Math 6770 - Mathematical Biology (Stochastic Processes in Physiology)
Time: T Th 10:45-12:00 am LCB 222
Prerequisite: Some knowledge of probability, partial differential Equations, and rudiments of physiological processes
Gardiner, Handbook of Stochastic Methods
Allen, An Introduction to Stochastic Processes with Application to Biology Van Kampen, Stochastic Processes in Physics and Chemistry Berg, Random Walks in Biology Lawler, Introduction to Stochastic Processes Grimmett and Stirzaker, Probability and Random Processes Fall, Marland, Wagner, and Tyson, Computational Cell Biology Kloeden and Platen, Numerical Solution of Stochastic Differential Equations
Stochastic processes: Markov Processes, Random walks, Brownian Motion - Weiner-Einstein process, Poisson processes, Ornstein-Uhlenbeck process.
Stochastic equations: Ito equation, Master equations - Chapman-Kolmogorov, Langevin equation, Fokker Planck equations.
Exit time probabilities
Numerical methods: Monte Carlo, Gillespie and Metropolis algorithms
Biology - Applications: Chemotaxis via biased random walks, Birth-Death processes, Binding-unbinding dynamics: Kramer's law, kinetic rate theory (Evans and Ritchey, etc.), Ion channel kinetics -stochastic ML, HH, potassium, sodium channels (fast transitions), calcium sparks (fast dynamics - Hinch), Stochastic calcium dynamics - sparks and waves, IFT (IntraFlagellar Transport), Noisy neurons, stochastic resonance (noisy IF neurons, noisy phase oscillators-Nesse), neural filtering (hair cells ala Mike Reed), noisy thresholds, molecular motors and brownian ratchets - flashing ratchet, microtubule growth and collapse.
Homework assignments will be posted and updated regularly at this .pdf file.
In addition to homework problems, students (in groups of 2-3) will be expected to do a project, which will consist of reading a paper or two, developing and studying a model of some biological relevance. Proposals for each project must be submitted by Sept. 29, and the Final Project report will be due at the end of the class. In addition, each student group will be required to give an in-class presentation describing the results of their research project.
Invariant manifolds and Ion channel kinetics.
Notes on Off-Rates and the Kramer-Bell Formula.
Two state channel Matlab code (Gillespie algorithm).
Stochastic Lotka Volterrra simulation Matlab code (Gillespie algorithm).
Adiabatic Reduction of Master Equations.
For more information contact J. Keener, 1-6089