Computer Simulation of Metabolism
In considering radiolabeling kinetics, we have seen that the rate of uptake of an exogenous precursor, E, can be made to be proportional to the quantity of E remaining in the medium, by invoking a constant, k. Thus, in the scheme shown
below, the rate of uptake of E (rate A4) is equal to E2*k, where E2 is the pool size of the exogenous precursor. As the pool of E2 declines due to uptake, so does the uptake rate, A4.
Fig. 25. Radiolabeling kinetics assuming fixed rates of metabolism
(Pool sizes (E2, A2, B2, C2 and D2) are given in units of nmol.gfw^{1}, rates of uptake (A4) or synthesis (A3, B3, C3, D3) are given in units of nmol.min^{1}.gfw^{1}) and specific activities (E1, A1, B1, C1, D1, F1) are given in units of nCi.nmol^{1}. The left hand graph panel shows radioactivity in each pool; E1*E2, A1*A2, B1*B2, C1*C2 and D1*D2. The right hand graph panel shows pool sizes; E2, A2, B2, C2 and D2.
[Visual Basic program code used for simulation shown in Fig. 25]
Download an enhanced version of the Visual Basic program illustrated above, including a color key. To run this program you must have Visual Basic 5.0 (or greater) installed on your computer.
In this specific model, because the rates A3, B3, C3 and D3 are fixed (at 0.05 nmol.min^{1}.gfw^{1}), the pool of A (A2) expands as a result of rapid influx of E from the medium (see blue line
of the right hand graph panel of Fig. 25).
[A Java applet version of the above program is available. This applet should function with any Javaenabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above. Output from the model is in graphical format].
It is straight forward to make the rates B3, C3 and D3 linearly responsive to their respective substrate pool sizes (A2, B2 and C2, respectively), much like the radiolabeled precursor uptake kinetics, as follows (see Fig. 26):
B3 = A2*k2
C3 = B2*k3
D3 = C2*k4
Fig. 26. Radiolabeling kinetics assuming rates of metabolism are linearly responsive to precursor pool sizes
(Note that the constants k1, k2, k3 and k4, convert a pool size (nmol.gfw^{1}) into a rate (nmol.min^{1}.gfw^{1}) and must therefore be expressed in units of
time^{1} (i.e. min^{1}). In this example, we have chosen values of k1 to k4 which each give rates of approx. 0.05 nmol.min^{1}.gfw^{1} when multiplied by the appropriate starting pool size).
[Visual Basic program code used for simulation shown in Fig. 26]
Download an enhanced version of the Visual Basic program illustrated above, including a color key. To run this program you must have Visual Basic 5.0 (or greater) installed on your computer.
In this scenario, the rapid uptake of E from the medium causes a transient rise in the pool of A. This rise in the pool of A in turn results in an increased rate of utilization of A in the synthesis of B, causing the pool of B to expand, and the pool of A to return to its original steadystate level. The expansion of pool B in turn promotes increased synthesis of C, etc. Thus, a wave of transient pool expansions is propagated down the pathway (see center panel of Fig. 26). The rates of the reactions are linearly related to their respective substrate pool sizes, as shown in the right hand panel of Fig. 26. Such a model may suffice for most radiotracer experiments in which relatively small pool size perturbations take place as a result of rapid influx of a supplied radiolabeled precursor, and where enzyme reaction rates in vivo are operating well below their maximum velocities.
[A Java applet version of the above program is available. This applet should function with any Javaenabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above. Output from the model is in graphical format].
It is important to emphasize that the pool sizes are not strictly equivalent to substrate concentrations in an enzyme reaction system in vitro, since the exact volume in which the pools are dissolved in the tissue cannot be known with certainty. The k values (k1, k2, k3 and k4) assumed in Fig. 26 are not equivalent to Km values of the enzymes catalyzing the reactions A > B, B > C, and C > D, respectively, because the k values are expressed in units of time^{1}, while Km values are expressed in units of concentration! Nevertheless, one can make the reaction rates in the simulation model behave in a MichaelisMenten type fashion, by introducing an additional
variable for each of the reaction rates, thus:
A4 = 1/((1/(E2*k1) + (1/Vm1))
B3 = 1/((1/(A2*k2) + (1/Vm2))
C3 = 1/((1/(B2*k3) + (1/Vm3))
D3 = 1/((1/(C2*k4) + (1/Vm4))
where the Vm values are essentially maximal velocities of the reactions in vivo (nmol.min^{1}.gfw^{1}), and are therefore somewhat analogous to the Vmax's of enzyme reactions in vitro.
