The previous Sections of this chapter were concerned mainly with models of enzymes in isolation. Specifically, the concern was how to derive steady-state rate equations for various mechanisms and how to relate the unitary rate constants of these mechanisms to the steady-state parameters, the latter usually being the only parameters that have been measured experimentally. Thus, having described how to model individual enzymes we now turn to the simulation of metabolic pathways. As a simple example, a model of the urea cycle (Figure 3.6) is developed.

The model presented here was originally developed® to study the possible effects on metabolite concentrations of changing various kinetic parameters of the enzymes of the urea cycle. The concentrations of the metabolic intermediates are known to be affected in inborn errors of the enzymes of the urea cycle. The most dramatic clinical signs arise from an overall slowing of flux through the cycle and hence of a buildup of free ammonia in the body. The high ammonia concentrations lead to nausea, vomiting, loss of consciousness, convulsions, and ultimately death.

The model as it was conceived was the first to attempt to simulate the kinetics of a metabolic pathway that is subjected to an inborn error of one of the enzymes. At the time it was developed, the use of unitary rate constants led to very slow simulations of metabolic outcomes; in fact, using a Univac 1108 that had 128 K of RAM, the simulation of 10 min of a time course took 10 min of central processing unit (cpu) time. How times change! But as was shown in 1977, for most purposes the urea cycle can be simulated by using only the steady-state equations; this avoids the stiffness (see Section 1.6) that consideration of the pre-steady-state phases of the enzymic reactions imposes on the computation.

The urea cycle model consists of four enzyme reaction schemes: arginase, ornithine carbamoyl transferase, argininosuccinate lyase, and argininosuccinate synthetase. Rate equations and kinetic parameters for the first two reactions have been determined above in Sections 3.4 and 3.5. The mechanisms of the latter two enzymes are given in the exercises at the end of this chapter; and it is left as an exercise for you the reader to verify the rate equations and kinetic parameters used in the following model.

Q: Assemble the differential equations for each of the enzymes of the urea cycle and simulate a time course of the operation of the cycle for a period of 1 h.

A: The model is based on the urea cycle that operates in human liver. The steady-state output of urea is set to the rate that is found for the 'average' adult human, namely, 6.6 x 10-5 mol L-1 s-1. Since the steady-state concentrations of the intermediates of the cycle develop rapidly, the efflux rate of other metabolites that are peripheral to the cycle must ultimately have this value as well.

To simulate this system we first define the rate equations for the four main enzymes.® Ornithine carbamoyl transferase (OCT)

k-1,oct |
= 63; |

k2 ,oct |
= 2.1 x 106 ; |

k-2,oct |
= 1.0 x 103 |

k3,oct |
= 3 x 103 ; |

k-3,oct |
= 9.0 x104 |

k4,oct |
= 2.6 x 103 ; |

k-4,oct |
= 5.0 x 105 ; |

Voct[t_] := eoct —--(cp[t] o[t] kl,oct k2,oct k3,oct k4/oct -

denonioct c[t] p[t] k-4,oct k-3,oct k-2,oct k-1,oct );

denonioct : = c [ t ] k_3,oct k-2, oct k_i,oct + P [ t ] (k-4, oct k-2, oct k-1, oct +

c [t] (k-4,oct k-3,oct k-2,oct + k-4,oct k-3,oct k-1,oct) + k-4,oct k-1,oct k3,oct + o [t] (c[t] k-4,oct k-3,oct k2,oct + k-4,oct k2,oct k3,oct)) + k-2,oct k-1,oct k4 ,oct + k-1,oct k3 ,oct k4,oct + O [t] k2,oct k3,oct k4,oct +

cp[t] (c[t] k-3,oct k-2,oct k1,oct + k-2,oct k1,oct k4,oct + k1,oct k3,oct k4,oct + o[t] (c[t] k-3,oct k1,oct k2,oct + k1,oct k2,oct k3,oct + k1,oct k2,oct k4,oct))

