Probabilistic methods for eigenvalue estimates

373

Mathematics and Computers in SimulationXXVI (1984)373-376 North-Holland

PROBABILISTIC

METHODS

Gabriella de1 GROSSO, Istituto

Some

Matematico

Anna GERARD1

“G. Castelnuouo”,

probabilistic

FOR EIGENVALUE

Uniuersitb

techniques

ESTIMATES

and Federico MARCHETTI

di Roma,

are discussed

Roma,

Italy

regarding

their use in estimating

the

Such estimates are useful when studying principal eigenvalue of an elliptic operator. models for biochemical reactions governed by lateral diffusion.

1.

In this paper we summarize

recent results

developed

biological

to study certain

on some mathematical

techniques

that were

models.

INTRODUCTION

In the study of the structure

of the cell mem-

only approximately,

brane an important topic is the description of erratic movements of molecules on the cellular surface and their chemical reactions. movements are usually called "lateral and are commonly modelled process

ter the smaller gion.

These diffusion"

as a diffusion

In the following obtained

a number

(cfr. (Z), (3), (4),

(5), (6)) both from a theoretical tal point of view. reactions

it will be shown how estimates

of a suitable

can be

of the principal

elliptic

operator.

2.

THE MATHEMATICAL

MODEL

and experimen-

The main goal of these

is the computation

re-

of papers were devoted

to this class of phenomena

papers

sections,

exponential

via the computation

eigenvalue In recent years,

being bet-

(cfr. (7)).

an asymptotic

(1).

the approximation

the size of the capturing

Let M be a smooth manifold

of the rate of the

Consider

studied.

a regular

first hitting

with boundary,

diffusion

aM.

on M up to the

time of aM, T, which we denote by

x(t). Most of these models ignore the geometry of the problem, treating it as a plane diffusion

The operator

governing

neglecting

differential

operator

possible

effects

due to the curvature

of the membrane.

boundary

conditions

contraction In some recent papers,

the present

investigated

these problems

lar surfaces

as a compact

Specifically,

Riemannian

let us consider

fusing on M until This is a model reaction,

it reaches

on M.

semigroup

Dirichlet

It will generate

a

Tt on Cb(M).

authors have

treating

and the movements of the molecules brownian motion on M.

x(t), L, is an elliptic with homogeneous

the cellumanifold

It is known that

M,

as a llTtll= sup

a particle a trapping

difregion.

for a diffusion-controlled

e.g. when a ligand is captured

ITt lM (x)1

= sup P

xQ4

fiM

1 11

(T > t) x

and that the upper bound of the spectrum

of L

(when M is compact

this is

the principal

and L is self-adjoint

eigenvalue)

is given by:

by the

membrane and diffuses until it reacts with a receptor molecule (cfr. (1)). To determine the forward reaction rate we thus need the distribution of the time of first arrival of the particle in the region. Strictly speaking, a reaction rate is well defined if the hitting time distribution is exponential. In the present case this is true

-Xo=lim t++-

+

log llT+ll

[ 21

-

(cfr. (8), (9)). 111 and [ 21 illustrate the relation between A and the distribution of T. In order to estimzte this distribution it may be useful to recall that, if u(x) := EAT is finite, it satisfies

0378-4754/84/$3.000 1984,IMACS/Elsevier SciencePublishersB.V. (North-Holland)

G. del Grosso et al. / Eigenunlue estimates

314

Lu=-1

in M

[ 31

ul =0 aM

In the model under consideration, M represents the cellular surface with some deleted regions, representing

the traps.

From the discussion

When the problem one-dimensional

above it follows

reduced

to a

with a single exit

point, as in the case just mentioned, exit times, and hence principal eigenvalues, can becomputed by direct comparison of the drifts. (cfr.(l2)). 3.2 Lower Bounds Exit Time

through Estimates

of the Mean

that in

order to compute the forward reaction estimates on ho are needed.

rate,

The following

section will be devoted

to some

probabilistic

techniques

Upper bounds

on the solution

automatically

lower bounds

eigenvalue, that can be applied

proven

because

of [3] yield

on the principal

of the following

estimate,

in (13)

to

this end. 3.

can be effectively process,

[ 41

S sup Ex~

ESTIMATION

?EM

TECHNIQUES Thus whenever

To be more specific, complete Riemannian

our manifold manifold

will be a

with a number of

variables

reduced whose

We have studied asymptotic corresponding 3.1 Comparison

principal

for the AE as E + 0. 0

differential

manifolds

only by a first order

i.e. n-dimensional with a global metric

ds2 = dr2 + $2(r) d S

L1=L2+b'V

dS2 is a metric

of the corresponding

exit

times are related via the Cameron-Martin-Girsanov formula.

2 r E (0,R) r E (0,R)

Taking L as the Laplace-Beltrami

0

of the form:

on the (n-1)-dimensional

Ms as the region

E < r
sphere.

operator

the result

and

is

n-l

A >VOlS

If

to so-cal-

Riemannian

$(O) = $(R) = 0, Jl(r) > 0 Q'(r) + 1 as r -* 0 $'(r) + -1 as r+R

term:

the distributions

equation,

has to be estimated.

