The Effect of Recurrent Mutations on Genetic Diversity in a Large Population of Varying Size

Abstract

Recurrent mutations are a common phenomenon in population genetics. They may be at the origin of the fixation of a new genotype, if they give a phenotypic advantage to the carriers of the new mutation. In this paper, we are interested in the genetic signature left by a selective sweep induced by recurrent mutations at a given locus from an allele $A$ to an allele $a$, depending on the mutation frequency. We distinguish three possible scales for the mutation probability per reproductive event, which entail distinct genetic signatures. Besides, we study the hydrodynamic limit of the $A$- and $a$-population size dynamics when mutations are frequent, and find non trivial equilibria leading to several possible patterns of polymorphism.

Appendix: Technical Results

In this last section, we recall some results on birth and death processes whose proofs can be found in Lemma 3.1 in [44] and in [1], pp. 109 and 112.

Proposition A.1

Let $Z=(Z_{t})_{t \geq 0}$ be a birth and death process with individual birth and death rates $b$ and $d$. For $i \in \mathbb{Z}_{+}$, $T_{i}=\inf \{ t\geq 0, Z_{t}=i \}$ and $\mathbb{P}_{i}$ (resp. $\mathbb{E}_{i}$) is the law (resp. expectation) of $Z$ when $Z_{0}=i$. Then

• For $(i,j,k) \in \mathbb{Z}_{+}^{3}$ such that $j \in (i,k)$,

  $$ \mathbb{P}_{j}(T_{k}< T_{i})= \frac{1-(d/b)^{j-i}}{1-(d/b)^{k-i}}. $$

  (A.1)

• If $d\neq b \in \mathbb{R}_{+}^{*}$, for every $i\in \mathbb{Z}_{+}$ and $t \geq 0$,

  $$ \mathbb{P}_{i}(T_{0}\leq t )= \biggl( \frac{d(1-e^{(d-b)t})}{b-de^{(d-b)t}} \biggr)^{i}. $$

  (A.2)

• If $0< d< b$, on the non-extinction event of $Z$, which has a probability $1-(d/b)^{Z_{0}}$, the following convergence holds:

  $$ T_{N}/\log N \mathop{\to }_{N \to \infty } (b-d)^{-1}, \quad \textit{a.s.} $$

  (A.3)

The next lemma quantifies the time spent by a birth and death process with logistic competition in a vicinity of its equilibrium size. It is stated in [9], Theorem 3(c).

Lemma A.1

Let $b,d,c$ be in $\mathbb{R}_{+}^{*}$ such that $b-d>0$. Denote by $(W_{t},t \geq 0)$ a density dependent birth and death process with birth rate $bn$ and death rate $(d+c n/K)n$, where $n \in \mathbb{N}$ is the current state of the process and $K \in \mathbb{N}$ is the carrying capacity. Fix $0 < \eta_{1} < (b - d)/c$ and $\eta_{2} > 0$, and introduce the stopping time

$$\begin{aligned} \mathcal{S}_{K} = \inf \biggl\{ t \geq 0: W_{t} \notin \biggl[ \biggl(\frac{b - d}{c}- \eta_{1} \biggr)K, \biggl( \frac{b - d}{c}+\eta_{2} \biggr) \biggr] \biggr\} . \end{aligned}$$

Then, there exists $V > 0$ such that, for any compact subset $C$ of $](b - d)/c - \eta_{1}, (b - d)/ c + \eta_{2} [$,

$$ \lim_{K \to \infty } \sup_{k/K \in C} \mathbb{P}_{k}\bigl( \mathcal{S}_{K}< e^{KV} \bigr) = 0. $$

(A.4)

We end this section with a coupling of birth and death processes with close birth and death rates.

