Most undirected random graphs are amplifiers of selection for Birth-death dynamics, but suppressors of selection for death-Birth dynamics

We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-death updating, where an individual is selected to reproduce with probability proportional to its fitness and one of its neighbors is replaced by that offspring at random. If we instead focus on death-Birth updating, in which a random individual is removed and its neighbors compete for the empty spot, then the same ensemble of graphs consists of almost only suppressors of selection for which the fixation probability is decreased for advantageous mutants and increased for disadvantageous mutants. Thus, the impact of population structure on evolutionary dynamics is a subtle issue that will depend on seemingly minor details of the underlying evolutionary process

Counterintuitive properties of the fixation time in network-structured populations

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations

Comparison of Bd and dB update in the well-mixed population.

<p>Even in a well-mixed population, Birth-death and death-Birth processes do not lead to the same fixation probability, cf. Eqs. (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.e003" target="_blank">3</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.e006" target="_blank">5</a>). This implies that we have to choose a different reference case in order to infer whether graph-structured populations are suppressors or amplifiers of selection. All fixation probabilities are expected to pass through the shaded region. For Bd updating, the fixation probability of amplifiers passes through region 1, whereas for dB updating, it passes through 1 or 3. The fixation probability of suppressors for dB passes through 2, whereas for Bd it passes through 2 or 3. Interestingly, when we directly compare the two well-mixed update processes, dB is a suppressor of selection compared to Bd—and Bd is an amplifier of selection compared to dB.</p

Comparison of the cycle and the well-mixed population.

<p>Top: The cycle for small <i>N</i> is neither an amplifier nor a suppressor of selection. It decreases the fixation probability compared to the well-mixed population for both advantageous and disadvantageous mutants. Bottom: The difference between the fixation probability in the well-mixed population and the cycle increases with <i>N</i> for advantageous mutants. For disadvantageous mutants, all fixation probabilities tend to zero as the graph size increases.</p

Bd and dB update on directed random graphs.

<p>500 directed random graphs are classified into amplifiers or suppressors of selection according to their fixation probability for one randomly placed mutant. The red line represents the proportion of graphs having at least one node with no incoming links, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.e021" target="_blank">Eq. (16)</a>, which is equivalent to the sum of the disconnected, multi-rooted and one-rooted graphs. Technically, one-rooted graphs are also suppressors of selection, because a randomly placed mutant has a fixation probability of 1/<i>N</i> in such a graph. We visualize them in orange to distinguish them from the other suppressors. Multi-rooted graphs, given in grey, have a fixation probability of zero. The seven “unclassified” graphs we found for dB updating of size <i>N</i> = 4 are the same as the undirected cycle, in this case given by eight directed links instead of four undirected links. As shown above, the cycle is neither an amplifier nor a suppressor of selection.</p

Bd and dB update on undirected random graphs.

<p>Fixation probability for the Moran process on undirected random graphs for varying probability of link connection <i>p</i> in Erdős-Rényi graphs. The black line depicts the proportion of connected graphs, given by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.e011" target="_blank">Eq. (9)</a>. In the left panels, the blue line yields the proportion of isothermal graphs. Since the isothermal theorem does not hold for dB [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.ref007" target="_blank">7</a>], in the right panels the blue line depicts only the complete graphs, whereas the white line gives the proportion of cycles for <i>N</i> = 4. Left: For Birth-death updating, most non-trivial graphs are amplifiers of selection and only a tiny fraction are either suppressors of selection or remain unclassified. Right: If the population size is sufficiently large for death-Birth updating, we only find suppressors of selection. The only “unclassified” graph of size <i>N</i> = 4 is the cycle, which is neither an amplifier nor a suppressor of selection for dB, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004437#pcbi.1004437.e017" target="_blank">Eq. (14)</a>.</p