Acta Biotheoretica 40: 121-129, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.
APPLICATION OF THE METABOLIC CONTROL THEORY TO THE STUDY OF THE DYNAMICS OF SUBSTRATE CYCLES
F. Fassy #, J.-F. Hervagault N, T. Letellier ~, J.P. Mazat §, C. Reder I and P. Villalobos #
§ lnstitut de Biochimie Cellulaire et Neurochimie - Centre National de la recherche Scientifique & Universit6 de Bordeaux I1, 146 rue l.,6o-Saignat, 33076 Bordeaux Cedex, France. ¶ Math6matiques et haformatique, Universit6 de Bordeaux I, 33405 Talence Cedex, France. # URA n ° 1442 du Centre National de la Recherche Scientifique, Universit6 de Compi~gne, BP 649, 60206 Compi~gne Cedex, France.
ABSTRACT Substrate cycles are ubiquitous structures of the cellular metabolism (e.g. Krebs cycle, fatty acids /3-oxydation cycles, etc...). Moiety-conserved cycles (e.g. adenine nucleotides and NADH/NAD, etc...) are also important. The role played by such cycles in the metabolism and its regulation is not clearly understood so far. However, it was shown that these cycles can generate multistationarity (bistebility), irreversible transitions, enhancement of sensitivity, temporal oscillations and chaotic motions (Hervagault & Canu, 1987; Hervagault & Cimino, 1989; Reich & Sel'kov, 1981; Ricard & Souli6, 1982).
v~
v~ c(
v'=aS°aI ~
-'~
[A]
P
v,
~
•
~
P
v,=a(so.S) 4,~______~
V4= a- P
[B]
Fig. 1: Scheme of the open binary substrate cycle under study. The substrate S is converted into P with a net rate v v Substrate P is converted in m m into S with a not rate v~. Step v2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input of S at rates o~0(v0 and first order outputs of S and P at rates c~S and aP(v,), respectively.
122 The metabolic control theory (see also Fell, 1990), which shows how a metabolic network reacts to small perturbations in the vicinity of a steady state, and is formulated with the so-called "control coefficients", was applied to such a cycle in order to get a better knowledge on the importance of each step at the regulatory point of view. The behaviour of a binary substrate cycle (fig. 1) in which one of the enzymes may be subjected to inhibition by excess of its substrate (vz) was studied theoretically. The flux and concentration control coefficients were calculated for various steady states of the system. The evolution of the different control coefficients is compared to the evolution of the steady states. We mainly focused our study on situations for which the steady states are stable.
1. P R E S E N T A T I O N
OF THE
MODEL
The model cycle used in this study was introduced by Hervagault & Cimino (1989): substrate S is converted into P with a net rate v2. Substrate P is converted in turn into S with a net rate v 3. Step v 2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input and first order output o f S at rates c~S0 and c~S(vl) and first order output o f P at rate c~P(v4), respectively. Its dynamic behaviour is fully described by the following set of differential equations, set to zero at steady-state,
"ai" = v3 - v2 +
°e(S°-S)
dP --~
otP
=
v 2
-
v 3
-
(1)
with v 2 --- VM2
S
S2
Ks+S +
'
v3 = VM3
P Kp+P
(2)
Kss w h e r e V ~ and V m are the maximal activities, and Ks, Kp and Kss are the Michaelis and inhibition constants, respectively. The varying S and P steady-state concentrations, and the corresponding values o f fluxes J1, J2 and J3 are depicted in figure 2. It is noticable on this figure that, according to the value o f So, one or two steady states coexist (with one intermediary instable steady state) depending upon the s o value. Hysteretic phenomena can then be observed (see Hervagault & Cimino, 1989, for further discussion). The S O values for which bistability occurs are located between 3.477 and 5.0. The lower stable branch of the hysteresis correspond to low values of S and the upper one to high values, that is when the inhibition by excess o f substrate has a great influence on the rate v 2.
2. C A L C U L A T I O N OF THE CONTROL COEFFICIENTS Elasticity coefficients are easily deduced from the rate equations:
123 J J3 .07 .06
//" ....................... J2
.05 .04 .03 .02 .01 0
9 8 7 6 5 4
3 2 1 o o
{
4
;
;
~
~
~
~So
S
9 ~ So
P 8 7 6 5
3 2 1 0 0
1
2
3
4
5
6
7
Fig. 2.: Evolution of the J's fluxes steady-state values, and of the S and P substrate concentrations when varying the S O input concentration.
