Narrow Escape Problems in Three-Dimensional DomainsDaniel Gomez and Alexey F. Shevyakov

Department of Mathematics and Statistics, University of Saskatchewan

Motivation/Application

Numerous biological processes involve the transport of particles from a cell through its membrane:

I RNA transport through nuclear pores.

I Passive diffusion of molecules (e.g. CO2 and O2) through cell membrane.

I Diffusion of ions through protein channels (e.g. Na-K-Cl co-transporter in blood cells).

Typical size of transport regions is ∼0.1% relative to overall cell size.Biological Cells

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Cell Membrane

Retrieved Aug. 13, 2013 from:http://library.thinkquest.org/C004535/cell membranes.html

Red Blood Cells

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The Narrow Escape Problem (NEP)

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NARROW ESCAPE FROM A SPHERE 837

the method of matched asymptotic expansions to study the narrow escape problemin a certain three-dimensional context.

In a three-dimensional bounded domain Ω, it is well known (cf. [19], [35], [38]) thatthe MFPT v(x) satisfies a Poisson equation with mixed Dirichlet–Neumann boundaryconditions, formulated as

v = − 1

D, x ∈ Ω ,(1.1a)

v = 0 , x ∈ ∂Ωa =

N⋃

j=1

∂Ωεj , j = 1, . . . , N ; ∂nv = 0 , x ∈ ∂Ωr .(1.1b)

Here D is the diffusivity of the underlying Brownian motion, and the absorbing setconsists of N small disjoint absorbing windows, or traps, ∂Ωεj for j = 1, . . . , N eachof area |∂Ωεj | = O(ε2). We assume that ∂Ωεj → xj as ε → 0 for j = 1, . . . , N andthat the traps are well separated in the sense that |xi −xj| = O(1) for all i = j. Withrespect to a uniform distribution of initial points x ∈ Ω for the Brownian walk, theaverage MFPT, denoted by v, is defined by

(1.2) v = χ ≡ 1

|Ω|

∫

Ω

v(x) dx ,

where |Ω| is the volume of Ω. The geometry of a confining sphere with traps on itsboundary is depicted in Figure 1.1.

Fig. 1.1. Sketch of a Brownian trajectory in the unit sphere in R3 with absorbing windows onthe boundary.

There are only a few results for the MFPT, defined by (1.1), for a bounded three-dimensional domain. For the case of one locally circular absorbing window of radius εon the boundary of the unit sphere, it was shown in [41] (with a correction as notedin [44]) that a two-term expansion for the average MFPT is given by

(1.3) v ∼ |Ω|4εD

[1 − ε

πlog ε + O (ε)

],

where |Ω| denotes the volume of the unit sphere. This result was derived in [41] byusing the Collins method for solving a certain pair of integral equations resulting froma separation of variables approach. A similar result for v was obtained in [41] for the

I Three-Dimensional domain Ω.

I Boundary traps: ∂Ωεj (j = 1, ..., N).

I Brownian motion.

I Mean first passage time (MFPT): v(x).

I Average MFPT: v ≡ 1

|Ω|

∫

Ωv(x) d3x.

I Dirichlet-Neumann Boundary Value Problem [3]:

∆v(x) = − 1

D, x ∈ Ω;

∂nv(x) = 0, x ∈ ∂Ω \⋃j ∂Ωεj ;

v(x) = 0, x ∈ ⋃j ∂Ωεj.

(1)

Asymptotic Solutions

I The problem (1) does not admit a known analytic solution.

I It is difficult to solve numerically due to its highly heterogeneous boundary conditions.

An asymptotic approximation is beneficial because it offers fast computation times, and gives properties ofthe solution that would otherwise be hidden by numerical data. One considers asymptotic expansion ofthe form

v(x) ∼ ε−1v0(x) + v1(x) + ε log

(ε

2

)v2(x) + εv3(x) + ....

where ε is the trap size parameter. Using the method of matched asymptotic expansions one then obtainsthe average MFPT of the form

v(x) = v +

N∑

j=1

kjGs(x;xj), (2)

where kj are particular constants. The function Gs(x;xj) is known as the surface Neumann Green’sfunction and it satisfies the boundary value problem

∆Gs(x;xj) =1

|Ω|, x ∈ Ω;

∂nGs(x;xj) = δs(x− xj), x ∈ ∂Ω;∫

ΩGs(x;xj)d3x = 0.

(3)

The Unit Sphere

When Ω is the unit sphere with N holes of radii εaj located at xj respectively, it was found in [1] that

v =|Ω|

2πεDNc

[1 + ε log

(2

ε

)∑Nj=1 c

2j

2Nc+

2πε

Ncpc(x1, ...,xN )− ε

Nc

N∑

j=1

cjκj + O(ε2 log ε)

]. (4)

In this expression pc(x1, ...,xN ) depends on the hole sizes and their relative distances, the constants κjdepend only on the hole radii aj, and the cj are known as the trap capacitance and are given by

cj =2ajπ. (5)

Such a high order asymptotic expansion for the average MFPT is made possible by the explicit knowledgeof the surface Neumann Green’s function for the sphere. Unfortunately this is not the case fornon-spherical domains.

