Chapter 3 Population Genetics for Large Populations The diversity of life is a fundamental empirical fact of nature. Casual observation alone confirms the great variety of species. But diversity also prevails within single species. As in human populations, the individuals of a species vary considerably in their outward traits—size and shape, external markings, fecundity, disease resis- tance, etc. This is called phenotypic variation, and, since phenotypes are shaped by genes, it reflects an underlying, genotypic diversity. The achievements of genetics and molecular biology, as described in Chapter 1, have allowed us to measure and confirm the genotypic variability of populations down to the molecular level. The science of genotypic variation in interbreeding populations is called popu- lation genetics. Its goal is understanding how genetic variation changes under the influence of selection, mutation, and the randomness inherent in mating, as one generation succeeds another. To achieve this, population geneticists combine what is known about the mechanisms of heredity—how DNA carries genetic information, how chromosomes function and give rise to Mendel’s laws, how mutations arise— with hypotheses about mating and selective advantage, to propose mathematical equations for the evolution of genotype frequencies. By comparing the solutions of these equations to field data, they can then test hypotheses and make inferences about genealogy and evolution. This chapter is an introduction to elementary, population genetics models for large populations and simple genotypes. ‘Large’ is a vague term, but here it means the limit as the population size tends to infinity. The use of this limit is called the “infinite population assumption.” When it is imposed in a model, genotype frequen- cies become deterministic functions of time, and evolve as deterministic dynamical systems, even though random factors influence how each generation is produced from the previous one. The mathematics employed in this chapter are some elementary probability and difference equations (also called recursion equations). For the required probability— 1 2 CHAPTER 3. POPULATION GENETICS I random sampling, the law of large numbers, simple calculations using independence and conditioning—the student should read Section 2.1. Linear difference equations are treated briefly in section 3.1, and a graphical technique for nonlinear difference equations is described at the beginning of Section 3.4. Difference equations appear in this chapter as models for the evolution of genotype frequencies. Population genetics is an excellent introduction to the art of probabilistic mod- eling. Like any art, it can only be learned by doing, and this chapter is written to encourage active participation. The student is asked to carry out many steps of the exposition and to construct extensions of models, through guided exercises embedded in the text. It is important to treat these exercises as an integral part of reading the chapter. 3.1 Difference Equations; Introduction The usual notation for a generic sequence is (x1, x2, x3, . , xn,...). In this chapter, we instead use the notation (x(0), x(1), x(2) ...), and we denote a generic term in a sequence by x(t). We use t as the index because it will always represent a time, and we write x(t) rather than xt so that subscripts may be used for other purposes. In general, our convention is that the first term of a sequence denotes a value at time t = 0, and we begin sequences with x(0). Often (x(t))t≥0, or simply {x(t)}, is used to abbreviate (x(0), x(1),...). A difference equation is a recursive equation that determines a sequence of num- bers. Some simple examples are x(t+1) = αx(t), t ≥ 0 (3.1) x(t+2) = x(t+1) + x(t), t ≥ 0 (3.2) x(t+2) = x(t+1)x(t), t ≥ 0 (3.3) A solution to one of these difference equations is any sequence satisfying the given relationship for all integers t ≥ 0. In each case, it is easy to see that there is a unique solution once appropriate starting values are given. Consider equation (3.1), for example. If {x(t)} is a solution, then x(1) = αx(0), x(2) = αx(1) = α2x(0), and x(3) = αx(2) = α2x(0), for starters. Clearly, by continuing this procedure, we find that x(t) = αtx(0) for all t ≥ 0. Thus (3.1) determines a unique sequence for each initial value x(0). In the case of equation (3.2), the first equation, for t+0, is x(2) = x(1) + x(0), which