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Equation 21 · Part 18 · Einstein's Random Walk and the Mathematics of Genetic Drift

Denominator: 2N

∂ϕ(p,t)∂t=12 ∂2∂p2[V(p) ϕ(p,t)],V(p)=p(1−p)2N(Kimura, 1955/1962)\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)}
2N2N

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

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

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

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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