124 S
Vi
R I R2 R3 R4
P
0 2
S 0
+1
-1
P
-~ Es
0
V3
0
Ep
V4
0
+~
-1
3
Matrice N Matrice E'
~/D
APPLICATION OF THE METABOLIC CONTROL THEORY TO THE STUDY OF THE DYNAMICS OF SUBSTRATE CYCLES
F. Fassy #, J.-F. Hervagault N, T. Letellier ~, J.P. Mazat §, C. Reder I and P. Villalobos #
§ lnstitut de Biochimie Cellulaire et Neurochimie - Centre National de la recherche Scientifique & Universit6 de Bordeaux I1, 146 rue l.,6o-Saignat, 33076 Bordeaux Cedex, France. ¶ Math6matiques et haformatique, Universit6 de Bordeaux I, 33405 Talence Cedex, France. # URA n ° 1442 du Centre National de la Recherche Scientifique, Universit6 de Compi~gne, BP 649, 60206 Compi~gne Cedex, France.
ABSTRACT Substrate cycles are ubiquitous structures of the cellular metabolism (e.g. Krebs cycle, fatty acids /3-oxydation cycles, etc...). Moiety-conserved cycles (e.g. adenine nucleotides and NADH/NAD, etc...) are also important. The role played by such cycles in the metabolism and its regulation is not clearly understood so far. However, it was shown that these cycles can generate multistationarity (bistebility), irreversible transitions, enhancement of sensitivity, temporal oscillations and chaotic motions (Hervagault & Canu, 1987; Hervagault & Cimino, 1989; Reich & Sel'kov, 1981; Ricard & Souli6, 1982).
v~
v~ c(
v'=aS°aI ~
-'~
[A]
P
v,
~
•
~
P
v,=a(so.S) 4,~______~
V4= a- P
[B]
Fig. 1: Scheme of the open binary substrate cycle under study. The substrate S is converted into P with a net rate v v Substrate P is converted in m m into S with a not rate v~. Step v2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input of S at rates o~0(v0 and first order outputs of S and P at rates c~S and aP(v,), respectively.
122 The metabolic control theory (see also Fell, 1990), which shows how a metabolic network reacts to small perturbations in the vicinity of a steady state, and is formulated with the so-called "control coefficients", was applied to such a cycle in order to get a better knowledge on the importance of each step at the regulatory point of view. The behaviour of a binary substrate cycle (fig. 1) in which one of the enzymes may be subjected to inhibition by excess of its substrate (vz) was studied theoretically. The flux and concentration control coefficients were calculated for various steady states of the system. The evolution of the different control coefficients is compared to the evolution of the steady states. We mainly focused our study on situations for which the steady states are stable.
1. P R E S E N T A T I O N
OF THE
MODEL
The model cycle used in this study was introduced by Hervagault & Cimino (1989): substrate S is converted into P with a net rate v2. Substrate P is converted in turn into S with a net rate v 3. Step v 2 is inhibited by excess of the substrate, S. In addition, the cycle operates under open conditions, that is zero-order input and first order output o f S at rates c~S0 and c~S(vl) and first order output o f P at rate c~P(v4), respectively. Its dynamic behaviour is fully described by the following set of differential equations, set to zero at steady-state,
"ai" = v3 - v2 +
°e(S°-S)
dP --~
otP
=
v 2
-
v 3
-
(1)
with v 2 --- VM2
S
S2
Ks+S +
'
v3 = VM3
P Kp+P
(2)
Kss w h e r e V ~ and V m are the maximal activities, and Ks, Kp and Kss are the Michaelis and inhibition constants, respectively. The varying S and P steady-state concentrations, and the corresponding values o f fluxes J1, J2 and J3 are depicted in figure 2. It is noticable on this figure that, according to the value o f So, one or two steady states coexist (with one intermediary instable steady state) depending upon the s o value. Hysteretic phenomena can then be observed (see Hervagault & Cimino, 1989, for further discussion). The S O values for which bistability occurs are located between 3.477 and 5.0. The lower stable branch of the hysteresis correspond to low values of S and the upper one to high values, that is when the inhibition by excess o f substrate has a great influence on the rate v 2.
2. C A L C U L A T I O N OF THE CONTROL COEFFICIENTS Elasticity coefficients are easily deduced from the rate equations:
123 J J3 .07 .06
//" ....................... J2
.05 .04 .03 .02 .01 0
9 8 7 6 5 4
3 2 1 o o
{
4
;
;
~
~
~
~So
S
9 ~ So
P 8 7 6 5
3 2 1 0 0
1
2
3
4
5
6
7
Fig. 2.: Evolution of the J's fluxes steady-state values, and of the S and P substrate concentrations when varying the S O input concentration.
124 S
Vi
R I R2 R3 R4
P
0 2
S 0
+1
-1
P
-~ Es
0
V3
0
Ep
V4
0
+~
-1
3
Matrice N Matrice E'
~/D
