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Necessity is the mother of invention

Yoav Ram
January 01, 2020

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.

Yoav Ram

January 01, 2020
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  1. Necessity is the
    mother of invention
    Yoav Ram
    School of Computer Science
    IDC Herzliya
    TAU Theory-Fest, 1 Jan 2020
    1

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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
    4

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

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  6. Variation
    Fidelity
    Balance
    6

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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
    8

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  9. Migrating
    Homing
    Balance: Migration
    9

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  10. Innovating
    Imitating
    Balance: Learning
    10

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  11. Exploration
    Exploitation
    Balance: Learning
    11

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  12. Models
    Ram, Altenberg, Liberman & Feldman, TPB 2018
    12

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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
    13

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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
    14

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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)
    15

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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
    16

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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 − −
    17

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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)
    18
    ,
    = 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.
    19

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  20. Models
    • Mutation: single locus
    • Mutation: multilocus
    • Migration
    • Learning

    20

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  21. Equilibrium
    Looking for the equilibrium:
    ,
    ∗∗ = − + ∗

    22

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  22. 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

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  23. Long-term mean fitness
    ∗: stable equilibrium frequency vector
    ,
    ∗: stable equilibrium population mean fitness
    ,
    ∗∗ = − + ∗

    Globally stable
    24

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  24. 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

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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 ,


    ,


    = ,
    ∗ −
    26

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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 ,


    ,


    = ,
    ∗ −
    27

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  27. 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

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  28. 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

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  29. 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

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  30. 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

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  31. Increased transition from below-average
    individuals increases the population
    mean fitness…
    But will it evolve?
    34

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  32. 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

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  33. 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

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  34. Math details…
    ,
    / = − −
    ,
    / = − ′ − ′
    Set to equilibrium (, ) = (∗, ) (B is absent)
    Check external stability of (∗, ) to increase in g
    Using eigenvalue of Jacobian of system at (∗, )
    37

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  35. 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

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  36. Result 2:
    Evolution of increased genetic variation
    Invader allele B invades the population if it
    increases transition from types with below-
    average fitness.
    39

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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.
    j

    /

    k >
    = ,
    ∗ −
    40

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  38. Summary
    • Increased transition from
    below-average types:
    • Increases population mean
    fitness
    • Expected to evolve
    • Assuming M is irreducible!
    • Applications to mutation,
    migration, learning…
    42

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  39. 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

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  40. Ram Lab @ IDC
    66
    [email protected]
    @yoavram
    www.yoavram.com
    Now recruiting
    grad students
    and postdocs Interdisciplinary Center Herzliya

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