Some questions related to modeling in cellular biology - ENS Biologie

J. fixed point theory appl. Online First c 2010 Birkh¨

auser/Springer Basel DOI 10.1007/s11784-010-0146-1

Journal of Fixed Point Theory and Applications

Some questions related to modeling in cellular biology D. Holcman and I. Kupka Mathematics Subject Classification (2010). Keywords.

Abstract. Several years ago, we decided to switch our main focus of interest toward the field of modeling cellular biology. Several reasons motivated this move: first cellular and molecular biology offer a fantastic new source of physical and mathematical problems. Second, to understand the function of cellular microdomains, modeling and computer simulations are necessary tools to organize and structure experimental observations. We review here some questions we have started to address.

Introduction In the past 60 years, major discoveries in biology have changed the direction of science. From the study of the sexual life of snails and oysters, which was in some sense boring for the previous generations, biology has become today the Queen of Science in amount of funding, number of researchers and social and medical impact. Surprisingly, all hardcore fields, such as physics, mathematics, chemistry, and computer science are now necessary for the big adventure of unraveling the secrets of life and conversely, the mathematical sciences are all now enthusiastically inspired by biological concepts, to the extent that more and more physicists and mathematicians are interacting with biologists. Actually, it is not an exaggeration to say that modern biology has had the effect of reinvigorating many classical fields of mathematics. What is today the role of a theorist among the biologists, eager to incorporate new concepts? An important part of biology, besides amassing new experimental information, is the explanation and prediction of new phenomena by applying the quantitative laws of physics, chemistry and quantifying phenomena in mathematical terms, not merely fitting curves with Numerical. Theory is more than a description of the reality: it gives us a framework to apply mathematical methods to biology. The understanding of the puzzle of life begins

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D. Holcman and I. Kupka

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with the study of proteins, microstructures, cells, networks behaviour and finally, the life of a complex living organism. In this article we will discuss three areas of biology where we believe that mathematical modeling and analysis can lead to important progress: the fields of 1) development, 2) molecular trafficking in cells, and 3) learning and memory. But before, let us start with three striking examples from the bygone century where theory made a big impact in biology: The first is the discovery of the helicoidal structure of the DNA molecule, which came out from the analysis of Xrays crystallographic picture by Watson–Crick. This discovery was made possible by the recognition of the Fourier transform of a helix, which was quite a novel way of thinking in the 50’s. The second striking example comes from A. Turing, who introduced, in his 1952 paper, the idea of and developed the reaction-diffusion equations to model the spread of morphogens across cells. Morphogens moving from cell to cell specify the cell position and ultimately lead to their physiological identity. A. Turing shows that this process can generate morphogenetic gradient and complex patterns, which ultimately leads to cell differentiation and specialization. The last example comes from the Hodgkin–Huxley model, also published in 1952, which shows that opening and closing of channels can generate a wave of depolarization across an axon. The lesson drawn from the Hodgkin– Huxley model is that the propagation equation of an action potential can be derived 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 we can ask are: how does it function, how are cells organized and what are the rules and the mechanisms involved at a molecular level to make cells work? The efficient working of organisms indicates that cells in the body are very well ordered, organized and subtly orchestrated. But, what kind of molecular mechanisms control the cell behavior? In a pluricellular organism such 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 construction and the organization of a complex organism. Each cell is dedicated to a precise task. For example in the brain’s early development, some cells are involved in the cortical construction, while other groups are devoted to the skin layer, and so on. Small errors can occur in the distribution of tasks and if for example, the number of cells dedicated to build a cortical region is not sufficient, then it may result in severe impairment of the brain function [4]. It is a challenging question to understand how cells get their instruction to build a specific region and not another. Obviously, positional information needs to be exchanged between cells in order to identify where they are and thus to activate the necessary fraction of the genetic code [12]. Positional

