Evolution as a Stochastic Process

by Prof. Hirotaka Sugawara (Sokendai, Graduate University for Advanced Studies, Hayama)

Thursday Feb 15, 2007, 3:00 PM → 4:30 PM Pacific/Honolulu

Rm 112 (UHM - Watanabe Hall)

Let me consider evolution as a Markovian process. The most general Markovian process can be described by the Chapman-Kolmogorov equation and it is non-linear.　I would like to investigate the legitimacy of the Fokker-Planck equation (which is linear) utilized by Wright or Kimura in their description of the evolution process based on the general Markovian view point. I appeal to the well known fact that the Fokker-Planck equation is equivalent to the Langevin equation. The result shows a rather peculiar property of the Wright-Kimura equation. I, then, turn back to the original Chapman-Kolmogorov and start from a natural assumption that the system is large. It is well known ( at least to physicists) that this case leads to the so called Hamilton-Jacobi equation. My own expectation had been that this equation would become linear only for the case of neutral evolution and so the well known fluctuation-dissipation theorem of the linear transport theory would apply resulting in the famous equality of evolution velocity to the mutation frequency by Kimura.　
To my surprise, my expectation turns out to be wrong. The neutral evolution as defined by Kimura does not satisfy linear equation but rather satisfies a solvable non-linear equation. I am wondering if I had not made a rather trivial error in my computation. Anyway, my purpose is not to criticize the theoretical structure of the population genetics but rather to contemplate on the possible theoretical framework which is suitable in the era of bio-informatics.　 For this purpose in mind, I pay attention to the fact　that the most general Markovian process can be written in the form of path integral. It is well known to physicists that the path integral can be formulated in terms of certain operators. This gives me a motivation to try an operator formalism for the life system.　I divide the life system into the sexually reproducing species and the rest. I consider only the former to start with ( the latter is easier but important when we treat the somatic case). Very naturally, mating is the fundamental process in this case and mutation, copying, crossing over etc. etc. are treated as accompanying processes. Nucleotides are regarded as the basic unit but we are naturally lead to what I call the gene approximation. Here the alleles play the role of basic unit replacing the nucleotides. What we call the classical approximation in quantum theory　will be justified in this case when the system is large which often is the case in the living system.
Finally I would like to make a remark that it is crucial to have a deep understanding of population of a life form as a system. I also feel obliged to emphasize that life is incredibly diverse but it is not complex but rather it is simple and beautiful at least from the eye of a physicist.

J.G. Learned