Eur. J. Biochem. 195. 109-113 (1991) 0FEBS 1991 001429569100012F
Irreversible transitions in the 6-phosphofructokinase/fructose 1,6-bisphosphatase cycle Wolfgang SCHELLENBERGER' and Jean-Franqois HERVAGAULT'
' Institut fur Biochemie, Karl-Marx-Universitat, Leipzig, Federal Republic of Germany Unite de Recherche Associee no. 523 du Centre National de la Recherche Scientifique, Universitt de Compiegne, France (Received June 25,1990)
-
EJB 90 0734
The dynamics of the fructose 6-phosphate/fructose-1,6-bisphosphate cycle operating in an open and homogeneous system reconstituted from purified enzymes was extensively studied. In addition to 6-phosphofructokinase and fructose-l,6-bisphosphatase, pyruvate kinase, adenylate kinase and glucose-6-phosphate isomerase were involved. In that multi-enzyme system, the main source of non-linearity is the reciprocal effect of AMP on the activities of 6-phosphofructokinase and fructose-I ,6-bisphosphatase. Depending upon the experimental parameter values, stable attractors, various types of multiple states and sustained oscillations were shown to occur. In the present report we show that irreversible transitions are also likely to occur for realistic operating conditions. Two parameters of the system, that is the adenylate energy charge of the influx and the fructose-1,6bisphosphatase maximal activity, are potential candidates to provoke such irreversible transitions from one steady state to the other: (a) when varying the maximal activity of fructose-l,6-bisphosphatase, the system can jump irreversibly from a low to a high stable steady state, and (b) when the adenykdte energy charge of the influx is the changing parameter, irreversible transitions occur from a high stable steady state to a stable oscillatory state (limit cycle motion). This behavior can be predicted by constructing the loci of limit points and Hopf bifurcation points.
An important feature of the cellular metabolism is the existence of moiety-conserved cycles [l, 21. The two best known groups of metabolites participating in such operations are ATP/ADP and NAD(P)/NAD(P)H. The role played by cycles in the regulation of metabolism is still a matter of controversy. However, from experimental data and theoretical considerations, it is likely that substrate cycling is involved in thermogenesis [3,4], in the orientation of fluxes, in the control of concentrations [5], and in the amplification of sensitivity [6 - 91. When dealing with cycles subjected to a destabilising factor (product activation, substrate inhibition, etc.), more sophisticated dynamics may be observed, such as multiple steady states, damped and sustained oscillations or even chaotic motions [2, lo]. When dealing with bistability, the transition between the two steady states might be irreversible. By irreversible transitions, we mean that, contrary to what occurs in a classical bistable system, the switching from one branch of steady states to the other is, at the most, possible only once. These irreversibility features are in no way due to a bounded domain of variation of one parameter (experimental constraints) but are an intrinsic property of the highly nonlinear system considered. As a minimum system, a simple substrate cycle involving two antagonist enzymes, one of them being inhibited by excess Correspondence to J. F. Hervagault, U.R.A. no. 523 duC.N.R.S., UniversitC de Compiegne Boite Postale no. 649, F-60206 Compiegne, France Abbreviations. Fru6P, fructose-6-phosphate; Glc6P, glucose 6phosphate; Fru(l,6)P2, fructose 1,6-bisphosphate; PPrv, phosphoenolpyruvate; [AECIIN,influx adenylate energy charge; Fru(l,6)P,ase, fructose-l,6-bisphosphatase;Glc6Pase, glucose-6-phosphate isomerase. Enzymes. 6-Phosphofructokinase (EC 2.7.1.11); pyruvate kinase (EC 2.7.1.40); fructose-1,6-bisphosphatase(EC 3.1.3.11); glucose-6phosphate isomerase (EC 5.3.1.9); adenylate kinase (EC 2.7.4.3).