Modeling in cellular biology

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information tells the cell what to do, and this process determines its identity. Ordering an ensemble of cells can be accomplished through the generation of morphogenetic gradients, where a substance travels from cell to cell, and each time a cell is crossed, the morphogen concentration decreases. Cells are labeled by gradients, but more specifically, certain genes are activated at a given morphogen concentration [44, 45]. However, how a cell can read a gradient remains an open question. Nevertheless, it is now clear that the cell specialization and function depend on the portion of the activated genetic code [44, 45], [24, 25, 26]. Fly embryos have been a very useful model to study gradient formation and patterning [29]. Many models based on reaction-diffusion equations have been developed to predict complex and patterns not at all intuitive. These efforts 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, interactions between inhibitor and excitatory molecules are the basis of early patterning and monotonic gradients. Recently, using the concept of morphogen, we have analyzed the precision of the boundary between regions [19, 12]. The accuracy of the boundary is crucial for the stability of an organism or in the case of the cortex, for the cognitive function. Interestingly, we reported in [19, 12] that there is a 4% percent fluctuation, inherent to the process underlying the cellular organization and the construction of the morphogenetic regions. Much more remains to be understood in patterning. Finally, it would be interesting to clarify the geometrical organization of a cellular organism. It is unclear how the blue print for the construction of an organism from a single 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 is a 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 they cease to be functional, they must be replaced. Interestingly, it has been observed experimentally that this is indeed the case all the time. This regulation process ensures that the cell remains in an equilibrium state (homeostasis), however, it is difficult to understand and interesting to quantify the number of misplacements that take place when we know that the molecules of our body are being replaced every month. Motion in the cytoplasm Intracellular motion of macromolecules in cellular compartments is required for numerous processes, including transport phenomena, DNA-protein interactions, metabolites signaling and pathogen infection. The diffusion of small solutes is relevant in drug delivery. The motion of larger molecules, such as nucleic 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 and the time that it hits its target can be drastically different, depending on the

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specific characteristics (size, geometry, charge...) of the solute, the microrheology of the surrounding environment, the localization and the number of targets. Studying and modeling the course of macromolecule motion in a confined biological microdomain requires the derivation of explicit quantities such as the rate of success, as a function of fundamental parameters such as the geometry of the cell. Now, we shall present some recent tools developed for the study of cytoplasmic trafficking. These tools are based on homogenization procedures for stochastic equations [22], asymptotic methods for mixed boundary value problems and on the small hole theory [31]. In contrast to macroscopic models, using these technics [8, 22], we can study intracellular trafficking of plasmid DNAs and viruses, and predict their trajectories. It is remarkable that the probability of infection and the mean time for a single virus to reach its target (a nuclear pore) can be formulated in terms of partial differential equations that are relatively simple and their analysis is feasible. There are several applications of these results, such as predicting cytoskeleton network disruption, plasmid DNA compaction into nanosphere and finally estimating how nuclease inhibition can affect the mean arrival time for a virion to the nucleus. Small holes and Mean First Passage Time of a polymer to a small hole Cellular microdomains are regulated by chemical reactions involving a small number of molecules that have to find their targets in a complex and crowded environment. 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 small window, through which it can escape [35, 36, 37, 31]. This problem, known as the narrow escape problem in diffusion theory (also called the Narrow Escape Time (NET)), goes back to Lord Rayleigh: the small hole often represents a small target on a cellular membrane, such as a protein channel for ions, a narrow neck in the neuronal spine [3] for calcium ions, and so on. The motion of a Brownian particle in a force field can be described by the overdamped Langevin equation (known as the Smoluchowski limit): √ 1 ˙ (1) x˙ − F (x) = 2D w, γ where D=

kB T , γ

(2)

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


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

|∂Ωa | 1, |∂Ω|

(3)

the escape time can be estimated asymptotically [34]. For this purpose, we use the probability density function (pdf) pε (x, t) of the trajectories of (1) which is the probability per unit volume (area) of finding the Brownian particle at the point x at time t prior to its escape. The pdf satisfies the Fokker–Planck equation ∂pε (x, t) 1 = D∆pε (x, t) − ∇ · [pε (x, t)F (x)] = Lpε (x, t), ∂t γ

(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 is initially located at position y) and the mixed Dirichlet–Neumann boundary conditions for t > 0

D

pε (x, t) = 0

for x ∈ ∂Ωa ,

(6)

∂pε (x, t) pε (x, t) − F (x) · n(x) = 0 ∂n γ

for x ∈ ∂Ωr .

