●
Epistasis: the effect of the mutation depends on the genetic background in which it happens
●
Diminishing returns: the fitter the genotype, the hardest it is to find beneficial mutations
●
(limited) Open-endedness:
novel genotypes can occur indefinitely
Which properties are needed for biological realism?
From Wikipedia
From Wiser et al, 2013
From Tenaillon, 2014 Individual occupy a multi-dimensional
phenotype space (here n=2)
There is an optimal phenotype, and fitness (vertical axis on the plot) depends on the distance to this optimum:
Scaled size of the mutation r∗
√
n2∗d
Enters Fisher Geometrical Model (FGM)
2 / 35
Which properties do we get in FGM?
●
Epistasis: ✔ (the same mutation --black arrow-- has a different effect from the green dot than
from the blue dot
●
Diminishing returns: ✔ (the fraction of
mutations of a given size that are beneficial decreases as we approach the optimum)
●
Weak Open-endedness: ✔ (infinite number of phenotypes)
●
And also: mutations of bigger size are more
likely to be detrimental (Fisher’s idea)
Insisting on diminishing returns in FGM: the distribution of the fitness effects of mutations
4 / 35
from Perfeito et al, 2007
Experimental data on the
distribution of fitness effects of beneficial mutations fixed during
experimental evolution
Medium population sizeLarge population size
Consistent with expectations from the FGM:
Many beneficial mutations of small effects,
beneficial mutations of large effects are rare
Conclusion of 2 nd lecture, homework
●
You now have a broad, implementable knowledge of evolution, and start discovering more or less advanced computational techniques around it
●
(Before 2020-11-17) read this text from Laurent Duret on neutral evolution and the controversy around it
●
(Optional) you can read further on this topic on quanta
●
(Optional) read this review from Barrick & Lenski on experimental evolution
●
(Suggested) read some of the articles pointed in this lecture
●
Questions?
6 / 35Part II:
Modeling and simulating life and
evolution in space and time
Fixed population size with overlapping and non-overlapping generations
Wright-Fisher process: same thing, but with non-overlapping (synchronous) generations (at each generations, all individuals are replaced)
Moran process:
overlapping generations (picture from
Chalub & Souza, 2008)
8 / 35
Fixed population size with overlapping and non-overlapping generations
Wright-Fisher process: same thing, but with non-overlapping (synchronous) generations (at each generations, all individuals are replaced)
Moran process:
overlapping generations (picture from
Chalub & Souza, 2008)
●
These processes intrinsically integrate stochasticity of the evolutionary processes (genetic drift)
●
They are discrete regarding the individuals → biological realism:
there can not be 0.03 individuals of a given type
●
Mathematically tractable without mutations and selection
●
When adding mutations and selection, we may need to simulate.
Temporal complexity O(N*Δt)
Strengths and weaknesses of Moran and Wright-Fisher processes
9 / 35
Variation and regulation of population size
dN
dt =r∗N N (t )=N
0∗e
r∗t● Unregulated population growth:
where r is the net growth rate = birth rate – death rate
● Regulation of growth (eg due to nutrient or space limitation): logistic equation
Image from Khanacademy
Variation and regulation of population size
dN
dt =r∗N∗(1− N
K ) N (t )= K 1+ K −N
0N
0∗e
−r∗tdN
dt =r∗N N (t )=N
0∗e
r∗t● Unregulated population growth:
where r is the net growth rate = birth rate – death rate
● Regulation of growth (eg due to nutrient or space limitation): logistic equation
where K is the carrying capacity (equilibrium population size)
Image from Khanacademy
10 / 35
Models of evolution with continuous growth and continuous time
A B
r
Ar
Bμ * r
Bμ * r
ADifferential equations are an attractive option:
What if growth is limited (carrying capacity?)
dA
dt = A∗r
A− A∗r
A∗μ+ B∗r
B∗μ dB
dt =B∗r
B−B∗r
B∗μ+ A∗r
A∗μ
Models of evolution with continuous growth and continuous time
A B
r
Ar
Bμ * r
Bμ * r
ADifferential equations are an attractive option:
What if growth is limited (carrying capacity?)
dA
dt = A∗r
A− A∗r
A∗μ+ B∗r
B∗μ dB
dt =B∗r
B−B∗r
B∗μ+ A∗r
A∗μ
dA
dt =( A∗r
A− A∗r
A∗μ+B∗r
B∗μ)∗(1− A+B K )
And what about death? Is net growth rate sufficient to
describe the dynamics?
