Earlier we considered Michaelis-Menten kinetics, specifying a K_m and a V_max for each reaction rate. When an additional variable, corresponding to a Hill coefficient, is introduced for each reaction rate it is possible to consider allosteric behavior; specifically cooperative substrate binding, yielding a sigmoidal relationship between reaction rate and substrate concentration [pool size].

Consider, for example, the metabolic scheme in Fig. 35 (below) where each rate is a function of a substrate pool size (E2, A2, B2 or C2), a V_max (Vm1, Vm2, Vm3 or Vm4), a K_m (Km1, Km2, Km3 or Km4), and a H value (H1, H2, H3 or H4) [where ^ indicates "raised to the power of"]:

(Km values are given in units of nmol.gfw^-1 and Vm values are given in units of nmol.min^-1.gfw^-1).

In this example, all H values are set to 1, and therefore all rates show classical Michaelis-Menten kinetics (see right hand graph panel of Fig. 35 which shows rates as a function of substrate pool size). When the Hill coefficient for rate B3 is set to 2 [i.e. H2 = 2], however, rate B3 becomes a sigmoidal function of the substrate pool size (A2), with the pool size of A2 giving half maximal velocity at the specified Km value [Km2 = 20 nmol.gfw^-1] (see right hand graph panel of Fig. 36).

When the Hill coefficient for rate B3 is increased to 3 [i.e. H2 = 3], more pronounced substrate cooperativity results (see right hand graph panel of Fig. 37):

Fig. 38 (below) shows the effects of further increasing the Hill coefficient for rate B3 to 4 [i.e. H2 = 4]:

Note that in all the above examples, the half-maximal velocity for rate B3 (i.e. 0.1 nmol.min^-1.gfw^-1) is reached at a pool size of A2 corresponding to the Km2 value assumed in the model (i.e. Km2 = 20 nmol.gfw^-1).

When the uptake rate (A4) is considered to be allosteric with respect to the pool size of the exogenous precursor (E2), this facilitates accommodation of non-Michaelis-Menten uptake kinetics, i.e. situations where the pool of exogenous precursor may not be completely exhausted from the medium during the labeling time-course [see e.g. Fig. 39].

The model of course permits consideration of cooperative substrate binding at each of the rates in the metabolic sequence, as simulated in Fig. 40, where the Hill coefficient is set to 2 for all of the rates.

[A Java Applet version of the above program is available for consideration of cooperative substrate binding. This applet should function with any Java-enabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above].