(7)

The function Z

∞

Z

uε (y) =

pε (x, t | y) dt,

dx Ω

(8)

0

where pε (x, t | y) is the pdf conditioned on the initial position, represents the mean conditional sojourn time in Ω for a particle starting at y. It is the solution of the boundary value problem [30] L∗ uε (y) , D∆uε (y) +

1 F (y) · ∇uε (y) = −1 γ uε (y) = 0 ∂uε (y) =0 ∂n

for y ∈ Ω,

(9)

for y ∈ ∂Ωa ,

(10)

for y ∈ ∂Ωr .

(11)

The survival probability is Z Sε (t) =

pε (x, t) dx,

(12)

Ω

where Z pε (x, t | y)ρ0 (y) dy.

pε (x, t) = Ω

(13)

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

∞ X

ai (ε)ψi,ε (x)ψi,ε (y)e−λi (ε)t ,

(14)

i=0

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), Z ai (ε) = ρ0 (y)ψi, (y) dy. (15) Ω

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 Ω and the geometry of the small opening ∂Ω [34, 35, 36, 37]. When the geometry of a hole is regular, the escape time uε (y) is given for ε 1 by [34]  A 1 for dim Ω = 2,   πD ln ε + O(1) |Ω| (17) uε (y) =  for dim Ω = 3,  L(0)+N (0) 4aD 1 + a log a + o(a log a) 2π where a is the radius of the hole assumed to be a geodesic disk located on the surface of the domain Ω. A (resp V ) is the surface (resp. volume) of the domain Ω, and L(z) and N (z) are the principal curvatures of ∂Ω at z. The function uε (y) does not depend on the initial position y, except in a small boundary layer near ∂Ωa [16, 35, 36, 37, 34]. This formula can be extended to the case of several windows. We shall consider only the case of two windows. For a regular planar domain Ω with two absorbing arcs of lengths 2ε and 2δ (normalized by the perimeter |∂Ω|) in its boundary and such the Euclidean distance of the middles is ∆ = ε+∆0 +δ, and for a three-dimensional d = 3 domain Ω with two absorbing circular windows of small radii a and b, such that the Euclidean distance of the centers is ∆ = a + ∆0 + b, we obtained for the NET τ¯ε [18, 17]:  2 2  1 − 16ab 4π|a+∆+b|  |Ω|   for dim Ω = 3,  1  4(a + b)D 1 − 8ab a+b 2π|a+∆+b| (18) τ¯ε =  log 1δ log 1ε − (log |ε + ∆ + δ|)2 |Ω|   for dim Ω = 2,   πD log 1 + log 1 |ε+∆+δ|  1 + 2 log ε δ log 1 +log 1 δ

ε

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


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estimate the NET of viruses to small nuclear pores. We shall now discuss some 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 with coordinates FN = (x1 , . . . , xN ), where the consecutive beads are connected linearly by a spring of constant k and each bead is subjected to isotropic random forces. We are interested in several questions such as the mean time for any one of the beads to hit a small hole ∂Ωa ⊂ ∂Ω or the mean first time for a single specific bead to reach the target (a nuclear pore). This computation generalizes the small hole computations, which gave the mean first 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 constant D/N . Thus it takes more time to reach the small hole, but when N is large enough, the polymer occupies a certain fraction of the space and thus the volume per bead becomes so small that the MFPT of a single bead hits the small hole decreases. Thus this intuitive analysis shows that there is a maximum value for the MFPT as a function of N . In a first part, we present a physical model of the polymer motion, and in a second part we obtain asymptotic estimates of the MFPT to a small hole, as a function of N , the diffusion 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 overdamped medium, such as water or a biological fluid, can be described by a system of first order stochastic differential equations. Neglecting possible hydrodynamic effects, the Smoluchowski limit of the Langevin equation can be written as r ∂U 2 ˙ 1, x˙ 1 + = w ∂x1 γ r ∂U 2 ˙ 2, x˙ 2 + = w ∂x2 γ ... (19) r ∂U 2 ˙ n, x˙ n + = w ∂xn γ . .r . ∂U 2 ˙ N, x˙ N + = w ∂xN γ ˙ 1, . . . , w ˙ N are N inwhere = kT /m and γ is the friction coefficient and w 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