Lemma A.2

Let $Z_{1}$ and $Z_{2}$ be two birth and death processes with initial state 1 and respective individual birth and death rates $(b_{1},d _{1})$ and $(b_{2},d_{2})$ belonging to a common compact set $D$ in $\mathbb{R}_{+}^{2}$. Then we can couple $Z_{1}$ and $Z_{2}$ in such a way that

$$ \sup_{t \geq 0} \mathbb{E} \bigl[ \bigl( Z_{1}(t)e^{-(b_{1}-d_{1})t}-Z _{2}(t)e^{-(b_{2}-d_{2})t} \bigr)^{2} \bigr]\leq c \bigl(|b_{2}-b_{1}|+|d _{2}-d_{1}|\bigr), $$

(A.5)

where the positive constant $c$ only depends on $D$.

Proof

For the sake of simplicity, we assume in the proof than $b_{1}< b_{2}$ and $d_{1}< d_{2}$, but other cases can be treated similarly. Let $B(ds,d\theta )$ and $D(ds,d\theta )$ be two independent Poisson random measures with intensity $ds d\theta$. We can construct the two processes $Z_{1}$ and $Z_{2}$ with respect to the measures $B$ and $D$. For $i \in \{1,2\}$, introduce

$$\begin{aligned} Z_{i}(t)= 1+ \int_{0}^{t} \int_{\mathbb{R}_{+}} \mathbf{1}_{\{ \theta \leq b_{i} Z_{i}(s^{-}) \}}B(ds,d\theta ) - \int_{0}^{t} \int_{\mathbb{R}_{+}}\mathbf{1}_{\{ \theta \leq d_{i} Z_{i}(s^{-}) \}}D(ds,d \theta ). \end{aligned}$$

We also introduce an auxiliary birth and death process, $Z_{3}$, with individual birth and death rates $(b_{3},d_{3})=(b_{2},d_{1})$, which will allow us to compare $Z_{1}$ and $Z_{2}$:

$$\begin{aligned} Z_{3}(t)= 1+ \int_{0}^{t} \int_{\mathbb{R}_{+}} \mathbf{1}_{\{ \theta \leq b_{2} Z_{3}(s^{-}) \}}B(ds,d\theta ) - \int_{0}^{t} \int_{\mathbb{R}_{+}}\mathbf{1}_{\{ \theta \leq d_{1} Z_{3}(s^{-}) \}}D(ds,d \theta ). \end{aligned}$$

As $b_{1}< b_{2}$ and $d_{1}< d_{2}$, we have the following almost sure inequalities:

$$ Z_{1}(t)\leq Z_{3}(t)\quad \text{and} \quad Z_{2}(t)\leq Z_{3}(t) \quad \text{a.s.} $$

(A.6)

Applying Itô’s Formula yields that $M_{i}(t)= Z_{i}(t)e^{-(b_{i}-d _{i})t}$, $i \in \{1,2,3\}$ are martingales, and we can express the quadratic variation of their differences. Using (A.6) we get:

$$\begin{aligned} \langle M_{3}-M_{1} \rangle_{t} =& \int_{0}^{t} \bigl((b_{1} +d_{1}) Z _{1}(s) \bigl(e^{-(b_{1}-d_{1})s}-e^{-(b_{2}-d_{1})s} \bigr)^{2} \\ &{}+ \bigl( b_{2} Z_{3}(s)- b_{1} Z_{1}(s)\bigr)e^{-2(b_{2}-d_{1})s}+ \bigl( d_{1} Z _{3}(s)- d_{1} Z_{1}(s)\bigr)e^{-2(b_{2}-d_{1})s} \bigr)ds. \end{aligned}$$

Then, by taking the expectation, we obtain

$$\begin{aligned} \mathbb{E} \bigl[ (M_{3}-M_{1})^{2}(t) \bigr] =& \int_{0}^{t} \bigl((b_{1} +d _{1}) \bigl(e^{-(b_{1}-d_{1})s}-2e^{-(b_{2}-d_{1})s}+e^{-(2b_{2}-b_{1}-d _{1})s} \bigr) \\ &{} + b_{2}e^{-(b_{2}-d_{1})s} - b_{1} e^{-(2b_{2}-d_{1}-b_{1})s}+ d_{1}e ^{-(b_{2}-d_{1})s} - d_{1} e^{-(2b_{2}-d_{1}-b_{1})s} \bigr)ds \\ \leq& (b_{1} +d_{1}) \biggl(\frac{1}{b_{1}-d_{1}}- \frac{2}{b_{2}-d_{1}}+\frac{1}{2b _{2}-b_{1}-d_{1}} \biggr) \\ &{} + \frac{b_{2}}{b_{2}-d_{1}} - \frac{b_{1}}{2b_{2}-d_{1}-b_{1}}+ \frac{d _{1}}{b_{2}-d_{1}} - \frac{d_{1}}{2b_{2}-d_{1}-b_{1}} \\ \leq& c (b_{2}-b _{1}), \end{aligned}$$