of its substrate was studied theoretically [Ill. In addition to classical bistability (with respect to both the maximal activity of one of the enzymes and the sum of the metabolite concentrations), irreversible transitions were also found to occur, but only with respect to the sum of the metabolite concentrations. The model was extended in order to investigate the changes in the dynamic properties which may arise when such a cycle is open to mass transfers [12]. In particular, the coexistence of both reversible and irreversible transitions could be observed when, for instance, the level of exchanges with the surroundings are altered. Recently, it was shown experimentally [13] that irreversible transitions could actually be observed in such a model minimum substrate cycle (ATP/ADP, with enzymes phosphofructokinase and pyruvate kinase from rabbit muscle). Yeast glycolysis is a prototype of a nonlinear dynamical system. Damped and sustained oscillations, as well as chaotic behaviour, were observed experimentally in yeast cells and cell-free yeast extracts [14, 151. There is general concern that, in yeast glycolysis, the source of nonlinear dynamic phenomena is the 6-phosphofructokinase/fructose-l,6-bisphosphatase cycle [14-171. There is evidence for a regulation of 6phosphofructokinase and fructose-l,6-bisphosphatase by allosteric and epigenetic mechanisms [15]. With respect to the allosteric regulation, the effects of AMP and fructose 2,6bisphosphate are regarded as most important. These metabolites, acting reciprocally, have effects on the activities of 6phosphofructokinase and fructose-l,6-bisphosphatase:both are efficient activators of 6-phosphofructokinase and inhibitors of fructose-l,6-bisphosphatase[16, 171. While AMP is directly involved in the cellular energy metabolism, synthesis and degradation of fructose 2,6-bisphosphate is accomplished by a cyclic bypass of glycolysis, catalyzed by phosphofructokinase-2 and several fructose-2,6-bisphosphate-degradingenzymes [17].
110 Influx of metabolites and enzymes
Glc6P
Fru6P
ATP
Fru (1,6)
-*
.. . ADP
uvate kinase for the regeneration of ATP from phosphoenolpyruvate. Quasi-equilibria of the adenine nucleotides and the hexose monophosphates are maintained by high concentrations of adenylate kinase and glucose-6-phosphate isomerase, respectively. The reaction network operates under conditions far from equilibrium by using a stirred flowthrough reactor. The system is maintained with adenine nucleotides, fructose 6-phosphate and phosphoenolpyruvate, as well as with the enzymes by different channels. In the reaction chamber the metabolites are converted by the enzymes involved. The efflux contains the metabolites according to their actual concentration in the reaction chamber. The cooperation of 6-phosphofructokinase, Fru( 1,6)P,ase, pyruvate kinase, adenylate kinase and Glc6P isomerase is governed by a set of coupled differential equations for the metabolite concentrations. These equations take into account the influx and efflux processes as well as the enzymatic conversions. According to the stoichiometric structure of the system, conservation equations hold for the total concentrations of the metabolite pools: [FrubP]
Efflux
Fig. 1. Cooperation of 6-plzosph~f~uctokinase (PFK), Fru(l,6)P2ase, pyruvate kinasse ( P K l , adenylate kirzase I A K ) and Glc6P isomerase in an open real system alimented with adenine nucleotides { ([ATP] + [ADP] + [AMP])IN},hexosemonophosphates ([Fru6PjlNj and phosphoenolpyruvute ( [ P P ~ v ] ~Substrates ~). and enzymes are pumped into the reaction chamber (5 = time of residence). The metabolite concentrations are measured in the efflux
In the open systems reconstituted from purified enzymes containing yeast phosphofructokinase as the central allosteric enzyme, multiple stationary states and oscillations could be observed. The experimental data were found to be predictable by mathematical models based on the kinetic properties of the enzymes involved [18]. Recently, the dynamics of a partial glycolytic reaction sequence, which converts glucose 6-phosphate to triose phosphates, was studied in cell-free extracts of yeast. Multiple stationary states and a hysteretic cycle formed by reversible transitions between unique and stable states could be established. When the activity concentrations of the enzymes involved were taken into account, the dynamics of the reaction sequence was predictable by a variant of the models used for investigating reconstituted enzyme systems ~91. In this paper the dynamics of the reconstituted enzyme system are investigated with respect to the occurrence of irreversible transitions between alternative stationary solutions. For the analysis, dynamic structures were investigated with respect to the maximum activity of fructose-1,6-bisphosphatase and the influx adenylate energy charge. The influx energy charge characterizes the energy supply of the system. Yeast fructose-I ,6-bisphosphatase is a strong candidate for epigenetic regulation in that it changes significantly during the cell cycle and growth of yeast [20,21].
