## Lectures 2011-2012 : Stochastic analysis and applications to molecular dynamics in the cell nucleus

### Class : Stochastic analysis and applications to molecular dynamics in the cell nucleus

David Holcman

**WHEN :** Oct-Feb 2011-2012

Wed. 10h30-13h30.
**
Starting date : ** Sept. 29 2011

**WHERE** "Salle Conference" : 46 rue d’Ulm, 75005 Paris

**General description : **

Eukariotic cell are characterized by an envelope that separates the genetic material from the other organelles. Chromosomes are highly dynamics structures, but we still do not understand the associated physical rules. The goal of the class is to present the physical rules and the underlying mathematical analysis, used to explain and predict the chromosomal organization in various conditions.

The first part of the class will be based on stochastic analysis and partial differential equations in parallel with some basic statistical physics considerations.

In the second part, we will focus on telomere organization (ends of the chromosomes) and introduce a model of aggregation-dissociation to describe telomere clustering.

Finally, we will study the motion of mARN or fundamental proteins transported in the nucleus, some models of gene activation and some physical principle associated with DNA repair.

**General theory**

**Part I**

Brownian motion, Ito calculus. Dynkin’s equation, Fokker-Planck equation, Short and long time asymptotics, ray methods.

Mean first passage time Equations, conditional MFPT, distribution of exit points. -Escape in dimension 1 : Smooth and sharp potential barrier.

Jump processes. Approximation by continued fraction.

The small hole theory, no potential, with an attracting and a repulsive potential. The case of one and several holes.

Minimization of MFPT, consequence hole distribution. MFPT in random environments.

Narrow escape time for a switching particle. Homogenisation theory with many small holes.

Direstrait time and conformal mappings. sum of MFPTs : Green’s function identities (Dendrititc spine laws : micro chambers).

Modeling diffusion of shaped object, law of reflection, polymer dynamics using Rouse model.

**Part II and III**

Aggregation-dissociation of telomeres, formation and stability of the telomere cluster. Mean encounter time, Markov description of clusters. Dissociation scenario of cluster.

Recurrent time of 2 telomeres, dissociation time from a cluster. Asymptotic estimations.

Dynamics of diffusion of chromosomes and polymers. Stochastic dynamics of anisotropic objects in confined microdomains.

Gene activation by transcription factors.

Dynamics of the double strand DNA break repair.

### Requirements : notions of partial differential equations, probability, cellular biology.

**Evaluation : ** small projects.

#### References :

**Basics :**

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

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

Schuss, Z., Theory and Applications of Stochastic Processes by Zeev Schuss (Hardback, 2009) Springer ; 1st Edition. (December 21, 2009)

**Advanced :**

J. Reingruber D. Holcman, Transcription factor search for a DNA promoter in a three-state model, PRE short com 000900® (2011)

D. Holcman N. Hoze Z.schuss, Narrow escape through a funnel and effective diffusion on a crowded membrane, PRE, 001900 (2011)2011

G. Malherbe D. Holcman, Stochastic modeling of gene activation and application to cell regulation, J. Th.Bio 2010.

G. Malherbe D. Holcman, Search for a DNA target site in the nucleus, Phys. Lett. A. 374,3,2010, 466-471

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 A. Singer, Z. Schuss, Narrow escape and leakage of Brownian particles. PRE 78:051111 (2008).

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

Reingruber D. Holcman, Narrow escape for a switching state Brownian particle, PRL 2009.

T. Lagache E. Dauty D. Holcman, Physical principles and models describing intracellular virus particle dynamics, Current Opinion in Microbiology, 12,4 (2009).

D. Coombs and R. Straube M Ward Diffusion on a Sphere with Localized Traps : Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points (SIAM J. Appl. Math., Vol. 70, No. 1, (2009), pp. 302-332.)

S. Pillay, A. Peirce, and T. Kolokolnikov, M. Ward, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems : Part I : Two-Dimensional Domains (SIAM Multiscale Modeling and Simulation, (March 2009), 28 pages.)

A. Cheviakov and R. Straube M. Ward, An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems : Part II : The Sphere (SIAM Multiscale Modeling and Simulation, (March 2009), 32 pages.)