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I am reading from the classical textbook Principles of Population Genetics, Hartl and Clark (pdf here).

Introduction

Let $f(p,x,t)$ denote the distribution of allele frequency $x$ at time $t$ knowing that at time $t=0$ the frequency was $p$. One can model the change in this distribution with time using the Kolmogorov forward equation

$$\frac{\partial f(p,x,t)}{\partial t} = -\frac{\partial [M(x) \cdot f(p,x,t)]}{\partial x}+\frac12\,\frac{\partial^2[V(x) \cdot f(p,x,t)]}{\partial x^2}$$

where $M(x)$ is the "drift parameter", which represents natural selection (thus, in the absence of the selection $M(x)=0$), and $V(x)$ is the "diffusion parameter", which represents genetic drift. The diffusion parameter is $V(x) = \frac{x(1-x)}{2N}$, where $N$ is the population size.

Question

Why is it true that $V(x) = \frac{x(1-x)}{2N}$?

I welcome intuitive explanations and mathematical proof.

Thoughts

I would think it derivates from the Wright-Fisher model of genetic drift where the distribution of allele frequency at the next generation is given by a binomial distribution.

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  • $\begingroup$ What is the definition of N in V(x)? $\endgroup$ – ddiez Dec 23 '15 at 5:32
  • $\begingroup$ $N$ is the population size. Thanks for noticing. Question edited. $\endgroup$ – Remi.b Dec 23 '15 at 5:38
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It does indeed proceed from the Wright-Fisher model, specifically from its diffusion process approximation.

If the population is size $N$, then at generation $t$ the number of alleles is $A(t)$, and so the frequency of alleles is $x(t) = A(t)/(2N)$, assuming the diploid case. Then Wright-Fisher says that: $$ A(t+1)\mid A(t) \sim \text{Bin}(2N,x(t)) $$ So the distribution of the next generation's count, given the last one, is binomially distributed (under random mating). One can find that, for a binomially distributed random variable $b\sim \text{Bin}(m,p)$, the mean and variance are given by: $$ \mathbb{E}[b]=mp\;\;\;\;\;\;\&\;\;\;\;\;\;\mathbb{V}[b]=mp(1-p) $$ Thus, we get that $$ \mathbb{E}[A(t+1)\mid A(t)] = 2N x(t) \;\;\;\;\;\;\&\;\;\;\;\;\; \mathbb{V}[A(t+1)\mid A(t)] = 2N x(t)[1-x(t)] $$ Applying $x(t)=A(t)/(2N)$, we see that \begin{align} \mathbb{E}[x(t+1) \mid x(t)] &= \frac{1}{2N}\mathbb{E}[A(t+1)\mid A(t)]= x(t) \\ \mathbb{V}[x(t+1)\mid x(t)] &= \frac{1}{(2N)^2}\mathbb{V}[A(t+1)\mid A(t)] = \frac{x(t)[1-x(t)]}{2N} \end{align} using the fact that $\mathbb{V}[cX]=c^2\mathbb{V}[X]$.

We can relate this to forward Kolmogorov as follows. Recall that the binomial distribution can be approximated by a normal distribution, with mean $\mu$ and variance $\sigma^2$ given by the mean and variance of the binomial. This tells us that: $$ x(t+\delta t)\mid x(t) \sim \mathcal{N}(x(t),(\delta t)x(t)[1-x(t)]/(2N)) $$ The properties of the normal distribution then imply that $$ \Delta x_t=x(t+\delta t) - x(t)\sim\mathcal{N}(0,\sigma^2(x_t)\delta t) $$ where $\sigma^2(x_t) = x(t)[1-x(t)]/(2N)$. This implies the following equality (in distribution): $$ \Delta x_t = \sigma(x_t)\Delta W_t $$ where $\Delta W_t\sim \mathcal{N}(0,\delta t)$. As $\delta t\rightarrow 0$, we get a stochastic differential equation, $$ dx_t = \sigma(x_t)dW_t $$ where the solutions are Markov random processes (unlike e.g. ODEs where there is only one solution and it is a deterministic path; think of it as a noisy ODE), specifically in this case an Ito diffusion. Note the SDE has no $dt$ component since the increment mean was $0$. If we then consider the density function for the random process $p(x,t)$, it must satisfy the Fokker-Planck (Forward Kolmogorov) equation: $$ \partial_t p(x,t) = -\partial_{xx}[p(x,t)\sigma^2(x)/2] $$ which gives a probability distribution over the allelic frequency value at each time point (given some initial value I've not specified here). Note that $V(x)=\sigma^2(x)$.


In terms of intuition, I'm not really sure. Essentially $V(x)$ measures how much perturbation in the allelic frequency you can expect to happen each generation due to purely random effects, i.e. genetic drift. Notice that there is no perturbation when $x=0$ or $x=1$, i.e. no random changes can happen if no one or everyone has the allele. Notice also that this variance is exactly the variance of a Bernoulli distribution. It's like we've distilled the individual-level model to a population-level model that simply looks at the frequency of the binary choice of allele presence, I guess. The variance (noise) is maximal when the frequency is $1/2$. It's sort of pushing the allele away from the middle, by increasing noise when one goes there; one might expect that (if run long enough) any such model will hit and get stuck at $0$ or $1$ (not sure if this is true). I looked a bit if there were any other interesting interpretations of the sde considered here (e.g. in physics), but couldn't find any. Basically, it would be equivalent to a heat equation, spreading under some potential function controlled by $\sigma$.


Your question is closely related to this one. My answer largely follows Tataru et al, Statistical Inference in the Wright–Fisher Model Using Allele Frequency Data.

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