Institute for Biological Physics, University of Cologne, Köln, Germany.
J Math Biol. 2023 Feb 15;86(3):46. doi: 10.1007/s00285-023-01882-z.
A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph [Formula: see text], where [Formula: see text] is the number of genetic loci and the allele graph [Formula: see text] encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in [Formula: see text] displays a transition from 0 to a positive value at a threshold [Formula: see text] for the fitness difference between the initial and final genotype. We derive a lower bound on [Formula: see text] for general [Formula: see text] and show that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs.
适应景观是将离散基因型映射到实数的一种映射。适应景观中的一条路径是由单突变步骤连接的基因型序列。如果路径上遇到的基因型的适应值单调增加，则称该路径是可访问的。我们研究了纸牌屋类型的随机适应景观上的可访问路径，其中适应值是独立的、同分布的、连续的随机变量。基因型空间被取为笛卡尔乘积图[公式：见正文]，其中[公式：见正文]是遗传位点的数量，等位基因图[公式：见正文]编码一个位点上可能的等位状态和突变跃迁。在距离线性[公式：见正文]的两个基因型之间存在可访问路径的概率在初始和最终基因型之间的适应值差异的阈值[公式：见正文]处从 0 变为正。我们为一般的[公式：见正文]推导出了[公式：见正文]的下界，并表明对于一大类等位基因图，这个下界是紧的。我们的结果推广了二倍体超正方体上可访问性渗流的先前结果，并与多等位基因汉明图的已发表数值结果进行了比较。
