A classic example for examining a metabolic pathway in its evolutionary
context is Dean and co-workers' analysis of lactose catabolism in
Escherichia coli
(Dean, et al., 1986). This is a basic linear pathway involving the diffusion of
lactose through the outer cell membrane followed by active transport through the
periplasmic membrane by permease activity. The lactose is subsequently broken down
to glucose and galactose by 
-galactosidase. The flux rates giving rise to
glucose and galactose can usually be correlated to bacterial growth rate by a
linear relation. Thereby we can causally relate metabolic flux to relative growth
rate. Relative growth rate is also the variable by which fitness is measured in
bacterial population ecology. Consequently it is possible to place the metabolic
pathway in its evolutionary context by relating the underlying physiological
characteristics (in the form of enzyme activities and kinetic parameters) to a
relevant phenotype (metabolic flux) and hence fitness.
The relative fitness (i.e. relative growth rate) of a mutant strain can be expressed as
where 
is the relative fitness of the mutant, 
is the flux rate of the mutant,
J is the flux rate of the wild type strain and Y is the linear yield
coefficient for growth. Using the derivation of pathway flux developed by
Kacser and Burns (Kacser and Burns, 1973) and independently by Heinrich and
Rapoport (Heinrich and Rapoport, 1974), the pathway flux can be expressed as:
where D is the diffusion constant across the outer membrane,
is the equilibrium constant for the permease reaction,
and 
are the dissociation constants and maximum
velocities respectively, with the subscripts 
or 
designating permease or 
-galactosidase as the respective enzymes.
The symbol 
is an abbreviation for the summation of kinetic ratios
in the denominator and, as a component of flux, can be considered a physiological
phenotype. In effect equation (1) can be re-written using (2) as:
(Dykhuizen, et al., 1987). The kinetic parameters in this derivation are based on the Michaelis-Menten conception of enzymatic reactions represented by:
where 
is the linear rate coefficient for each reaction (i=1,...,4), E
is the enzyme, S is the substrate, ES is the enzyme-substrate complex and P
the product.
Consider a mutation affecting the maximal rate 
of an enzyme. Assuming that 
is negligible the mutation is actually affecting the underlying kinetic
parameter 
, such that the mutant enzyme's maximal rate 
is expressed by:
where 
is the change in 
caused by the mutation.
[2]
Note that mutations affecting regulation of enzyme levels can be
modeled as those changing [E].
For a haploid organism eq. (4) can be substituted into (3) such that:
This equation predicts a fitness surface as a function of 
and 
.