This is the second lecture, in a series of three.
For an abstract see the linked pdf-file.
abstract:
We consider the evolution of biological populations under the joint action
of mutation and selection. The genotypes are linear arrangements of N sites,
each of which may take the value +1 or -1; the evolution of an infinite
population is then described by a dynamical system for a probability
distribution on {-1,+1}^N.
We first show that this model is exactly equivalent to an Ising quantum
chain.
In this picture, the fitness of a configuration corresponds to the
interaction energy of the spins within the chain, whereas mutation
corresponds to interaction of the spins with a transversal field. This
equivalence is exploited to obtain exact equilibrium solutions for
representative fitness landscapes, with special emphasis on phase
transitions.
We then reinterpret the approach in the framework of branching processes.
This admits a plausible biological interpretation of the quantum-mechanical
observables that appeared `enigmatic' in the previous approach. The key to
the connection is the time-reversed Markov process and its stationary
distribution (i.e., the type distribution of the individuals that are
ancestral to today's population).