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
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2. Collaborators
   2
   University of Hawai‘i
   Lee Altenberg
   Stanford University
   Marcus W. Feldman
   Tel Aviv University
   Lilach Hadany
   Uri Liberman
   

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3. My main interest
   Evolution of mechanisms for
   generation and transmission
   of phenotypic and genetic variation
   

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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
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5. Produce
   individual
   differences
   Make the
   child like its
   parents
   Balance
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6. Variation
   Fidelity
   Balance
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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
   

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8. Beneficial
   mutations
   Deleterious
   mutations
   Balance: Mutation
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9. Migrating
   Homing
   Balance: Migration
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10. Innovating
    Imitating
    Balance: Learning
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11. Exploration
    Exploitation
    Balance: Learning
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12. Models
    Ram, Altenberg, Liberman & Feldman, TPB 2018
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13. General Model
    • Types A1
    , A2
    , …, An
    (mutants, sites, behaviors)
    • Frequencies f1
    , f2
    , …, fn
    • Fitness w1
    , w2
    , …, wn
    A3
    A4
    A1
    A2
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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
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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)
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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
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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 − −
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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)
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    ,
    = K
    1 − , =
    , = + 1
    0, ℎ
    ,
    = K
    1 − , =
    , = ± 1
    0, ℎ
    

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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.
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20. Models
    • Mutation: single locus
    • Mutation: multilocus
    • Migration
    • Learning
    …
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21. 21
    Results
    

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22. Equilibrium
    Looking for the equilibrium:
    ,
    ∗∗ = − + ∗
    ∗
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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…
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24. Long-term mean fitness
    ∗: stable equilibrium frequency vector
    ,
    ∗: stable equilibrium population mean fitness
    ,
    ∗∗ = − + ∗
    ∗
    Globally stable
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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 ,
    ∗
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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 ,
    ∗
    
    ,
    ∗
    
    = ,
    ∗ −
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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 ,
    ∗
    
    ,
    ∗
    
    = ,
    ∗ −
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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
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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.
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30. Fisher’s Reproductive value
    Relative contribution to long-term
    population
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    Fisher, 1930 pg 27
    Hermisson et al, TPB 2002
    Appendix B of Ram et al., TPB 2018
    

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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 ,
    ∗
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32. Increased transition from below-average
    individuals increases the population
    mean fitness…
    But will it evolve?
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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
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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
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35. Math details…
    ,
    / = − −
    ,
    / = − ′ − ′
    Set to equilibrium (, ) = (∗, ) (B is absent)
    Check external stability of (∗, ) to increase in g
    Using eigenvalue of Jacobian of system at (∗, )
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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
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37. Result 2:
    Evolution of increased genetic variation
    Invader allele B invades the population if it
    increases transition from types with below-
    average fitness.
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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 >
    = ,
    ∗ −
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39. Summary
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40. Summary
    • Increased transition from
    below-average types:
    • Increases population mean
    fitness
    • Expected to evolve
    • Assuming M is irreducible!
    • Applications to mutation,
    migration, learning…
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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
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42. Ram Lab @ IDC
    66
    yoav@yoavram.com
    @yoavram
    www.yoavram.com
    Now recruiting
    grad students
    and postdocs Interdisciplinary Center Herzliya
    

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