Evolutionary graph theory: Difference between revisions

In the following, we study the simplest possible question: what is the probability that a single mutant generates a lineage that takes over the entire population? This fixation probability determines the rate of evolution. The higher the correlation between the mutant's fitness and its probability of fixation, the stronger the effect of natural selection; if fixation is largely independent of fitness, drift dominates. Interestingly, it turns out that a large class of population structures maintains a characteristic balance between selection and random drift and leaves the fixation probability unchanged, i.e. identical to unstructured populations. Those graphs are called circulations. However, other graphs can have tremenduous effects on the fixation probability of mutants, ranging from complete suppression of selection to complete suppression of random drift, i.e. amplification of selection.

[[Image:Moran graph (lattice).png|thumb|300px|On lattices the [[spatial Moran process]] results in the same fixation probability of a single (or several) mutants as the original [[Moran process]] in unstructured populations.]]

Population structure can be modelled by arranging individuals on a graph such that each individual is represented by a vertex and its neighbourhood is defined through links connecting it to other individuals. Interestingly, for the [[spatial Moran process]] the fixation probability remains unaffected for a large class of population structures (graphs known as circulations), i.e. is the same as for the original [[Moran process]] in unstructured populations. For a single mutant this is

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[[Image:Evolutionary suppressor (chain).png|thumb|300px|Evolutionary suppressors are structures that reduce selection and enhance random drift, i.e. the fixation probability of advantageous (deleterious) mutants is decreased (increased) as compared to unstructured populations. The simplest example is a linear chain with directed links all pointing in the same direction.]]

The characteristic balance between selection and drift in Moran graphs can tilt to either side for graphs that are not circulations. For example, suppose \(N\) individuals are arranged in a linear chain. Each individual places its offspring into the position immediately to its right. The leftmost individual is never replaced. The mutant can only reach fixation if it arises in the leftmost position, which happens with probability \(1/N\), but then it will eventually reach fixation with certainty. Clearly, for such one-rooted graphs the fixation probability of a single, randomly placed mutant is  

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[[Image:Evolutionary amplifier (star).png|thumb|300px|Evolutionary amplifiers are structures that enhance (suppress) the fixation probability of advantageous (deleterious) mutants as compared to unstructured populations. The simplest example is the star graph, which consists of a single, central hub that is (bidirectionally) connected to all other vertices.]]

Interestingly, it is also possible to create population structures that amplify selection and suppress random drift. For example, on the star graph, where all nodes are connected to a central hub and vice versa, the fixation probability of a single, randomly placed mutant becomes

===Press & News===

# Guimerà, R. & Sales-Pardo, M. (2006) Form follows function: the architecture of complex networks ''Molecular Systems Biology'' '''2''' 42 [http://dx.doi.org/10.1038/msb4100082 doi: 10.1038/msb4100082].