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Equation 28 · Einstein's Random Walk and the Mathematics of Genetic Drift

What does this equation mean?

u(p)=1−e−4Nsp1−e−4Nsu(p) = \frac{1-e^{-4Nsp}}{1-e^{-4Ns}}

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Start with1-e^-4Nsp
Divide by1-e^-4Ns
This relates tou(p)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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uu

Symbol u

u is part of the quantity the equation computes from the expression on the right.

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pp

Symbol p

the starting from frequency.

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e−4Nspe^{-4Nsp}

Symbol e^-4Nsp

e−e^-4Nsp occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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e−4Nse^{-4Ns}

Symbol e^-4Ns

e−e^-4Ns occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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1−e−4Nsp1-e^{-4Nsp}

Numerator: 1-e^-4Nsp

The complete quantity above the fraction bar.

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1−e−4Ns1-e^{-4Ns}

Denominator: 1-e^-4Ns

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

That machinery pays off in the cleanest result in the theory of drift. 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)=1−e−4Nsp1−e−4Nsu(p) = \frac{1-e^{-4Nsp}}{1-e^{-4Ns}}. as verified directly from his derivation [ 7 ] . Let the selective advantage vanish — the allele is neutral, indistinguishable from its alternatives in its effect on survival or reproduction — and the exponentials cancel term by term, leaving

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