Lecture Class :Stochastic analysis, Asymptotics of Partial Differential Equation and Applications to Big Data in molecular and cellular dynamics
WHEN : Oct-Feb 2014-2015
Starting date : Wed September. 24 2014
WHERE "Salle Conference" : 46 rue d’Ulm, 75005 Paris
see also Bionewmetrics online class
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 million by million about the mean distances between any two locus on the chromatin or 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 physical modeling and recent mathematical analysis, used to explain and predict datas. 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 applications to understand the nucleus organization, synapses and neural network in neuroscience.
Holcman’s class on applied mathematics and application in neuroscience and modeling cellular microdomains (September 25 2013).
General description of the class : from modeling in cellular biology to mathematical equations and methods.
2- Introduction dynamical systems
General considerations on dynamical systems. Lecture Part 1
Linearization neat a critical point. Lecture Part 2
Limit cycle, recurent sets. Lecture Part 3
Discussion of the Hopf’s Bifurcation. Lecture Part 4
Exercices from Control switching. Lecture Part 5
3- Statistical physics approach to Langevin’s equation
Derivation of Langevin equation from Gas theory (Chandasekar derivation)
4- Stochastic processes
Space of non-anticipating function
Ito’s Example : \int w^2(s)ds
Characterization of the drift and tensor : Feller’s formula.
Conditional process : stochastic equations.
Derivation of the exit point equation from Ito’s formula. Distribution of points in the dimensional disk. Poisson Kernel. Lecture Distribution of exit points
Derivation of the MFPT from Ito’s formula : Lecture Mean Firt Passage Time
Derivation of the FPE from Ito’s formula. Lecture Fokker-Planck equation
Boundary conditions : Dirichlet, Neuman and mixed boundary conditions
Derivation from discret brownian simulations : law of reflection, boundary layer analysis for a Random walk : jump condition.
Reflection in simulations : accross a surface, shaped objects.
Rod motion in a slab : stochatic dynamics, Perrin dynamics for stochatic ellispses.
5-Asymptotics estimation of the MFPT
Introduction to Laplace’s method
conformal mappings : principle, exp, Mobius, transfomartion of a Hopf-field, exp (1/z).Schwartz-Christofel : triangle and the probability of an encounter before exit.
Hopf’s dynamical system transformed by a Mobius map.
Greens’ functions : A) R^n, S^n, bounded domain, singular parts, regular parts, Neumann function, the factor 2, approximation. Lectture-Green-part1 B) Explicit expression. Ward’s analysis.
Lecture Additive formula for the MFPT. Derivation from Green’s function.
Escape from an attractor : asymptotics using boundary layer analysis
Narrow escape : general Introduction on the narrow escape : mean time for a Brownian particle to escape from a small hole. Physical description and mathematical formula as a mixed boundary value problem. lecture general introduction NET
Narrow escape : dimension 3.
Narrow escape with a drift. Application with a potential well. Motion near confined to surface (application to motion of vesicle in the soma for a growing neuron).
Direstrait Time in dim 2 : time to escape from a singular cusp. Application to the rotation of a stochastic needle in a band.
Narrow Escape : many holes.
Classical Activation escape : -generic case. -accross a boundary. -Oscillation escape.
Probability with killing : -killing measure Lecture Killing-1 -Survival probability Lecture Killing-2->http://www.youtube.com/watch?v=dult…] -Examples : delta-Dirac and homogenous. Application to diffusion of ions in microdomains.
Theory : Markov equation. Kramers-Moyal approximation, takas equation.
Application to Telomere dynamics.
Search process in the nucleus
Theory of aggregation with a finite number of particle
7-Applications to model cellular microdomains
Stochatic chemical reactions : modeling stochastic reactions occuring in microdomains. Derivation of the Master equations. Equation for the first time to a threshold. Steady state analysis. 2 lectures : Lecture SCR Part I and Lecture SCR Part II.
Virus Dynamics : -homogenization of the drift.
-Arrival probability to the nucleus.
Diffusion in dendritic spines : modeling and introduction to diffusion in composite domain. Derivation of diffusion laws of escape from a head connected to a cylinder. Application of the narrow escape formula. Lecture Diffusion Spines.
Telomere organization in the nucleus : theory, simulation and data analysis.
8-Analysis of superresolution data -empirical estimation of the Langevin’s equation parameters. -Reconstruction of a surface from projection of large of short stochastic trajectories -Simulations in empirical domains
8-Polymer models to study nuclear organization -Looping laws in free and confined domains. -Chavel-Feldman theory for perturbed spectrum. -Extraction of information from Chromosomal-Capture data. -Beta-model theory. -Interaction recovery theory.
Basics : 1-Molecular Biology of the Cell, B. Alberts, et al,4rd ed., 2002.
2-Z. Schuss D. Holcman, The dire strait time, SIAM Multicale Modeling and simulations, 2012.
3-D. Holcman Z. Schuss, the Narrow Escape Problem, SIAM Rev 56 no. 2, 213–257 2014 4-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). 5-Z. Schuss, Brownian Dynamics at Boundaries and Interfaces, Springer series on Applied Mathematics Sciences, vol.186 (2013). 6-Schuss, Z., Theory and Applications of Stochastic Processes (Hardback, 2009) Springer ; 1st Edition. (December 21, 2009)
• 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. 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 eff
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
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.)
Description of the live class : Syllabus :
The goal of the class is to present methods in applied mathematics, statistical physics and simulations to study computational problems in cell and neurobiology. The class will also introduce method in data analysis.
In brief, 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 part II 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, in the third part, we will study synaptic transmission, network properties, Up-down state dynamics and neuron-glia interaction.
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.
Explicit computation of the spectrum of the (non-selfadjoint operator) Fokker-Planck operator, distribution of exit time. Application in neurobiology.
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
Modeling diffusion of shaped object, law of reflection, polymer dynamics using Rouse model.
Part II : dynamics of the cell nucleus
Modeling telomere length dynamics using Markov jump processes : law of the shortest telomere, mean time to threshold and scenecence (Takas equation)
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 : simulation and and mean first looping time in confined environement. Analysis of Chromosome-capture data.
Search process in the nucleus by transcription factors.
Dynamics of the double strand DNA break repair. Analysis of a locus motion. Extraction feature from live cell imaging data.
Part III : synaptic properties
Mean-field model for synaptic depression. Distribution of times in the Up-state.
Modeling synaptic transmission : simulations, approximation, receptor trafficking, diffusion in narrow domains
Neuron-gli interaction, modeling.
Diffusion in dendrititc spine laws.
Electro-diffusion in spines.
Requirements : notions of partial differential equations, probability, cellular biology.
Evaluation : small projects.