Some questions related to modeling in cellular biology · Some questions related to modeling in...

J. fixed point theory appl. Online Firstc© 2010 Birkhauser/Springer Basel

DOI 10.1007/s11784-010-0146-1

Journal of Fixed Point Theoryand Applications

Some questions related to modeling incellular biology

D. Holcman and I. Kupka

Mathematics Subject Classification (2010).

Keywords.

Abstract. Several years ago, we decided to switch our main focus of in-terest toward the field of modeling cellular biology. Several reasons mo-tivated this move: first cellular and molecular biology offer a fantasticnew source of physical and mathematical problems. Second, to under-stand the function of cellular microdomains, modeling and computersimulations are necessary tools to organize and structure experimentalobservations. We review here some questions we have started to address.

Introduction

In the past 60 years, major discoveries in biology have changed the directionof science. From the study of the sexual life of snails and oysters, which wasin some sense boring for the previous generations, biology has become to-day the Queen of Science in amount of funding, number of researchers andsocial and medical impact. Surprisingly, all hardcore fields, such as physics,mathematics, chemistry, and computer science are now necessary for the bigadventure of unraveling the secrets of life and conversely, the mathematicalsciences are all now enthusiastically inspired by biological concepts, to the ex-tent that more and more physicists and mathematicians are interacting withbiologists. Actually, it is not an exaggeration to say that modern biology hashad the effect of reinvigorating many classical fields of mathematics. Whatis today the role of a theorist among the biologists, eager to incorporate newconcepts? An important part of biology, besides amassing new experimentalinformation, is the explanation and prediction of new phenomena by apply-ing the quantitative laws of physics, chemistry and quantifying phenomenain mathematical terms, not merely fitting curves with Numerical. Theory ismore than a description of the reality: it gives us a framework to apply math-ematical methods to biology. The understanding of the puzzle of life begins

2 D. Holcman and I. Kupka JFPTA

with the study of proteins, microstructures, cells, networks behaviour andfinally, the life of a complex living organism.

In this article we will discuss three areas of biology where we believethat mathematical modeling and analysis can lead to important progress:the fields of 1) development, 2) molecular trafficking in cells, and 3) learningand memory. But before, let us start with three striking examples from thebygone century where theory made a big impact in biology: The first is thediscovery of the helicoidal structure of the DNA molecule, which came outfrom the analysis of Xrays crystallographic picture by Watson–Crick. Thisdiscovery was made possible by the recognition of the Fourier transform of ahelix, which was quite a novel way of thinking in the 50’s.

The second striking example comes from A. Turing, who introduced, inhis 1952 paper, the idea of and developed the reaction-diffusion equationsto model the spread of morphogens across cells. Morphogens moving fromcell to cell specify the cell position and ultimately lead to their physiologi-cal identity. A. Turing shows that this process can generate morphogeneticgradient and complex patterns, which ultimately leads to cell differentiationand specialization.

The last example comes from the Hodgkin–Huxley model, also pub-lished in 1952, which shows that opening and closing of channels can generatea wave of depolarization across an axon. The lesson drawn from the Hodgkin–Huxley model is that the propagation equation of an action potential can bederived at the molecular level from channel dynamics.

1. Some general concepts in the theory of development

The smallest living unit in biology is the cell and the central questions wecan ask are: how does it function, how are cells organized and what arethe rules and the mechanisms involved at a molecular level to make cellswork? The efficient working of organisms indicates that cells in the bodyare very well ordered, organized and subtly orchestrated. But, what kind ofmolecular mechanisms control the cell behavior? In a pluricellular organismsuch as mammalian cells, the specialization depends on the cell location,which is also part of its identity. The field of morphogenesis [24, 44, 45]consists precisely in identifying the rules used to orchestrate the constructionand the organization of a complex organism. Each cell is dedicated to aprecise task. For example in the brain’s early development, some cells areinvolved in the cortical construction, while other groups are devoted to theskin layer, and so on. Small errors can occur in the distribution of tasks andif for example, the number of cells dedicated to build a cortical region is notsufficient, then it may result in severe impairment of the brain function [4].