11 / 35Properties of these models based on differential equations
●
No stochasticity (absence of genetic drift)
●
No discrete representation of the individuals → lack of biological realism for small population sizes
●
May be mathematically tractable
●
If not, numerical integration: temporal complexity O(|Eq|*Δt) (depends
on the number of equations but not on population size)
Simple equations do not always lead to simple population dynamics: thresholds and breakpoints
from Robert M. May, 1977
Population of herbivores at constant population size H Feed on vegetation with biomass V(t)
Consumption of vegetation per herbivore is c(V) Growth of vegetation in absence of grazing is G(V)
dV
dt =G(V )−H∗c (V )
Realistic assumptions for c(V) and G(V):13 / 35
Looking for the fixed points:
when is the derivative null?
dV
dt =G(V )−H∗c (V )
Stable and unstable fixed points depending on the value of H
15 / 35
Model of evolution with continuous growth and synchronous (discrete) time: difference equations
N
t +1=r∗N
t∗[1−N
t/ K ]
Difference equation:Eg growth with saturation:
General form: transformation to get
→ single parameter
N
t +1= F( N
t)
X
t+1= p∗X
t∗[1−X
t]
p=3.414 p=2.707
y=x
What happens when no stable fixed point (eg p=3.414)?
Let’s look at
F2 can have stable fixed points even when F has no stable fixed point
→ periodicity!
Same reasoning for all Fn
→ larger periodicity
F
2( X)=F (F ( X ))
p=3.414
17 / 35
p=3.414 p=2.707
Bifurcation
From bifurcations...
Convergence points (obtained by simulation) depending on the parameter value 19 / 35
From bifurcations... … to chaos
(infinite number of values with no cyclic behaviour)
Continuous growth is not always realistic
A B
r
Ar
Bμ * r
Bμ * r
ASuch equations can not accurately represent small population sizes:
- how could there be 0.01 mutant in a population?
- when the last individual dies, a population is extinct → no spontaneous generation
dA
dt = A∗r
A− A∗r
A∗μ+ B∗r
B∗μ dB
dt =B∗r
B−B∗r
B∗μ+ A∗r
A∗μ
→ We need a discrete growth model
Several available methods, inspired by stochastic chemistry
20 / 35Discrete simulations of time-continuous asynchronous processes
A B
r
Ar
Bμ * r
Bμ * r
AEach process (reaction) has a probability of occurring per individual per time interval:
A→ A + A occurs at a rate rA(1-μ) A → A + B occurs at a rate rA*μ
…
These rates do not need to be constant (so we could have population size regulation as before)
The probability of occurrence of each
Discrete simulations: rejection sampling
A B
r
Ar
Bμ * r
Bμ * r
AA→ A + A occurs at a rate rA(1-μ) A → A + B occurs at a rate rA*μ ...
Choose a small value for Δt (the time interval for the simulations)
While time < final_time:
time = time + Δt
for each reaction R with rate r:
draw a random number x uniformly distributed in [0,1]
if r*Δt < x
reaction R occurs (→ update the population variables)
Note: the reaction rates r already take into account Problem: this is inefficient, because many
computations step with no reaction occurring
22 / 35
Discrete simulations: Gillespie algorithm (aka SSA)
A B
r
Ar
Bμ * r
Bμ * r
AA→ A + A occurs at a rate rA(1-μ) A → A + B occurs at a rate rA*μ
Notice that the waiting time for the next occurrence of each reaction R with rate r is exponentially distributed:
The global waiting time to the next reaction (among N different reactions) is thus also exponentially
distributed with parameter rtot = r1+r2+...+rN
→ draw waiting time Δt to this next reaction from this distribution
→ decide which of the possible reaction occurs at t+Δt by random sampling with probabilities r/rtot
p (tnext≤Δt)=1−e−r∗Δt
●
Chemical Master Equation: describe all possible states of the systems and the transition probability between them (Markov jump process), impossible to solve in practice
●
Gillespie with “Next Reaction Method”: Gibson & Bruck, 2000
●
Adaptive tau-leap: Gillespie, 2001
●
More generally, how can we mix processes well described by continuous growth (birth and death in a large population) and by
discrete events (mutagenesis)? → still a quite open research question
Discrete simulations: advanced variants
24 / 35
●
Moran or Wright-Fisher process on a 2D lattice → individual-based model
●
Fixed population size, or allow empty spots to permit varying population size
●
Competition is local = dead individual replaced by one of the neighbours
●
Fitness of the individuals can depend on phenotype of the neighbours to represent interactions between individuals (competition for
resources, but also public good or bioweapons)
More realistic scenarios for evolution in space
Cooperative and competitive interactions
Respirators (high K, low r)
Fermenters (high r, low K)
Resources
dN
1dt =r∗N
1∗( 1− N
1+ N
2K ) dN
2dt =r∗N
2∗( 1− N
1+ N
2K )
Two genotypes in the same environment sharing the same resources: growing fast or efficiently? → Cooperative dilemma
In well-mixed environment, individuals growing efficiently loose the competition against those growing fast.
