Necessity is the mother of invention

I present our work on the "modified mean fitness principle", which suggests that if individuals with below-average fitness transition (e.g. mutate, migrate) to a different type, then the population mean fitness increases. We show this is the case whenever an individual is likely to increase his reproductive value due to a transition, and use a modifier model to analyse evolutionary stability of such a trait.

Transcript

1. Necessity is the mother of invention Yoav Ram School of Computer Science IDC Herzliya TAU Theory-Fest, 1 Jan 2020 1
2. Collaborators 2 University of Hawai‘i Lee Altenberg Stanford University Marcus W. Feldman Tel Aviv University Lilach Hadany Uri Liberman
3. My main interest Evolution of mechanisms for generation and transmission of phenotypic and genetic variation
4. Generation of variation “Some authors believe it to be as much the function of the reproductive system to produce individual differences… as to make the child like its parents.” -- Charles Darwin On the Origin of Species, 1872 4
5. Produce individual differences Make the child like its parents Balance 5
6. Variation Fidelity Balance 6
7. “favorable mutations... The only raw material for evolution.” “high frequency of new mutant genes that cause an appreciable reduction in viability” Balance: Mutation -- A.H. Sturtevant Quar Rev Biol, 1937 7
8. Beneficial mutations Deleterious mutations Balance: Mutation 8
9. Migrating Homing Balance: Migration 9
10. Innovating Imitating Balance: Learning 10
11. Exploration Exploitation Balance: Learning 11
12. Models Ram, Altenberg, Liberman & Feldman, TPB 2018 12
13. General Model • Types A1 , A2 , …, An (mutants, sites, behaviors) • Frequencies f1 , f2 , …, fn • Fitness w1 , w2 , …, wn A3 A4 A1 A2 13
14. General Model • Type Ak transition probability is Ck • Ak transitions to Aj with probability Mj,k # → % = C( ⋅ %,# A3 A4 A1 A2 * Type transmission is vertical & uni-parental 14
15. General Model The change in f=(f1 , f2 , …, fn ) is given by , / = − + where D and C is a positive diagonal matrices: = 8 0 0 0 ⋱ 0 0 0 ; = 8 0 0 0 ⋱ 0 0 0 ; M is an irreducible column-stochastic matrix, , is a normalizing factor to ensure ∑#>8 ; # / = 1 (1) 15
16. Mutation Model 1 n possible alleles of a specific locus, A1 , …, An wk fitness of allele Ak Ck mutation rate with allele Ak , = 1/: mutations are equally probable to any allele 16
17. Mutation Model 2 Ak : individual with k deleterious mutations wk fitness with k deleterious mutations Ck mutation rate with k deleterious mutations Mutations are deleterious or beneficial with probability δ and β: E,EF8 = , E,EH8 = , E,E = 1 − − 17
18. Migration Model Ak individual in deme (site) k wk fitness in deme k Ck probability of leaving deme k Different migration schemes can apply (Karlin 1982) 18 , = K 1 − , = , = + 1 0, ℎ , = K 1 − , = , = ± 1 0, ℎ
19. Learning Model Ak phenotype\behavior k e.g. number of hours to invest in foraging, etc. wk fitness of phenotype k Ck exploration rate of phenotype k i.e. 1-Ck exploitation rate Exploration breadth is modeled with M: Mj,k is the probability that an exploring individual with phenotype j will switch to phenotype k. 19
20. Models • Mutation: single locus • Mutation: multilocus • Migration • Learning … 20
21. 21 Results
22. Equilibrium Looking for the equilibrium: , ∗∗ = − + ∗ ∗ 22
23. Math details… , ∗∗ = − + ∗ • , ∗ and ∗ are eigenvalue and eigenvector of − + • … which is non-negative primitive matrix • So , ∗ and ∗ exist, unique, non-negative • Perron-Frobenius theory • So… 23
24. Long-term mean fitness ∗: stable equilibrium frequency vector , ∗: stable equilibrium population mean fitness , ∗∗ = − + ∗ ∗ Globally stable 24
25. Result 1: Modified mean fitness principle If: fitness wk of Ak is below the mean fitness , ∗ Then: increasing Ck transition from Ak will increase mean fitness , ∗ 25
26. Result 1: Modified mean fitness principle If: fitness wk of Ak is below the mean fitness , ∗ Then: increasing Ck transition from Ak will increase mean fitness , ∗ , ∗ = , ∗ − 26
27. Result 1: Modified mean fitness principle If: fitness wk of Ak is below the mean fitness , ∗ Then: increasing Ck transition from Ak will increase mean fitness , ∗ , ∗ = , ∗ − 27
28. Math details… Analysis uses Caswell’s formula for sensitivity of the population growth rate to changes in life history parameters ^ = = ⇒ = Caswell, TPB 1978 Hermisson et al, TPB 2002 Reproduced in appendix A of Ram et al., TPB 2018 28
29. Stress-induced mutation Increasing the mutation rate Ck of individuals with below average fitness wk increases the population mean fitness , Ram & Hadany, Evolution 2012 Ram & Hadany, PRSB 2014 Modified mean fitness principle in action. 29
30. Fisher’s Reproductive value Relative contribution to long-term population 30 Fisher, 1930 pg 27 Hermisson et al, TPB 2002 Appendix B of Ram et al., TPB 2018
31. Corollary 2: Reproductive value principle If: fraction of long-term population descending from Ak will increase, on average, from transitions Then: increasing Ck transition from Ak will increase mean fitness , ∗ 32
32. Increased transition from below-average individuals increases the population mean fitness… But will it evolve? 34
33. Evolutionary genetic stability* • Modifier locus that modifies Ck • Start with resident allele b with {C1 , …, Cn } • Introduce invader allele B with {C’1 , …, C’n } • Can allele B increase in frequency and invade? • Allele b that cannot be invaded is evolutionary stable *Liberman, JMB 1988 35
34. Modifier model , / = − − , / = − ′ − ′ is the frequency vector for resident allele b is the frequency vector for resident allele B , is the total population mean fitness 36
35. Math details… , / = − − , / = − ′ − ′ Set to equilibrium (, ) = (∗, ) (B is absent) Check external stability of (∗, ) to increase in g Using eigenvalue of Jacobian of system at (∗, ) 37
36. Reduction principle If transition rate is uniform: Ck =C doesn’t depend on k Then according to the Reduction principle*: Invader allele B invades the population if and only if it decreases transition rate C. * Altenberg, Liberman & Feldman, PNAS 2017 38
37. Result 2: Evolution of increased genetic variation Invader allele B invades the population if it increases transition from types with below- average fitness. 39
38. Result 2: Evolution of increased genetic variation Invader allele B invades the population if it increases transition from types with below- average fitness. j / k > = , ∗ − 40
39. Summary 41
40. Summary • Increased transition from below-average types: • Increases population mean fitness • Expected to evolve • Assuming M is irreducible! • Applications to mutation, migration, learning… 42
41. Outlook Cultural transmission Frequency-dependent transmission , / = − − Preliminary result in Liberman, Ram, Altenberg & Feldman, TPB 2019 Recombination and sex Preliminary result in Ram & Hadany, AmNat 2019 Transmission of social traits 43
42. Ram Lab @ IDC 66 yoav@yoavram.com @yoavram www.yoavram.com Now recruiting grad students and postdocs Interdisciplinary Center Herzliya