It is a challenging question to understand how cells get their instructionto build a specific region and not another. Obviously, positional informationneeds to be exchanged between cells in order to identify where they are andthus to activate the necessary fraction of the genetic code [12]. Positional

Modeling in cellular biology 3

information tells the cell what to do, and this process determines its identity.Ordering an ensemble of cells can be accomplished through the generationof morphogenetic gradients, where a substance travels from cell to cell, andeach time a cell is crossed, the morphogen concentration decreases. Cells arelabeled by gradients, but more specifically, certain genes are activated ata given morphogen concentration [44, 45]. However, how a cell can read agradient remains an open question. Nevertheless, it is now clear that the cellspecialization and function depend on the portion of the activated geneticcode [44, 45], [24, 25, 26].

Fly embryos have been a very useful model to study gradient formationand patterning [29]. Many models based on reaction-diffusion equations havebeen developed to predict complex and patterns not at all intuitive. Theseefforts originate in the early work of A. Turing, followed by L. Wolpert,H. Meinhart [44, 45], [24, 25, 26] and many others. In these models, interac-tions between inhibitor and excitatory molecules are the basis of early pat-terning and monotonic gradients. Recently, using the concept of morphogen,we have analyzed the precision of the boundary between regions [19, 12]. Theaccuracy of the boundary is crucial for the stability of an organism or in thecase of the cortex, for the cognitive function. Interestingly, we reported in[19, 12] that there is a 4% percent fluctuation, inherent to the process un-derlying the cellular organization and the construction of the morphogeneticregions. Much more remains to be understood in patterning. Finally, it wouldbe interesting to clarify the geometrical organization of a cellular organism.It is unclear how the blue print for the construction of an organism from asingle fertilized cell is encoded in the latter and later on decoded.

2. Cellular biology: Dynamics in the cytoplasm and the nucleus

Let us go back to the field of cellular biology where the elementary unit isa protein or a molecule. In order to guarantee the functionality of a cell,molecules and proteins have to be located at the right place. When theycease to be functional, they must be replaced. Interestingly, it has been ob-served experimentally that this is indeed the case all the time. This regulationprocess ensures that the cell remains in an equilibrium state (homeostasis),however, it is difficult to understand and interesting to quantify the numberof misplacements that take place when we know that the molecules of ourbody are being replaced every month.

Motion in the cytoplasm

Intracellular motion of macromolecules in cellular compartments is requiredfor numerous processes, including transport phenomena, DNA-protein inter-actions, metabolites signaling and pathogen infection. The diffusion of smallsolutes is relevant in drug delivery. The motion of larger molecules, such asnucleic acids, is important in gene therapy and RNA interference (RNAi).The time between the release of a given molecule or a particle in a cell andthe time that it hits its target can be drastically different, depending on the


specific characteristics (size, geometry, charge...) of the solute, the micro-rheology of the surrounding environment, the localization and the numberof targets. Studying and modeling the course of macromolecule motion in aconfined biological microdomain requires the derivation of explicit quantitiessuch as the rate of success, as a function of fundamental parameters such asthe geometry of the cell.

Now, we shall present some recent tools developed for the study of cy-toplasmic trafficking. These tools are based on homogenization proceduresfor stochastic equations [22], asymptotic methods for mixed boundary valueproblems and on the small hole theory [31]. In contrast to macroscopic mod-els, using these technics [8, 22], we can study intracellular trafficking of plas-mid DNAs and viruses, and predict their trajectories. It is remarkable thatthe probability of infection and the mean time for a single virus to reachits target (a nuclear pore) can be formulated in terms of partial differentialequations that are relatively simple and their analysis is feasible. There areseveral applications of these results, such as predicting cytoskeleton networkdisruption, plasmid DNA compaction into nanosphere and finally estimatinghow nuclease inhibition can affect the mean arrival time for a virion to thenucleus.