But with spatial structure? 26 / 35
Cooperative and competitive interactions
Respirators (high K, low r)
Fermenters (high r, low K)
Resources
Trade-off between rate and yield → cooperative dilemma Pfeiffer, Schuster & Bonhoeffer,
dN
1dt =r∗N
1∗( 1− N
1+ N
2K ) dN
2dt =r∗N
2∗( 1− N
1+ N
2K )
Two genotypes in the same environment sharing the same resources: growing fast or efficiently? → Cooperative dilemma
In well-mixed environment, individuals growing efficiently loose the competition
●
Many definitions of
cooperation in biology.
Let’s focus on the game theory point of view: a
behaviour is cooperative if it leads to a
prisoner’s dilemma
●
Examples: r/K selection (previous slide), secretion of public good molecules (eg enzymes acting extra- cellularly), eusociality,
programmed cell death…
C D
C R S
D T P
Prisoner dilemma:
T>R>P>S and R>(T+S)/2
Nash equilibrium, ESS Pareto optimal
Cooperation
27 / 35
●
Many definitions of
cooperation in biology. Let’s focus on the game theory point of view: a behaviour is
cooperative if it leads to a prisoner’s dilemma●
Examples: r/K selection
(previous slide), secretion of public good molecules (eg enzymes acting extra-
cellularly), eusociality, programmed cell death…
In well-mixed populations,
C D
C R S
D T P
Prisoner dilemma:
T>R>P>S and R>(T+S)/2
Nash equilibrium, ESS Nowak, 2006:
expectation in mixed
populations Pareto optimal
Cooperation
● Many definitions of cooperation in
biology. Let’s focus on the game theory point of view: a behaviour is
cooperative if it leads to a prisoner’s dilemma
● Examples: r/K selection (previous slide), secretion of public good
molecules (eg enzymes acting extra- cellularly), eusociality, programmed cell death…
● In well-mixed populations, cooperation is expected to be counter-selected
● Thought to evolve by kin / group selection, where individuals of the same types interact more often
● Many possible mathematical
frameworks, and much controversy around it
Chuang et al, 2009:
framework based on Simpson paradox
C D
C R S
D T P
Prisoner dilemma:
T>R>P>S and R>(T+S)/2
Nash equilibrium, ESS Nowak, 2006:
expectation in mixed
populations Pareto optimal
Cooperation
27 / 35
Competition
● From Fleming, 1929:
(some) Antibiotics are weapons produced by microbes to kill other microbes
● Evolution of virulence due to competition? Frank 1992, Brown et al 2009
(want to know more about
Modeling and simulating evolution with interactions
1 2 3
4 6
8
7 9
5
1 2 3
4 6
8
7 9
Competition 7’
between the neighbours for reproduction Death of an
individual randomly chosen
●
In space: rare to have analytical
expressions (but see vanBaalen 2000) → usually simulations
29 / 35
Modeling and simulating evolution with interactions
1 2 3
4 6
8
7 9
5
1 2 3
4 6
8
7 9
Competition 7’
between the neighbours for reproduction Death of an
individual randomly chosen
●
In space: rare to have analytical
expressions (but see vanBaalen 2000) → usually simulations
●
No space?
Replicator equation
x is the frequency of A
fB = cx + d(1 − x) is the payoff of B fA = ax + b(1 − x) is the payoff of A
Part III:
Confronting models with data Parameters estimation
30 / 35
Estimating parameter of deterministic temporal
models (example with ODEs)
Estimating parameter of deterministic temporal models (example with ODEs)
A classical optimization problem for which you can apply your favorite approaches (Simulated annealing? Particle swarm? ...)
But what about stochastic biological processes,
eg our old problem of mutation rate estimation? 31 / 35
Estimating parameter in stochastic models: the
example of mutation rate estimation
… the maximum likelihood approach
33 / 35
… but what if the PDF can not be expressed analytically?
Questions?
35 / 35