A solvable model of the evolutionary loop

– A model for the evolution of a finite population in a rugged fitness landscape is introduced and solved. The population is trapped in an evolutionary loop, alternating periods of stasis to periods in which it performs adaptive walks. The dependence of the average rarity of the population (a quantity related to the fitness of the most adapted individual) and of the duration of stases on population size and mutation rate is calculated. The simplest conceivable evolutionary situation is a population of asexually reproducing individuals set in a fixed environment. The reproductive power of an individual is measured by its fitness, i.e., by a quantity proportional to the expected number of its offspring [1, 2]. In model building, one often assumes that the fitness is determined by the genotype, and that the genotype itself is transmitted identically from parent to offspring, apart from mutations. The role of mutations is favorable if the overall fitness of the members of the population is low, because they allow the population to find genotypes with higher fitness, i.e., to adapt. This can be represented, following Sewall Wright [3], by saying that the population approaches a fitness peak. On the other hand, mutations become pernicious when the population is located on such a peak, since they may let the population lose contact with it. In fact, the two effects have different relevance depending on population size: if the population size is large, and the fitness small, adaptation dominates; but if the population is small, and the fitness peak, no matter how lofty, is narrow, mutations have a negative effect. This situation has been described by two classes of models: 1. In the quasispecies model [4] one takes the infinite population size limit from the outset, obtaining an equation (akin to a Master Equation) for the genotype distribution in the (∗) Permanent address. Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS population. It is interesting that, if the fitness peaks are narrow enough, this equation exhibits a transition (the error threshold) between an adaptive regime, and a regime in which adaptation is irrelevant. Nevertheless, the description of the non-adaptive regime is not satisfactory within this class of models. 2. The adaptation process has been described as a special kind of random walk, the adaptive walk, by Kauffman and Levine [5] and others. In this model, the fitness can only increase, and mutations only have positive effects. An “annealed” version of this model has been exactly solved by Flyvbjerg and Lautrup [6]. The stochastic escape of a finite population from a narrow fitness peak has been discussed by Higgs and Woodcock [7]. They find that, in the same limit in which the error threshold appears in its fullest glory in the quasispecies model, a finite population eventually loses contact with the adaptation peak. Building on this observation and on numerical simulations, Woodcock and Higgs [8] have been led to describe the behavior of a population evolving in a rugged fitness landscape (i.e., in a situation where even slight changes of the genotype lead to arbitrarily large changes in the fitness) as an evolutionary loop: 1. If the fitness of the population is low, favorable mutations get fixed in the population, which thus performs an adaptive walk, reaching a local fitness peak. 2. The population can be evicted from the adaptation peak by stochastic escape, and start a new adaptive walk from a random, usually low, fitness value. In this Letter, I introduce a solvable model that exhibits such a behavior. The model is a slight generalization of the Annealed Adaptive Walk Model introduced and solved by Flyvbjerg and Lautrup [6], and allows for stochastic escape. It depends on only two parameters, namely population size and mutation rate. I argue that it should describe any mutation-selection model in the strong selection limit, i.e., when the fitness distribution is broad, provided that the correct variables are identified and the correct scaling of the parameters is performed. This conclusion is borne out by simulations of a slightly more realistic model which I report at the end of the Letter. I consider a population of M individuals (M is fixed) evolving in a rugged fitness landscape. At each generation, for each member α of the new population, its parent α is chosen among the old population. The probability Wγ that the individual γ is chosen to reproduce is given by Wγ = Fγ