Small holes and Mean First Passage Time of a polymer to a small hole

Cellular microdomains are regulated by chemical reactions involving a smallnumber of molecules that have to find their targets in a complex and crowdedenvironment. To estimate the mean time of a molecule to reach its target,we have studied the dynamics of a Brownian particle (molecule, protein)confined in a compartment with a reflecting boundary, except at a smallwindow, through which it can escape [35, 36, 37, 31]. This problem, known asthe narrow escape problem in diffusion theory (also called the Narrow EscapeTime (NET)), goes back to Lord Rayleigh: the small hole often representsa small target on a cellular membrane, such as a protein channel for ions, anarrow neck in the neuronal spine [3] for calcium ions, and so on. The motionof a Brownian particle in a force field can be described by the overdampedLangevin equation (known as the Smoluchowski limit):

x− 1γF (x) =

√2D w, (1)

where

D =kBT

γ, (2)

γ is the friction coefficient, F (x) the force per unit of mass, T is absolutetemperature, kB is Boltzmann’s constant. w is an independent δ-correlatedGaussian white noise, representing the effect of the thermal motion. Thederivation of the Smoluchowski equation (1) is given in [30] for the three-dimensional motion of a molecule in a solution, where Einstein’s formula (2)can be applied.


Now we recall the analysis that leads to the estimates of a mean sojourntime of a Brownian particle in a bounded domain Ω, before it escapes througha small absorbing window ∂Ωa in its boundary ∂Ω. The remaining part ofthe boundary ∂Ωr = ∂Ω − ∂Ωa is reflecting for the particle. The reflectionmay also be represented by a high potential barrier on the boundary, or bean actual physical impenetrable obstacle. When the volume ratio is small,

ε =|∂Ωa||∂Ω|

1, (3)

the escape time can be estimated asymptotically [34]. For this purpose, we usethe probability density function (pdf) pε(x, t) of the trajectories of (1) whichis the probability per unit volume (area) of finding the Brownian particle atthe point x at time t prior to its escape. The pdf satisfies the Fokker–Planckequation

∂pε(x, t)∂t

= D∆pε(x, t)−1γ∇ · [pε(x, t)F (x)] = Lpε(x, t), (4)

with the initial condition

pε(x, 0) = ρ0(x), (5)

where ρ0(x) is the initial pdf (e.g, ρ0(x) = δ(x − y), when the molecule isinitially located at position y) and the mixed Dirichlet–Neumann boundaryconditions for t > 0

pε(x, t) = 0 for x ∈ ∂Ωa, (6)

D∂pε(x, t)∂n

− pε(x, t)γ

F (x) · n(x) = 0 for x ∈ ∂Ωr. (7)

The function

uε(y) =∫

Ω

dx

∫ ∞0

pε(x, t |y) dt, (8)

where pε(x, t |y) is the pdf conditioned on the initial position, represents themean conditional sojourn time in Ω for a particle starting at y. It is thesolution of the boundary value problem [30]

L∗uε(y) , D∆uε(y) +1γF (y) · ∇uε(y) = −1 for y ∈ Ω, (9)

uε(y) = 0 for y ∈ ∂Ωa, (10)

∂uε(y)∂n

= 0 for y ∈ ∂Ωr. (11)

The survival probability is

Sε(t) =∫

Ω

pε(x, t) dx, (12)

where

pε(x, t) =∫

Ω

pε(x, t |y)ρ0(y) dy. (13)


The density pε(x, t | y) can be computed using the eigenfunction expansion

pε(x, t | y) =∞∑i=0

ai(ε)ψi,ε(x)ψi,ε(y)e−λi(ε)t, (14)

where λi(ε) (resp. ψi,ε) are the eigenvalues (resp. normalized eigenfunctions)of the Fokker–Planck operator Lε with the associated boundary conditions(10)–(11) and the coefficients ai(ε) depend on the initial function ρ0(y),

ai(ε) =∫

Ω

ρ0(y)ψi,ε(y) dy. (15)

