Baake Ellen, Georgii Hans-Otto
Faculty of Technology, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany.
J Math Biol. 2007 Feb;54(2):257-303. doi: 10.1007/s00285-006-0039-5. Epub 2006 Oct 31.
We consider the evolution of populations under the joint action of mutation and differential reproduction, or selection. The population is modelled as a finite-type Markov branching process in continuous time, and the associated genealogical tree is viewed both in the forward and the backward direction of time. The stationary type distribution of the reversed process, the so-called ancestral distribution, turns out as a key for the study of mutation-selection balance. This balance can be expressed in the form of a variational principle that quantifies the respective roles of reproduction and mutation for any possible type distribution. It shows that the mean growth rate of the population results from a competition for a maximal long-term growth rate, as given by the difference between the current mean reproduction rate, and an asymptotic decay rate related to the mutation process; this tradeoff is won by the ancestral distribution. We then focus on the case when the type is determined by a sequence of letters (like nucleotides or matches/mismatches relative to a reference sequence), and we ask how much of the above competition can still be seen by observing only the letter composition (as given by the frequencies of the various letters within the sequence). If mutation and reproduction rates can be approximated in a smooth way, the fitness of letter compositions resulting from the interplay of reproduction and mutation is determined in the limit as the number of sequence sites tends to infinity. Our main application is the quasispecies model of sequence evolution with mutation coupled to reproduction but independent across sites, and a fitness function that is invariant under permutation of sites. In this model, the fitness of letter compositions is worked out explicitly. In certain cases, their competition leads to a phase transition.
我们考虑在突变和差异繁殖（即选择）的联合作用下种群的演化。种群被建模为连续时间的有限类型马尔可夫分支过程，并且相关的谱系树在时间的正向和反向都被观察。反向过程的平稳类型分布，即所谓的祖先分布，结果成为研究突变 - 选择平衡的关键。这种平衡可以以变分原理的形式表达，该原理量化了繁殖和突变对于任何可能的类型分布的各自作用。它表明种群的平均增长率源于对最大长期增长率的竞争，这由当前平均繁殖率与与突变过程相关的渐近衰减率之间的差异给出；这种权衡由祖先分布胜出。然后我们关注类型由字母序列（如核苷酸或相对于参考序列的匹配/错配）确定的情况，并询问通过仅观察字母组成（由序列中各种字母的频率给出），上述竞争还能看到多少。如果突变率和繁殖率可以以平滑的方式近似，那么随着序列位点数量趋于无穷大，由繁殖和突变相互作用产生的字母组成的适应性在极限情况下得以确定。我们的主要应用是具有与繁殖耦合但位点间独立的突变的序列演化的准物种模型，以及在位点置换下不变的适应度函数。在这个模型中，字母组成的适应性被明确计算出来。在某些情况下，它们的竞争会导致相变。
