## Class 2008-2009

**Introduction to modeling in cellular biology and synaptic transmission
**

*where :* Salle Conference, 46 rue d’Ulm, 75005 Paris.

*When :* Wed. 9h-12h.

**Starting date :** October 2008 (to be confirmed)

**Syllabus :**

Overview

Brownian motion, Ito calculus.

Dynkin’s equation, Fokker-Planck equation, Short and long time asymptotics.

Mean first passage time.

The small hole theory with a attracting (resp. repuslive) potential.

Homogenisation theory with many small holes.

Stochastic chemical reactions in a microdomain.

Modeling synaptic transmission, synaptic weight.

Receptor trafficking, synaptic current, Dwell time of a receptor at the synapse. Calcium dynamics in a dendritic spine.

Cellular trafficking.

Modeling of Neuron-Gli interactions.

Summary : Toward a quantitative approach in cellular biology.

**Requirements :** notion of partial differential equations, probability,some notions of cellular biology.

**Evaluation :** small projects.

**References :**

Molecular Biology of the Cell, B. Alberts, et al,4rd ed., 2002.

D. Holcman, Z. Schuss. 2004. Diffusion of receptors on a postsynaptic membrane:exit through a small opening, J. of Statistical Physics, 117, 5/6 191-230.

A. Singer Z, Schuss, D. Holcman, Narrow Escape I, II and III, in J. of Statistical Physics, 2006, Vol 122, N. 3, p 437 - 563.

D. Holcman, Z. Schuss. 2005. A theory of stochastic chemical reactions in confined microstructures, Journal of Chemical Physics 122, 114710, 2005.

D. Holcman A. Marchevska Z. Schuss, The survival probability of diffusion with trapping in cellular biology Phys.Rev E Stat Nonlin Soft Matter Phys. 2005,72 : 031910.

Schuss, Z., 1980, Theory and Applications of Stochastic Differential, New York, Wiley. (Book)

**Description :** The precise function of biological microdomains, and specifically neurobiological microstructures such as synapses, is still unclear despite the great efforts of the past 20 years to unravel the molecular pathways responsible for the maintenance and modulation of cellular functions, and ultimately, to extract fundamental principles. The cytoplasm of eukaryotic cells is a complex environment where dynamical organelles, cytoskeletal network and soluble macromolecules are organized in heterogeneous structures and local microdomains. These sub-micron domains may contain only a small number of molecules, of the order between just a few and up to hundreds.

The goal of this class is to present modeling methods to study cellular microdomains and in particular synapses, which are local micro contacts between neurons. Because synapses are still unavailable to direct experimental recordings, mathematical modeling and simulations have been used to analyze their functions.

In the class, we will describe stochastic tools to study receptor trafficking, which are the fundamental molecular components underlying the synaptic current. We will introduce the partial differential equations necessary to estimate the mean time a receptor stays inside a synapse. We will derive the explicit dependence of quantities such as the decay rate of the population of receptors or the forward chemical reaction rate constant on the geometry of the domain and present general methods to study the dynamics of a Brownian particle (ion, molecule, protein) confined to a bounded domain (a microcompartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The mean escape time from the domain diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.

The present methods are general and can be applied to various microdomains where there is a low molecular number and thus most of the existing models based on continuum concepts, in which the medium is assumed homogeneous cannot be applied. In that case, to obtain quantitative information about chemical processes, modeling and simulations seem to be inevitable to reconstruct the microdomains environment and to obtain precise quantitative information about the molecular dynamics.

**
!!Warning to students !!**
The modern approach of modeling cellular biological systems requires many branches of theoretical physics, mathematics and chemistry. A minimal package includes some knowledge in stochastic processes, partial differential equations, statistical physics, differential equations, dynamical systems, asymptotics analysis, some numerical analysis and some differential geometry. In addition, it is necessary to include to the list the biological fields of interest and some knowledge of theoretical chemistry. Finding principle in biology using theory is not easy, although it looks very attractive. There are no hopes to enter seriously into this new field without going through the previous materials. I encourage motivated students to start studying the fields mentioned above. Again, they are no shortcuts.

**General references**

References Books

Stochastic equations and applications

Schuss, Z., Theory and Applications of Stochastic Differential 1980, John Wiley.

