Computer Simulation of Metabolism
Considering Feedback Inhibition
In an earlier page we have seen that it is possible to introduce MichaelisMenten kinetics into metabolic simulation models by specifying a K_{m} and a V_{max} for each reaction rate. Once this is accomplished it is straightforward to introduce feedback inhibition control within the pathway by allowing the pool size of the endproduct to competitively or noncompetitively inhibit one of the reactions involved in its own synthesis.
Consider, for example, the metabolic scheme in Fig. 30 (below) in which no feedback control is initially envisaged [i.e. option "None" of the Inhibition option menu is selected]. Rate B3 does not respond to the endproduct pool size, but rather only to the substrate pool size (A2), thus:
Rate B3 = (Vm2*A2)/(Km2+A2)
Fig. 30. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics with no feedback inhibition by endproduct D on rate B3
(Km values are given in units of nmol.gfw^{1} and Vm values are given in units of nmol.min^{1}.gfw^{1}).
When the "Competitive" Inhibition option is selected (see Fig. 31) rate B3 becomes a function not only of the substrate pool size (A2) but also the pool size of the endproduct (D2) determined by the Ki for the feedback inhibitor [i.e. the concentration of pool D (nmol.gfw^{1}) giving 50% inhibition], as follows:
Rate B3 = (Vm2*A2)/((Km2+(Km2*D2/Ki))+A2)
Note that D2 modulates the apparent K_{m} of reaction rate B3, not the V_{max} of this reaction rate.
Fig. 31. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics with competitive feedback inhibition by endproduct D on rate B3 assuming Ki = 40 nmol.gfw^{1}
When the "Noncompetitive" Inhibition option is selected (see Fig. 32) rate B3 again becomes a function of the substrate pool size (A2), the pool size of the endproduct (D2), and the Ki of reaction B3 for the endproduct [the concentration of pool D (nmol.gfw^{1}) giving 50% inhibition], as follows:
Rate B3 = ((Vm2/(1+(D2/Ki)))*A2)/(Km2+A2)
But note here that D2 modulates the apparent V_{max} of rate B3 rather than the K_{m} of this reaction rate.
Fig. 32. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics with noncompetitive feedback inhibition by endproduct D on rate B3 assuming Ki = 40 nmol.gfw^{1}
In the simulations shown in Figs. 31 and 32 (above), a Ki of 40 nmol.gfw^{1} is assumed. Shown in Figs. 33 and 34 (below) are simulations in which the Ki is reduced to 10 nmol.gfw^{1}. Note that in these scenarios, both the noncompetitive (Fig. 33) and the competitive (Fig. 34) modes of inhibition lead to substantial reduction in the flux of radiolabel to endproduct D, and expansion of the pool of A, as expected.
Fig. 33. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics with noncompetitive feedback inhibition by endproduct D on rate B3 assuming Ki = 10 nmol.gfw^{1}
Fig. 34. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics with competitive feedback inhibition by endproduct D on rate B3 assuming Ki = 10 nmol.gfw^{1}
Note that there is greater inhibition of the radiolabel incorporation into pools B, C, and D in the noncompetitive inhibition model (Fig. 33) than the competitive inhibition model (Fig. 34). This is a consequence of the expansion of the pool of A. In the competitive inhibition model the elevated pool of A competes with the endproduct (D) and so alleviates some of the feedback inhibition. In the noncompetitive model, feedback inhibition is unaffected by substrate concentration.
Because rates C3 and D3 are not feedback sensitive, some continued synthesis of endproduct D can occur from the unlabeled pools of C (C2) and (D2) during the simulation timecourse. Consequently, despite substantial inhibition of radiolabel incorporation into pool D in the noncompetitive model when Ki = 10 nmol.gfw^{1} [see left hand graph panel of Fig. 33] the pool size of D continues to rise [see center graph panel of Fig. 33]. This expansion of the pool of D occurs at the expense of pools B and C.
[Visual Basic program code used for simulations shown in Figs. 30  34]
[Java Applet versions of the above program are available with either competitive feedback inhibition or noncompetitive feedback inhibition. These applets should function with any Javaenabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above].
