## Exercises

(1) Adjust the values of Vmax and Km in the worked example (question) in Section 3.2.1 to observe what happens to the shape and the duration of the simulated time course for both substrate and product. (2) Next, assume that this reaction produces two molecules of P from each molecule of A. What happens to the shape of the progress curves for both A and P? Hint: change the conservation of mass equations.

(1) For the arginase reaction (see the first question in Section 3.2.4), try substituting the values of the unitary rate constants back into the steady-state expressions to check that they do indeed return the original steady-state values. (2) Suppose that the temperature of an assay mixture is raised so that the turnover number of the enzyme is increased by a factor of 2. If all the other steady-state parameters remain unchanged, calculate a set of unitary rate constants that are consistent with these values.

The reaction mechanism for the urea cycle enzyme, argininosuccinate lyase, is where AS, F, and A denote argininosuccinate, fumarate, and arginine, respectively. Derive the steady-state rate equation for this enzyme using the function RateEquation as in Section 3.5.

Use the methods described in Section 3.5 to derive the expressions for the standard steady-state kinetic parameters for arginase that has the mechanism shown in Eqn [3.17].

In Section 3.7 we examined the effects of pH on the kinetics of an enzyme. For the worked example in this section, (1) alter the range of pH values for which the Michaelis-Menten equation is plotted. Notice the extent to which Vmax is likely to be able to be accurately inferred from the graphs; and (2) what happens to the apparent Km as the pH is varied?

Verify that the unitary rate constants used for argininosuccinate lyase that were used in the model of the urea cycle presented in Section 3.8 are consistent with the following steady-state parameters and the overall equilibrium constant for the reaction: k{at = 70 molL-1 s-1; Km,AS = 5 x 10-5 molL-1 ; Km,F = 1 x 10-4 molL-1 ; Km,A = 1 x 10-4 mol L- 1 ; Keq = 3.2 x 10-3 molL-1 . The reaction mechanism for argininosuccinate lyase is given in Exercise 3.3.

The reaction mechanism for argininosuccinate synthetase is where C, ATP, ASP, PP, AMP, and AS denote citrulline, ATP, aspartate, pyrophosphate, AMP, and argininosuccinate, respectively. Verify that the unitary rate constants used in the model of the urea cycle model presented in Section 3.8 are consistent with the following steady-state kinetic parameters and the overall equilibrium constant for the reaction:

- 1.0 x 10-4 mol L-1; K^atpamp - 3.5 x 10-4 mol L-1

Clinically, arginase deficiency leads to profound hyperammonemia. It has been found that in the patients who survive early infancy, the maximal activity of the enzyme is around 3% of the normal value. Assuming that this implies that there is only 3% of the normal enzyme concentration, simulate the operation of the human urea cycle under this pathological condition. Comment on the concentrations of the intermediates and whether they attain steady states. In the event of the latter not occurring, speculate on the likely outcome for the particular metabolite(s) and hence the clinical status of the patient.

There are basically two major clinical variants or types of inborn errors of argininosuccinate lyase deficiency: (1) one condition in which there are early signs of the severe onset of hyperammonemia that leads to death in infancy; and (2) a milder course of disease with survival into adolescence. The former appears to result from a low concentration of normal enzyme, while the latter patients probably have an enzyme with lower substrate affinity.

(1) Suppose that in the first case [E]0 is 3% of its normal value. Simulate the operation of the urea cycle for a period of 1 h. Comment on dramatic features of the simulated time course.

(2) Simulate the operation of the urea cycle in which the affinity of the enzyme for argininosuccinate lyase is reduced to 3% of its normal value. Again, comment on any dramatic findings. Does the system attain a new steady state of metabolite concentrations?

### 3.10.10

This exercise will be able to be completed only after reading Chapters 4 and 5. Apply MCA (Chapter 5) to the model of the urea cycle described in Section 3.8 and determine the values of the flux control coefficients for each enzyme. Comment on the occurrence of metabolic resistance in these simulations; this occurs in situations where the affinity of the enzyme for one of its substrates is diminished and the steady-state concentration of the substrate(s) rises to a new value thus overcoming the blockage of the pathway. 