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.

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):

• 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 $T_1$,

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

• 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

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

• Phase 3: $t\in [T_2,T_3]$ Initial conditions:

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

• 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).