Equation 21 · Part 9 · Einstein's Random Walk and the Mathematics of Genetic Drift

derivative

\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)}

derivative

What this part means

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

Its job in the formula

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

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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 derivative describes how quickly one quantity changes as another changes. It is the slope of a curve at a particular point.

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Sources cited in the surrounding passage

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