We have applied the analytically solvable model23 for diffusing diffusivity of Granick et al.21,24 and Slater et al.22 to the problem of diffusion in confinement. We show that for larger values of ρ
∑
1 3/2 1 2 τ + τ + ··· 4 4
1 1 = 2 4 γ0n
which may be used in eq 25. We are interested in the small τ behavior of U(τ). This means that we have to find the large p behavior of U(p). For this, we use the asymptotic behavior of Iν(z), given as equation 9.7.1 in the handbook by Abramowitz and Stegun,28 to get
(24)
The expression presented in the above equation requires a little bit of explanation. At short times (τ ≪ 1), variation of the diffusion coefficient with time is not important but its distribution is. Thus, averaging eq 9 over all possible values of D (with the average diffusion coefficient given by D0 = 2F/ ω) leads to the expression in eq 24. This expression for survival probability at small values of τ is in good agreement with the exact summation given in eq 23, as shown in Figure 2b. It may
∑ n
(γ02n
1 1 1 1 1 ∼ 1/2 − − − + ··· 3/2 4p + p) 16p2 2p 16p
for large p
(27)
Using this in eq 25 and taking the inverse transform, we get 9219
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B 4 1/2 1 3/2 1 2 τ +τ+ τ + τ + ··· π 3 π 8
U(τ ) ≈ 1 −
■
2 L̂N Yp(Ω) = −pYp(Ω)
where p = l(N + l − 2), with l = 0, 1, 2, ..., and functions Yp(Ω) are known as hyperspherical harmonics. The survival probability for a given initial position (r0, Ω0) is given by
(28)
for τ ≪ 1
(34)
APPENDIX B : GENERALIZATION TO N DIMENSIONS
:(r0 , Ω 0 , T ) =
One Dimension
The diffusion equation is
∫0
R
∫ dΩ P(r , Ω, T |r0 , Ω0, 0)
r N − 1 dr
(35)
2
∂ ∂ P(r , t ) = D 2 P(r , t ) ∂t ∂r
After performing integration over angular variables, the only modes surviving correspond to p = 0. Therefore, we get
(29)
The solution to this equation, with the absorbing boundaries at r = ±R and under the initial condition, P(r, t = 0) = δ(r − r0), is
:(r0 , Ω 0 , T ) =
⎡ ⎛ nπr ⎞ ⎛ nπr ⎞ 1 ⎟ P(r , T |r0 , 0) = ∑ ⎢sin⎜ 0 ⎟ sin⎜ R n=1 ⎣ ⎝ R ⎠ ⎝ R ⎠
∞
2
N /2 − 1 JN /2 (λ 0nR ) n = 1 λ 0nr0
R
∞
(36)
⎛ (2n − 1)πr ⎞ −(2n − 1)2 π 2DT /4R2⎤ ⎥ cos⎜ ⎟e ⎝ ⎠ 2R ⎦
S(R , T ) =
:(r0 , T ) =
n+1
∞
∞
S(τ ) =
(31)
Here, is a partial differential operator on unit sphere SN−1, involving angle co-ordinates Ω = (θ1, θ2, ..., θN−2, ϕ). The solution of this equation, analogous to eq 3, is
The derivation of this result is simple and follows the method given in Appendix A. Note that for odd N the series terminates after a finite number of terms. We illustrate this for (a) N = 1 and (b) N = 3. For N = 1, γ0n = (2n − 1) π/2; therefore, the 1 1 Laplace transform of eq 39 gives S ̅ (p) = p − 3/2 tanh(p1/2 ),
J1/2
Y *p (Ω 0)
p
which is exact. The large p behavior of this function is 1 1 S ̅ (p) ∼ p − 3/2 . Therefore, the small τ behavior for the p
1
2
J2 (λ R ) 1/2 (N − 2)2 + 4p + 1 pn 2
(N − 2)
(λ r ) + 4p pn
r N /2 − 1
Yp(Ω)
(40)
+ ···
P(r , Ω, T |r0 , Ω 0 , 0)
e−λpnDT
(39)
N (N − 1) 2N 1/2 τ + τ π 2 N (N − 1)(N − 3) 3/2 N (N − 1)(N − 3) 2 − τ − τ 6 π 16
(32)
(λpnr0)
2N −γn2τ e γn2
S (τ ) ∼ 1 −
L̂ 2N
r0N /2 − 1
(38)
