Evolutionary dynamics on graphs

Transcription

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

2 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

3 How does spatial population structure change the dynamics of evolution? Empirical Theoretical Well-mixed = 1 1 r 1 1 r N 1-d lattice 2-d lattice A. W. Nolte et al. Proc. R. Soc. B (2005) Laura Hindersin Evolutionary dynamics on graphs 2 / 40

4 The Moran Process Discrete time stochastic process M := {M n }, 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

5 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

6 The Moran Process 1 r Replacement 1 r 1 1 r 1 r r Birth Death Laura Hindersin Evolutionary dynamics on graphs 5 / 40

7 The Moran Process 1 r Replacement 1 r 1 1 r 1 r r 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

8 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: t + i := P (M n+1 = i + 1 M n = i) = t i := P (M n+1 = i 1 M n = i) = ri ri + N i N i N 1 N i ri + N i i N 1 t 0 i := P (M n+1 = i M n = i) = 1 t + i t i. for 0 i N. Laura Hindersin Evolutionary dynamics on graphs 6 / 40

9 Success Probability of the Mutants : Φ N i is the probability to reach state N from state i. 1 r r r 1 r r 1 r r Laura Hindersin Evolutionary dynamics on graphs 7 / 40

10 Success Probability of the Mutants Φ N i = t i ΦN i 1 + t+ i ΦN i+1 + (1 t i t + i )ΦN i where Φ N 0 = 0; ΦN N = 1. i 1 n t j Solving the recursion: Φ N i = n=0 j=1 t + j N 1 n t j n=0 j=1 t + j. For the Moran process in a well-mixed pop.: Φ N i = 1 1 r i 1 1 r N. Laura Hindersin Evolutionary dynamics on graphs 8 / 40

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

12 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

13 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

14 Moran Process on Graphs Extinction Initial State Fixation Laura Hindersin Evolutionary dynamics on graphs 11 / 40

15 Reference Case Well-mixed population Complete graph Laura Hindersin Evolutionary dynamics on graphs 12 / 40

16 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

17 Transition matrix Renumber the s = t + a states. The transition matrix now has the following canonical form: ( ) Qt t R T s s = t a. 0 a t I a a Call F = n=0 Qn = (I Q) 1 the fundamental matrix of the Markov 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

18 Transition matrix Probability of absorption in state j after starting in state i: Φ j i = (FR) i,j. (1) Conditional fixation time 1 : τ N i = N 1 j=1 ( Φ N j Φ N i F i,j ). 1 W.J. Ewens. Theoretical Population Biology (1973) Laura Hindersin Evolutionary dynamics on graphs 15 / 40

19 Popular Examples Time M. Frean et al. Proc. R. Soc. B (2013) Laura Hindersin Evolutionary dynamics on graphs 16 / 40

20 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

21 There are six different connected graphs of size four: complete diamond ring shovel star line Laura Hindersin Evolutionary dynamics on graphs 18 / 40

22 States of the Moran process on the complete graph: V I II III IV complete ring Laura Hindersin Evolutionary dynamics on graphs 19 / 40

23 States of the Moran process on the complete graph: V I II III IV complete V I II III IV complete ring And on the ring: ring Laura Hindersin Evolutionary dynamics on graphs 19 / 40

24 Links s consider the structure Outlook: with Metapopulations five links again. The states I,II,...,IX are shown in figure 3.8. e are several mutant-node-configurations possible in this structure, which is why the states o longer just determined by the number of mutants on the graph. They are also determined e degree of the respective node. 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

25 Question 2: Does the removal of a link always lead to a higher fixation time? 12 Mean conditional fixation time Fitness of mutants Laura Hindersin Evolutionary dynamics on graphs 21 / 40

26 Question 2: Does the removal of a link always lead to a longer fixation time? 12 Mean conditional fixation time Fitness of mutants Laura Hindersin Evolutionary dynamics on graphs 21 / 40

27 Question 2: Does the removal of a link always lead to a longer fixation time? 12 Mean conditional fixation time Fitness of mutants Answer: No! Laura Hindersin Evolutionary dynamics on graphs 21 / 40

28 Is this effect still present in larger networks? average conditional fixation time ring ring plus one link r Figure : Influence of the extra link in rings of size four, six and eight. A mutant fitness r = 1.5 is used. Laura Hindersin Evolutionary dynamics on graphs 22 / 40

