Biomathematics 3

The syllabus will depend in part on student interests; a preliminary outline (with the need to make some cuts) is as follows.

[NB: The "background" portion could be distributed throughout the course, introducing the tools as needed for the course.]

· Part I: Background (3 weeks)

o Review of elementary probability
o Review of stochastic processes (and their simulation); equilibrium approach.
o Foundations of statistical mechanics and statistical physics: ensembles, partition function, Maxwell and Boltzmann distributions
o Entropy
o Monte Carlo method

· Part II: Evolution (6 weeks)

o Carrying capacity, competition for resources, population dynamics.
o Fitness
o Replication equations, gene frequencies and their evolution.
o Mutations; fixation of genetic traits in a limited population. Temporal aspects (Kimura theory); the coalescent.
o Mutations in complex genomes; fitness landscape. Error classes and simple models. Quasispecies.
o Simulation of random evolution for simple models; discrepancy between mean-field theory and numerical experimental results (and its explanation)
o Sharp Peak model; the error catastrophe. Adaptation to a variable environment; antiviral strategies.
o Coevolution; example: viruses and the immune system.
o Simple models and viral evolution.
o Evolution with spatial structure
o Evolution and information
o Variations on a theme: survival of the fittest, survival of the fattest, survival of the simpler.

· Part III: Complex systems (3 weeks)

o Compartmentalization
o Complex systems; free energy in a realistic (disordered) fitness landscape
o Random Energy model
o The Perceptron: from neural network modeling to AI, via fitness landscape.