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of two terms k 1 |xk − xk+1 |2 − l0 |xk − xk+1 | , γ 2 k 1 |xk − xk−1 |2 − l0 |xk − xk−1 | . Uk,k−1 (xk , xk−1 ) = γ 2 Uk,k+1 (xk , xk+1 ) =

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

(22) (23)

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

(24)

and the total potential is given by U N (x) =

N X

Uk (xk−1 , xk , xk+1 ).

(25)

k=1

Equivalently, U N (x) = U (x) − (N − 1)l02

k 2γ

(26)

where U (x) =

N k X (|xk − xk−1 | − l0 )2 . 2γ

(27)

k=2

When the polymer FN is confined in a microdomain Ω, each bead is reflected at the boundary except in a small patch ∂Ωa , where any bead can be absorbed. We present here the case where any one of the beads can hit the small hole. To estimate the mean time any of the beads reaches the small 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 element QN dVx = 1 (xk + dxk ) in the (N dim Ω)-space ΩN = Ω × · · · × Ω, | {z }

(28)

N times

p(x1 , . . . , xN , t)dVx = Pr{X 1 (t) ∈ x1 + dx1 , . . . , X N (t) ∈ xN + dxN }. (29) p is the solution of the Fokker–Planck equation (FPE) (see [31]) ∂p(x, t) = D∆x p(x, t) + ∇x [∇U N (x)p(x, t)] ∂t p(x, 0) = p0 (x) for x ∈ ΩN ,

for x ∈ ΩN ,

where D = /γ, and p0 (x) is the initial polymer distribution Z p0 (x) dx = 1, ΩN

(30)

(31)


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∆x is the Laplace operator in N dim Ω variables (x1 , . . . , xN ). The boundary conditions associated with the FPE are given by p(x1 , . . . , xN , t) = 0

for (x1 , . . . , xN ) ∈ ∂ΩN a ,

(32)

J(x1 , . . . , xN , t) = 0

∂ΩN r ,

(33)

for (x1 , . . . , xN ) ∈

where J(x1 , . . . , xN , t) = D

∂p (x1 , . . . , xN , t) + ∇U N (x).np(x1 , . . . , xN , t), (34) ∂nx

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

∂ΩN a =

n=1

Ω × · · · × ∂Ωa × · · · × Ω |{z}

(35)

n

and the reflective part is N N ∂ΩN r = ∂Ω − ∂Ωa .

(36)

Outside the boundary layer of the small hole, we can look for the solution using the ansatz uη (x) = Cη e−U

N

(x)/D

;

(37)

we get Cη ≈ e+U

N

(Q)/D

1 DN |∂Ωa | |Ω|N −1

Z N (Q, x) dSx

(38)

∂ΩN a

and N is the Neumann function, solving the boundary value problem ∆x N (x, ξ) = −δ(x − ξ) ∂N (x, ξ)) 1 =− ∂n(x) |∂ΩN |

for x, ξ ∈ ΩN ,

(39)

for x ∈ ∂ΩN , ξ ∈ ΩN .

(40)

We obtained for two-dimensional domains [15] the expression Z τ¯η (N ) = u(x) dVx ΩN Z Z N d1,N wq−1 1 +U N (Q)/D ≈ e dSQ e−U (x)/D dx, ln N N −1 N D|∂Ωa | |Ω| δ ∂ΩN ΩN a (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 small diffusion limit, it is possible to obtain from (41) the precise dependence of the mean time on the number of beads N (equivalently the length of the polymer). Interestingly, we expect that there is a range of values for k for which τ¯η (N ) has a maximum as a function of N .