Simulations of this scenario are shown in Fig. 27, assuming Vm values of 0.2 nmol.min^{1}.gfw^{1} for each of the reaction rates B3, C3 and D3, and the uptake rate, A4, and maintaining the same values of k1
to k4 assumed earlier.
Fig. 27. Radiolabeling kinetics assuming rates of metabolism approximating to MichaelisMenten kinetics
(Vm values are expressed in units of nmol.min^{1}.gfw^{1})
[Visual Basic program code used for simulation shown in Fig. 27]
As can be seen in this example, the uptake rate, A4, and rates of metabolism (A3, B3 and C3) are now hyperbolically (rather than linearly) related to their corresponding substrate pool sizes (see right hand graph panel of Fig. 27). This becomes
particularly evident when a large dose (40 nmol.gfw^{1}) of exogenous radiolabeled precursor, E2, is supplied (Fig. 28) (cf. Fig. 27 where only 4 nmol.gfw^{1} was supplied).
Fig. 28. Radiolabeling kinetics assuming rates of metabolism approximating to MichaelisMenten kinetics
The latter type of model may be useful in evaluating radiotracer labeling kinetics where large pool size perturbations are encountered, and/or where in vivo fluxes may approach their maximum values.
Inspection of Fig. 28 indicates that the uptake rate, A4, is at half maximal velocity (i.e. 0.1 nmol.min^{1}.gfw^{1}) at a pool size of E2 = 16 nmol.gfw^{1}; thus the apparent
Km for this uptake system is 16 nmol.gfw^{1}. Rate B3 is at half maximal velocity at a pool size of A2 = 24 nmol.gfw^{1}; thus the apparent Km for the step A > B is 24
nmol.gfw^{1}. It follows that the apparent Km values for the reactions are equal to the Vm values divided by the corresponding k values; e.g. for rate B3, Km2 = Vm2/k2 (i.e. k2 = Vm2/Km2).
The rate equation for reaction B3 (A > B) in the model shown in Figs. 27 and 28 is:
B3 = 1/((1/(A2*k2)) + (1/Vm2))
If we substitute Vm2/Km2 for k2 in this equation we obtain:
B3 = 1/((Km2/(Vm2*A2))+(1/Vm2))
1/B3 = (Km2/(Vm2*A2)) + (1/Vm2)
[note that this is very similar to the LineweaverBurk plot of 1/V versus 1/S, where: 1/V = (Km/(Vmax*[S])) + (1/Vmax)]
1/B3 = (1/Vm2)*((Km2/A2) + 1)
Vm2/B3 = (Km2/A2) + 1
Vm2 = (B3*Km2/A2) + B3
Vm2*A2 = (B3*Km2) + (B3*A2)
Vm2*A2 = B3*(Km2 + A2)
B3 = (Vm2*A2)/(Km2 + A2)
where B3 is the reaction velocity, Vm2 is the maximal velocity, Km2 = substrate pool size at half maximal velocity, and A2 is the substrate pool size.
This is analogous to the MichaelisMenten equation:
V = (Vmax*[S])/(Km + [S])
where V = the reaction velocity, Vmax = the maximal velocity, Km = substrate concentration at half maximal velocity, and [S] = substrate concentration. Not surprisingly then, the model shown in Fig. 28 gives identical results to a model in which Km and Vmax are independently specified, and rates are as follows (Fig. 29):
A4 = (Vm1*E2)/(Km1 + E2) B3 = (Vm2*A2)/(Km2 + A2) C3 = (Vm3*B2)/(Km3 + B2)
D3 = (Vm4*C2)/(Km4 + C2)
Fig. 29. Radiolabeling kinetics assuming rates of metabolism have MichaelisMenten kinetics, specifying Km and Vmax values for each rate
(Km values are given in units of nmol.gfw^{1} and Vm values are given in units of nmol.min^{1}.gfw^{1}).
[Visual Basic program code used for simulation shown in Fig. 29]
Download an enhanced version of the Visual Basic program illustrated above, including a color key. To run this program you must have Visual Basic 5.0 (or greater) installed on your computer.
[A Java applet version of the above program is available. This applet should function with any Javaenabled browser, including Microsoft Internet Explorer 3.0 or above, or Netscape Navigator 3.0 or above. Output from the model is in graphical format].
