Math 6780 - Mathematical Biology (Stochastic Processes in Physiology)
Time: TH (LCB121) 12:25 - 1:45pm
Overview: This course will give an introduction to the study of stochastic processes with specific applications drawn from physiology. This course will cover a variety of topics, including Markov Processes (birth-death processes chemical reactions), Poisson processes, Random walks and Brownian Motion - Weiner-Einstein process, Ornstein-Uhlenbeck process. We will discuss stochastic differential equations including the Ito equation, Shmolukowski equation, Chemical Master equations - Chapman-Kolmogorov equation, Langevin equation, Fokker Planck equation (and maybe Black-Scholes equation, but maybe not). as well as various approximation methods (system size expansion, adiabatic reduction). Applications will include chemotaxis via biased random walks, chemical binding-unbinding dynamics: (Kramer's law, kinetic rate theory), Ion channel kinetics -stochastic sodium channels and calcium release (sparks), polymerization and gelation, microtubule growth and collapse, IntraFlagellar Transport, flagellar construction, Noisy neurons, stochastic resonance, neural filtering by hair cells, molecular motors and brownian ratchets - flashing ratchets.
We will also discuss important simulation techniques, including Monte Carlo, Gillespie and Metropolis simulations as well as hidden Markov methods for estimating parameters.
This should be enough to fill the term.
Texts include
Gardiner, Stochastic Methods, 4th edition, Springer
with frequent reference to other books such as
vanKampen, Stochastic Processes in Physics and Chemistry,3rd edition, Elsevier
Wilkinson, Stochastic Modelling for Systems Biology, Chapman and Hall
Bressloff, Stochastic Processes in Cel Biology, Springer
Keener and Sneyd, Mathematical Physiology, 2nd edition, Springer
Schuss, Theory and Application of Stochastic Processes, Springer.
Homework assignments will be posted and updated regularly at this .pdf file.
Notes: Class notes are constantly evolving and are not guaranteed to be in the best of shape. Nonetheless, here are some notes for some of the material that has been covered in class. If you find a mistake, or if there is something you would like to see covered in the notes, please let me know and I will do something about it.
Here is a file for stochastic simulation of the simple exponential decay process, using the Gillespie algorithm: simple_decay_process.m.
Here is a file for stochastic simulation of a Markov model of a Michaelis Menten enzyme kinetics, using the Gillespie algorithm: michaelis_menten_discrete.m.
Here is a file for stochastic simulation of a Markov model of a potassium channel, using the Gillespie algorithm: n_state_potassium.m.
For more information contact J. Keener E-mail: keener@math.utah.edu