Equation 21 · Part 4 · Einstein's Random Walk and the Mathematics of Genetic Drift

Symbol p^2

\frac{\partial \phi(p,t)}{\partial t} = \frac{1}{2}\,\frac{\partial^{2}}{\partial p^{2}}\Big[V(p)\,\phi(p,t)\Big], \quad V(p) = \frac{p(1-p)}{2N} \qquad \text{(Kimura, 1955/1962)}

p^{2}

What this part means

the square of p; the starting from frequency.

Its job in the formula

$p^2$ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Full expressionSymbol p^2Article meaning

Where the article explains it

Kimura’s 1962 formula for the probability that a gene with selective advantage s eventually fixes, starting from frequency p in a population of effective size N , is u(p) = $\frac{1-e^{-4Nsp}}{1-e^{-4Ns}}$ as verified directly from his derivation [ 7 ] .

The passage around this formula

Motoo Kimura, at Japan’s National Institute of Genetics, made the borrowing exact rather than approximate. His 1955 paper in the Proceedings of the National Academy of Sciences solved the full time-dependent diffusion equation derived from the Wright-Fisher model, recovering not just the eventual resting distribution of an allele’s frequency but its whole transient shape as it evolves generation by generation [ 6 ] . His 1962 paper in Genetics then posed the general fixation problem as a diffusion boundary-value equation and named the lineage precisely: the question of a mutant gene’s ultimate fate was “first treated quantitatively by Fisher (1922),” with “equivalent results” reached…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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