[ATP]
+ [Glc6P] + [Fru(l,6)P2]= ( [ F ~ U ~ P ] )(1)~ ~
+ [ADP] + [AMP] = ([ATPI + [ADP] + [AMP])m
(2)
+
[PPrv] [Prv] = [PPrv],, (3) where the subscript IN indicates influx solution. The reversible and fast reactions catalyzed by adenylate kinase and glucose-6-phosphate isomerase give rise to substrate equilibration. Therefore it is suficient to consider the time evolution of the hexose monophosphates, ATP and phosphoenolpyruvate (Eqns 4- 6). d((Fru6PI
{[ F ~ u ~ P I I([FrufjPI N
+ [Glc6P])/dt =
+ [Glc6Pl)}l~ ~ ~ P F+K I ~ F H P (4) ~ ~ ~ -
+
d(2[ATP] [ADP])/dt = {(2[ATPl + [ADPI)IN-(~[ATP] [ADPl))/T-Upm
+
+ UPK
(5)
d[PPrv]/dt = ([PPrv],,-[PPrv])/z -upK. (6) These equations contain expressions for the activities of 6-phosphofructokinase (upFK), Fru( 1,6)P2ase (oFBpasc)and pyruvate kinase (up& which were taken from the kinetic investigation of the individual enzymes [19,20]. The concentrations of metabolites in the influx, the adenylate energy charge of the influx solution, the time of residence (7) and the maximum activities of the enzymes can be adjusted experimentally and are the control parameters of this system. The steady-state curves as a function of one control parameter (bifurcation diagrams) and the continuation of limit points and Hopf bifurcation points (parametric planes) were computed by using the software package AUTO [22]. The time evolution of the metabolite concentrations was obtained by a semi-implicite Gear algorithm [23]. For the sake of presentation and simplification, instead of the metabolite concentrations ([ATPI, [PPrv] and [Fru6P] + [Glc6P]) the norm of the concentration vector (Eqn 7) was used: Norm = [ATPI’ ([Fru6P] [ G ~ c ~ P ] [PPrvI2 )~ ([ATPI [ADP] [AMPI)& [Fru6P]k [PPrv]?,
+
+
+
r2.
