Stochastic models of evolution driven by Spatial Heterogeneity R H, Utrecht University, Department of Biology, Theoretical Biology Group

Three research lines Physiology of bacterial during steady- state exponential growth: Bulk properties highly predictable 1 yet huge cell-to-cell fluctuations 2 1. Bacterial growth Heterogeneous environments Non-Poissonian mutation processes 2. Stochastic models of evolution 1 You et al (2013), 2 Kiviet et al (2014), 3 (The evolution of) fratricide in Streptococcus pneumoniae

Outline I. Evolutionary population genetics basic notions and two examples II. Evolution in heterogeneous environments A.The staircase model many connected compartments Intermezzo: The source-sink model B. 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 Moran 1 process Structure of the model: population of N individuals stochastic reproduction … … coupled to death of random individual fitness = reproduction rate Question: Assume that fitness of red mutant is 1 + s times that of the blue ones. What is its fixation probability? 1 P. Moran (1958) time reproduction death “selection coefficient”

In physics language, this Moran model is…

Mutation-fixation probability

Example 2: The wave of advance of a beneficial mutation 1 R.A. Fisher, The genetical theory of natural selection (1930) “logistic growth” “carrying capacity”

Outline I. Evolutionary population genetics basic notions and two examples II. Evolution in heterogeneous environments A.The staircase model many connected compartments Intermezzo: The source-sink model B. 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

Processes and rates: Mutation Migration Death Logistic growth above staircase Landscape:          0      Mutants (increasing resistance) Compartments (increasing drug level)

Mesa-shaped fitness landscape

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. How long does it take before the population adapts?

Monte Carlo simulations again show two regimes

Monte Carlo simulations show the two regimes

Mean first arrival time calculation We find: Collapse on master curve:

Two regimes: mutation- and migration- limited

Path faster than path This means: Adaptation usually originates as a neutral mutation in patch 1 (“Dykhuizen–Hartl effect”)

Cost of adaptation... patches genotypes 0 Assume that the mutant has a fitness defect in patch 1 Does this suppress adaptation?...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.

Large density: deterministic, ‘mean-field’ treatment (Similar to Fisher-KPP wave!)

Asymptotic speed can be calculated analytically Numerical solutions confirm this. Adaptation rate increases monotonously with slope a. No (finite) optimal slope?? Using theory of front propagation into unstable states: Van Saarloos, “Front propagation into unstable states” Physics Reports 386 (2003)

Beyond the mean field: stochastic simulations

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

Discrete, stochastic simulations Low density High density

Stochastic simulations do show an optimum

What’s wrong with the mean field equation?

Take-away messages 1. In the presence of environ- mental gradients, adaptation and range expansion can go hand in hand. 3. This mode of adaptation can be fast. 2. This is relevant for the evolution of drug resistance and virulence. 5. The dynamics show two regimes: a mutation-limited and a migration-limited one. 4. A fitness cost has little effect. 6. An optimal gradient exists – for any finite carrying capacity. Be careful with mean-field limit!

Acknowledgements Veni Grant (2012) J. Barrett Deris Terence Hwa Hwa lab at the UC San Diego: Cees Dekker Department of Bionanoscience, TU Delft