The survival probability is exponentially distributed [31],

Sε(t) ≈ e−λ0(ε)t for t 1/λ1(ε). (16)

The asymptotic solution of equations (9)–(11) depends on the dimension of Ωand the geometry of the small opening ∂Ω [34, 35, 36, 37]. When the geometryof a hole is regular, the escape time uε(y) is given for ε 1 by [34]

uε(y) =

AπD ln 1

ε +O(1) for dim Ω = 2,|Ω|

4aD[1 + L(0)+N(0)

2π a log a+ o(a log a)] for dim Ω = 3, (17)

where a is the radius of the hole assumed to be a geodesic disk located onthe surface of the domain Ω. A (resp V ) is the surface (resp. volume) of thedomain Ω, and L(z) and N(z) are the principal curvatures of ∂Ω at z. Thefunction uε(y) does not depend on the initial position y, except in a smallboundary layer near ∂Ωa [16, 35, 36, 37, 34].

This formula can be extended to the case of several windows. We shallconsider only the case of two windows. For a regular planar domain Ω withtwo absorbing arcs of lengths 2ε and 2δ (normalized by the perimeter |∂Ω|) inits boundary and such the Euclidean distance of the middles is ∆ = ε+∆′+δ,and for a three-dimensional d = 3 domain Ω with two absorbing circularwindows of small radii a and b, such that the Euclidean distance of thecenters is ∆ = a+ ∆′ + b, we obtained for the NET τε [18, 17]:

τε =

|Ω|

4(a+ b)D

1− 16ab(

24π|a+∆+b|

)21− 8ab

a+b1

2π|a+∆+b|for dim Ω = 3,

|Ω|πD(log 1

ε + log 1δ

) log 1δ log 1

ε − (log |ε+ ∆ + δ|)2

1 + 2 log |ε+∆+δ|log 1

δ+log 1ε

for dim Ω = 2,

(18)

as a, b, ε, δ,∆′ → 0. Here r = r(∆′, ε, δ) is a function of ∆′, ε, δ that variesmonotonically between 0.6 and 1 as ∆′ goes from 0 to ∞. Recently, in acollection of papers [40, 6], Ward and co-workers have obtained the zero orderterm for the asymptotic expansion of the NET, using the explicit regular partof the Green function in a sphere. This approach allows one to compute theeffect of many holes on the NET. Similar formulas for stochastic dynamicscontaining a drift term are still missing. Indeed, it would be important to


estimate the NET of viruses to small nuclear pores. We shall now discusssome extension of the small hole theory to polymer dynamics.

Small hole problem for a polymer

We model a polymer FN as an ordered collection of N random beads withcoordinates FN = (x1, . . . , xN ), where the consecutive beads are connectedlinearly by a spring of constant k and each bead is subjected to isotropicrandom forces. We are interested in several questions such as the mean timefor any one of the beads to hit a small hole ∂Ωa ⊂ ∂Ω or the mean firsttime for a single specific bead to reach the target (a nuclear pore). Thiscomputation generalizes the small hole computations, which gave the meanfirst passage time (MFPT) of a single Brownian molecule to a small hole[16, 34, 35, 36, 37].

Let us describe some results: for small N , the MFPT increases with N ,because the center of mass of the polymer moves with a diffusion constantD/N . Thus it takes more time to reach the small hole, but when N is largeenough, the polymer occupies a certain fraction of the space and thus thevolume per bead becomes so small that the MFPT of a single bead hitsthe small hole decreases. Thus this intuitive analysis shows that there is amaximum value for the MFPT as a function of N . In a first part, we presenta physical model of the polymer motion, and in a second part we obtainasymptotic estimates of the MFPT to a small hole, as a function of N , thediffusion constant and the geometrical parameters of the hole and the domain.