(A.7)

where for the first inequality we have used that the square of a martingale is a submartingale, and in the last one the continuity of the functions involved. We obtain similarly

$$\begin{aligned} \langle M_{3}-M_{2} \rangle_{t} =& \int_{0}^{t} \bigl(\bigl(b_{2} Z_{2}(s)+d _{2}Z_{2}(s) \wedge d_{1} Z_{3}(s) \bigr) \bigl(e^{-(b_{2}-d_{1})s}-e^{-(b_{2}-d _{2})s} \bigr)^{2} \\ &{} + b_{2}\bigl( Z_{3}(s)- Z_{2}(s) \bigr)e^{-2(b_{2}-d_{1})s} \\ &{} + \mathbf{1}_{\{d_{1} Z_{3}(s)< d_{2}Z_{2}(s) \}} \bigl( d_{2} Z_{2}(s)- d _{1} Z_{3}(s)\bigr)e^{-2(b_{2}-d_{2})s} \\ &{} + \mathbf{1}_{\{d_{1} Z_{3}(s)> d_{2}Z_{2}(s) \}} \bigl( d_{1} Z_{3}(s)-d _{2} Z_{2}(s)\bigr)e^{-2(b_{2}-d_{1})s} \bigr)ds. \end{aligned}$$

But applying (A.6) we get that

$$\begin{aligned} \mathbf{1}_{\{d_{1} Z_{3}(s)< d_{2}Z_{2}(s) \}} \bigl( d_{2} Z_{2}(s)- d _{1} Z_{3}(s)\bigr) \leq& (d_{2}-d_{1})Z_{2}(s)+d_{2} \bigl(Z_{2}(s)- Z_{3}(s)\bigr) \\ \leq& (d_{2}-d_{1})Z_{2}(s) \end{aligned}$$

and that

$$\begin{aligned} \mathbf{1}_{\{d_{1} Z_{3}(s)> d_{2}Z_{2}(s) \}} \bigl( d_{1} Z_{3}(s)-d _{2} Z_{2}(s)\bigr)\leq d_{2}\bigl( Z_{3}(s)- Z_{2}(s)\bigr), \end{aligned}$$

which yields

$$\begin{aligned} \langle M_{3}-M_{2} \rangle_{t} \leq& \int_{0}^{t} \bigl(\bigl(b_{2} Z_{2}(s)+d _{2}Z_{2}(s) \bigr) \bigl(e^{-(b_{2}-d_{1})s}-e^{-(b_{2}-d_{2})s}\bigr)^{2} \\ &{} + b_{2}\bigl( Z_{3}(s)- Z_{2}(s) \bigr)e^{-2(b_{2}-d_{1})s} + (d_{2}-d_{1})Z _{2}(s)e^{-2(b_{2}-d_{2})s} \\ &{} + d_{2}\bigl( Z_{3}(s)- Z_{2}(s) \bigr)e^{-2(b_{2}-d_{1})s} \bigr)ds. \end{aligned}$$

Taking the expectation and reasoning similarly as before give for every positive $t$,

$$\begin{aligned} \mathbb{E} \bigl[ (M_{3}-M_{2})^{2}(t) \bigr] \leq c (d_{2}-d_{1}). \end{aligned}$$

Using that for $a,b,c\geq 0$, $(a-c)^{2} \leq 2(a-b)^{2}+2(b-c)^{2}$ and adding (A.7) end the proof of Lemma A.2. □