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An introduction to modeling evolutionary dynamics John von Neumann In mathematics you don't understand things. You just get used to them.

Types of models in evolutionary biology Conceptual models  What determines what will happen? When does natural selection overwhelm genetic drift? When is recombination important? When will sex evolve? Predictive models  What will happen? Which strain of influenza will be dominant next year? What selection differential must be applied to increase milk yield by 10%? How quickly will insecticide resistance spread in the European Corn Borer? Statistical inferential models  What did happen? Has Influenza hemaglutinin evolved in response to natural selection or drift? Did speciation in Heliconius occur in sympatry or allopatry? Was differential pollinator visitation responsible for stabilizing selection?

Types of models in evolutionary biology Conceptual models Predictive models Statistical inferential models Many parameters and variables  Greater accuracy? Few parameters and variables  Simple equations

Conceptual models The goal is conceptual insight, not precise quantitative prediction This requires a simple model yielding analytical tractable equations This in turn, requires considering only a subset of variables and parameters

How to build a conceptual model “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” John von Neumann The challenge is to decide which four build the elephant!

How to choose the parameters and variables that matter Develop a simple and well defined question For example: What types of selection maintain polymorphism at a single locus? Does directional selection favor increased recombination? Be willing to make risky or even obviously incorrect assumptions For example: Infinite population size Free recombination Random mating No selection

Treat modeling as an ongoing process Identify the minimal set of parameters and variables needed to address your question Develop a mathematical model based on this set Analyze the model and develop testable predictions Develop simulations based on a more complete and realistic model Test predictions using simulations Test predictions using empirical data (experiments, field studies, literature surveys) Develop a specific question Answer to question qualitatively correct Answer to question qualitatively incorrect add parameters or variables Answer to question qualitatively correct Nobel Prize Answer to question qualitatively incorrect add parameters or variables

Example 1: Modeling the evolutionary dynamics of sickle cell mediated malarial resistance Malaria in red blood cells A ‘sickled’ red blood cell GenotypePhenotype AANormal red blood cells, malaria susceptible AaMostly normal red blood cells, malaria resistant aaMostly sickled cells, very sick

Empirical background An Example: Sickle cell and Malaria resistance. Genotype (s AA =.11, s SS =.8) Fitness Two alleles, A and S that differ at only a single amino acid position AA Individuals are susceptible to Malaria AS Individuals are resistant to Malaria and have only mild anemia SS Individuals have severe anemia.

Develop simple, well-defined questions Will genetic polymorphism be maintained? How much genetic polymorphism will exist at equilibrium? At equilibrium, what proportion of the population will experience sickle cell anemia?

Make risky or even incorrect assumptions Infinite population size  Need to follow expectations only (higher moments disappear) Random mating  Can utilize Hardy-Weinberg Equilibrium (1 dynamical equation) No mutation  Saves a parameter; yields simpler equations No gene flow  Can consider only local dynamics (1 dynamical equation) Constant population size  R 0 can be used as an index of fitness

Develop consistent notation W X = The fitness of genotype X p S = The frequency of the sickle allele S p i ´= The frequency of the sickle allele S in the next generation = The mean fitness of the population = The equilibrium frequency of the sickle allele S

Write down dynamical equations using the notation

Solve for equilibria

What do these equilibria tell us biologically?

Next time we will use local stability analyses to answer the remainder of our questions

What is fitness? Fitness – The fitness of a genotype is the average per capita lifetime contribution of individuals of that genotype to the population after one or more generations* Genotype R0R0 * Note that R 0 is a good measure of an organisms fitness only in a population with a stable size. Things are more complicated in growing populations!

Overdominant selection on single loci Genotype s 1 =.11, s 2 =.8 (Stabilizing selection/Overdominance) Fitness Generations Frequency of sickle cell allele, p Predicted evolutionary trajectories The actual frequency of the A allele is in the ballpark of our estimate of.879 Frequency of the S allele in African populations

An introduction to conceptual models The goal is conceptual insight, not precise quantitative prediction “Truth is much too complicated to allow anything but approximations” John von Neumann “There's no sense in being precise when you don't even know what you're talking about”

An introduction to modeling evolutionary dynamics John von Neumann In mathematics you don't understand things. You just get used to them.