Class 2017-2018

Stochastic analysis, asymptotic of Partial Differential Equation, application to Big Data in molecular and cellular dynamics and neuroscience

David Holcman

WHEN : Oct-Jan 2017-18

Wed. 16h30-19h30. Starting date : beginning of October

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

Youtube class google

General description : A large amount of data are now generated in molecular and cellular biology using recent techniques such as superresolution microscopy for trajectories of single molecular particles, chromosomal capture, leading to matrix of millions by millions about the mean distances between any two locus on the chromatin. Other examples are multi-electrode array to record signal from 10 000 electrodes. How to make sense of such signals and extract information ?

The goal of the class is to present modeling and recent mathematical analysis, used to explain and extract features hidden in large data. The first part of the class will be based on stochastic analysis and partial differential equations and statistical physics. In the second part, we will focus on several applications such as the nucleus organization, synapses and neural networks in neuroscience.

This class (in english) is based on the Holcman’s Cambridge lecture and e-class presented in… and Youtube contact : david.holcman


Part I

Stochastic processes, Fokker-Planck equation, jump process

Recovering a stochastic from noisy trajectories (formula Feller+Hoze PRE 2015)

Short time asymptotics

• Dim 1 and n

• Escape for the fastest particle dim 1 (result in dimension 3)

Exit problem and boundary layer for linear PDE for Mean First Passage Time Equations.

Small hole theory : search for a small target.

• Two holes and many

• Direstrait Non-selfadjoint Fokker-Planck and the full spectrum External Greens’function.Application to sensing

• Dim 2 and 3

• Hybrid simulations (stochastic analytical)

Model of electro-diffusion, asymptotic and singularities.

• General model : gap junction, channels, pumps, glial cell neurons and interactions

• Ball PNP

• Cusp funnel, dim 2 and 3.

• Injection of a current.

Deconvolution of time series (voltage dye)

• Causal signal

• Electro-diffusion with PNP

Introduction to projection of microscopy data. New Nonlinear PDE.

Application to superresolution data analysis.

Nucleus organization :

• HI-C

• Modeling polymer dynamics using Rouse model

• RLC-polymer model and statistical properties

• Looping time

• Search for a target

Part II and III

Synaptic transmission and plasticity.

• Model of the current. synaptic current.

• Modeling synaptic transmission : homogenization of the Robin constant for small cluster a receptor.

• Modeling synaptic transmission, synaptic weight. • Diffusion in the Synaptic cleft.

Diffusion in microdomains :

• Calcium dynamics in a dendritic spine.

• Molecular and vesicular trafficking.

• Hybrid (Markov and mass-action) model of reaction-diffusion.

Analysis of nucleus organization.

• Search time (asymptotic formula) for modeling DNA break repair. • Recurrent time of 2 telomeres,

• dissociation time from a cluster. Asymptotic estimations.

Aggregation-dissociation with a finite number of particles in confined microdomains. Application to Virus assembly and telomere organization. Evaluation : small projects.

References :

D. Holcman Z. Schuss, Stochastic Narrow Escape : theory and applications, Springer 2015

D. Holcman, Z. Schuss, Asymptotics of Singular Perturbations and Mixed Boundary Value Problems for Elliptic Partial Differential Equations, and their applications, Springer (in press) 2017

Basics :

Z. Schuss D. Holcman, The dire strait time, SIAM Multicale Modeling and simulations, 2012. D. Holcman Z. Schuss, the Narrow Escape Problem, SIAM Rev 56 no. 2, 213–257 2014 D. Holcman, Z. Schuss Control of flux by narrow passages and hidden targets in cellular biology, Reports on Progress in Physics 76 (7):074601. (2013). Z. Schuss, Brownian Dynamics at Boundaries and Interfaces, Springer series on Applied Mathematics Sciences, vol.186 (2013). Schuss, Z., Theory and Applications of Stochastic Processes (Hardback, 2009) Springer ; 1st Edition. (December 21, 2009)

Advanced :

• N.Hoze, N. Deepak, E. Hosy, C. Sieben, S. Manley, A. Herrmann, JB Sibarita, D. Choquet, D. Holcman, Stochastic analysis of receptor trajectories from superresolution data, PNAS doi:10.1073/pnas.1204589109 2012. • N. Hoze Z. Schuss D. Holcman, Reconstruction of surface and stochastic dynamics from a planar projection of trajectories, SIAM Journal on Imaging Sciences 2013 • Z. Schuss D. Holcman, The dire strait time, SIAM Multicale Modeling and simulations, 2012. • Dao Duc, D. Holcman, Computing the length of the shortest telomere across cell division, Phys. Rev Lett. 111, 228104 (2013). Spotlight of Exception Research Physics, 6, 129, 2013 • 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.)