as the Monod function); and
3 the Ivlev equation.Fig. 1.7 Growth rate of the diatom Asterionella japonica as a function of available
nitrate (squares), fitted by the hyperbolic Michaelis–Menten relationship.
(^) (Data and curve parameters from Eppley & Thomas 1969.)
(^) In some instances there is little to choose among these representations, since the
scatter of the data is usually great. The choice is made on the basis of convenience to
the application. We will develop the list above so you have a reference. Two linear
segments can usually be fitted by eye. These will represent the two basic parameters
of the relation: the asymptotic growth rate and the slope of the initial response to
increase of the limiting factor.
(^) The Michaelis–Menten equation is borrowed from biochemistry. Enzymes are often
characterized in terms of their reaction kinetics. The data are measures of reaction
velocity at various substrate concentrations. Reaction rate takes the form shown in
Fig. 1.7.
(^) Suppose there are a number of active sites on an enzyme to which a substrate can
bind, and let the concentration of those sites be [E]. Binding is the slowest (limiting)
step in conversion of substrate to product. The reaction is E + S ES → Product. The
dissociation constant for the enzyme–substrate complex, ES, is ks = ([E] − [ES])
[S]/[ES]. Solving for [ES]:
(^) Since ES will be transformed to product at a rate proportional to [ES], we have