Diploid Dominance Selection Order

Abstract

Diploid dominance changes the rare-allele selection signal from second to first order.

Theorem 1.1 (Recessive selection is quadratic and exposed selection is linear).

$$\begin{gathered} \forall s,h\in \mathbb{R},s\ne 0\Rightarrow \\ \mathrm{let}(\mathrm{meanFitness}(h:\mathbb{R},x:\mathbb{R}):=(1-x{)}^2+2\cdot 1-x\cdot x\cdot 1-h\cdot s+x^2\cdot 1-s, \\ \mathrm{selectedAlleleMass}(h:\mathbb{R},x:\mathbb{R}):=x^2\cdot 1-s+1-x\cdot x\cdot 1-h\cdot s, \\ \mathrm{updatedFrequency}(h:\mathbb{R},x:\mathbb{R}):=\frac{\mathrm{selectedAlleleMass}(h,x)}{\mathrm{meanFitness}(h,x)}, \\ \mathrm{selectionChange}(h:\mathbb{R},x:\mathbb{R}):=\mathrm{updatedFrequency}(h,x)-x) \\ (\forall x\in \mathbb{R},\mathrm{meanFitness}(0,x)\ne 0\Rightarrow \mathrm{selectionChange}(0,x)=\frac{-s\cdot 1-x\cdot x^2}{1-s\cdot x^2})\wedge \\ \mathrm{IsBigOAtZero}(x,\mathrm{selectionChange}(0,x)--s\cdot x^2,x^3)\wedge \\ \mathrm{analyticOrderAt}(\mathrm{selectionChange}(0),0)=2\wedge \\ \mathrm{IsBigOAtZero}(x,\mathrm{selectionChange}(h,x)--h\cdot s\cdot x,x^2)\wedge \\ (h\cdot s\ne 0\Rightarrow \mathrm{analyticOrderAt}(\mathrm{selectionChange}(h),0)=1). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S0/Asymptotics/EscapeProbability/DiploidDominanceSelectionOrder.diploid_dominance_selection_order (✓ std3). ∎

Source. Repository-derived.

Commentary.

Mean fitness and selected allele mass are constructed from the aa, ab, and bb genotype frequencies with fitnesses 1, 1-hs, and 1-s.

At h=0 the exact frequency change has a quadratic leading term and a cubic remainder. A nonzero product hs supplies a nonzero linear leading term, so the analytic order drops from two to one.

References

• Truth anchor: D5/S0/Asymptotics/EscapeProbability/DiploidDominanceSelectionOrder.diploid_dominance_selection_order
• Dependency: D5/S0/Asymptotics/EscapeProbability/CompletelyRecessiveSelectionOrder