The recovery of a recessive allele in a Mendelian diploid model

Abstract

We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. Neukirch and Bovier (J Math Biol 75:145–198, 2017) proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they would in populations reproducing asexually. In the present paper we prove that this indeed opens the possibility that individuals with a pure genotype can reinvade in the population after the appearance of further mutations. We thus expose a rigorous description of a mechanism by which a recessive allele can re-emerge in a population. This can be seen as a statement of genetic robustness exhibited by diploid populations performing sexual reproduction.

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Notes

  1. 1.

    The article Neukirch and Bovier (2017) only state that survival occurs up to time \(K^{1/4-{\alpha }}\). However, taking into account that it is really only the survival of the aA population that needs to be ensured, one can easily improve this to \(K^{1/2-{\alpha }} \).

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Correspondence to Loren Coquille.

Additional information

We acknowledge financial support from the German Research Foundation (DFG) through the Hausdorff Center for Mathematics, the Cluster of Excellence ImmunoSensation, and the Priority Programme SPP1590 Probabilistic Structures in Evolution. L.C. has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) through the Exploratory Project CanDyPop, as well as by the Swiss National Science Foundation through the Grant No. P300P2_161031, and by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement - École Polytechnique - Muséum National d’Histoire Naturelle - Fondation X. We would like to thank Pierre Collet and Vincent Beffara for their help on the theory of dynamical systems and fruitful discussions.

Appendix

Appendix

We collect in this appendix all the important definitions of times separating phases or subphases of the process, and provide a proof summary with all the implications. Recall Definition 4.1, and Figs. 5 and 6. We write \(N_\delta (x)\) for the \(\delta \)-neighbourhood of \(x\in {\mathbb R}^6\). We consider

$$\begin{aligned} {\Delta }>\delta>{\varepsilon }_0>{\varepsilon }>0. \end{aligned}$$
(A.1)

and

$$\begin{aligned} T_1&:=T_{{\varepsilon }_0}^{aa+aA+aB+AB+BB},\\ T_=&:=T^{aA=aB},\\ T_2&:=T^{aA=\delta AA}\wedge T^{aB=\delta AB}\wedge T^{aa= aA\wedge aB},\\ T_3&:=T^{aa}_{{\bar{n}}_a-{\varepsilon }^{{\gamma }/2}}. \end{aligned}$$

where \(\gamma :=2/(1+\eta {\bar{n}}_B-\Delta )\) is the order of magnitude of \(n_{AA}\) at \(T_2\) : \(n_{AA}(T_2)={\Theta }({\varepsilon }^\gamma )\).

We summarise below the detailed structure of the proof (we abbreviate Lemma, Proposition and Theorem by L,P, and T respectively):

  1. Phase 1:

    \(t\in [0,T_1]\) Initial conditions: \((n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})=({\Theta }({\varepsilon }^2),{\varepsilon },{\bar{n}}_A\pm {\Theta }({\varepsilon }), 0,{\varepsilon }^3, 0)\). With those initial conditions the following bounds hold:

    $$\begin{aligned} n_{AA}\le {\bar{n}}_A,\quad n_{BB}\le {\bar{n}}_B, \quad n_{aB}\le n_{AB}\quad \text {(Proposition 4.3) } \end{aligned}$$

    Here are the main steps of the proof of Propostion 4.4 and Corollary 4.5. Until the time \(T_1\),

