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)
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.
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.
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.
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