The effects of a substitution in a two locus, two allele haploid system can be redefined as
and
where the epistatic influence of locus B on locus A is defined as
If the value of 
is considered a physiological phenotype, it can be equated with the genotypic
values delineated in the previous section such that
and
Thereafter by using the definition that 
we have
Similarly,
which with the appropriate substitutions and simplifications gives
From eq. (9) and (7) we see that
In other words the strength of epistatic interactions between locus A (i.e. 
) and
B (i.e. 
) with respect to 
(flux denominator) is zero. Furthermore, with appropriate substitutions and rearrangements,
epistasis with respect to fitness can also be shown to be zero. This is an interesting result
since it shows that a non-linear genotype-phenotype map per se
does not necessarily imply epistasis.
In our example the explanation for zero epistasis can be
found at two levels. First as a technical property of the equation
and secondly, as a consequence of the assumptions that underlie this
model of a metabolic pathway. The technical reason is apparent if
the structure of 
in (2) is examined. Essentially there is no
interaction term between the two enzymes and consequently the effect
of changing either of the two 
parameters is independent.
As long as temperature is constant, the condition of zero epistasis
will hold, since according to thermodynamic principles 
will remain constant.
For the more subtle representational explanation we have to refer
back to the original derivation of the flux equations by Kacser and
Burns (1973). Although the details of the derivation will not be
treated here, an important assumption that went into these equations
was that all enzymes in the pathway are far from saturation. It was
also assumed that changes due to mutations were small enough that
they would not lead to saturation. It is not too difficult to
visualize the converse scenario in which at least one enzyme such
as 
-galactosidase would be at saturation and would thereby be
rate limiting. Epistasis is a natural consequence in such a scenario.