Class 2009-2010
Oct-Feb 2009-2010
Wed. 12h30-16h.
Starting date : October 7 2009
Salle Conference :
46 rue d’Ulm, 75005 Paris
Syllabus :
– Overview
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.
– MFPT in random environments.
– Narrow escape time for a switching particle.
– Homogenisation theory with many small holes.
– Modeling diffusion of shaped object, law of reflection, polymer dynamics using Rouse model.
Application to chemical reactions :
– Stochastic chemical reactions in a microdomain.
– Theory of threshold.
– Clustering in confined environment, application to telomere organization in the nucleus
Application to cellular microdomains :
– 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.
– Phototransduction and sensory cells.
– Neurite growth.
– Introduction to morphogenesis and the boundary formation in cellular tissue.
Toward a quantitative cellular virology, cellular trafficking.
– Cellular trafficking : diffusion, active transport by molecular motors.
– Drift homogenisation
– First arrival time of the first virus to a nuclear pore.
– Escape from endosome.
Application to neurobiology :
– Modeling synaptic transmission, synaptic weight. Synaptic cleft.
– Modeling of Neuron-Gli interactions.
-Summary : Toward a quantitative approach in cellular biology and open questions (signaling in virus trafficking, growth cone dynamics, DNA breaks, and many others).
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.
Advanced :
– 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
– 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.
– 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.)
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 will be to present methods to study synapses, which are local micro contacts between neurons, underlying their direct communication. Because synapses are still unavailable to direct experimental recordings, they have been studied by means of modeling and simulations.
We will present here 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 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.