2. Modeling the motion of a polymer in a microdomain

The motion of a linearly coupled chain of N beads FN in an overdampedmedium, such as water or a biological fluid, can be described by a system offirst order stochastic differential equations. Neglecting possible hydrodynamiceffects, the Smoluchowski limit of the Langevin equation can be written as

x1 +∂U

∂x1=√

2εγw1,

x2 +∂U

∂x2=√

2εγw2,

. . . (19)

xn +∂U

∂xn=√

2εγwn,

. . .

xN +∂U

∂xN=√

2εγwN ,

where ε = kT/m and γ is the friction coefficient and w1, . . . , wN are N in-dependent 2 or 3-dimensional independent Brownian motions. By definition,the potential Uk generated by two springs adjacent to the k bead is the sum


of two terms

Uk,k+1(xk, xk+1) =k

γ

(12|xk − xk+1|2 − l0|xk − xk+1|

), (20)

Uk,k−1(xk, xk−1) =k

γ

(12|xk − xk−1|2 − l0|xk − xk−1|

). (21)

The potential Uk is, for 0 < k < N ,

Uk(xk−1, xk, xk+1) = Uk,k+1(xk, xk+1) + Uk,k−1(xk, xk−1), (22)

U0(x1, x2) = U12(x1, x2), (23)

UN (xN−1, xN ) = UN,N−1(xN , xN−1) (24)

and the total potential is given by

UN (x) =N∑k=1

Uk(xk−1,xk,xk+1). (25)

Equivalently,

UN (x) = U(x)− (N − 1)l20k

2γ(26)

where

U(x) =k

2γ

N∑k=2

(|xk − xk−1| − l0)2. (27)

When the polymer FN is confined in a microdomain Ω, each bead is re-flected at the boundary except in a small patch ∂Ωa, where any bead canbe absorbed. We present here the case where any one of the beads can hitthe small hole. To estimate the mean time any of the beads reaches thesmall patch, we consider the joint probability density function p(x1, . . . , xN , t)for the chain (X1, . . . , XN ) (k = 1, . . . , N) to be inside the volume elementdVx =

∏N1 (xk + dxk) in the (N dim Ω)-space

ΩN = Ω× · · · × Ω,︸︷︷︸N times

(28)

p(x1, . . . ,xN , t)dVx = PrX1(t) ∈ x1 + dx1, . . . ,XN (t) ∈ xN + dxN.(29)

p is the solution of the Fokker–Planck equation (FPE) (see [31])

∂p(x, t)∂t

= D∆xp(x, t) +∇x[∇UN (x)p(x, t)] for x ∈ ΩN , (30)

p(x, 0) = p0(x) for x ∈ ΩN ,

where D = ε/γ, and p0(x) is the initial polymer distribution∫ΩN

p0(x) dx = 1, (31)


∆x is the Laplace operator in N dim Ω variables (x1, . . . ,xN ). The boundaryconditions associated with the FPE are given by

p(x1, . . . ,xN , t) = 0 for (x1, . . . ,xN ) ∈ ∂ΩNa , (32)

J(x1, . . . ,xN , t) = 0 for (x1, . . . ,xN ) ∈ ∂ΩNr , (33)

where

J(x1, . . . ,xN , t) = D∂p

∂nx(x1, . . . ,xN , t) +∇UN (x).np(x1, . . . ,xN , t), (34)

nx is the exterior unit normal vector to the absorbing part of the boundary

∂ΩNa =N⋃n=1

Ω× · · · × ∂Ωa︸︷︷︸n

× · · · × Ω (35)

and the reflective part is

∂ΩNr = ∂ΩN − ∂ΩNa . (36)

Outside the boundary layer of the small hole, we can look for the solutionusing the ansatz

uη(x) = Cηe−UN (x)/D; (37)

we get

Cη ≈ e+UN (Q)/D 1DN |∂Ωa| |Ω|N−1

∫∂ΩNa

N(Q,x) dSx (38)

and N is the Neumann function, solving the boundary value problem

∆xN(x, ξ) = −δ(x− ξ) for x, ξ ∈ ΩN , (39)

∂N(x, ξ))∂n(x)

= − 1|∂ΩN |

for x ∈ ∂ΩN , ξ ∈ ΩN . (40)