determines x(2) from x(1) and x(0), but imposes no relationship between x(0) and x(1). Thus both x(1) and x(0) must be prescribed to get a unique solution. Consider what happens when x(1) = x(0) = 1. Then x(2) = 1 + 1 = 2, x(3) = x(2) + x(1) = 2 + 1 = 3, x(4) = x(3) + x(2) = 5, etcetera. This is just the definition of the famous Fibonacci sequence, in which each term is the sum of the previous two. In Exercise 3.1.3 you will derive an explicit formula for x(t) as a function of t. 3.1. DIFFERENCE EQUATIONS 3 As a simple exercise, the reader should solve equation (3.3) with initial conditions x(0) = x(1) = 1. This solution is very simple! A slightly more challenging problem is to solve the equation when x(0) = 1 and x(1) = 2. (Hint: express the answer in terms of the Fibonacci sequence.) Difference equations of the sort x(t+1) = φ(t, x(t)), t ≥ 0, where φ is some given function of (t, x), are called first-order difference equations. Equations of the sort x(t+2) = ψ(t, x(t+1), x(t)), where ψ is function of (t, x, y), are called second-order difference equations. In the same way, one can define third, fourth, and higher order difference equations. We will only encounter first and second order equations. When x(t+1) = φ(x(t)), or x(t+2) = ψ(x(t+1), x(t)), that is, when the right-hand side does not depend on t explicitly, the difference equation is said to be autonomous. We will deal exclusively with autonomous equations. Two major goals in the study of any difference equation are: to find a solution in closed form, i.e., a formula expressing the solution x(t) as an explicit function of t; and to analyze the limiting behavior of x(t) as t → ∞. Of course, if you can solve the first problem, the solution to the second usually follows as an easy consequence. Now, except for special cases, finding an explicit solution is usually hopeless. However, it is often possible to deduce the long-time, limiting behavior, even when a closed form solution is not known. We shall study one technique for doing so in in Section 3.4. One class of difference equations that do admit explicit solutions is linear dif- ference equations. A difference equation is said to be linear if it is linear in all the variables x(t), x(t + 1), x(t + 2), etc., that appear in it. For example, (3.1) and (3.2) above are linear difference equations, but (3.3) is non-linear. The general, autonomous, linear, first order difference equation is x(t+1) = αx(t) + β. (3.4) The general, autonomous, linear, second order difference equation is x(t+1) = αx(t) + γx(t−1) + β. (3.5) If β = 0, these equations are said, in addition, to be homogeneous, and when β 6= 0, they are called inhomogeneous. These equations are important in elementary population genetics, and we will devote the rest of this section to formulas and methods for their solution. Consider first-order, linear equations. By solving equation (3.1), we have already seen that the general solution to (3.4) with β = 0, is x(t) = Aαt. Here, we use A in place of x(0), to indicate it can be any constant. To solve (3.4) when β 6= 0, we will take advantage of the linearity of the equation. Suppose {z(t)} is a given solution to (3.4), and {x(t)} is any other solution. Then x(t+1) − z(t+1) = αx(t) + β − [αz(t) + β] = α[x(t) − z(t)]. 4 CHAPTER 3. POPULATION GENETICS I Thus {x(t) − z(t)} is a solution of the linear, homogeneous version of (3.4), and so x(t) − z(t) = Aαt for some constant A. Therefore, given one, particular solution {z(t)} of (3.4), any other solution has the form , x(t) = Aαt + z(t), (3.6) which means the right-hand side is a general solution. The trick is now to guess a particular solution {z(t)}. Suppose we can find a fixed point of the equation (3.4). This is a value b such that b = αb + β. Then the constant sequence, z(t) = b for all t is a particular solution, because z(t+1) = b = αb + β = αz(t) + β. But, as long as α 6= 1, b = αb + β has the unique solution b = β/(1 − α), and we find a constant particular solution, which can be inserted in (3.6) to find the general solution. The explicit form of this solution, its properties as t → ∞, and what happens when α = 1 are all summarized in the next result. Proposition 1 (i) If α 6= 1, the general solution to (3.4) is β x(t) = Aαt + , t ≥ 0, where A is an arbitrary constant, (3.7) 1 − α and the solution to (3.4) satisfying the initial condition x(0) = x0 is β β x(t) = x − αt + , t ≥ 0.