Evolutionary Graph Theory

Uses of graphical framework Graph can represent relationships in a social network of humans E.g. “Six degrees of separation”

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It can also analyze the effect of population structure on evolutionary dynamics… Uses of graphical framework

The Basic Idea Vertices – individuals in a population Edge – competitive interaction w ij = Probability that an offspring of i replaces j Edge can go both directions and so describes a digraph ij w ij ij

The Basic Idea Label all the individuals in the population with i=1,2,…,N Represent each with a vertex At each time step, choose a random individual for reproduction Determine direction of edge Every edge has weight = w ij –w ij >0 → an edge from i to j –W ij =0 → no edge from i to j Hence process determined by W=[w ij ]; 0< w ij <1 The matrix W defines a weighted digraph

Moran Process Consider a homogeneous population of size N consisting of residents (white) and mutants (black).

Moran Process At each time step, choose an individual for reproduction with a probability proportional to its fitness Here, a resident is selected for reproduction

Moran Process A randomly chosen individual is eliminated Here, a mutant is selected for death

Moran Process The offspring replaces the eliminated individual.

Why Moran Process? Represents the simplest possible stochastic model to study selection in a finite population Where 2 individuals are chosen at each time step One for reproduction & one for elimination Offspring of first replaces the second Total population size, N, is strictly constant

Moran Process Represented by a complete graph with identical weights An unstructured population is given by a complete graph: an edge btw any 2 vertices Evolutionary process is equivalent to the Moran process

Moran Process Fixation probability: probability that a mutant invading a population of N -1 residents will produce a lineage that takes over the whole population Fixation probability α evolution rate Suppose  all resident individuals are identical and one new mutant is introduced  new mutant has relative fitness r, as compared to the residents, whose fitness is 1 Fixation probability of the mutant is then given by:  R = (1-1/r)/(1-1/r N )

Evolutionary Suppressors Line Burst

The line 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. Mutant can only reach fixation if it arises in the leftmost position, which happens with probability 1/N. Fixation probability = 1/N, independent of r

The Burst A new mutant can only reach fixation if it arises in the center: Probability that a randomly placed mutant originates in the center = 1/N Hence fixation probability is again independent of r, the relative fitness of the new mutant Represent suppressors of selection All mutants – irresp of their fitness – have the same fixation probability as a neutral mutant in the Moran process

Evolutionary Amplifiers StarSuperstar

The Star For a large N, a mutant with a relative fitness r has a fixation probability ρ = (1-1/r 2 )/(1-1/r 2N ). So, a relative fitness r on a star is equivalent to a relative fitness r 2 in the Moran process Thus, a star is an amplifier of selection

The Superstar l= no. of leaves m= no. of loops in a leaf k= the length of each loop for sufficiently large N, a super-star of parameter k satisfies: Fixation probability The superstar amplifies a selective difference r to r k A powerful amplifier of selection! l=5 k=3 m=5

References Nowak, Martin A. “Evolutionary Graph Theory.” Evolutionary Dynamics: exploring the equations of life. Cambridge, Massachusetts, and London : Belknap Press of Harvard University Press, 2006: Graph images from: Lieberman, Erez and Hauert, Christoph and Nowak, Martin A. “Evolutionary dynamics on graphs.” Nature. 433 (2005): Image in slide #3 from: on on