    $$\begin{aligned} \text {(P4.3)}\Rightarrow \quad&n_{BB}\le n_{AB}^2 \quad \text {(1)}\\ \text {(1)}\Rightarrow \quad&n_{aB}\le n_{aA}n_{AB} \quad \text {(2)}\\ \text {(1)(2)}\Rightarrow \quad&n_{aa}\le n_{aA}^2 \quad \text {(3)}\\ \text {(1)(2)(3)}\Rightarrow \quad&T_1= T^{AB}_{{\varepsilon }_0}={\Theta }(\log ({\varepsilon }_0/{\varepsilon }^3)^{1/(\Delta -{\Theta }({\varepsilon }_0))})\quad \text {(Proposition 4.4)}\\ \text {(P4.4)}\Rightarrow \quad&{\dot{n}}_{AB}={\Theta }(\Delta ) n_{AB} \quad \text {(Corollary 4.5(1))}\\ \text {(P4.4)}\Rightarrow \quad&{\bar{n}}_A-{\Theta }({\varepsilon }) \le \Sigma _5\le {\bar{n}}_A+2\Delta {\varepsilon }_0 \quad \text {(4)}\\ \text {(4)(P4.4)}\Rightarrow \quad&n_{aB}\le {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}{\varepsilon }_0), n_{aA}\le {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}),\\&n_{aa}\le {\Theta }({\varepsilon }^{2-{\Theta }({\varepsilon }_0)}), n_{AA}={\bar{n}}_A\pm {\Theta }({\varepsilon }_0) \quad \text {(Corollary 4.5(3)))}\\ \end{aligned}$$
  2. Phase 2:

    \(t\in [T_1,T_2]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&=({\Theta }({\varepsilon }^{2-{\Theta }({\varepsilon }_0)}),{\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}),\\&{\bar{n}}_A\pm {\Theta }({\varepsilon }_0 ), {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}{\varepsilon }_0),{\varepsilon }_0, {\Theta }({\varepsilon }_0^2)). \end{aligned}$$

    As the process stays uniformly bounded in time, until \(T_2\) we have

    $$\begin{aligned} n_{aa},n_{aA},n_{aB}\le {\Theta }(\delta )\qquad (*) \end{aligned}$$
    $$\begin{aligned} (*)&\Rightarrow \quad (n_{AA},n_{AB},n_{BB})=(n^{up}_{AA},n^{up}_{AB},n^{up}_{BB})+{\Theta }(\delta ) \quad \text {(Lemma 4.6)}\\ \text {L4.6,}(*)&\Rightarrow \quad \Sigma _5={\bar{n}}_B-{\Delta }n_{AA}/{c{\bar{n}}_B}+ {{\Theta }({\Delta }^2n_{AA})} \quad \text {(Proposition 4.7)}\\ \text {P4.7,}(*)&\Rightarrow \quad {\dot{\Sigma }}_{aA,aB}\ge -{\Theta }({\Delta })\Sigma _{aA,aB} \quad \text {(Lemma 4.8)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{aa}={\Theta }(\Sigma _{aA,aB}^2) \Rightarrow T_2=T^{aA=\delta AA}\wedge T^{aB=\delta AB} \quad \text {(Lemma 4.9)}\\ \text {P4.7,L4.9}&\Rightarrow \quad \text {Until }T_=,\quad n_{aB}\le n_{aA}={\Theta }({\varepsilon }) \quad \text {(Proposition 4.10)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{aB}\le \frac{n_{AB}+2n_{BB}+{\textstyle {2\Delta \over c}}}{n_{AB}+2n_{AA}}n_{aA} \quad \text {(Lemma 4.11)}\\ \text {P4.7,L4.9,P4.10,}&\Rightarrow \quad T_=<T_2 \quad \text {(Lemma 4.12)}\\ \text {L4.11}&\Rightarrow \quad n_{AA}(T_=)\le n_{BB}(T_=)+{\Theta }(\Delta ) \quad \text {(Lemma 4.12)}\\ \text {P4.7,}(*)&\Rightarrow \quad \max _{t\in [T_1,T_2]}n_{AB}=:n_{AB}(T^{max}_{AB})={\bar{n}}_B/2+{\Theta }(\Delta ) \quad \text {(Proposition 4.14)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{AA}(T^{max}_{AB}),n_{BB}(T^{max}_{AB})={\bar{n}}_B/4+{\Theta }(\Delta ) \quad \text {(Proposition 4.14)}\\ \text {L4.9,P4.7,}(*)&\Rightarrow \quad n_{aA}\le n_{aB}\vee {\Theta }({\varepsilon }) \quad \text {(Proposition 4.15)}\\ \text {L4.8,4.9,P4.7,}(*)&\Rightarrow \quad n_{aA}= \Sigma _{aA,aB} {\Theta }(n_{AB}+2n_{AA}) \quad \text {(Lemma 4.16)}\\ \text {L4.9,P4.7,}(*)&\Rightarrow \quad {\dot{\Sigma }}_{aA,aB}=n_{aA}(\eta n_{BB}-{\Theta }(\Delta ))\quad \text {(Proposition 4.17)}\\ \text {L4.16,P4.14,4.17,}&\Rightarrow \quad T_2={\Theta }\left( {\varepsilon }^{1/(1+\eta {\bar{n}}_B-\Delta )}\right) \quad \text {(Theorem 4.19)}\\ \text {L4.9,T4.19}&\Rightarrow \quad T_2=T^{aA=\delta AA}\quad \text {(Propostion 4.20)}\\ \end{aligned}$$
  3. Phase 3:

