Stochastic models of
evolution driven by
Spatial
Heterogeneity
R
Group
H
, Utrecht University, Department of Biology, Theoretical Biology
Three research lines
1. Bacterial growth
2.
Stochastic models of evolutio
Physiology of bacterial during steadystate exponential growth:
• Bulk properties highly predictable1
• Heterogeneous environments
• Non-Poissonian mutation
processes
• yet huge cell-to-cell fluctuations2
3 (The evolution of)
fratricide
in Streptococcus pneumoniae
1 You
et al (2013), 2 Kiviet et al (2014),
Outline
I.
Evolutionary population genetics
basic notions and two examples
II. Evolution in heterogeneous environments
A.
The staircase model
many connected compartments
Intermezzo:
B.
The source-sink model
The ramp model: continuous space
Classical theoretical population genetics
Founders:
• Ronald A. Fisher (1890 – 1962)
• Sewall B.S. Wright (1889 – 1988)
• John B.S Haldane (1892 – 1964)
Some of the fundamental questions:
• How do stochastic forces interfere with (natural)
selection?
• What determines the time scales of evolution?
• Migration, sexual reproduction, linkage?
• How do social interactions evolve?
(cooperation, competition, spite, …)
Example 1: Continuous-time Moran1
process
death
Structure of the model:
• population of N individuals
• stochastic reproduction …
• … coupled to death of
random individual
• fitness = reproduction rate
“selection coefficient”
What is its fixation probability?
1 P.
Moran (1958)
time
Question:
Assume that fitness of red
mutant is 1 + s times that of the
blue ones.
reproduction
In physics language, this Moran model is…
• a continuous-time Markov process
• a one-step process with two absorbing boundaries
• described by a Master equation with transition rates:
𝑁−𝑛
𝑊𝑛,𝑛+1 = 1 + 𝑠 𝑛
𝑁
𝑛
𝑊𝑛,𝑛−1 = 𝑁 − 𝑛
𝑁
(Here, 𝑛 is the number of red individuals.)
Calculating the mutation-fixation probability:
Determine the “splitting probabilities” for initial condition:
𝑃𝑛 (0) = 𝛿𝑛,1
Mutation-fixation probability
For small s:
𝑃fix 𝑠, 𝑁
𝑁−1 𝑠
≈1+
+ 𝑂(𝑠 2 ).
𝑃fix 0, 𝑁
2
Conclusion:
• Selection has a negligible effect unless (𝑁𝑠) is of order 1.
• Selection is a stronger force in larger populations.
Example 2:
The wave of advance of a beneficial
mutation
• Population in 1D space
• Mutation occurs carrying selection
coefficient s
• How rapidly does this mutation spread?
• Mean-field (Fisher-KPP)
equation1:
“logistic growth”
𝜕𝑛
𝜕2𝑛
= 𝑠 𝑛 1 − 𝑛/𝐾 + 𝐷 2 .
𝜕𝑡
𝜕𝑥
“carrying capacity”
• Compact initial distributions 𝑢(𝑥, 0) converge
to traveling wave front with speed 𝑣0 =
2 𝑠𝐷.
1
Fisher, The genetical theory of natural selection (1930)
• R.A.
Example
of a pulled front.
Outline
I.
Evolutionary population genetics
basic notions and two examples
II. Evolution in heterogeneous environments
A.
The staircase model
many connected compartments
Intermezzo:
B.
The source-sink model
The ramp model: continuous space
Mutations can affect a species’ range
Biotic and abiotic factors
vary in space
They limit a species
range
Mutations that extend
one’s range can be
successful, even if they
confer no benefit within
the original range
Illustration: Giraffes and Trees
Usual population genetics models
Scenario considered here
Relevant for resistance and virulance
evolution
Evolution of drug / antibiotic resistance
(heterogeneities in body or between individuals)
Evolution of virulence
(ability of an organisms to invade a host’s tissue)
A.
The staircase model
Many connected compartments
Landscape:
Processes and rates:
Mutants (increasing resistance)
Mutation
𝜇f


