A computational view of population genetics

A Computational View of Population Genetics (preliminary version) Yuval Rabanit Yuri Rabinovicht Alistair Sinclair] This paper contributes to the study of nonlinear dynamical systems from a computational perspective. These systems are inherently more powerful than their linear counterparts (such as Markov chains), which have had a wide impact in Computer Science, and they seem likely to play an increasing role in future. However, there are as yet no general techniques available for handling the computational aspects of discrete nonlinear systems, and even the simplest examples seem very hard to analyze. We focus in this paper on a class of quadratic systems that are widely used as a model in population genetics and also in genetic algorithms. These systems describe a process where random matings occur between parental chromosomes via a mechanism known as “crossover”: i.e., children inherit pieces of genetic material from different parents according to some random rule. Our results concern two fundamental quantitative properties of crossover systems: 1. We develop a general technique for computing the rate of convergence to equilibrium. We apply this technique to obtain tight bounds on the rate of convergence in several cases of biological and computational interest. In general, we prove that these systems are “rapidly mixing”, in the sense that the convergence time is very small in comparison with the size of the state space. 2. We show that, for crossover systems, the classical quadratic system is a good model for the behavior of finite populations of small size. This stands in sharp contrast to recent results of Arora et al and Pudlak, who show that such a correspondence is unlikely to hold for general quadratic systems. tD~p~~t~ent Of Computer Science, University of TorontO, Toronto, Ontario M5S 1A4, Canada. Email: {rabani ,yuri}Q cs. toronto. edu. t Computer Science Division, University of California, Berkeley CA 94720-1776, U.S.A. Email: sinclair@cs .berkeley. edu. Perrmssion to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyri ht notice and the % title of the publication and Its dat~ appear, an notice IS gwen that copyin is by permission of tne Association of Computing ? Machinery o copy otherwise, or to republish, requires a fee andlor specific permission. STOC” 95, Las Vegas, Nevada, USA @ 1995 ACM 0-89791 -718-9/95/0005 .$3.50