Evolutionary dynamics on graphs Laura Hindersin May 4th 2015 - PowerPoint PPT Presentation

Evolutionary dynamics on graphs Laura Hindersin May 4th 2015 Max-Planck-Institut für Evolutionsbiologie, Plön

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Evolutionary dynamics Main ingredients: Fitness: The ability to survive and reproduce. Selection emerges when two or more individuals reproduce at different rates. Mutation: One type can change into another. Neutral drift: A finite population of two types will eventually consist of only one type. Laura Hindersin Evolutionary dynamics on graphs 1 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations How does spatial population structure change the dynamics of evolution? Theoretical Empirical Well-mixed 1 − 1 ρ = r 1 1 − rN 2-d lattice 1-d lattice A. W. Nolte et al. Proc. R. Soc. B (2005) Laura Hindersin Evolutionary dynamics on graphs 2 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations The Moran Process Discrete time stochastic process M := { M n } , n ∈ ◆ 0 . Birth-death process on a well-mixed population of N individuals. Here, the initial state of the population is: N − 1 wild type individuals with fitness 1 1 mutant with fitness r > 0 Laura Hindersin Evolutionary dynamics on graphs 3 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations The Moran Process One reproductive event at each time step: Select one individual for birth at random, but with probability proportional to its fitness. This individual produces one clonal offspring. Randomly choose an individual to be replaced by the new offspring. Laura Hindersin Evolutionary dynamics on graphs 4 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations The Moran Process 1 1 r r Replacement 1 1 r r r 1 Birth Death Laura Hindersin Evolutionary dynamics on graphs 5 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations The Moran Process 1 1 r r Replacement 1 1 r r r 1 Birth Death Markov process on the number of mutants. State space S = { 0 , 1 , 2 , . . . , N } with initial state M 0 = 1 . Assumption: no further mutations. Therefore, the states 0 and N are absorbing. Laura Hindersin Evolutionary dynamics on graphs 5 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Transition Probabilities for the Moran Process The probability to increase or decrease the number of mutants, or to stay with i mutants at the next time step are: ri + N − i · N − i ri t + := P ( M n +1 = i + 1 | M n = i ) = i N − 1 N − i i t − := P ( M n +1 = i − 1 | M n = i ) = ri + N − i · i N − 1 t 0 = 1 − t + i := P ( M n +1 = i | M n = i ) i − t − i . for 0 ≤ i ≤ N . Laura Hindersin Evolutionary dynamics on graphs 6 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Success Probability of the Mutants Fixation probability: Φ N is the probability to reach state N from state i . i 1 r r r r 1 r 1 r r Laura Hindersin Evolutionary dynamics on graphs 7 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Success Probability of the Mutants i − 1 + t + i − t + Φ N i = t − i Φ N i Φ N i +1 + (1 − t − i )Φ N i where Φ N 0 = 0 ; Φ N N = 1 . i − 1 n t − j � � t + n =0 Φ N j =1 Solving the recursion: i = j . N − 1 n t − j � � t + n =0 j =1 j 1 − 1 Φ N ri For the Moran process in a well-mixed pop.: = rN . i 1 − 1 Laura Hindersin Evolutionary dynamics on graphs 8 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Conditional Fixation Time The expected time until absorption into the state N starting from one single mutant, given that it will succeed: N − 1 k k Φ N t − � � � τ N l m 1 = . t + t + m l k =1 l =1 m = l +1 Laura Hindersin Evolutionary dynamics on graphs 9 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Moran Process on Graphs Let G := ( V, E ) define a graph , consisting of a set of vertices V and edges E. Individuals inhabit the nodes of a graph and reproduce into their adjacent nodes. Laura Hindersin Evolutionary dynamics on graphs 10 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Moran Process on Graphs Let G := ( V, E ) define a graph , consisting of a set of vertices V and edges E. Individuals inhabit the nodes of a graph and reproduce into their adjacent nodes. Birth death Replacement Wild-type: fitness 1 Mutant: fitness r Laura Hindersin Evolutionary dynamics on graphs 10 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Moran Process on Graphs Extinction Initial State Fixation Laura Hindersin Evolutionary dynamics on graphs 11 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Reference Case Well-mixed population Complete graph Laura Hindersin Evolutionary dynamics on graphs 12 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Methods Different approaches for calculating the fixation probability and time in graphs: Individual-based simulations Transition matrix for up to 2 N states. Laura Hindersin Evolutionary dynamics on graphs 13 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Transition matrix Renumber the s = t + a states. The transition matrix now has the following canonical form: � � Q t × t R t × a T s × s = . 0 a × t I a × a n =0 Q n = ( I − Q ) − 1 the fundamental matrix of the Markov Call F = � ∞ chain. The entry F i,j is the expected sojourn time in state j , given that the process starts in transient state i . C.M. Grinstead & J.L. Snell Introduction to Probability (1997) Laura Hindersin Evolutionary dynamics on graphs 14 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Transition matrix Probability of absorption in state j after starting in state i : Φ j i = ( FR ) i,j . (1) Conditional fixation time 1 : N − 1 � � Φ N � j τ N = · F i,j . i Φ N i j =1 1W.J. Ewens. Theoretical Population Biology (1973) Laura Hindersin Evolutionary dynamics on graphs 15 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Popular Examples Time M. Frean et al. Proc. R. Soc. B (2013) Laura Hindersin Evolutionary dynamics on graphs 16 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Questions Question 1: Does every undirected graph that differs from the well-mixed population increase the fixation time of advantageous mutants? Question 2: Given any population structure, does the removal of one link always lead to a higher fixation time? Laura Hindersin Evolutionary dynamics on graphs 17 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations There are six different connected graphs of size four: complete diamond ring shovel star line Laura Hindersin Evolutionary dynamics on graphs 18 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations States of the Moran process on the complete graph: Laura Hindersin Evolutionary dynamics on graphs 19 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations States of the Moran process on the complete graph: And on the ring: Laura Hindersin Evolutionary dynamics on graphs 19 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations States of the Moran process on the diamond: III I VI IX IV VIII II VII V Laura Hindersin Evolutionary dynamics on graphs 20 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Fixation Time Question 2: Does the removal of a link always lead to a higher fixation time? 12 10 Mean conditional fixation time � � 8 � � � � � � � � 6 4 2 0 0 1 2 3 4 5 Fitness of mutants Laura Hindersin Evolutionary dynamics on graphs 21 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Fixation Time Question 2: Does the removal of a link always lead to a longer fixation time? 12 � � 10 Mean conditional fixation time � � � � � � � � 8 � � � � � � � � � � 6 4 2 0 0 1 2 3 4 5 Fitness of mutants Laura Hindersin Evolutionary dynamics on graphs 21 / 40

The Moran Process Fixation Time Fixation probability Outlook: Metapopulations Fixation Time Question 2: Does the removal of a link always lead to a longer fixation time? 12 � � 10 Mean conditional fixation time � � � � � � � � � � � � � � � � 8 � � � � � � � � � � � � 6 4 2 0 0 1 2 3 4 5 Fitness of mutants Answer: No! Laura Hindersin Evolutionary dynamics on graphs 21 / 40