𝜇b
Migration

𝜈

𝜈
Death
𝛿


0
Logistic growth above staircase
Compartments (increasing drug level)
𝛾𝑗 𝑁𝑗


𝛾𝑗 𝑁𝑗 = 𝜆(1 − 𝑁𝑗 /𝐾)
Simulations (kinetic Monte Carlo)
Adaptation is fast and shows two regimes
At the front, only 2x2 tiles matter
Intermezzo:
A source–sink model
Two patches
Source-sink model: two somewhat isolated
patches
Wild type cannot
reproduce in patch 2
Mutant can.
Logistic reproduction:
𝛾𝑖𝑗
𝑁𝑖
𝑁𝑖 = 𝜆 1 −
𝐾
Death rate 𝛿
Mutant has no
advantage in patch 1!!
How long does it take before the population adapts?
Monte Carlo simulations again show two
regimes
Mean first arrival time calculation
Collapse on master curve:
We find:
𝑐𝑇≈
𝜋 Γ 𝜅
2 Γ 𝜅+1
2
where 𝑐 ≈
𝜈(𝛿 + 𝜈)
𝜇f 𝐾
𝜅≈
2(𝛿 + 𝜈)
Two regimes: mutation- and migrationlimited
Mutation-limited regime:
if 𝜇f 𝐾 ≪ 𝛿 + 𝜈:
Migration-limited regime: if 𝜇f 𝐾 ≫ 𝛿 + 𝜈:
𝛿/𝜈
𝑇≈
𝜇f 𝐾
𝑇≈
𝜋/2𝜈
𝜇f 𝐾
Path
faster than path
This means:
Adaptation usually originates
as a neutral mutation
in patch 1
(“Dykhuizen–Hartl effect”)
Cost of adaptation...
genotypes
Assume that the mutant has a
fitness defect in patch 1
Does this suppress
adaptation?
Cost negligible unless
0
patches
𝑠≳ 𝜈 𝛿
½
Even when 𝜈/𝛿 = 10−2 we need
𝑠 ≳ 0.1, which is a large cost.
...is relevant only if it is large.
Back to the staircase:
At the front, source-sink dynamics
Theory very similar to source-sink model
explains rate of propagation of the staircase
model
Cost of adaptation hardly affects rate
(Simulations, and theory similar to source-sink model.)
Defeating the Superbugs –2012
B.
The ramp model
Continuous space & quantitative traits
Is there such a thing as a ‘just right’ gradient?
Many traits are continuous
Many biological characteristics are, for all practical purposes,
continuous, due to the involvement of many genes, e.g.:
•
•
•
•
•
•
fat content in plants
body size
skin color
neck length
minimal inhibitory concentration
…
Continuous “ramp” model
Individuals:
• diffuse in real space,
• diffuse in trait space
(mutation),
• die,
• reproduce logistically when
above the ramp.
Result: Population wave front that expands above the line.
Q: What is the asymptotic speed of that wave?
Q: In particular: how does it depend on the slope 𝑎?
Large density: deterministic, ‘mean-field’
treatment
𝜕𝑡 𝑛 = 𝐷x 𝜕𝑥𝑥 𝑛 + 𝐷y 𝜕𝑦𝑦 𝑛 + 𝜆 𝑛 1 −
𝑛
d𝑦 𝐻 𝑦 − 𝑎𝑥 − 𝛿 𝑛,
𝐾
Numerical solutions indeed
show a propagating pulse.
What is its asymptotic
speed?
Define:
𝑣𝑦 = rate of adaptation
𝑣𝑥 = rate of invasion
trait value 𝑦
(Similar to Fisher-KPP wave!)
position 𝑥
Asymptotic speed can be calculated
analytically
Using theory of front
propagation into unstable
states:
𝛿
= 0,
𝜆
𝛿
= 0.1
𝜆
𝛿
= 0.5
𝜆
𝛿
= 0.7
𝜆
𝑣𝑦
𝑎
𝑣y = 2
𝜆 − 𝛿 𝐷x 𝐷y 𝑎2
𝐷y + 𝐷x 𝑎2
Numerical solutions confirm
this.
Adaptation rate increases
monotonously with slope a.
No (finite) optimal
slope??
Van Saarloos, “Front propagation into unstable states” Physics Reports 386 (2003) 29-222
Beyond the mean field: stochastic