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Double strand DNA breaks, rough modeling of a polymer motion in a confined environment To motivate our analysis, we shall now present the underlying biology: cellular radiation can generate two types of DNA breaks: single or double strands DNA breaks (dsDNAb). In the first case, the DNA molecule is not fully disrupted as only one branch is cut and thus the break can relatively easily be 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 break can be associated to a loss of base pairs. This is a severe lesion that needs to be repaired, otherwise, the cell can generate genetic mutations and/or degenerates into a cancer cell. In an extreme case, at high concentration of breaks generated by high level cellular irradiation, it has been observed that the cell triggers apoptosis or cell death. The ability to repair these breaks is thus an essential process for survival. In a reasonable range of break concentration, to repair double-strand DNA breaks, two mechanisms have been identified: the first is the Homologous Recombination where the DNA molecule uses unaffected DNA-strands to repair itself, by searching for base pair complement at the correct location. This process relies on the possibility that moving fragments not only find the complementary genetic information, but glue themselves back together correctly [47]. If these fragments were to move freely under a Brownian motion, the probability that these moving fragments would glue together correctly becomes extremely low, suggesting that a complex and unknown process might exist to prevent incorrect recombination. The second process is known as non-homologous end-joining and consists of the joining of two DNA breaks by simple physical and direct interactions. We focus here on this second process. The process of correct dsDNA repair by non-homologous end-joining is 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 free ends. Incorrect ligation would lead to loss of genetic information (deletions). Despite the current knowledge about the repair machinery, we are still lacking a kinetic view of this process. We now present some preliminary results about the dynamics of dsDNA repair, which uses a biophysical model of DNA motion in a constraint environment [48]. So far, no direct experiments have revealed the dynamics of repair. Whatever the details of the mechanism, two DNA broken ends have to find each other, and this presumably must happen via random motion. If the DNA strands are floating freely in the nucleus, without any physical constraint, and do not meet quickly enough, it will be extremely unlikely that they will ever meet in a reasonable time (on the scale of the cell). Therefore, it is likely that surrounding structures that restrict the movement of the DNA play a major role in determining whether two DNA strands will meet. To identify the physical phenomena involved in DNA repair, we model the dynamics of an isolated DNA break by a search mechanism of the two cut strands (ends) in a neighborhood of the break.


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Using the Brownian dynamics, we analyzed the motion of two attached polymers moving in a confined environment. In our analysis, gluing, which is the most sensitive part of the repair process, is achieved when the two polymer chains 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 later steps 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 of the polymer (see previous section), represented as a collection of beads connected by springs of constant k. The mean length between the beads is l0 and is usually known as the persistence length. Each bead can move according to a Langevin equation, driven by a potential field, generated by the spring action of its immediate neighbors. To account for the geometry of the DNA and nucleus organization, we shall assume that the DNA molecule can only move in a restricted strip of length L and thus no beads can exit from the strip. To account for the nucleosome organization, we fix the initial position of a bead. Interestingly for small L (result not shown here), most of the DNA breaks can be repaired. To obtain some quantitative analysis, we consider the drastic simplification where the motion of the DNA molecule tip can be approximated as a one-dimensional Brownian motion. We consider the motion of 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 coalesce into a single one subjected to a Brownian motion. The probability PM that the two particles meet before one of them hits the boundary of the interval can be obtained as a function of the initial positions a < x1 < x2 < b; we found [14] that ω(Z − a) −2 √ =m log P , (43) PM (x1 , x2 ) = π L 8 where =m denotes the imaginary part, P is the Weierstrass elliptic function P02 = 4P3 − g2 P − g3 ,

√

with parameters g2 = 1 and g3 = 0, L = b − a, Z = x2 + −1x1 , and Z +∞ dx ω= = 5.244115106. [x(x − 1)]3/4 1

(44)

(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 comes from conformal mapping. A word about the method The dynamics of each particle is given for i = 1, 2 by p dXi = 2Df dwi

(46)

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

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meet before one of them exits the interval [a, b]. If we consider the two random times τ1 = inf{t > 0 : X1 (t) = a or X2 (t) = b, X1 (0) = x1 and X2 (0) = x2 , x1 < x2 }, τ2 = inf{t > 0 : X1 (t) = X2 (t), X1 (0) = x1 and X2 (0) = x2 , x1 < x2 }, then for x = (x1 , x2 ), the probability PM (x) = Pr{τ2 < τ1 | x}

