Computer Simulation of Metabolism
Modeling Allosteric Behavior  Cooperative Substrate Binding
Earlier we considered MichaelisMenten 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"]:
Rate A4 = (Vm1*(E2^H1)/((Km1^H1)+(E2^H1))
Rate B3 = (Vm2*(A2^H2)/((Km2^H2)+(A2^H2))
Rate C3 = (Vm3*(B2^H3)/((Km3^H3)+(B2^H3))
Rate D3 = (Vm4*(C2^H4)/((Km4^H4)+(C2^H4))
Fig. 35. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics; i.e. all Hill coefficients = 1
(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 MichaelisMenten 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).
Fig. 36. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics except for rate B3 which shows cooperative substrate binding with a Hill coefficient = 2
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. 37. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics except for rate B3 which shows cooperative substrate binding with a Hill coefficient = 3
Fig. 38 (below) shows the effects of further increasing the Hill coefficient for rate B3 to 4 [i.e. H2 = 4]:
Fig. 38. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics except for rate B3 which shows cooperative substrate binding with a Hill coefficient = 4
Note that in all the above examples, the halfmaximal 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 nonMichaelisMenten uptake kinetics, i.e. situations where the pool of exogenous precursor may not be completely exhausted from the medium during the labeling timecourse [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.
[Visual Basic program code used for simulations shown in Figs. 35  40. ]
[A Java Applet version of the above program is available for consideration of cooperative substrate binding. This applet should function with any Javaenabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above].
