Narrow Capture Problems: Volume Traps in the Unit SphereJason Gilbert and Alexey Shevyakov

Department of Mathematics and Statistics, University of Saskatchewan

Motivation & Application Areas

First passage time: the time required for a random walker to encounter a barrier.

I General interest: first passage time problems are found in a variety of fields, from economics to biology.

I A specific application: narrow escape and narrow capture problems are relevant to biophysics andcellular biology, specifically intracellular interactions and diffusion through cellular membranes.

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Brownian Motion

Narrow capture problems concern the time required for a particle undergoing Brownian motion to firstencounter some trap which stops the motion of the particle. This time is called the First-Passage Time(FPT).

I Particles suspended in a fluid undergo erratic,seemingly random, movement called Brownianmotion.

I Brownian motion is described mathematically bya random process, called the Wiener process.

I The random nature of Brownian motion meansthere are many paths leading to a trap a particlecan take, each with its own FPT.

I The expected time a particle will wander beforebeing captured is given by the MeanFirst-Passage Time (MFPT).

𝝏𝛀𝝐𝟏

𝝏𝛀𝝐𝟐

𝝏𝛀𝝐𝟑

𝝏𝛀𝝐𝟒

𝒙

Narrow Capture in 2D Paths of a particle

Narrow Capture in 3D

The Narrow Capture Problem

The MFPT is given by a Poisson equation with Neumann-Dirichlet boundary conditions:

∇2v(x) = − 1

D, x ∈ Ω/Ωa ;

∂nv = 0, x ∈ ∂Ω; v = 0, x ∈ ∂Ωa = ∪Nj=1∂Ωεj ,

(1a)

(1b)

where

I v(x) is the MFPT for a particle initially located at x

I D is the diffusion coefficient

I Ω is the domain

I ∂Ωεj is an absorbing trap boundary (j = 1, ..., N), N is the number of traps in the domain

The Asymptotic Approximation

The MFPT boundary value problem (1) does not have a closed-form solution for non-trivial trapconfigurations.

To study the MFPT for more complicated trap configurations, we usean asymptotic approximation, which:

I Considers the space occupied by a trap to be negligible;

I Allows much faster computation of MFPT;

I Gives insight into optimal configurations.

The MFPT is dependent on the Neumann problem Green’s Function. For a trap located at ξ and a pointx in the unit sphere, this is given by

G(x, ξ) =1

4π|x− ξ|+

1

4π|x||x′ − ξ|+

1

4πlog

(2

1− |x||ξ| cos θ + |x||x′ − ξ|

)+

1

6|Ω|(|x|2 + |ξ|2)− 7

10π. (2)

In terms of the Green’s Function, under the assumptions that

I Traps are small, ∼ ε 1;

I Distance between traps is large relative to traps;

I Traps are not near the boundary of the domain.

one may derive an asymptotic approximation for the MFPT, and Average MFPT (AMFPT) [1]:

The Mean First-Passage Time (MFPT):

v(x) =|Ω|

4πNcDε

1 − 4πε

N∑j=1

cjG(x; ξj) +4πε

Ncpc(ξ1, ..., ξN ) + O(ε2)

(3)

The Average Mean First-Passage Time (AMFPT):

v =

∫Ωv(x)dΩ =

|Ω|4πNcDε

[1 +

4πε

Ncpc(x1, ..., xN ) + O(ε2)

], (4)

where ε is determined by the size of the trap, and c is the weighted capacitance determined by the sizeand shape of the traps.

I The asymptotic expression for the AMFPT can be used to determine an optimal configuration.

The Accuracy of the Asymptotic Approximation

For a single spherical trap of radius ε, located at the center of the unit sphere, an exact solution to theboundary value problem (1) is given by

ve(r) =1

6D

[ε3 + 2

ε− r3 + 2

r

]. (5)

Due to the spherical symmetry of the configuration, the MFPT depends only on the distance from thecenter of the unit sphere.

The accuracy of the asymptotic approximation can be determined by comparing it to the exact solution.

δv =|v − ve|ve

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5

5.5

6

6.5

710-7

Relative Error in MFPT vs Distance from Originε = 0.01

δv =|v − ve|ve

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

710-4

Relative Error in AMFPT vs Trap Volume

Numerical Approximation of the Mean First-Passage Time

I For configurations of more than one trap, closed-form solutions to (1) are extremely hard to obtain.

I Numerical methods must be used to approximate the MFPT for these configurations.

I As the numerical approximation is more accurate than the asymptotic approximation, comparison ofthe two gives a measure of the accuracy of (3) and (4).

Numerical approximations were obtained using COMSOL Multiphysics 5.1 software.

I Finite element PDE solver.

I Free tetrahedral meshing.