METHODS
[--
The Fru6P/Fru(l,6)P2 cycle operating in an open and homogeneous system reconstituted from purified enzymes is depicted in Fig. 1. In addition to 6-phosphofructokinase and fructose-1,6-bisphosphatase[Fru(l ,6)Pzase], it contains pyr-
RESULTS In Fig. 2 is shown a bifurcation diagram of the enzyme system with respect to the influx adenylate energy charge,
+
+
+
+
(7)
111 5 4
$ 3 m
P
2 1
0 0.5
1.o
2.0
1.5
2.5
3.0
[A=],,
1
2
[Adenylate energy charge],,
0.4
0.3
f
0.2
P 0.1
0
0.4
1
1
I
1
I
0.5
0.6
0.7
0.8
0.9
1.0
[AECI,,
0
1
2
3
Fig. 2. Bifurcation diagrams in the [ A ECjIN/vFBPose parametric plane. Qualitative properties of stationary solutions obtained by bifurcation analysis for [AECIlN = 0.5-3.0 (A) and 0.4-1.0 (B). Control parameters: ([ATP] [ADP] [AMP])IN= 3 mM, [PPrv],, = 6.3 mM, ([Fru6P]),, = 6 mM, upFK = 1 U/ml, upK = 10 U/ml and z = 40 min. The light and the heavy curves mark limit points and Hopf bifurcations, respectively. Domains: (A, E) Unique stable states (S, and SH); (B) unique unstable states (UL);(C) bistability (SH, saddle point and UL);(D, F) bistability (SH, saddle point, SL);(F) complex dynamic behavior
Fig. 3. Dependence of stationary states on the influx energy charge. vFBPase = 2 U/ml and the other parameters correspond to Fig. 2. (a) Concentration of metabolites; (b) scaled norm of the concentrations. (0)Limit points; (m) Hopf bifurcation points. The broken parts of the curves refer to unstable stationary solutions. The full curves correlate to stable stationary states. The limit points cover a domain of input adenylate energy charges in which three alternative stationary solutions exist
[AECIIN,and the maximum activity of Fru(l,6)P2ase. The other parameters fit to the experimental conditions used in [18] for which the occurrence of multistability and of sustained oscillations was proved experimentally and shown to be in good agreement with model calculations. The light solid curves in the parameter domain connect limit points of the system and hence limit a domain in which multiple steady states exist. The heavy solid curve indicates Hopf bifurcations. Accordingly, seven parameter domains can be distinguished. In A and E, unique and stable stationary states occur, respectively, characterized by low and high, respectively, levels of the adenylate energy charge and phosphoenolpyruvate and are denoted as low, SL, and high, SH, energy states respectively. In the domain B, the single stationary solutions are unstable and correspond to the emergence of stable oscillations (checked by numerical integration). In C, three stationary solutions coexist, a stable attractor (S"), a saddle point and an unstable low-energy state (U,). In the domain D, the low-energy state becomes stable (S,) and relates to stable or unstable states. Domain F too is a region of classical bistability. In a rather narrow region (domain G) the dynamic behavior is much more complex (i.e. SH,UL, fivefold
stationarities). Which of the low or high energy attractor applying in domains C, D, F (and G) is actually reached, depends on the initial conditions (history of the system). In this way, two basins of attraction are defined, from which the trajectories lead to the states SHor SL (U,) respectively. Fig. 3 shows steady-state curves in dependence on the influx adenylate energy charge, [AECIIN.The control parameter was selected so that the graphs relate to a horizontal line in Fig. 2 (uFBPase= 2 Ujml), which crosses the domains A, B and C. Fig. 3 a shows the concentrations of the hexose monophosphates, ATP and phosphoenolpyruvate and in Fig. 3 b the scaled norm of the concentration vector is indicated. The graphs give the steady-state curves in a region of [AECIIN between 0-3, although values above unity are without any physical meaning. At [AECIIN= 1, two stationary solutions coexist (SH,U,). By decreasing [AECIIN,a limit point is passed at which the branch of high-energy states disappears. Hence, decreasing [AECIINbelow the limit point will induce a transition to the low-energy branch of stationary states. The other limit point however, corresponds to [AECIIN> 1 and can not be reached by changes of the adenylate energy charge. Hence, the transitions SH--t SL induced by decreasing [AECIINare irreversible.
