Multilocus behavior in random environments. I. Random Levene models
Genetics, 82 (1):
123-137 (January 1976)Abstract
In this paper the consequences of natural selection acting on several loci simultaneously in a spatially fluctuating environment are described. The fitnesses of the genotypes are assumed to be additive both within and between loci. The environment is assumed to be made up of a very large (effectively infinite) number of patches in which fitnesses are assigned at random. The resulting deterministic model is called a Random Levene Model and its properties are approximate by a system of differential equations. The main equilibrium properites are that (1) the linkage disequilibrium is zero and (2) the correlations in fitness between alleles at different loci are the principle determinants of the dynamic inter-locus interactions. Although there is no epistasis as conventionally defined, the equilibrium state at the two loci are highly interdependent, the governing principle being that two alleles at different loci whose fitness are negatively correlated across environments have a higher overall fitness due to the reduction in their variance in fitness through the negative correlation. When a large number of loci are considered, they naturally fall into correlation groupings which lead to an enhanced likelihood for polymorphism over that predicted by single-locus theory.
Discussion
The most exciting aspect of the results reported here concerns the role of the
correlations in fitness between alleles at different loci in the final genetic makeup
of the population. To put the results in the context of previous studies, it is
important to note that the work on constant fitness models has led to a belief that
in additive models two things invariably hold: (1) there is zero linkage disequi-
librium at the dynamic equilibrium and (2) the equilibria at the separate loci
are exactly those predicted from single-locus considerations.
Our model is also additive, and in common with the constant fitness model
D = 0 at equilibrium. But unlike the constant fitness model, the final genetic
makeup depends on a complex multi-locus behavior which is caused by the
correlations in fitnesses across environments. Surprisingly, the ultimate (i.e.,
equilibrium) expression of this complex behavior is not in the non-random
association of alleles on chromosomes, but rather in the allele frequencies. This
is most dramatically illustrated in ( 2 4 ) .
The principle operating in this model is a simple one: if two alleles at t w o
different loci have fitnesses, U,j (n) which are negatively correlated across
patches, their overall fitnesses will be enhanced over what they would be if there
were no such correlation. The enhancement comes from the reduction in the
variance in fitness of these two alleles across environments. It is natural to
inquire whether these postulated correlations exist in natural populations. Many
lines of thought all point to their probable existence, if environmental fluctua-
tions do, in fact, affect fitnesses. The simplest instances involve any environ-
mental parameter which is likely to affect a broad category of enzymes. Temper-
ature, humidity, mineral and hydrogen ion concentrations, etc.,
potential differential affect on different alleles at all loci and will lead naturally
to a high correlation between loci due to the ordered nature of the parameters.
Less obvious are factors which may affect enzymes participating in a single
pathway. For example, in the oxidation of alcohols it is easy to envision that one
allele in an alcohol dehydrogenase may work on one category of alcohols, the
other allele on another category. The same may be true for a n aldehyde oxidase
which works on the product of the alcohol dehydrogenase reaction with the same
categorization of compounds. Thus fluctuations in the qualitative nature of
alcohols in the environment will lead to highly correlated fluctuations in fitnesses
of the alleles at the two loci. Similar reasoning will lead to many other ways in
which the correlations could crop up. The two cases mentioned, however, illus-
trate two extreme situations-one where we would expect a very large number
of loci to be involved, the other a very few. The former case has a greater conse-
quence in the genetic structure of populations since it leads to the categorization
of alleles into sets of alleles within which the fitnesses are positively correlated
and between which they are negatively correlated. Using the results of the
previous section, we could then characterize the polymorphic state in this situa-
tion as one where the more common alleles are more likely to be positively corre-
lated in their fitnesses than the less common alleles. This prediction is certainly
open to experimental verification if the relevant environmental parameter can
be identified.