## Lectures 2012-2013

**Asymptotic analysis and computational methods : Applications to molecular, cellular biology and to neurobiology (synaptic transmission and plasticity) (6 ECTS)
**

David Holcman, École Normale Supérieure

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

**Wed. 9h30-12h30. Starting date : Sept. 27 2012**

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

**Syllabus :**

** General theory :**
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.
The small hole theory, no potential, with an attracting and a repulsive potential.
Small hole in a narrow domain.
The case of one and several holes.
Minimization of MFPT, consequence hole distribution.
Narrow escape time for a switching particle.
Modeling diffusion of shaped object, law of reflection, polymer dynamics using Rouse model.
Dire strait time : time to reach a hole in a funnel or at a cusp.

Application to chemical reactions : Stochastic chemical reactions in a microdomain. Theory of threshold.

Application to neurobiology and synaptic transmission : Receptor trafficking, synaptic current, Resident time of a receptor at synapses (with a small and a large amount of scaffolding molecules) Calcium dynamics in a dendritic spine. Modeling synaptic transmission : homogenization of the Robin constant for small cluster a receptors. Modeling synaptic transmission, synaptic weight. Synaptic cleft. Modeling of Neuron-Gli interactions. Stochatic methods to analyze superresolution data.

**Summary :** Toward a quantitative approach in cellular biology.

**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.
D. Holcman, Computational challenges in synaptic transmission, AMS Contemporary Mathematics : Proceedings of the Conference on Imaging Microstructures : Mathematical and Computational Challenges, Edited by H. Ammari and H. Kang. 2009
Schuss, Z., 1980, Theory and Applications of Stochastic Differential, New York, Wiley.
Schuss, Z., Theory and Applications of Stochastic Processes (Hardback, 2009) Springer ; 1st Edition. (December 21, 2009)

** Advanced :**
D. Holcman Z. Schuss, Brownian needle in dire straits : Stochastic motion of a rod in very confined narrow domains, Phys. Rev. E 85, 010103® (2012) 2012.
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.
D. Holcman N. Hoze Z.schuss, Narrow escape through a funnel and effective diffusion on a crowded membrane, PRE, 001900 (2011)
K. Dao Duc D. Holcman, Threshold activation for stochastic chemical reactions in microdomains, PRE 1, 81,2010.
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
A. Taflia D. Holcman, The Optimal PSD of synaptic transmission
J. Reingruber and D. Holcman, Diffusion in narrow domains and application to phototransduction, Phys Rev E 2009 ;79(3 Pt 1):030904.
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.)

### 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 synaptic compartment 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 course is to present the mathematical and computational methods to study synapses, which are local micro contacts between neurons, underlying their direct communication. The course will give the basis to combine theoretical method with experimental data. The class starts with stochastic tools to study trafficking of receptor channels, which are the fundamental molecular components underlying the synaptic current. We will also 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. The methods allow studying 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.