Diploid Selection Reveal Order

Abstract

Nonzero heterozygote effect changes the rare-allele signal from second to first order.

Theorem 1.1 (Complete recessivity is quadratic and nonzero dominance is linear).

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

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

Source. Repository-derived.

Commentary.

Mean fitness, selected allele mass, updated frequency, and selection change are the source’s diploid genotype formulas with fitnesses 1, 1-hs, and 1-s.

For nonzero s, complete recessivity has the displayed exact change, a cubic remainder after its quadratic leading term, and analytic order two. Under h greater than zero, the exposed change has a quadratic remainder after its linear leading term. Whenever hs is nonzero, the selection reveal order is one.

References

• Truth anchor: D5/S0/Asymptotics/EscapeProbability/DiploidSelectionRevealOrder.diploid_selection_reveal_order
• Dependency: D5/S0/Asymptotics/EscapeProbability/DiploidDominanceSelectionOrder