Quantitative approximation of the discrete Moran process by a Wright–Fisher diffusion

Abstract

The Moran discrete process and the Wright–Fisher model are the most popular models in population genetics. The Wright–Fisher diffusion is commonly used as an approximation in order to understand the dynamics of population genetics models. Here, we give a quantitative large-population limit of the error occurring by using the approximating diffusion in the presence of weak selection and weak immigration in one dimension. The approach is robust enough to consider the case where selection and immigration are Markovian processes, whose large-population limit is either a finite state jump process, or a diffusion process.

Appendis: Discrete Wright–Fisher model and its approximating diffusion

Let consider the Wright–Fisher discrete model with selection and immigration. The population still consists of two species, immigration and selection are still the same. But the Markovian process $X^n_J$ evolves according to the following probability:

$$\begin{aligned} \mathbb {P}\left( X^J_{n+1}=\frac{k}{J}|X^J_{n}=x\right) = \left( \begin{array}{c} J\\ k\end{array}\right) P_x^k(1-P_x)^{J-k} \end{aligned}$$

with $P_x=mp+(1-m)\frac{(1+s)x}{1+sx}$.

At each step, all the population is renewed, so this process runs J times faster than the Moran process. And we usually, in the case of weak selection and immigration, it is usual to use the diffusion $\{Y_t\}_{t>0}$ defined by the following generator to approach this discrete model, when the population goes to infinity.

$$\begin{aligned} L=\frac{1}{2J}x(1-x)\frac{{\partial }^2}{\partial {x}^2}+ \left( sx(1-x)+m(p-x)\right) \frac{\partial }{\partial {x}} \end{aligned}$$

Theorem 4

Let f be in $C^5(I)$ then there exist a, $b\in \mathbb {R}^+$, depending on $m'$ and $s'$ (but not on f), and a constant K which satisfy when J goes to infinity:

$$\begin{aligned} \Vert \mathbb {E}_x \Big [f(X_{n}^{J} )\Big ]-\mathbb {E}_x \Big [f(Y_{n}^{J} ) \Big ] \Vert _J \le&a(e^{bt}-1)\left( \sum \limits _{i=1}^3\frac{\Vert f^{(i)\Vert _J} }{J}+K\frac{\sum \limits _{i=1}^5\Vert f^{(i)} \Vert _{J}}{J^2}\right) . \end{aligned}$$

Proof

Even if the structure of the proof is the same as for the discrete Moran model, however the difference of scale (in $\frac{1}{J}$ now) causes some small differences. Mainly, the calculation of the $\{\gamma _j\}_{j \in \{1,2,3,4\}}$ is a bit different. Note that we need to have $f \in C^5$ in the present theorem, which is stronger than for the Moran process. The main explanation comes from the calculation of $E[(X_{n+1}^J-x)^k|X_{n}^J=x]$, for which for the Wright–Fisher discrete process it is no longer of the order of $\frac{1}{J^k}$. Let us give some details.

First consider the moments $\{ E[\big (X_{n+1}^J-x\big )^k|X^J_n=x]\}_{k\leqslant 5}$:

$$\begin{aligned} E[X_{n+1}^J-x|X^J_n=x]=&m(p-x)+\frac{sx(1-x)}{1+sx}\\ E[\big (X_{n+1}^J-x\big )^2|X^J_n=x]=&\frac{1}{J}x(1-x)+\frac{1}{J}\left( m(p-x)+\frac{sx(1-x)}{1+sx}\right) \\&+\left( m(p-x)+\frac{sx(1-x)}{1+sx}\right) ^2+O\Big (\frac{1}{J^3}\Big )\\ E[\big (X_{n+1}^J-x\big )^3|X^J_n=x]=&x(x-1)(2x-1)\frac{1}{J^2}\\&-\frac{1}{J}3x(x-1)\big (m(p-x)+sx(1-x)\big )+O\Big (\frac{1}{J^3}\Big )\\ E[\big (X_{n+1}^J-x\big )^4|X^J_n=x]=&\frac{1}{J^2}3x^2(1-x)^2+O\Big (\frac{1}{J^3}\Big )\\ E[\big (X_{n+1}^J-x\big )^5|X^J_n=x]=&O\Big (\frac{1}{J^3}\Big ). \end{aligned}$$

To get a quantity of the order of $\frac{1}{J^3}$ we need to go to the fifth moment of $X_{n+1}^J-x$, so in the Taylor development we need to have f in $C^5$. Then,

$$\begin{aligned} L^{1}f(x)=&\;\frac{x(1-x)}{2J}f^{(2)}(x)+sx(1-x)+m(p-x)f^{(1)}(x)\\ L^{2}f(x)=&\;\frac{(x(1-x))^2}{4J^2}f^{(4)}(x)\\&+ \Big [ \frac{2(1-2x)x(1-x)}{4J^2}+\frac{2x(1-x)(sx(1-x)+m(p-x)}{2J}\Big ]f^{(3)}(x)\\&+ \Big [ \frac{-2x(1-x)}{4J^2} \\&+\frac{2x(1-x)(s(1-2x)-m)+(1-2x)(sx(1-x)+m(p-x))}{2J}\\&+\frac{(sx(1-x)+m(p-x))^2}{4J^2}\Big ]f^{(2)}(x) \\&+ \Big [\frac{-2sx(1-x)}{2J}\\&+(s(1-2x)-m)(sx(1-x)+m(p-x))\Big ]f^{(1)}(x)\\ L^{3}f(x)=&\;O\Big (\frac{1}{J^3}\Big ). \end{aligned}$$

We are now able to give the expression of the $\{\gamma _j\}_{j \in \{1,2,3,4\}}$, as in the Lemma 1.

Lemma 5

It exists bounded functions of x, $\{\gamma _j\}_{j \in \{1,2,3\}}$ such as when J is big enough,

$$\begin{aligned} \Vert (S_1-T_1)f\Vert _{J}\le \vert \gamma _1^{J}\vert \Vert f^{(1)} \Vert _{J} +\vert \gamma _2^{J}\vert \Vert f^{(2)} \Vert _{J}+\vert \gamma _3^{J}\vert \Vert f^{(3)} \Vert _{J}+\frac{K_{1}}{J^3} \end{aligned}$$

where for $i=1,\ldots ,3$, $\vert \gamma _i^{J}\vert$ is of the order of $\frac{1}{J^2}$.

Proof

The proof of this lemma is exactly the same as in lemma 1. Just the calculations are a little bit more tedious:

$$\begin{aligned}&\gamma _1^J=\frac{-sx(1-x)}{J} +(s(1-2x)-m)(sx(1-x)+m(p-x))\\&\gamma _2^J= \frac{-x(1-x)}{4J^2} +\frac{xs(6x^2-7x+1)+m(4x^2-2xp-x-p)}{4J}+O\Big (\frac{1}{J^3}\Big )\\&\gamma _3^J=\frac{x(x-1)(2x-1)}{12J^2}+O\Big (\frac{1}{J^3}\Big )\\&\gamma _4^J=O\Big (\frac{1}{J^3}\Big ) \end{aligned}$$

$\square$

The end of the proof follow exactly the same pattern. $\square$

So The Wright–Fisher dynamics requires more tricky calculations than the discrete Moran model but the spirit of the proof is the same. All the methods studied in this paper can be adapted in order to treat the Wright–Fisher model.