A simplified model of cell metabolism, consisting of a series of linked reversible enzymatic reactions dependent on the concentration of a single external substrate has been developed. The general mathematical solution for this system of reactions is presented. This general solution confirms the concept of a rate-limiting step, or “master reaction”, in biological systems as first proposed by Blackman. The maximum rate of such a process is determined by, and equal to, the maximum rate of the slowest forward reaction in the series.
Of practical interest in modeling the growth rate of cells are three cases developed from the general model. The simplest special case results in the Monod equation when the maximum forward rate of one enzymatic reaction in the cell is much less than the maximum forward rate of any other enzymatic reactions.
More realistic is the case where the maximum forward rates of more than one enzymatic reaction are slow. When two slow enzymatic reactions are separated from each other by any number of fast reactions that overall can be described by a large equilibrium constant, the Blackman form results:
A third case is that in which two slow enzymatic steps are separated by an equilibrium constant that is not large. Unlike the Monod and Blackman forms, which contain only two arbitrary constants, this model contains three arbitrary constants:
The Monod and Blackman forms are special cases of this three constant form.
In comparing equations with two arbitrary constants the Monod equation gave poorer fit of the data in most cases than the Blackman form. It is concluded that workers modeling the growth of microorganisms should give a t least as much consideration to the Blackman form as is given to the Monod equation.