29 Is this effect still present in larger networks? Laura Hindersin Evolutionary dynamics on graphs 23 / 40

30 (a) 4000 Non-periodic boundary conditions 3000 A 2000 B 1000 C (b) r = Periodic boundary conditions 3000 D 2000 E 1000 F 0 r = Population Size size Mean Conditional conditional fixation Fixation time Time Mean Conditional conditional Fixation fixation time Time L. Hindersin & A. Traulsen. Proc. R. Soc. Interface (2014) Laura Hindersin Evolutionary dynamics on graphs 24 / 40

31 What makes it so hard to answer Question 1 Question 1: Does every graph that differs from the well-mixed population increase the fixation time of advantageous mutants? Problems: Non-trivial relationship of the fixation probability and time, because the fixation time depends on the probability. For non-isothermal graphs, the transition matrix does not have a tridiagonal shape, since it is generally not a simple birth-death process, can be very large. Laura Hindersin Evolutionary dynamics on graphs 25 / 40

32 Popular Examples Time M. Frean et al. Proc. R. Soc. B (2013) Laura Hindersin Evolutionary dynamics on graphs 26 / 40

33 Amplifier and Suppressor of Selection fixation probability Graph G is an amplifier of selection if r > 1 ρ G > ρ mix and r < 1 ρ G < ρ mix. G is a suppressor of selection if r > 1 ρ G < ρ mix and r < 1 ρ G > ρ mix. ρ star = 1 1 r r ρ mix = r 2N r N ρ directedline = 1 N Laura Hindersin Evolutionary dynamics on graphs 27 / 40

34 Amplifiers of Selection E. Lieberman et al. Nature (2005) Laura Hindersin Evolutionary dynamics on graphs 28 / 40

35 Node Properties Let G = (V, E) be an undirected graph with N nodes. The degree k i of a node i V is defined by the number of its neighbors: k i := {e i,j : e i,j E}. The temperature T i of a node i V is defined by the sum over all incoming links, weighted by their degree: T i := N j=1 e j,i k j. Laura Hindersin Evolutionary dynamics on graphs 29 / 40

36 Graph Properties A graph G = (V, E) is called isothermal if T i = T j for all i, j V. Denote the fixation probability of a single mutant in a well-mixed population as ρ mix := Φ N 1. A population structure represented by a graph G, where one mutant has fixation probability ρ G = ρ mix is called ρ-equivalent to the well-mixed population. Laura Hindersin Evolutionary dynamics on graphs 30 / 40

37 Isothermal Theorem Theorem A graph G is ρ-equivalent iff it is isothermal. A proof can be found in 2. 2 Lieberman et al. [2005]: Evolutionary dynamics on graphs. Nature, 433, Pages , Supplementary Notes. Laura Hindersin Evolutionary dynamics on graphs 31 / 40

38 Observation For size N = 4, all non-regular graphs are amplifiers of selection Fitness of mutants Laura Hindersin Evolutionary dynamics on graphs 32 / 40

39 Underlying Update Mechanisms Birth Bd death death db Birth Laura Hindersin Evolutionary dynamics on graphs 33 / 40

40 Procedure Generate an Erdős-Rényi random graph G for given N and p. Calculate the fixation probability ρ G. Compare it to the fixation probability ρ mix. Classify as amplifier or suppressor if possible. Laura Hindersin Evolutionary dynamics on graphs 34 / 40

41 Bd db L. Hindersin & A. Traulsen. arxiv: (2015) Laura Hindersin Evolutionary dynamics on graphs 35 / 40

42 Summary Changing the structure can have counterintuitive effects on the fixation time. Bd: almost all random networks are amplifiers. db: almost all random networks are suppressors. Mean conditional fixation time Fitness of mutants Bd db Laura Hindersin Evolutionary dynamics on graphs 36 / 40

43 Applications Biological: Social: Experimental evolution: Using an amplifier graph increases fixation probability, but the time is increased as well. Biological networks, like protein or gene regulatory networks are often scale-free 3. Scale-free networks can amplify selection. Social networks: The spreading of ideas can be very likely, but may take a long time. Scientific collaborations networks are often scale-free. 3 R. Albert & A.-L. Barabási, Review of Modern Physics (2002) Laura Hindersin Evolutionary dynamics on graphs 37 / 40

44 Next Step Consider subpopulations at the nodes. The links determine the migration paths. Laura Hindersin Evolutionary dynamics on graphs 38 / 40

45 Next Step A. W. Nolte et al. Proc. R. Soc. B (2005) Laura Hindersin Evolutionary dynamics on graphs 39 / 40

46 The Moran Process Acknowledgements Department of Evolutionary Theory Arne Traulsen Julia n Garcia, Monash University Max-Planck-Institut für Evolutionsbiologie, Plön Laura Hindersin Evolutionary dynamics on graphs 40 / 40