104 James F. Crow
place, Okazaki. There is a museum and a public statue. For a more detailed
biography and account of Kimura’s work, see[Crow, 1997].
2 POPULATION GENETICS THEORYSoon after beginning graduate work at the University of Wisconsin, Kimura wrote
one of his most influential papers. He was particularly adept at using diffusion
equations. Starting with a population with a specified frequency of two neutral
alleles, he worked out the probability of every allele frequency for all future times
until one allele is eventually lost from the population. He not only constructed
the entire process for the first time, but in doing so he demonstrated the great
power and convenience of a diffusion approximation. He soon extended this to
three alleles, and on to any arbitrary number. He then added mutation, selection,
and migration. His general procedure was to use the Kolmogorov forward equation
greatly extending the work of Wright[Wright, 1945]. This work was all done in his
first year of graduate work. He was invited to the Cold Spring Harbor Symposium
of 1955, where this material and much more was presented[CSH, 1955]. He also
introduced an elementary derivation of the Fokker-Planck equation, widely used
by physicists but coming into use by evolutionists. The paper was atour de
force; but it was doubly difficult to understand, both because of its mathematical
complexity and because of Kimura’s then-limited skill in English pronunciation.
In the discussion Sewall Wright, with his usual generosity, rose to say that only
those who had tried such problems could appreciate the enormity of Kimura’s
accomplishment.
After graduation in 1956, Kimura returned to Japan where he continued study-
ing stochastic processes. In particular, he made use of both the forward and
backward forms of the famous Kolmogorov equations[Kimura, 1964]. This en-
abled him to answer such questions as: the probability of ultimate fixation of a
mutant allele with arbitrary initial frequency and selective advantage, the distri-
bution of number of generations required for fixation or loss of a mutant allele, and
the number of individuals affected during the time the mutant allele is in transit.
Together with his colleague Takeo Maruyama, he discovered a way to obtain a
variety of functions of allele frequencies undergoing stochastic evolution[Kimura
and Maruyama, 1971].
Fisher’s “Fundamental Theorem of Natural Selection”[Fisher, 1930]has long
been a subject of controversy. It says that the instantaneous rate of change of
fitness is equal to the genic variance of fitness at the time, genic variance being
defined by least squares. It is exact only under highly simplified conditions, but as
an approximation tells a great deal about how selection works[Crow, 2002]. One
of Kimura’s accomplishments was to write explicit formulae, taking into account
dominance, epistasis, and variable selection coefficients[Kimura, 1958].
One of Kimura’s most interesting discoveries was that when there is directional
selection in a population, after a few generations it attains a state that he called
“quasi-linkage equilibrium”. In this state, the amount of linkage disequilibrium