Necessity is the
mother of invention
Yoav Ram
School of Computer Science
IDC Herzliya
TAU Theory-Fest, 1 Jan 2020
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Collaborators
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University of Hawai‘i
Lee Altenberg
Stanford University
Marcus W. Feldman
Tel Aviv University
Lilach Hadany
Uri Liberman
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My main interest
Evolution of mechanisms for
generation and transmission
of phenotypic and genetic variation
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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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Produce
individual
differences
Make the
child like its
parents
Balance
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Variation
Fidelity
Balance
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“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
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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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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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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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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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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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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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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Models
• Mutation: single locus
• Mutation: multilocus
• Migration
• Learning
…
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Results
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Equilibrium
Looking for the equilibrium:
,
∗∗ = − + ∗
∗
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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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Long-term mean fitness
∗: stable equilibrium frequency vector
,
∗: stable equilibrium population mean fitness
,
∗∗ = − + ∗
∗
Globally stable
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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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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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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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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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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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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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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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Increased transition from below-average
individuals increases the population
mean fitness…
But will it evolve?
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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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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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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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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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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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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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Summary
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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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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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Ram Lab @ IDC
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yoav@yoavram.com
@yoavram
www.yoavram.com
Now recruiting
grad students
and postdocs Interdisciplinary Center Herzliya