In (13) this method has been applied led models

that, if L1, L2 are two ellip-

differing

to an ordinary

solution

as in

is effectively

estimates

eigenvalue

Methods

It is well-known tic operators,

on M.

high symmetry,

the case we now discuss),one

now take as the Laplace-Beltrami

of

this is always possible

if M and L have a suitably

geodesic disks of radius s deleted, Mc, x(t) will be the diffusion generated by L, which we operator

[3 ]can be solved by separation

(for instance,

vol

M

‘p,(E)

+

Ob,W

where b(x) = @Log

B(x))

* a(x) f

for some smooth function

B(x) (here

a(x) is

the matrix of the principal symbol of Ll and L2) the stochastic integral in the CameronMartin-Grisanov formula can be eliminated using Its's formula . Reading the result as a Feynman-Kac formula, it is clear that the pro-

n=2

1 log l/E

[ 61

‘P,(E) = { (n-2) ~~~~

n>2

t

blem has been shifted to a comparison between operators differing by a potential term. It is

These result are actually sharp in the sense that first term in 151 is in fact the leading term in the asymptotic expansion of X as e7Q,

well-known comparison

as can be checked using the well-knog Riets variational principle.

that bounds on the potential imply a result for principal eigenvalue

Rayleigir

(cfr. (10)). 3.3 A Glueing Method As an application, the case of a sphere with a single disk deleted has been discussed in (11).

When the problem

lacks the necessary

symmetry

G. del Grosso et al. / Eigenvalue estimates

because

the presence

of many "holes"

destroys

the symmetry in the boundary conditions, the method described in the preceding subsection can still be implemented by Stroock-Varadhan

using a technique

REFERENCES

(1)

developed

(cfr. (14)).

(2) Roughly speaking, one can try and solve 131 locally around each hole and then "glue" the different

solutions

Berg, H.C., and Purcell, E.M., Physics of Chemioreception, Bioph. J., 20, (1977), 193-219. Vanderkooi, J.M. and Callis, A Probe of Lateral Diffusion phobic Region

are to be found in (15).

The idea

Lipids (i) First choose a locally finite atlas such that each chart contains the stopped

diffusion

Letters,

for M

at most one hole.

(4)

of Lateral

in Biological 28, n.2,

V., Aberlin,

using Pyrene

(iii) The "glueing"

Acta, (1979), 201-211. Nicolson, G.L. and Poste,

whole

technique

of M as a suitable

measures

of Stroock-Vara-

the diffusion inductive

(5)

Cell-surface

limit of

England

on the space of paths. this construction

to our problem

terms

can, at least in principle,be

from solutions

discussed

two cases have been explicitly

(7)

J. 26, (1979), l-22. Marchetti, F., Asymptotic

(8)

in (15): (9)

log tg ; A

1

a-

+o(

0 log

(ii)

cos

;

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1957).

Donsker,

M.D. and Varadhan,

0

> 2 log(2-& a

2

Operators,

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log

1/E

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Del Grosso, G., Gerardi, A. and Marchetti, F ., Lateral Diffusion and Eigenvalue

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Del Grosso, G., Gerardi, A. and Marchett!i, F ., A Diffusion Model for Patch Formation

Estimates,

In this case

1 --+0(-

S.R.S., On

of Elliptic

Second Order Differential

Bull. Math.

on Cellular

x

Eigenvalue

(10)

M

lattice of mesh a.

Exponentiality

to appear.

Analysis

the Principal

out that

295, (1976),

Hille,E., dence,

Me = S2 without two disks of radius (i) centered at an angular distance a. It turns

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Poo, Lam, J.W., Orida, N. and and Diffusion Chao, A.W., Electrophoresis in the Plane of the Cell Membrane, Biophis

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253-258.

Mu-Ming

computed

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G., The Cancer I, II, The New

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et Biophys.

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M.E. and

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Dembo, M., Glushko,

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generated

Diffusion

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13, n.19,

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The details

375

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(1981), (13)

yield precise bounds, the second case appears to be more delicate because of the structure of the mathematical model (cfr. (7), (16)).

(14)

Appl. Math. Optim.

Cap, Appl. Math. Optim.

of a 7,

137-139.

Del Grosso, G., and Marchetti, F., Asymptotic Estimates for the Principal Eigenvalue of the Laplacian in a Geodesic Ball, Appl. Math. Optim., to appear. Stroock, D.W. and Varadhan, S.R.S., Multidimensional Diffusion Processes (SpringerVerlag

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A check of these estimates using Rayleigh-Rietz suggests that these bounds are not even asymptotically sharp. While in the first case finer estimate of the integrals involved should

Surfaces,

Biol.

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New York,

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G. de1 Grosso et al. / Eigenualue

376

(16)

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estimates