Freidlin, M. I. Random perturbations of dynamical systems New York : Springer-Verlag, 1984

Ito, K McKean H., Diffusion processes and their sample paths / Kiyosi Ito, Berlin : Springer, c1965,
WIS library 519.233 ITO

Oksendal B. Stochastic Differential Equations : An Introduction With Applications,Springer-Verlag ; 2nd ed edition (November 22, 1989)

M. Rao, Brownian motion and classical potential theory, 1977, Lecture Notes series, no 47.

S. Karlin H. Taylor, First course in stochastic processes 1975

S. Karlin H. Taylor, Second course in stochastic processes,Academic Press (1981) 542 pages.

M. Tsuji, Potential theory in modern function theory,1959

M. Kac : probability, number theory, and statistical physics The MIT Press (1979)

M. Kac, Aspects probabilistes de la théorie du potentiel. (French) Séminaire de Mathématiques Supérieures (Été, 1968). Les Presses de l’Université de Montréal, Montreal, Que., 1970. 151 pp.

Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.
-M. Freidlin, Wentzell, A. Random perturbations of Hamiltonian systems. Mem. Amer. Math. Soc. 109 (1994), no. 523,

F. Spitzer, Principles of Random walks, Springer 1976
Analysis, partial differential equations

John, Fritz, Partial differential equations / [by] F. John New York : Springer-Verlag, 1971 , 515.353 JOH

Courant, Richard, Methods of mathematical physics / by R. Courant and D. Hilbert Methoden der mathematischen Physik, New York : Interscience Publishers, 1953-1962

Garabedian, Partial Differential Equations

Landkof NS, Foundations of modern potential theory,Berlin : Springer-Verlag 1972

Sneddon IN, Mixed boundary value problems in potential theory,Amsterdam : North-Holland Pub. Co., 1966

Folland G, Introduction to partial differential equations,Princeton, NJ : Princeton University Press, c1995

N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces / Providence, RI : American Mathematical Society, c1996

B. Helffer, Semi-classical analysis for the Schrödinger operator and applications (Lecture notes in mathematics)
by Springer-Verlag (1988) 107 pages

M. Costabel,M. Dauge, S. Nicaise, Boundary Value Problems and Integral Equations in Nonsmooth Domains (Lecture Notes in Pure and Applied Mathematics) Marcel Dekker1994.

M. Dauge, Elliptic Boundary Value Problems on Corner Domains : Smoothness and Asymptotics of Solutions (Lecture Notes in Mathematics, 1341), Springer (1988)

Kozlov, V. A. ; Maz’ya, V. G. ; Rossmann, J. Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs, 85. American Mathematical Society, Providence, RI, 2001

Kozlov, V. A. ; Maz’ya, V. G. ; Rossmann, J. Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, 52. American Mathematical Society, Providence, RI, 1997.

Maz’ya, Vladimir ; Nazarov, Serguei ; Plamenevskij, Boris Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I and II. Translated from the German by Plamenevskij. Operator Theory : Advances and Applications, 112. Birkhäuser Verlag, Basel, 2000. xxiv+323
Dynamical systems

Y. Kuznetsov, Elements of applied Bifurcation theory, Springer 112, 1994

C. Robinson,
Analysis,Markov Chains and Queueing

L. Kleinrock, Queueing System, vol I, II, Wiley 1975, 519.82 Kle

Saaty T, Elements of queueing theory, : With applications McGraw-Hill (1961)
Neurobiology and biochemistry

Principles of Neural Science by Eric R. Kandel (Editor), James H. Schwartz (Editor), Thomas M. Jessell (Editor)

Hille B, Ionic Channels of Excitable Membranes , 3rd ed., Sinaur, 2001

Molecular Biology of the Cell, B. Alberts, et al,4rd ed., 2002

Stryers L., Biochemestry published by W. H. Freeman and Company.

Biochemistry (3rd Edition), C. K. Mathews, K. E. van Holde, K. G. Ahern

Biochemistry by Donald Voet
Computational Biophysics

Biophysics of Computation : Information Processing in Single Neurons (Computational Neuroscience)
by Christof Koch (Hardcover - November 1, 1998)

Statistical physics Introduction to polymer dynamics, PG de Gennes, Cambridge Uni. Press 1990 Theory of polymer dynamics Doi Edwards, Oxford Science publication Scalling concepts in polymer physics, PG de Gennes, Cornell Univ. Press1979.

Review Articles

Research Articles

Maz\cprime ya, V. G. ; Nazarov, S. A. ; Plamenevski\u\i, B. A. Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small openings. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 347—371. (Reviewer : S. Z. Levendorski\u\i)