    \(t\in [T_2,T_3]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&=({\Theta }({\varepsilon }^2),{\Theta }({\delta }{\varepsilon }^{\gamma }),{\Theta }({\varepsilon }^\gamma ),\\&{\Theta }(\delta {\varepsilon }^{\gamma /2} ),{\Theta }({\varepsilon }^{\gamma /2}),{\bar{n}}_B-{\Theta }({\varepsilon }^{\gamma /2})). \end{aligned}$$
    $$\begin{aligned} \text {P4.22}&\Rightarrow \quad \Sigma _5={\bar{n}}_B\pm {\Theta }(n_{AB}) \quad \text {(Lemma 4.23)}\\ \text {L4.23}&\Rightarrow \quad n_{AA},n_{aA}\le {\Theta }(n_{AB}^2) \quad \text {(Lemma 4.24)}\\ \text {L4.23,4.24,4.26}&\Rightarrow \quad n_{AA},n_{aA}\ge {\Theta }(n_{AB}^2) \quad \text {(Lemmas 4.25 and 4.27)}\\ \text {L4.23,4.24,4.25,4.27}&\Rightarrow \quad n_{AB}\le {\varepsilon }^{\gamma /10} \quad \text {(Lemmas 4.26 and 4.28)}\\ \text {L4.24,4.27}&\Rightarrow \quad {\dot{n}}_{aa}\ge cn_{aa}\text { with } c>0 \quad \text {(Lemma 4.28)} \end{aligned}$$
  4. Phase 4:

    \(t\in [T_3,\infty ]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&={\bar{n}}_a-{\varepsilon }_0,{\Theta }({\varepsilon }^{\gamma /5}), {\Theta }({\varepsilon }^{\gamma /5}), \\&{\Theta }({\varepsilon }^{\gamma /10}), {\Theta }({\varepsilon }^{\gamma /10}),{\bar{n}}_B\pm {\Theta }({\varepsilon }^{\gamma /10}) ) . \end{aligned}$$

    T4.29 \(\Rightarrow \) For \(\eta <c\cdot 0.593644\), the fixed point \(p_{aB}=({\bar{n}}_a,0,0,0,0,{\bar{n}}_B)\) is stable and for \({\varepsilon },{\varepsilon }_0\) small enough, the system converges to it at speed 1 / t (Theorem 4.30).

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Bovier, A., Coquille, L. & Neukirch, R. The recovery of a recessive allele in a Mendelian diploid model. J. Math. Biol. 77, 971–1033 (2018). https://doi.org/10.1007/s00285-018-1240-z

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Keywords

  • Adaptive dynamics
  • Population dynamics
  • Mendelian reproduction
  • Diploid population
  • Nonlinear birth-and-death process
  • Genetic variability

Mathematics Subject Classification

  • 60K35
  • 92D25
  • 60J85