Evolutionary regime transitions in structured populations

Abstract

The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. Usually, the population adopts the form of an undirected graph where the place occupied by each individual is represented by a vertex and it is bidirectionally linked to the places that can be occupied by its offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any vertex. But there also are undirected graph structures acting as suppressors of selection where this probability is less than that of the same individual placed in a homogeneous population. Here, firstly, we present the distribution of these evolutionary regimes for all undirected graphs with N ≤ 10 vertices. Some of them exhibit transitions between different regimes when the mutant fitness increases. In particular, as it has been already observed for small-order random graphs, we show that most graphs of order N ≤ 10 are amplifiers of selection. Secondly, we describe examples of amplifiers of order 7 that become suppressors from some critical value. In fact, for graphs of order N ≤ 7, we apply computer-aided techniques to symbolically compute their fixation probability and then their evolutionary regime, as well as the critical values for which they change their regime. Thirdly, the same technique is applied to some families of highly symmetrical graphs as a mean to explore methods of suppressing selection. The existence of suppression mechanisms that reverse an amplification regime when fitness increases could have a great interest in biology and network science. Finally, the analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, with multiple transitions between different regimes, which have not been examined in detail until now.

Fixation probabilities in graph-structured populations under weak selection

“…Our results for uniform initialization are presented in Table 2 and Fig 4 . We find that the vast majority of small graphs (94%, in the case N = 10) are amplifiers of weak selection. This is consistent with previous numerical analyses [6,[9][10][11] which found that most small unweighted graphs are amplifiers for Bd with uniform initialization. In particular, the results for N = 6 and N = 7 agree with those from an exhaustive analysis performed by Cuesta et al [9], who computed fixation probabilities for arbitrary mutant fitness r.…”

“…We consider a well-studied model of natural selection [1][2][3][4][5][6][7][8][9][10][11][22][23][24][25][26][27][28][29][30]. There are two types of individuals: residents (or wild-types), which have fitness 1, and mutants, which have fitness r. The mutant is advantageous if r > 1, deleterious if r < 1, and neutral if r = 1.…”

“…Most previous works [1,3,4,6,7,9,10,[22][23][24][25]30] consider only uniform initialization, meaning that the mutant is equally likely to appear at each vertex, μ i = 1/N for all i. Uniform initialization corresponds to a biological assumption that heritable mutations arise primarily in adult individuals, with constant probability per unit time.…”

“…Due to its small size and symmetry, the fixation probability for this graph can be computed in closed form for arbitrary mutant fitness r. Employing an algorithm developed by Cuesta et al [7,9], adapted for temperature initialization, we obtained the fixation probability ρ G (r) as a ratio of 17th-degree polynomials (shown in S1 Text).…”

“…Determining fixation probability on an arbitrary graph is computationally intensive. Current methods [6,9,30,31] require solving a system of linear equations whose size grows exponentially with the graph size. This is prohibitive except for graphs that are small [6,7,[9][10][11]31], highly symmetric [1,[3][4][5][22][23][24][25], or have other special properties [32].…”