We obtained for two-dimensional domains [15] the expression

τη(N) =∫

ΩNu(x) dVx

≈ d1,Nwq−1

ND|∂ΩNa | |Ω|N−1ln(

1δ

)∫∂ΩNa

e+UN (Q)/DdSQ

∫ΩN

e−UN (x)/Ddx,

(41)

where

dp,N =1

(N(p+ 1)− 2)ωN(p+1)−1, (42)

where ωm is the volume of the unit sphere of dimension m. In the smalldiffusion limit, it is possible to obtain from (41) the precise dependence ofthe mean time on the number of beads N (equivalently the length of thepolymer). Interestingly, we expect that there is a range of values for k forwhich τη(N) has a maximum as a function of N .


Double strand DNA breaks, rough modeling of a polymer motion in a con-fined environment

To motivate our analysis, we shall now present the underlying biology: cellularradiation can generate two types of DNA breaks: single or double strandsDNA breaks (dsDNAb). In the first case, the DNA molecule is not fullydisrupted as only one branch is cut and thus the break can relatively easilybe repaired using the healthy DNA branch. However, in the second case,when the two strands are cut, they can drift apart and in addition, the breakcan be associated to a loss of base pairs. This is a severe lesion that needsto be repaired, otherwise, the cell can generate genetic mutations and/ordegenerates into a cancer cell. In an extreme case, at high concentration ofbreaks generated by high level cellular irradiation, it has been observed thatthe cell triggers apoptosis or cell death. The ability to repair these breaks isthus an essential process for survival.

In a reasonable range of break concentration, to repair double-strandDNA breaks, two mechanisms have been identified: the first is the Homolo-gous Recombination where the DNA molecule uses unaffected DNA-strandsto repair itself, by searching for base pair complement at the correct location.This process relies on the possibility that moving fragments not only find thecomplementary genetic information, but glue themselves back together cor-rectly [47]. If these fragments were to move freely under a Brownian motion,the probability that these moving fragments would glue together correctly be-comes extremely low, suggesting that a complex and unknown process mightexist to prevent incorrect recombination. The second process is known asnon-homologous end-joining and consists of the joining of two DNA breaksby simple physical and direct interactions. We focus here on this second pro-cess.

The process of correct dsDNA repair by non-homologous end-joiningis vital and turns out to be very complex [47]. A large number of proteins,such as enzymes, are involved to ensure the correct ligation of the two freeends. Incorrect ligation would lead to loss of genetic information (deletions).Despite the current knowledge about the repair machinery, we are still lack-ing a kinetic view of this process. We now present some preliminary resultsabout the dynamics of dsDNA repair, which uses a biophysical model of DNAmotion in a constraint environment [48]. So far, no direct experiments haverevealed the dynamics of repair. Whatever the details of the mechanism, twoDNA broken ends have to find each other, and this presumably must happenvia random motion. If the DNA strands are floating freely in the nucleus,without any physical constraint, and do not meet quickly enough, it will beextremely unlikely that they will ever meet in a reasonable time (on thescale of the cell). Therefore, it is likely that surrounding structures that re-strict the movement of the DNA play a major role in determining whethertwo DNA strands will meet. To identify the physical phenomena involved inDNA repair, we model the dynamics of an isolated DNA break by a searchmechanism of the two cut strands (ends) in a neighborhood of the break.


Using the Brownian dynamics, we analyzed the motion of two attached poly-mers moving in a confined environment. In our analysis, gluing, which is themost sensitive part of the repair process, is achieved when the two polymerchains meet again for the first time. Once the two strands are back together,although the repair might not be finished, it is conceivable that the latersteps will not be as difficult and less time consuming.

