## The diffusion approximation of genetic drift

The Markov chain model has discrete allelic states and time advances from the initial conditions in individual, discrete generations, as is the case in actual biological populations. This discrete step process can be approximated using mathematical expressions where time and allele frequency are continuous variables. This class of model is based on the processes of molecular diffusion and so is termed the diffusion approximation of genetic drift (often called the diffusion equation) first solved by Motoo Kimura (1955). The diffusion approximation is based on partial differential equations and advanced mathematical techniques beyond the scope of this text. However, the general principles behind diffusion equations can be understood, especially with the aid of a physical example. The goal of this section is to introduce the situation that diffusion equations model using a particle metaphor and then to cover some of the conclusions about the process of genetic drift that have been reached using the diffusion equation. This introduction to the physical process of diffusion relies heavily on Denny and Gaines (2000), to which readers can turn for more details and biological applications.

Diffusion is the process where particles, moving in random directions, spread out and eventually reach a uniform concentration within the physical boundaries that limit their movement. For example, imagine putting a drop of ink in the center of a Petri dish filled with water. Initially, the concentration of ink is very uneven but as time passes the ink will diffuse and eventually reach a uniform concentration everywhere in the Petri dish. The rate at which the ink spreads out depends on what is called a diffusion coefficient. To understand the diffusion coefficient, we have to examine random movement of particles in some detail. This will hopefully lead to an improved understanding of genetic drift, so please be patient.

Let's modify the ink-diffusion example by substituting a special Petri dish. In this imaginary dish the ink particles can move only to the left or to the right from their current position, a situation diagrammed in Fig. 3.12. The particles have a constant velocity, so each will move the same distance in a fixed amount of time. We can call this distance moved per unit time 8 (pronounced "delta") since it is the change in position to the left or right. The direction of particle movement is random, with equal probability of moving to the left (p = 1/2) or right (q = 1/2) at any moment in time. Let's pick a point of reference somewhere along this axis of movement, call it x, and then track the movement of the ink particles relative to that point. First, what is the average movement of N particles between two time points? There are p of the particles traveling toward x that each move the distance +8. There are also q of the particles traveling away from x that each move the distance -8. The average movement of the particles is then x = p(8) + q(-8) (3.18)

Since the particles have equal chances of moving left or right (p = q = 1/2):

which means that the average or net movement of particles is zero. To relate this to genetic drift, if populations are like the ink particles but moving randomly in the one dimension of allele frequency, Figure 3.12 An imaginary Petri dish that confines ink particles so that they can move only to the left or to the right from their current position. The particles have a constant velocity, so each will move the distance 8 in a fixed amount of time. If the direction of particle movement is random (equal probability of moving left or right at any moment in time), the mean position of particles does not change but the variance in particle position increases with time. The frequency of particles passing through an area, such as the plane at x, depends on the net balance of particles arriving minus those that are leaving, called the flux of particles. The flux is determined by both the rate of diffusion of particles and gradients in the concentration of particles (net movement of particles is from areas of higher concentration to areas of lower concentration). If the left and right boundaries capture particles, then the diffusion coefficient drops to 0 at those points and particles will accumulate. The process of diffusion for particles is analogous to the process of genetic drift for allele frequencies in an ensemble population where allele frequencies "diffuse" because of sampling error.

Figure 3.12 An imaginary Petri dish that confines ink particles so that they can move only to the left or to the right from their current position. The particles have a constant velocity, so each will move the distance 8 in a fixed amount of time. If the direction of particle movement is random (equal probability of moving left or right at any moment in time), the mean position of particles does not change but the variance in particle position increases with time. The frequency of particles passing through an area, such as the plane at x, depends on the net balance of particles arriving minus those that are leaving, called the flux of particles. The flux is determined by both the rate of diffusion of particles and gradients in the concentration of particles (net movement of particles is from areas of higher concentration to areas of lower concentration). If the left and right boundaries capture particles, then the diffusion coefficient drops to 0 at those points and particles will accumulate. The process of diffusion for particles is analogous to the process of genetic drift for allele frequencies in an ensemble population where allele frequencies "diffuse" because of sampling error.

then we expect equal numbers of populations to move toward fixation and toward loss. The average change in allele frequency among all populations is expected to be zero.

The next thing we could do is to describe the variance in the position of ink particles over time, a measure of how spread out the particles become. Intuition suggests that even though the average is zero the variance should not be zero: spreading out of particles is what occurs during diffusion, after all. Equation A.2 in the Appendix shows that the variance is the average squared deviation from the mean. We just showed the mean particle location is zero. So the variance in the location of particles is then just the average square of their positions after one time step. The location of one particle, call it particle i, at time t = 1 can be expressed as its location at time t = 0 plus the amount a particle moves during one time step:

To get an expression for the variance in particle location we need to start out by squaring this expression for particle location:

and expanding the right side to get x2(t=i) = x2(t=0) + 2x2(t=0)S + S (3.22)

Using equation 3.22, which is the squared position of one particle, we can average over all N particles to get the variance in particle position: