Yoav Ram
January 01, 2020
210

# 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.

January 01, 2020

## 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

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

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

∗ ∗ 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

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 [email protected] @yoavram www.yoavram.com Now recruiting

grad students and postdocs Interdisciplinary Center Herzliya