Argininosuccinate synthetase (ASS)

eass = |
4.0 x 10- |
6; |

k1,ass |
= 2.410A |
5; |

k-1,ass |
= 2.3; | |

k2,ass |
= 3.5 10 A |
5; |

k-2,ass |
= 10.0; | |

k3,ass |
= 4.810 A |
5; |

k-3,ass |
= 10.0; | |

k4 ,ass |
= 2.010 A |
1; |

k-4,ass |
= 8.910 |
A5 |

k5 ,ass |
= 5.0 10 a |
1; |

k-5,ass = 6.4 10 A5; k6,ass = 5.010A1; k-6,ass = 1.7 10 A5;

vass [t_] ass denomass

( k1 ,ass k2 ,ass k3 ,ass k4 ,ass k5 ,ass k6 ,ass c [t] atp[t] asp[t] -

k-1,ass k-2,ass k-3,ass k-4,ass k-5,ass k-6,ass PP[t] amp[t] as[t]);

denomass • = k-1,ass k-2 ,ass k5,ass k6,ass (k-3 ,ass + k4,ass ) + k1 ,ass k-2,ass k-3,ass k-4,ass k6 ,ass c [t] pp [t] + k1,ass k-2 ,ass k5 ,ass k6 ,ass (k-3,ass + k4,ass) c[t] + k-1,ass k3,ass k4 ,ass k5 ,ass k-6,ass asp [ t ] as [ t ] + k-1,ass k3,ass k4 ,ass k5 ,ass k6,ass asp [ t ] + k1 ,ass k2 ,ass k-3,ass k-4,ass k6 ,ass c[t] atp[t] pp[t] + k1, ass k2 ,ass k5,ass k6,ass (k-3,ass + k4,ass) c[t] atp[t] + k1 ,ass k-2,ass k-3,ass k-4,ass k-5,ass c[t] pp[t] amp[t] +

k1,ass |
k3 ,ass |
k4 ,ass |
k5,ass |
k6,ass c[t] asp[t] + |

k2,ass |
k3 ,ass |
k4 ,ass |
k5,ass |
k-6,ass atp [t] asp [ t] as [ t] + |

k2,ass |
k3 ,ass |
k4 ,ass |
k5,ass |
k6,ass atp[t] asp[t] + |

k-1,ass k3,ass k4,ass k-5,ass k-6,ass asp[t] amp[t] as[t] + k1,assk2,ass k3 ,ass ( k4 ,ass k5 ,ass + k4 ,ass k6 ,ass + k5 ,ass k6 ,ass ) c [t] atp [t] asp [t] + k1, ass k2 ,ass k3,ass k-4,ass k6,ass c [t] atp[t] asp[t] pp[t] + k-1,ass k-2,ass k-3,ass k-4,ass k6,ass pp[t] +

k1 ,ass k2 ,ass k3 ,ass k4 ,ass k-5,ass c [t] atp[t] asp[t] amp[t] +

k-1,ass k-2,ass k5 ,ass k-6,ass (k -3,ass + k4 ,ass ) as[t] +

k1,ass k2,ass k-3,ass k-4,ass k-5,ass c[t] atp[t] pp[t] amp[t] +

k-1,ass k-2,ass k-3,ass k-4,ass k-5,ass pp[t] amp[t] +

k2 ,ass k3 ,ass k4 ,ass k-5,ass k-6,ass atp [t] asp[t] amp[t] as [t] +

k-1,ass k-2,ass k-3,ass k-4,ass k-6,ass pp [t] as[t] +

k2,ass k-3,ass k-4,ass k-5,ass k-6,ass atp[t] pp[t] amp[t] as[t] + k-1,ass k-2,ass k-5,ass k-6,ass (k -3,ass + k4 ,ass ) amp[t] as[t] + k-1,ass k3,ass k-4,ass k-5,ass k-6,ass asp[t] pp[t] amp[t] as[t] + k-4,ass k-5,ass k-6,ass

(k-1 ,ass k-2,ass + k-1,ass k-3,ass + k-2,ass k-3,ass) pp[t] amp[t] as[t] + k1,ass k2,ass k3,ass k-4,ass k-5,ass c[t] atp[t] asp[t] pp[t] amp[t] + k2 ,ass k3 ,ass k-4,ass k-5,ass k-6,ass atp[t] asp[t] pp[t] amp[t] as[t]

Argininosuccinate lyase (ASL)

eas = 2.2 x 10-6 ; k1,as = 2.7 x 106 ; k-1,as = 7.0 x 101 ; k2,as = 7.5 x 101;

Vas [t ] : = ea k1,as k2,as k3,asas [t] - k-3 ,as k-2,as k-1,as a[t] f [t]

a [t] (f[t] k-3,ask-2,as + k-3,aSk-1,as + k-3,ask2,as) + k_1,ask3,as + k2 ,as k3 ,as + as [t] (f [t] k-2 ,as k1 ,as + k1 ,as k2 ,as + k1 ,as k3,as) ;

Arginase

earg = 8 |
.9 |
x 10-6 ; |

k1, arg = |
1. |
0 x 107 |

k-1,arg : |
=5 |
.4 x 104 |

k2 , arg = |
5. |
3 x 103 |

k3,arg = |
3. |
0 x 104 |

k-3,arg : |
=1 |
.0 x 107 |

denomarg denomarg := o[t] (k-3,arg k-1,arg + k-3,arg k2,arg) +

1,arg k3 ,arg + k2 ,arg k3 ,arg ^ a[t] (k1 ,arg k2 ,arg + k1 ,arg k3,arg)

Next we define the rate equations for the co-substrates and products that are peripheral to the cycle proper. For these peripheral reactions we assume simple first-order kinetics, as follows.