Asymptotic and Numerical MFPT for Unit Sphere with Six Identical Traps

Average MFPT

0.005 0.01 0.015 0.025

10

15

20

25

30

35

epsilon

Ave

rage

MF

PT

(1/

s)

Numerical (exact)Asymptotic

MFPT (epsilon = 0.02) MFPT (epsilon = 0.02)

Singer, Schuss, and Holcman Approximation

For non-spherical domains the surface Neumann-Green’s function is not explicitly known. Nevertheless, forsuch a domain, Ω, with exactly one circular trap of radius ε centred at x0 the average MFPT is given in[4] by

v ≡ |Ω|4εD

[1 +

H(x0)

πε log ε + O(ε)

]−1

, (6)

where H(x0) is the mean curvature of ∂Ω at x0. While this formula approximates the average MFPT foran arbitrary domain, its limitations are:

I Only valid for one absorbing window.

I Error bound is O(ε) as opposed to O(ε2 log ε) found for sphere.

Oblate Spheroid with One Trap

Average MFPT

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600

epsilon

Ave

rage

MF

PT

(1/

s)

Numerical (exact)Singer, Schuss, and Holcman Approximation

MFPT (epsilon = 0.02)

Multi-Trap Generalization

Comparison between the Singer, Schuss, and Holcman approximation (6) and the asymptotic formula forthe sphere (4) suggests the introduction of a modified trap capacitance

cj =2ajH(xj)

π,

where H(xj) is the mean curvature at the jth trap location. With the modified trap capacitance theproposed multi-trap average MFPT approximation takes on the form

v =|Ω|

4εD∑Nj=1 aj

[1 +

ε

2log ε

∑Nj=1 c

2j∑N

j=1 cj+ O(ε)

]−1

. (7)

Testing Procedure and COMSOL

Used oblate spheroid, prolate spheroid, and biconcave diskgeometries.

I Provide range of local curvatures.

I Represent different biological cells.

COMSOL Multiphysics 4.3b software used for numerical results.

I Finite element PDE solver.

I Tetrahedral mesh.

Numerical results for two and three traps of equal and different sizescompared to proposed multi-trap approximation in MATLAB.

COMSOL Mesh Refinement Example

Oblate Spheroid with Three Traps Prolate Spheroid with Three Traps Biconcave Disk with Three Traps

Results for Three Different Sized Traps

Oblate Spheroid

Average MFPT

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

100

120

140

160

epsilon

Ave

rage

MF

PT

(1/

s)

Numerical (exact)Three Trap Approximation

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

35

40

epsilon

Rel

ativ

e E

rror

(%

)

MFPT (epsilon = 0.02)

Prolate Spheroid

Average MFPT

0 0.01 0.02 0.03 0.04 0.050

50

100

150

200

250

300

350

400

450

500

epsilon

Ave

rage

MF

PT

(1/

s)

Numerical (exact)Three Trap Approximation

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

epsilon

Rel

ativ

e E

rror

(%

)

MFPT (epsilon = 0.02)

Biconcave DiskAverage MFPT

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

100

120

epsilon

Ave

rage

MF

PT

(1/

s)

Numerical (exact)Three Trap Approximation

0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

epsilon

Rel

ativ

e E

rror

(%

)

MFPT (epsilon = 0.02)

Conclusions

I A multi-trap generalization of (6) was proposed and tested.

I Relative error remained below 10% for ε ≤ 0.01 for tested geometries (numerical error ∼ 1%).

Future Research

I Comparison of numerical simulation to proposed formula for greater number of traps on more variedgeometries.

I Asymptotic methods for narrow escape problems with non-constant diffusivity.

I Determination of surface Neumann Green’s function and derivation of higher-order asymptoticexpansions for non-spherical domains.

I Study of dilute trap limit of homogenization theory for non-spherical domains [2].

References

[1] Alexei F. Cheviakov, Michael J. Ward, and Ronny Straube.An asymptotic analysis of the mean first passage time for narrow escape problems. II. The sphere.Multiscale Model. Simul., 8(3):836–870, 2010.

[2] Cyrill B. Muratov and Stanislav Y. Shvartsman.Boundary homogenization for periodic arrays of absorbers.Multiscale Model. Simul., 7(1):44–61, 2008.

[3] Zeev Schuss.Theory and applications of stochastic differential equations.John Wiley & Sons Inc., New York, 1980.Wiley Series in Probability and Statistics.

[4] A. Singer, Z. Schuss, and D. Holcman.Narrow escape and leakage of Brownian particles.Phys. Rev. E (3), 78(5):051111, 8, 2008.

Department of Mathematics and Statistics - University of Saskatchewan - Saskatoon, Saskatchewan