Numerical approximation of the MFPT for three traps in the domain: ε = 0.1

Asymptotic Error and Trap Separation

To determine the accuracy of (4) with respect to the distance between traps, the relative error betweenthe asymptotic and numerical approximations was computed for several distances between traps.

I A configuration of two traps fixed at the same distance from the origin was tested.

I Trap separation (d) is the smallest distance between the boundaries of the traps.

1 2 3 4 5 6 7 817

18

19

20

21

22

23

24

AsymptoticNumerical

AMFPT vs Trap Separationε = 0.01

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Relative Error in AMFPT vs Trap Separationε = 0.01

Asymptotic Error and Trap Position

To determine the accuracy of (4) with respect to the distance from a trap to the boundary of the domain,the relative error between the asymptotic and numerical approximations was computed for several trappositions.

I Distance between the trap and origin (r) was varied for a single trap.

I Error was found to be the result of interaction between the trap and its image reflected in the boundary.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 132

34

36

38

40

42

44

46

48

50

AsymptoticNumerical

Comparison of AMFPT vs Trap Positionε = 0.01

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Relative Error in AMFPT vs Image Separationε = 0.01

Optimal Trap Configurations

The optimal configuration of N traps:

I Is obtained when the traps are arranged to minimize the AMFPT.

I Can be thought of as the best way to arrange N traps to capture particles anywhere in the domain.

In order to determine the placement of traps which minimized the AMFPT, we minimize the interactionenergy, which is the sum of pair-wise Green’s Functions.

Interaction Energy (pc):

pc(ξ1, ..., ξN ) =

N∑i=1

N∑j=1

cicjG(ξi, ξj) (6)

Method of Optimization

I Minimize the interaction energy with respect to thecoordinates of the traps (r, θ, φ).

I To eliminate rotational symmetry, one trap was fixedalong the z-axis, another was fixed to the xz-plane.

I These constraints give an optimization problem of3N − 3 parameters.

I Due to high dimensionality of the problem, all optimalconfigurations discussed here are putative.

Optimization was achieved using a MATLAB interface fora Lipschitz-Continuous Global Optimizer (LGO). Importantvariables of this interface are:

I The number of locations in the parameter space whichare searched;

I The number of coordinate combinations about eachsearch location which are evaluated;

I How many poor combinations of coordinates LGO willencounter before abandoning a search location;

I The nominal configuration, or configuration whichLGO begins its search.

Nominal Configuration: 150 Traps

y = 3E-05x3 - 0.0064x2 - 0.2807x + 0.4393

-35

-30

-25

-20

-15

-10

-5

0

0 10 20 30 40 50 60

pc

N

Interaction energy as number of traps increasesequation of best-fit determined numerically

Qualities of Optimal Configurations

I It was previously thought that optimal configurations distributed traps over the surface of a sphere.

I It was found that traps tended to be located at discrete radial distances.

I Traps are located near, but not on, the surface of spheres within the domain of the unit sphere.

I As the number of traps increases, these spheres grow in size and number.

N = 5

0.593

0.5935

0.594

0.5945

0.595

0.5955

0.596

0 1 2 3 4 5 60.5925

0.593

0.5935

0.594

0.5945

0.595

0.5955

0.596

0.5965

N = 19

0.7255

0.726

0.7265

0.727

0.7275

0 2 4 6 8 10 12 14 16 18 20

j

0.725

0.7255

0.726

0.7265

0.727

0.7275

0.728

0.7285

0.729

r

N = 80N = 80r1 = 0.154 r

2 = 0.478 r

3 = 0.826

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Radial distribution of traps in the domain

Radial Distribution of Traps in Optimal Configurations

I Markers indicate the number of traps found about a common distance from the origin.

I Marker colour indicates common distance from origin.

I ∗ indicates a trap at the origin, indicates otherwise.

10 20 30 40 50 60

Total Traps (N)

0

10

20

30

40

50

60

Tra

ps o

n S

hell

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of traps about a common distance vs total number of traps

Conclusions & Future Research

I The applicability of the asymptotic approximate MFPT formulas was tested through comparisons with (special) exact andnumerical solutions. It was found the asymptotic approximation is valid outside the originally posed restrictions, inparticular, for substantial trap sizes, small trap separation distances, and traps located close to the boundary.

I Putative optimal 3D spherical trap arrangements for 21 ≤ N ≤ 100 were systematically computed for the first time. It wasobserved that optimal configurations tend to distribute traps discretely, about a common radial distance.

I Future work directions include the study of MFPT for non-spherical domains, and cases of non-equal traps.

References

[1] A.F. Cheviakov and M.J. Ward.Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps.Mathematical and Computer Modelling, 53:1394-1409, 2011

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