+
+
[Adenylate energy charge],,
112 I
I
1.0 [
I
1.0 I
I
I
I
I I I
I
I I I I I I 0 '
I
I
2
I
,
4
I
I
6
I
I
8
I
2
I
I
6
8
1
4
Time (rnin x l o 3 )
Time (rnin x l o 3 )
Fig. 4. Time e ~ d t r t i o nof the norm of the concentration vector. Parameter values as in Fig. 3. The time phases differ in the adenylate energy charge ([AEC],,) applied. Phases I, 111: [AEC],, = 1 and phase TI: [AECIIN= 0.5
1.0 I
0
I I
I
I
1
Fig. 5. Stutionary .stutes in dependence on the maximum activity qf Fru(l,6)P2ase, vFHPart,.[AECIIN= 1 and the other parameters as in Fig. 2. (n)Limit point; (0)critical uFBPase value below which the limit cycle surrounding the states U L becomes unstable. Broken and solid curves have the same meaning as in Fig. 3
In Fig. 4 is shown the respective computer simulation. In phase 1 ([AEC]" = 1) the metabolites evolve to a stable attractor (SH). In 11, [AEC]INis decreased down to 0.5. In accordance with Figs 2 and 3, the system evolves to a lowenergy stationary state. In phase 111, the adenylate energy charge is increased again to its upper positive level ([AEC],, = 1). This causes the appearence of sustained oscillations, the original state (phase I) cannot be reached again by any change of [AECIIN. In Fig. 5 a steady-state curve is shown, referring to a vertical intersection of the parameter plane shown in Fig. 2 ([AECIIN= 1). The curve crosses the regions B, C (and D) of the bifurcation diagram. For maximum activities of Fru(l,6)P2ase above a critical value (uFBPase = 0.26 Ujml), three stationary solutions exist (SH, saddle point, U,). In the plotted parameter region, the low-energy branch of the stationary solutions contains only unstable points and the HoDf bifurcation point, at which the low-energy _ _ branch becomes stable ( I I ~ = ~ 6.17 ~ ~ ~Ujml) , ~ is out of the frame of this
Fig. 6. Time evolution of' the norm qf the concentration vector. Parameter values as in Fig. 5. The time phases differ in the maximum activity of fructose-I ,6-bisphosphatase (uFBPase) applied. Phases I, IV, uFBPase = 20 U/ml; phase TI, uFBPase = 2 Uirnl; phase 111. tiFHPase = 1.75 Uiml
diagram. The high-energy branch of stationary solutions exists for all values of uFBPase,while the low-energy branch disappears at the limit point. Hence, when the stationary state correlates to the low-energy branch, decreases of u~~~~~~ cause a transition UL+ SH,which can not be reversed by increasing i)FRPasc again. A corresponding computer simulation is shown in Fig. 6. The computer experiment starts with a high value of uFBpase (phase I) : the system evolves to a stable low-energy stationary state (S,). Then in phase 11, the maximum activity of Fru(l,6)P2ase is decreased. After this, a Hopf bifurcation point is passed and sustained oscillations around a state U1 emerge. On further decreases of uFBPase,the limit cycle becomes unstable and the trajectories come into the domain of attraction of a state SH, which is approached in phase 111. Interestingly, the transition to the high-energy branch of the stationary solutions arises at levels of u F B which ~ ~ ~ are ~ significantly above the respective limit point ( u ~ ~= ~1.8 U/ ~ ~ ml). The critical instability of the limit cycles, however, is not obtained by a bifurcation analysis of the stationary solutions and the respective critical curve is not involved in the bifurcation diagram shown in Fig. 2. The states S,, exist for all possible values of uFBPasc. Hence, further increases of the maximum activity of Fru( 1,6)P2ase (phase IV) will not drive the system to switch to the branch of low-energy states: the transition SL+ S, is thus irreversible. DISCUSSION AND CONCLUSION The raison d'ctre of a bistable system is to allow dramatic reversible transitions between two different steady states. These transitions are observed when a parameter is varied back and forth, that is, the reversible switching from one steady-state concentration to another occurs at two different threshold values of the control parameter (limit points). The main consequences of this are (a) if the parameters cross the domain of bistability, a jump between two states and the reverse transition will proceed by different routes forming a dynamic hysteretic loop; (b) between the limit points, some of the variables of the system are insensitive to any fluctuation. a property which may be used by the cell to improve
