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

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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.

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Acknowledgements

The author would like to thank Helmut Pitters for fruitful discussions at the beginning of this work. She also wants to thank Jean-René Chazottes and Pierre Collet for advice and references on planar dynamical systems, as well as Pierre Recho for his help with the use of Mathematica, and two anonymous reviewers for their careful reading of the paper, and several suggestions and improvements. This work was partially funded by the Chair “Modélisation Mathémathique et Biodiversité” of Veolia Environnement—Ecole Polytechnique—Museum National d’Histoire Naturelle—Fondation X and the French national research agency ANR-11-BSV7-013-03.

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Correspondence to Charline Smadi.

Appendix: Technical Results

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. □

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Smadi, C. The Effect of Recurrent Mutations on Genetic Diversity in a Large Population of Varying Size. Acta Appl Math 149, 11–51 (2017). https://doi.org/10.1007/s10440-016-0086-x

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Keywords

  • Eco-evolution
  • Birth and death process with immigration
  • Selective sweep
  • Coupling
  • Competitive Lotka-Volterra system with mutations

Mathematics Subject Classification

  • 92D25
  • 60J80
  • 60J27
  • 92D15
  • 60F15
  • 37N25