where γn = λ0nR and is the root of the Bessel equation JN/2−1(γn) = 0. The survival probability for small values of τ is
⎧ 1 ∂ ⎛ N−1 ∂ ⎞ 1 ̂ 2⎫ ∂ ⎜r ⎟ + P(r , Ω, t ) = D⎨ N − 1 LN ⎬ ⎝ ⎠ ⎩ ∂t ∂r ∂r r2 ⎭ r
(N − 2)2 + 4p
∑ n=1
Here, λ0n is a solution of the equation J−1/2(λ0nR) = 0. N Dimensions. The diffusion equation in N (≥2) dimensions is29
∞ J 1/2 2 ∑ ∑ R2 p n = 1
2N 1 −λ02nDT e 2 ∑ R n = 1 λ 02n
with λ0n being a solution of the equation JN/2−1(λ0nR) = 0. Defining the dimensionless parameter as τ = DT/R2, we can rewrite the survival probability as
This can be rearranged in the following form
=
∫ dΩ0 :(r0 , Ω0, T )
r0N − 1 dr0
∞
S(R , T ) =
⎛ (2n − 1)πr0 ⎞ −(2n − 1)2 π 2DT /4R2 cos⎜ ⎟e ⎝ ⎠ 2R
P(r , Ω, t )
R
(37)
∫−R dr P(r , T |r0 , 0) = 2∑ (2n(−−1)1)π /2
J−1/2 (λ 0nr0) 2 2 e−λ0nDT −1/2 3/2 ∑ R n = 1 λ 0nr0 J1/2 (λ 0nR )
∫0
Here, VN(R) is the volume of a hypersphere of radius R, and integration over Ω0 gives the surface area of an N-dimensional hypersphere of unit radius. Finally, the survival probability is given as
n=1
:(r0 , T ) =
1 VN (R )
(30)
The survival probability, given that the particle started out at initial position r0, is given by ∞
2
e−λ0nDT
On comparing with eq 31, one realizes that this is valid even for N = 1. For a uniformly distributed initial position, the survival probability becomes
⎛ (2n − 1)πr0 ⎞ 2 2 2 e−n π DT / R + cos⎜ ⎟ ⎝ ⎠ 2R
R
JN /2 − 1(λ 0nr0)
−N /2 + 2 ∑
survival probability in one dimension becomes
S(τ ) ∼ 1 − (33)
2 1/2 τ π
Likewise, for the case of N = 3, γ0n = nπ and the Laplace 1 3 3 transform of eq 39 is S ̅ (p) = p − 3/2 coth(p1/2 ) + p2 .
where Yp(Ω) is the eigenfunction for angular operator L̂ 2N, with eigenvalue −p. That is,
p
9220
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B
Figure 4. Comparison of survival probabilties between the case of diffusing diffusivity and the case of normal diffusion in (a) one dimension (N = 1) and (b) three dimensions (N = 3). As the value of parameter ρ increases, the survival probabilty for diffusing diffusivity approaches that for normal diffusion. Moreover, for a larger value of N, survival is short-lived.
Taking the inverse transform of this expression in the large p limit gives the small τ behavior, that is,
These results are in agreement with eq 40. In the case of diffusing diffusivity, we can write the survival probability as ∞
∑ n=1
2N ̃ f (ρ , τ ) γ02n 0n
f0̃ n (ρ , τ )
=
⎛ ⎜ 1+ ⎝
2γ02n ρ2
2γ02n ρ
2
e
−ρ 2 τ ( 1 +
⎞2 ⎛ + 1⎟ − ⎜ 1 + ⎠ ⎝
2γ02n ρ2
2γ02n ρ2
− 1)
⎞ 2 − 2ρ 2 τ − 1⎟ e ⎠
1+
2γ02n ρ2
(42)
In this case, the short term behavior of survival probability is 1 N (N − 1)τ 2 1 1 − N (N − 1)(N − 3)τ 3/2 − N (N − 1) 8 8
̃ (ρ → 0, τ ) ∼ 1 − Nτ1/2 + Sdiff
(N − 3)τ 2 + ···
(43)
Effect of Dimensionality on S̃diff(ρ,τ). Figure 4 shows the plots of survival probability for two different values of N. With an increase in the value of N, the survival of a particle becomes shorter and shorter. As we go toward higher dimensions, the surface-to-volume ratio increases and therefore it becomes easier for the particle to find the absorbing surface, and survival is short-lived.