~
113 homeostasis; and (c) the sensitivity of the response to effectors, modulators or fluctuations may also be increased through bistability [24, 251. If however, one of the two limit points disappears or lies outside of the domain of physically attractible parameter values, irreversible transitions may arise, which will not allow any more cyclic changes between the alternative branches of stationary solutions. It follows that the stationary branch of which the correlated limit point is out of the range of accessibility has a definite preference in that transitions may drive the system to this branch, while the reverse transitions, and hence the formation of a hysteretic loop, is almost impossible. The irreversibility of transitions between alternative stationary solutions may be of principal importance for physiological properties such as memory, evolution, or the maintenance of homeostasis in the cellular metabolism. Although the classical references to bistable dynamic systems still refer to reversible transitions between the alternative branches of stationary states and the occurrence of hysteretic loops, irreversible transitions are not an exclusive property of nonlinear systems, but are very likely to occur in systems exhibiting bistability. Surprisingly, little attention has been paid so far to the existence and/or the significance of these irreversible transitions. (a) Studying analytically a Monod/Jacob model for induction and repression of synthesis of proteins, Babloyantz and Nicolis [26] have shown that the system is able to switch between two states of largely different enzyme concentration and, once coupled with the environment, it may be unable to trace back its previous history even when the constraints are released. (b) The analysis of isothermal chemical reactions (e. g. Michaelis-Menten mechanism) occurring in a volume bounded by a membrane and immersed in a reservoir of reactants (substrates) and products at fixed concentrations was made by Hahn et al. [27]. If the permeability of the membrane to a given species is taken to be a function of the concentration of that species or of another one, then the coupling between reaction and permeation provides feedback and irreversible transitions between stable branches of steady states can be induced. (c) A mathematical model for the glycolysis of human erythrocytes which takes into account ATP synthesis and consumption was proposed by Rapoport and Heinrich [28]. In the frame of this model, below a critical load of ATP consumption three steady states of the system exist: the physiological ATPproducing state, a saddle point and a stable non-energized steady state ([ATP] = 0). Above the critical level of energy consumption (high ATPase and diphosphoglycerate mutase activities) the system breaks down irreversibly to the nonenergized state. The physiological state can not be restored by decreasing the rate of ATP consumption. In this paper, irreversible transitions were shown to exist in a reconstituted enzyme system containing the Fru6Pl Fru( 1,6)P2 cycle. The irreversible character of transitions between alternative stationary solutions was shown to exist with respect to the adenylate energy charge input and the maximum activity of fructose-l,6-bisphosphatase. Varying the adenylate energy charge input, two limit points were shown to exist, one of which was not accessible ([AEC],, > 1). The existence of alternative stationary branches, even for uFBPase which tends to infinity, result in the occurrence of irreversible transitions with respect to the maximum activity of Fru(l,6)P2ase. Both irreversible transitions with respect to uFBPaseand [AECIIN might be of interest for the function of the 6-
phosphofructokinase/Fru(l ,6)P2ase cycle in vivo : irreversibility with respect to [AECIINcould refer to critical properties pertaining to the recovery of cellular energy metabolism. The irreversibility of transitions with respect to uFBPasf could be of significance for metabolic changes occurring during the yeast cell cycle. Experimental demonstrations of the occurrence of irreversible transitions in yeast cell extracts are presently under progress. This work was made in the frame of project BIO/CHE-4 of the French-GDR Program for Scientific and Technical Research.
