Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology Plön, 24306 Plön, August-Thienemann-Strasse 2, Germany.
BMC Evol Biol. 2012 Aug 30;12:160. doi: 10.1186/1471-2148-12-160.
In addition to selection, the process of evolution is accompanied by stochastic effects, such as changing environmental conditions, genetic drift and mutations. Commonly it is believed that without genetic drift, advantageous mutations quickly fixate in a halpoid population due to strong selection and lead to a continuous increase of the average fitness. This conclusion is based on the assumption of constant fitness. However, for frequency dependent fitness, where the fitness of an individual depends on the interactions with other individuals in the population, this does not hold.
We propose a mathematical model that allows to understand the consequences of random frequency dependent mutations on the dynamics of an infinite large population. The frequencies of different types change according to the replicator equations and the fitness of a mutant is random and frequency dependent. To capture the interactions of different types, we employ a payoff matrix of variable size and thus are able to accommodate an arbitrary number of mutations. We assume that at most one mutant type arises at a time. The payoff entries to describe the mutant type are random variables obeying a probability distribution which is related to the fitness of the parent type.
We show that a random mutant can decrease the average fitness under frequency dependent selection, based on analytical results for two types and simulations for n types. Interestingly, in the case of at most two types the probabilities to increase or decrease the average fitness are independent of the concrete probability density function. Instead, they only depend on the probability that the payoff entries of the mutant are larger than the payoff entries of the parent type.
除了选择，进化过程还伴随着随机效应，如不断变化的环境条件、遗传漂变和突变。通常认为，如果没有遗传漂变，有利的突变由于强烈的选择，会很快在单倍体种群中固定下来，并导致平均适应度的持续增加。这一结论是基于适应度不变的假设。然而，对于频率依赖的适应度，即个体的适应度取决于与种群中其他个体的相互作用，这种情况就不成立了。
我们提出了一个数学模型，可以理解随机频率依赖突变对无限大种群动态的影响。不同类型的频率根据复制者方程发生变化，而突变体的适应度是随机的且依赖于频率。为了捕捉不同类型之间的相互作用，我们采用了可变大小的收益矩阵，从而能够容纳任意数量的突变。我们假设一次最多只有一种突变类型出现。描述突变体类型的收益项是服从概率分布的随机变量，该概率分布与母体类型的适应度有关。
我们通过对两种类型的解析结果和对 n 种类型的模拟结果表明，在频率依赖选择的情况下，随机突变可以降低平均适应度。有趣的是，在最多只有两种类型的情况下，增加或降低平均适应度的概率与具体的概率密度函数无关。相反，它们仅取决于突变体的收益项大于母体类型的收益项的概率。