simulations
• Individuals diffusing randomly in continuous space.
• Stochastic death
• Stochastic reproduction at logistic rate:
𝜆 1−
𝑖
Ω 𝑥𝑖 − 𝑥
𝐾
𝐻(𝑦 − 𝑎 𝑥),
where Ω(𝑑) is a Gaussian interaction kernel with variance 𝜎 2 .
𝑦
𝑦=𝑎𝑥
𝑥
The simulation challenge:
Properties of the system:
• the reproduction rate of each organism depends on the location
of every organism in the system;
• each organism diffuses in continuous space and time;
• therefore rates fluctuate rapidly in time.
Problems:
• Cannot use a standard Gillespie algorithm
• Discrete time steps: require recalculation of interaction kernel
for each individual at each time step
The “Weep” Algorithm:
exact, event-driven simulation of time-, location- and
density-dependent reaction rates
The main idea:
• Add a dummy reaction to the system.
(I’ll imagine the organisms “weep”.)
• Choose the rate of the dummy process 𝑟w such that the total
rate becomes constant for each organism:
𝑟w = 𝜆
𝑖
𝑟w = 𝜆,
Ω 𝑥𝑖 − 𝑥
,
𝐾
(above the ramp)
below the ramp
so that 𝑟tot ≡ 𝑟b + 𝑟w + 𝑟d = 𝜆 + 𝛿 = is constant.
(The organisms tend to weep when conditions are tough.)
The “Weep” Algorithm:
exact, event-driven simulation of time-, location- and density-dependent
reaction rates
Then determine next reaction:
1. When? Draw next-reaction time 𝑡 + Δ𝑡 (as in Gillespie).
2. Where? Update the coordinates of all individuals.
(Greens function of diffusion for time Δ𝑡).
3. Who? Pick a random individual 𝑖.
4. What? Choose death with probability 𝛿/𝑟tot .
If it is not killed, it either reproduces or weeps.
• If 𝑦𝑖 < 𝑎 𝑥𝑖 , it weeps.
• If 𝑦𝑖 > 𝑎 𝑥𝑖 , calculate 𝑟w for organism 𝑖 only.
𝑟
It reproduces with probability b , and weeps
𝑟 +𝑟
b
otherwise.
5. Repeat.
w
Discrete, stochastic simulations
Low density
(𝐾 = 10)
High density
(𝐾 = 500)
Stochastic simulations do show an optimum
The optimal slope is:
𝑎≈
𝐷y 𝐷x
What’s wrong with the mean field equation?
• In “pulled” fronts, the leading edge determines the
speed.
• There, the density becomes arbitrarily small;
fractions of individuals reproduce exponentially.
• In this case, this leads to qualitatively misleading
results.
In the stochastic simulation, we’re waiting for unlikely
mutations/migrations to occur.
• Stochastic system does converge to mean-field
equation,
but not uniformly in 𝑎.
Take-away messages
1.
In the presence of environ- 2. This is
mental gradients, adaptation
relevant for the
and range expansion can go
evolution
hand in hand.
of drug resistance
and virulence.
3.
4.
A fitness cost has little
effect.
This mode of adaptation can be fast.
5.
The dynamics
show two regimes:
a mutation-limited and
a migration-limited
one.
6.
An optimal gradient exists –
for any finite carrying capacity.
Be careful with mean-field limit!
Acknowledgements
Hwa lab at the UC San Diego:
Terence Hwa
J. Barrett Deris
Department of Bionanoscience, TU Del
Cees Dekker
Veni Grant (2012)