Vcp [t_] : = kcp ampool [t] ; Vasp [t_] : = kasp asppool [t] ; Vamp [t_] : = kamp amp [ t] ; Vp [t_] := kp p [t] ;

Having defined the rate equations, we are now in a position to set up the system of differential equations which make up the model.

eqn! : = c' [t] % Voct [t] - Vass [t] ; eqn2 : = a' [t] % Vas [t] - Varg [t] ;

eqn3 : = u' [t] % Varg [t] ; eqn4 : = atp' [t] % Vatp [t] - Vass [t] ; eqn5 : = pp' [t] % -Vpp [t] + vass [t] ; eqn6 : = f' [t] % -Vf [t] + Vas [t] ; eqn7 : = as ' [t] % Vass [t] - Vas [t] ; eqn8 : = o' [t] % Varg [t] - Voct [t] ; eqn9 : = cp' [t] % Vcp [t] - Voct [t] ; eqn10 : = asp ' [t] % Vasp [t] - Vass [t] ; eqn11 : = amp' [t] % - Vamp [t] + Vass [t] ; eqn12 : = p' [t] % -Vp [t] + Voct [t] ;

In this model we assume that the pool concentrations of ATP, AMP, and aspartate are kept constant by some external processes. These concentrations can be thought of as 'external' parameters and they are assigned the following values:

atppool [t] = 1.0 x 10-4 ; ampool [t] = 1.0 x 10-4 ; asppool [t] = 1.0 x 10-4 ;

Now we solve this system of equations by using NDSolVe sol= NDSolVe[{eqn1, eqn2 , eqn3 , eqn4 , eqn5 , eqn6 , eqn7 , eqn , eqn9 , eqn10 , eqnn, eqn12 , c[0.] % 1.0 x 10 7 , as [0. ] % 1.0 x 10-5 , a [0. ] % 1.0 x 10-7 , o [0. ] % 4.5 x 10-4 , u [0. ] % 1.0 x 10-5, cp [0. ] % 1.0 x 10-4, atp [ 0. ] % 1.0 x 10-3 , asp [0. ] % 1.0 x 10-3, pp [0. ] % 1.0 x 10-5, amp[0. ] % 1.0 x 10-5, f [0.] % 1.0 x 10-5 , p[0.] % 1.0 x 10-5}, {c[t], a[t], u[t], atp[t] , pp[t], f[t], as[t] , o[t], cp[t] , asp [t] , amp [t] , p [t]} , {t, 0.0, 600} , AccuracyGoal 0 10] ;

and then plot the results as follows:

{t, 0, 600}, PlotRange-> {0, 0.00045}, AxesLabel 0 {"Time (s)", "[orn] (moles LA-1)"}];

0.0004

0.0003

0.0002

0.0001

100 200 300 400 500 600

Figure 3.7. Time course of ornithine concentration in the urea cycle simulation.

graph2 = Plot [Evaluate[c [t] /. sol] , {t, 0, 600}, PlotRange ^ All, AxesLabel ^ {"Time (s)", "[cit] (moles LA-1)"}];

100 200 300 400 500 600

Figure 3.8. Time course of citrulline concentration in the urea cycle simulation.

graph3 = Plot [Evaluate[a[t] /. sol] , {t, 0, 600}, PlotRange ^ All, AxesLabel ^ {"Time (s)", "[arg] (moles LA-1)"}];

3x10-6

Figure 3.9. Time course of arginine concentration in the urea cycle simulation.

graph4 = Plot [Evaluate[u[t] /. sol] , {t, 0, 600}, PlotRange ^ All, AxesLabel ^ {"Time (s)", "[ure] (moles LA-1)"}];

5x10

100 200 300 400 500 600

0.001

0.003

0.002

100 200 300 400 500 600

Figure 3.10. Time course of urea concentration in the urea cycle simulation.

Show[{graphl, graph2, graph3, graph4}];

0.0001

0.0004

0.0003

0.0002

0.0001

0.0004

0.0003

0.0002

100 200 300 400 500 600

Figure 3.11. Combined plot of the previous four graphs showing ornithine (upper curve), citrulline (lower curve), and urea (the middle curve which rapidly rises out of the envelope of the graph).

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