52 Sahotra Sarkar
16 possible genotypes from the frequencies of the gametesAB,Ab,aB andab.
It sufficed, therefore, only to consider the simpler problem of the change in the
gametic frequencies. In two papers, Robbins [1917; 1918a] generalized Jennings’s
results. In a third, and very important contribution, he turned to two loci and
proved two important results [Robbins, 1918b]:
(i) following Robbins’s notation, letrbe the proportion of gametes produced
without recombination compared to those that are produced with it. ris,
then, a measure of linkage.^6 Let mating be at random at both loci. Then,
ifpn,qn,snandtnare the frequencies of the four gametic types,AB,Ab,
aB andab, respectively, in then-th generation, then the quantity, ∆n=
qnsn−pntn, can be used to write down particularly simple relations for the
change in gametic frequencies:pn=pn− 1 +∆n− 1
1+r;qn=qn− 1 −∆n− 1
1+r;sn=sn− 1 −∆n− 1
1+r
;tn=tn− 1 +∆n− 1
1+r
;(ii) the quantity ∆nchanges by a factor ofr/(1 +r) each generation, that is,∆n=r
1+r
∆n− 1.The second result is important because it shows that, under the assumptions
of the model, ∆ngradually goes to 0 as the number of generations increases.
However, it is strictly equal to 0, and remains so, only if ∆ 0 = 0. It thus provides
an analog of the Hardy-Weinberg rule for two loci. Robbins’s results did not
receive the attention they deserved.^7
Both Jennings and Robbins published analyses of complete selection (that is,
complete elimination of the less fit type) which ignored the possibility of weaker
selection, measured by a fitness parameter. However, in unpublished work, for
(^6) In spite of Robbins’s notation,rshould not be confused with the recombination fraction. If
ρis the recombination fraction,ρ= 1/(1 +r). With this substitution, Robbins’s results assume
the standard form of Haldane [1926]. (For a modern appraisal, see Edwards [1977, 94–98]).
(^7) Apparently unaware of Robbins’s work, Haldane [1926) rederived the first of these two results
in the third part of “A Mathematical Theory of Natural and Artificial Selection”. The value of
the parameter ∆nwould not be fully appreciated until the 1950’s when -∆ncame to be called
“linkage disequilibrium” (or, alternatively, “gametic phase disequilibrium”). Robbins also gave
more complicated formulae when the linkage between the loci was not the same in the two sexes,
and in the final installment of his series of papers, discussed a model of “disassortative mating”
in which brother-sister mating was avoided [Robbins, 1918c].