Modeling semi-free DNA motion in a restricted strip

We model a DNA branch as a small polymer. Starting with the description ofthe polymer (see previous section), represented as a collection of beads con-nected by springs of constant k. The mean length between the beads is l0 andis usually known as the persistence length. Each bead can move accordingto a Langevin equation, driven by a potential field, generated by the springaction of its immediate neighbors. To account for the geometry of the DNAand nucleus organization, we shall assume that the DNA molecule can onlymove in a restricted strip of length L and thus no beads can exit from thestrip. To account for the nucleosome organization, we fix the initial positionof a bead. Interestingly for small L (result not shown here), most of the DNAbreaks can be repaired. To obtain some quantitative analysis, we consider thedrastic simplification where the motion of the DNA molecule tip can be ap-proximated as a one-dimensional Brownian motion. We consider the motionof two independent Brownian particles X1(t), X2(t) inside an interval [a, b](a < b) with the following rules: when the two particles meet, they coalesceinto a single one subjected to a Brownian motion. The probability PM thatthe two particles meet before one of them hits the boundary of the intervalcan be obtained as a function of the initial positions a < x1 < x2 < b; wefound [14] that

PM (x1, x2) =−2π=m log P

(ω(Z − a)L√

8

), (43)

where =m denotes the imaginary part, P is the Weierstrass elliptic function

P′2 = 4P3 − g2P− g3, (44)

with parameters g2 = 1 and g3 = 0, L = b− a, Z = x2 +√−1x1, and

ω =∫ +∞

1

dx

[x(x− 1)]3/4= 5.244115106. (45)

The probability distribution of a meeting of the particles can also be found.The role of the Weierstrass elliptic function is quite surprising here and comesfrom conformal mapping.

A word about the method

The dynamics of each particle is given for i = 1, 2 by

dXi =√

2Dfdwi (46)

where Df is the diffusion constant and w1, w2 are two Brownian motions ofunit variance. We are interested in the probability PM that the two particles


meet before one of them exits the interval [a, b]. If we consider the two randomtimes

τ1 = inft > 0 : X1(t) = a or X2(t) = b,X1(0) = x1 and X2(0) = x2, x1 < x2,τ2 = inft > 0 : X1(t) = X2(t), X1(0) = x1 and X2(0) = x2, x1 < x2,then for x = (x1, x2), the probability

PM (x) = Prτ2 < τ1 | x (47)

satisfies the Laplace equation (ch. 15, p. 192 of [38])

∆PM (x) = 0 for x ∈ T, (48)PM (x) = 1 for x ∈ D,PM (x) = 0 for x ∈ ∂T −D,

where T is a triangle with vertices a, b, b+a√−1, and D is the side joining a to

b+a√−1. To derive an explicit expression of the encounter probability P , we

have solved the equation (48) using the Schwarz–Christoffel transformationto map T onto the upper half-plane H. By using the explicit solution of theLaplace equation in H, we compute the solution of (48). It turns out thatthe Schwarz–Christoffel function is the log a Weierstrass function.

3. Brain organization, memory and synaptic plasticity

Although the notion of memory storage for computers is quite clear, we stilldo not know how and where memory is stored in the brain. Various scales areinvolved in memory encoding: molecular and cellular scales and the corticalnetwork. At the cellular level, neurons make micro-contacts, either directly ondendrites or on a structure called a dendritic spine (see Fig. 1) [32, 5]. It is still

Figure 1. This image shows the numerous dendritic spines,which are local protrusions, located on the dendrite of neu-rons.


intriguing that a single neuron can contain of the order of 100 000 of thesespines, although their function is still unknown. However, their mushroomshape geometry has attracted the attention of the neurobiological commu-nity, interested in quantifying the diffusion of molecules or studying theirelectrical properties [1, 13]. For example, many fundamental regulatory pro-cesses, such as calcium dynamics, occur in spines. It is still a matter of debateto quantify the amount of calcium that crosses a dendritic spine [21, 32], af-ter receptors have been activated. We would like to determine the number ofchemical bonds that calcium ions have made during their journey inside thespine. However, we are still limited experimentally by using exogenous buffersor fluorescent dye molecules that significantly perturb the chemical reactions[11]. The geometry and the spine organization might underlie synaptic plas-ticity and other complex learning functions that should be understood; inparticular, how spine uses its geometrical shape in encoding memory.