(47)

satisfies the Laplace equation (ch. 15, p. 192 of [38]) ∆PM (x) = 0

for x ∈ T,

PM (x) = 1

for x ∈ D,

(48)

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 transformation to map T onto the upper half-plane H. By using the explicit solution of the Laplace equation in H, we compute the solution of (48). It turns out that the Schwarz–Christoffel function is the log a Weierstrass function. PM (x) = 0

3. Brain organization, memory and synaptic plasticity Although the notion of memory storage for computers is quite clear, we still do not know how and where memory is stored in the brain. Various scales are involved in memory encoding: molecular and cellular scales and the cortical network. At the cellular level, neurons make micro-contacts, either directly on dendrites 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 neurons.


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intriguing that a single neuron can contain of the order of 100 000 of these spines, although their function is still unknown. However, their mushroom shape geometry has attracted the attention of the neurobiological community, interested in quantifying the diffusion of molecules or studying their electrical properties [1, 13]. For example, many fundamental regulatory processes, such as calcium dynamics, occur in spines. It is still a matter of debate to quantify the amount of calcium that crosses a dendritic spine [21, 32], after receptors have been activated. We would like to determine the number of chemical bonds that calcium ions have made during their journey inside the spine. However, we are still limited experimentally by using exogenous buffers or fluorescent dye molecules that significantly perturb the chemical reactions [11]. The geometry and the spine organization might underlie synaptic plasticity and other complex learning functions that should be understood; in particular, how spine uses its geometrical shape in encoding memory. Moreover, neuronal communication relies on micro-contacts called synapses. This communication induces in a change of the electrical activity, controlled by few channels (approximatively 50 to 100 only) that can in addition vary due to protein trafficking [7]. Thus in this context, it is not clear how the neuronal signal is stable over time, especially if synapses are memory checkpoints. An important question would be to estimate the number of receptors and how they are controlled. Thus, how the synaptic connections can be maintained for years, if the lifetime of receptors is about 24 hours? The proteins have to be replaced constantly and correctly. New scenarios inevitably appear with new mathematical models to explain the accuracy of such processes [9]. Recently, using asymptotic analysis of the mean first passage time equation for diffusing particles (receptors on the surface and calcium ions inside) in the spine, we estimated [10] the mean time for a diffusing particle (with diffusion constant D) to escape a thin spine neck. We obtained in general

τH ≈

 ΩH 1 L2 LΩH     D2a + πD ln a + 2D

in 2-dim,

 2    LΩH + ΩH + L Dπa2 4Da 2D

in 3-dim,

(49)

where ΩH is the spine head and L (resp. a) is the length (resp. the radius) of the spine neck. More complex chemical reactions should be taken into account in order to reveal the complex function of synapses. A lot more remains to be modeled and understood. Combining theory and experiments was already very beneficial to many fields of biology, it was used to unravel the organization of the visual cortex, where specific neurons fire in responses to visual stimuli. The presentation of a rotating bar induces a neuronal activity of neurons that also rotates around points, which are topological singularities, called pinwheel. It is interesting

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to find this type of typological singularities here. Noise seems to play a crucial role in the maintenance of the neuronal activity and in coding spatial information. To conclude this short and superficial survey, we shall add that this is only the early beginning where quantitative approaches and mathematical analysis are joining efforts to address questions related to cellular biology. A new generation of physicists, applied and pure mathematicians should be trained to crack biological challenges of tomorrow and to unravel the concepts behind the essence of life. There is much to benefit from mathematical training. In biology, there is no shortage of problems and we can compare our time with the gold rush of California in the mid XIXth century, where the rule was first come, first served. Today we do not have to bend much to reap, while there is also room for those who want to dig deeper. Now that the stock market is down, a new generation of theoreticians can either find new principles and rules to provide foundations of a future and hopefully stable economy, with out forgetting to put back the controllers or alternatively, this generation 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 contained in 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 penetrating 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 encoded? 8. How gradients and boundaries are made in early embryos? What is the variability due to this construction? In other words, once an egg is fertilized, with what accuracy a chicken is fashioned? What is the principle for making morphogenetic gradients? Can we estimate the fluctuation of 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 Program 0007/2006-C.


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