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 ones 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 hosts 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 sourcesink 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 (DykhuizenHartl 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 Whats 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