Moreover, neuronal communication relies on micro-contacts called syn-apses. This communication induces in a change of the electrical activity, con-trolled by few channels (approximatively 50 to 100 only) that can in additionvary due to protein trafficking [7]. Thus in this context, it is not clear howthe neuronal signal is stable over time, especially if synapses are memorycheckpoints. An important question would be to estimate the number of re-ceptors and how they are controlled. Thus, how the synaptic connectionscan be maintained for years, if the lifetime of receptors is about 24 hours?The proteins have to be replaced constantly and correctly. New scenarios in-evitably appear with new mathematical models to explain the accuracy ofsuch processes [9].

Recently, using asymptotic analysis of the mean first passage time equa-tion for diffusing particles (receptors on the surface and calcium ions inside)in the spine, we estimated [10] the mean time for a diffusing particle (withdiffusion constant D) to escape a thin spine neck. We obtained in general

τH ≈

LΩHD2a

+ΩHπD

ln1a

+L2

2Din 2-dim,

LΩHDπa2

+ΩH4Da

+L2

2Din 3-dim,

(49)

where ΩH is the spine head and L (resp. a) is the length (resp. the radius) ofthe spine neck. More complex chemical reactions should be taken into accountin order to reveal the complex function of synapses. A lot more remains tobe modeled and understood.

Combining theory and experiments was already very beneficial to manyfields of biology, it was used to unravel the organization of the visual cortex,where specific neurons fire in responses to visual stimuli. The presentation ofa rotating bar induces a neuronal activity of neurons that also rotates aroundpoints, which are topological singularities, called pinwheel. It is interesting


to find this type of typological singularities here. Noise seems to play a cru-cial role in the maintenance of the neuronal activity and in coding spatialinformation.

To conclude this short and superficial survey, we shall add that this isonly the early beginning where quantitative approaches and mathematicalanalysis are joining efforts to address questions related to cellular biology.A new generation of physicists, applied and pure mathematicians should betrained to crack biological challenges of tomorrow and to unravel the con-cepts behind the essence of life. There is much to benefit from mathematicaltraining. In biology, there is no shortage of problems and we can compareour time with the gold rush of California in the mid XIXth century, wherethe rule was first come, first served. Today we do not have to bend much toreap, while there is also room for those who want to dig deeper. Now that thestock market is down, a new generation of theoreticians can either find newprinciples and rules to provide foundations of a future and hopefully stableeconomy, with out forgetting to put back the controllers or alternatively, thisgeneration may want to join us to help and unravel the complex rules of life.

Some general questions

1. What is memory at a synaptic level? How much memory is containedin a spine, in a dendrite and a neuron?

2. How to quantify spine shapes?3. How to characterize the electrical properties of a dendritic spine?4. How to model cellular trafficking, how a protein knows where to go?

What defines its pathway to the final location?5. How is the address encoded where a molecule is sent to?6. How come viruses are so efficient in traveling the cytoplasm and pene-

trating the nucleus?7. Where the code that allows the development of an organism is located?

How is it activated and implemented? Where and how geometry is en-coded?

8. How gradients and boundaries are made in early embryos? What is thevariability due to this construction? In other words, once an egg is fer-tilized, with what accuracy a chicken is fashioned? What is the principlefor making morphogenetic gradients? Can we estimate the fluctuationof size of a morphogenetic region?

9. How are axons growing and with what precision?10. How cells read and quantify the morphogenetic gradient in which they

are immersed and what happens during cell divisions?

Acknowledgments

This research was supported by the grant Human Frontier Science Program0007/2006-C.


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D. HolcmanDepartement de Mathematiques et de Biologie

Ecole Normale Superieure46 rue d’Ulm75005 Paris, Francee-mail:

I. KupkaDepartement de Mathematiques et de Biologie

Ecole Normale Superieure46 rue d’Ulm75005 Paris, Francee-mail:

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