Transcript Lecture 1
PEES this week! An introduction to modeling evolutionary dynamics In mathematics you don't understand things. You just get used to them. John von Neumann 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 Statistical inferential models Conceptual models Few parameters and variables Simple equations Predictive models Many parameters and variables Greater accuracy? 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 Develop a specific question Identify the minimal set of parameters and variables needed to address your question Develop simulations based on a more complete and realistic model Develop a mathematical model based on this set Analyze the model and develop testable predictions Test predictions using simulations Answer to question qualitatively incorrect add parameters or variables Answer to question qualitatively correct Test predictions using empirical data (experiments, field studies, literature surveys) Answer to question qualitatively incorrect add parameters or variables Answer to question qualitatively correct Nobel Prize Example 1: Modeling the evolutionary dynamics of sickle cell mediated malarial resistance Malaria in red blood cells Genotype Phenotype AA Normal red blood cells, malaria susceptible Aa Mostly normal red blood cells, malaria resistant aa Mostly sickled cells, very sick A ‘sickled’ red blood cell Empirical background An Example: Sickle cell and Malaria resistance. (sAA = .11, sSS = .8) 1.1 Fitness 0.9 • Two alleles, A and S that differ at only a single amino acid position 0.7 0.5 0.3 0.1 AA • AA Individuals are susceptible to Malaria • AS Individuals are resistant to Malaria and have only mild anemia • SS Individuals have severe anemia. AS Genotype SS 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 R0 can be used as an index of fitness Develop consistent notation WX = The fitness of genotype X pS = The frequency of the sickle allele S pi´ = The frequency of the sickle allele S in the next generation W = The mean fitness of the population p̂S = 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* 60 50 40 R0 30 20 10 0 AA Aa aa Genotype * Note that R0 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 Predicted evolutionary trajectories (Stabilizing selection/Overdominance) 1.1 Fitness 0.9 0.7 0.5 0.3 0.1 AA AS SS Genotype Frequency of sickle cell allele, p s1 = .11, s2 = .8 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 Generations The actual frequency of the A allele is in the ballpark of our estimate of .879 Frequency of the S allele in African populations 200 An introduction to conceptual models “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” The goal is conceptual insight, not precise quantitative prediction An introduction to modeling evolutionary dynamics In mathematics you don't understand things. You just get used to them. John von Neumann