■
■
REFERENCES
(1) Rice, S. A. Diffusion-Limited Reactions; Elsevier: Amsterdam, The Netherlands, 1985. (2) Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J. Strange Kinetics. Nature 1993, 363, 31−37. (3) Yu, J.; Xiao, J.; Ren, X.; Lao, K.; Xie, X. S. Probing Gene Expression in Live Cells, One Protein Molecule at a Time. Science 2006, 311, 1600−1603. (4) Elf, J.; Li, G.-W.; Xie, X. S. Probing Transcription Factor Dynamics at the Single-Molecule Level in a Living Cell. Science 2007, 316, 1191−1194. (5) Bredt, D. S.; Nicoll, R. A. AMPA Receptor Trafficking at Excitatory Synapses. Neuron 2003, 40, 361−379. (6) Triller, A.; Choquet, D. Surface Trafficking of Receptors between Synaptic and Extrasynaptic Membranes: And Yet They Do Move! Trends Neurosci. 2005, 28, 133−139. (7) Holcman, D.; Marchewka, A.; Schuss, Z. Survival Probability of Diffusion with Trapping in Cellular Neurobiology. Phys. Rev. E 2005, 72, No. 031910. (8) Saxton, M. J. Lateral Diffusion in an Archipelago. Single-Particle Diffusion. Biophys. J. 1993, 64, 1766−1780. (9) Redner, S. A Guide to First-Passage Processes; Cambridge University Press: Cambridge, UK, 2001. (10) Risken, H. The Fokker-Planck Equation, 2nd ed.; SpringerVerlag: Berlin, 1989. (11) Bagchi, B. Molecular Relaxation in Liquids; Oxford University Press: USA, 2012. (12) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Oxford University Press: London, 1959. (13) Saxton, M. J. Lateral Diffusion in an Archipelago. The Effect of Mobile Obstacles. Biophys. J. 1987, 52, 989−997. (14) Saxton, M. J. Single-Particle Tracking: The Distribution of Diffusion Coefficients. Biophys. J. 1997, 72, 1744−1753. (15) Saxton, M. J.; Jacobson, K. Single-Particle Tracking: Applications to Membrane Dynamics. Annu. Rev. Biophys. Biomol. Struct. 1997, 26, 373−399. (16) Banks, D. S.; Fradin, C. Anomalous Diffusion of Proteins Due to Molecular Crowding. Biophys. J. 2005, 89, 2960−2971.
(41)
with
4 1+
ACKNOWLEDGMENTS
We thank Gary Slater, Ralf Metzler, and Arindam Chowdhury for useful discussions. We also thank the anonymous referee for the suggestions that inspired us to carry out calculations in N dimensions. The work of K.L.S. was supported by the J C Bose Fellowship of the Department of Science and Technology, Government of India.
6 1/2 S(τ ) ∼ 1 − τ + 3τ π
̃ (ρ , τ ) = Sdiff
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +91 8022932385. Notes
The authors declare no competing financial interest. 9221
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
Article
The Journal of Physical Chemistry B (17) Akabayov, B.; Akabayov, S. R.; Lee, S.-J.; Wagner, G.; Richardson, C. C. Impact of Macromolecular Crowding on DNA Replication. Nat. Commun. 2013, 4, No. 1615. (18) Bénichou, O.; Chevalier, C.; Klafter, J.; Meyer, B.; Voituriez, R. Geometry-Controlled Kinetics. Nat. Chem. 2010, 2, 472−477. (19) Chakrabarti, R.; Kesselheim, S.; Košovan, P.; Holm, C. Tracer Diffusion in a Crowded Cylindrical Channel. Phys. Rev. E 2013, 87, No. 062709. (20) Cherstvy, A. G.; Chechkin, A. V.; Metzler, R. Particle Invasion, Survival, and Non-Ergodicity in 2D Diffusion Processes with SpaceDependent Diffusivity. Soft Matter 2014, 10, 1591−1601. (21) Wang, B.; Kuo, J.; Bae, S. C.; Granick, S. When Brownian Diffusion is not Gaussian. Nat. Mater. 2012, 11, 481−485. (22) Chubynsky, M. V.; Slater, G. W. Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion. Phys. Rev. Lett. 2014, 113, 1− 5. (23) Jain, R.; Sebastian, K. L. Diffusion in a Crowded, Rearranging Environment. J. Phys. Chem. B 2016, 120, 3988−3992. (24) Wang, B.; Anthony, S. M.; Bae, S. C.; Granick, S. Anomalous yet Brownian. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 15160−15164. (25) Janakiraman, D.; Sebastian, K. L. Path-Integral Formulation for Lévy Flights: Evaluation of the Propagator for Free, Linear, and Harmonic Potentials in the over- and underdamped Limits. Phys. Rev. E 2012, 86, No. 061105. (26) Kamenev, A. Field Theory of Nonequilibrium Systems; Cambridge University Press: New York, 2011. (27) Bateman, H.; Erdélyi, A. Higher Transcendental Functions-Vol.2; McGraw Hill Book Company, Inc.: New York, 1953. (28) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications, Inc.: New York, 1972. (29) Al-Jaber, S. M. Hydrogen Atom in N Dimensions. Int. J. Theor. Phys. 1998, 37, 1289−1298.
9222
DOI: 10.1021/acs.jpcb.6b06094 J. Phys. Chem. B 2016, 120, 9215−9222