REFERENCES 1. Atkinson, D. E. (1977) Cellular energy metabolism and Its regulation, Academic Press, New York. 2. Reich, J. & Selkov, E. E. (1981) Energy metabolism of the cell: a theoretical treatise, Academic Press, New York. 3. Newsholme, E. A. & Crabtree, B. (1976) Biochem. Soc. Symp. 41, 61 - 109. 4. Hofmeyr, J. H. S., Kacser, H. & van der Merwe, K. J. (1986) Eur. J . Biochem. 155,631- 641. 5. Newsholme, E. A. & Start, C. (1973) Regulation of metabolism, John Wiley, London. 6. Stadtman, E. R. &Chock, P. B. (1978) Curr. Top. Cell. Regul. 13, 53 - 95. 7. Goldbeter, A. & Koshland Jr, D. E. (1981) Proc. Natl Acad. Sci. USA 78,6840 - 6844. 8. LaPorte, D. C. & Koshland Jr, D. E. (1983) Nature 305, 286290. 9. Cimino, A. & Hervagault, J. F. (1987) Biochem. Biophys. Res. Commun. 149, 615-620. 10. Ricard, J. & Soulit., J. M. (1982) J . Theor. Biol. 95, 105-121. 11. Hervagault, J. F. & Canu, S. (1987) J . Theor. Biol. 127,439-449. 12. Hervagault, J. F. & Cimino, A. (1989) J . Theor. Biol. 140, 399416. 13. Cimino, A. & Hervagault, J. F. (1990) FEBS Lett. 263,199-205. 14. Goldbeter, A. & Caplan, S. R. (1976) Annu. Rev. Biophys. Bioeng. 5,449 - 476. 15. Hess, B. & Markus, M. (1985) Ber. Bunsenges. Phys. Chem. 89, 642 - 651. 16. Hofmann, E. & Kopperschlager, G. (1982) Methods Enzymol. 90, 49 - 60. 17. Van Schaftingen, E. (1987) Adv. Enzymol. 59, 315-395. 18. Hofmann, E., Eschrich, K. & Schellenberger, W. (1985) Adv. Enzyme Regul. 23, 331 - 362. 19. Eschrich, K., Schellenberger, W. & Hofmann, E. (1990) Eur. J . Biochem. 188, 697 - 703. 20. Franqois, J., Eraso, P. & Gancedo, C. (1987) Eur. J . Biochem. 164, 369 - 373. 21. Van Doorn, J., Valkonburg, J. A. C., Scholte, M. E., Oehlen, L. J. W. M., Van Driel, R., Postma, P. W., Nanninga, N. & Van Dam, K. (1988) J . Bacterzol. 170,4808-4815. 22. Doedel, E. (1981) in Proceedings of the tenth Manitoba conference on numerical mathematics and computation, pp. 265 - 284, University of Manitoba, Winnipeg. 23. Gear, C. W. (1971) Numerical initial value problems in ordinary differential equutions, Prentice Hall, Englewood Cliffs NJ. 24. Goldbeter, A. & Koshland, D. E. Jr (1982) Q. Rev. Biophys. 15, 555 - 591. 25. Koshland, D. E. Jr & Goldbeter, A. (1982) Science 217, 220225. 26. Babloyantz, A. & Nicolis, G. (1972) J . Theor. Biol. 34, 185- 192. 27. Hahn, H. S., Ortoleva, P. J. & Ross, J. (1973) J . Theor. Biol. 41, 503 - 521. 28. Rapoport, T. A. & Heinrich, R. (1975) Biosystems 7, 120-129.
Irreversible transitions in the 6-phosphofructokinase/fructose 1,6-bisphosphatase cycle Wolfgang SCHELLENBERGER' and Jean-Franqois HERVAGAULT'
' Institut fur Biochemie, Karl-Marx-Universitat, Leipzig, Federal Republic of Germany Unite de Recherche Associee no. 523 du Centre National de la Recherche Scientifique, Universitt de Compiegne, France (Received June 25,1990)
-
EJB 90 0734
The dynamics of the fructose 6-phosphate/fructose-1,6-bisphosphate cycle operating in an open and homogeneous system reconstituted from purified enzymes was extensively studied. In addition to 6-phosphofructokinase and fructose-l,6-bisphosphatase, pyruvate kinase, adenylate kinase and glucose-6-phosphate isomerase were involved. In that multi-enzyme system, the main source of non-linearity is the reciprocal effect of AMP on the activities of 6-phosphofructokinase and fructose-I ,6-bisphosphatase. Depending upon the experimental parameter values, stable attractors, various types of multiple states and sustained oscillations were shown to occur. In the present report we show that irreversible transitions are also likely to occur for realistic operating conditions. Two parameters of the system, that is the adenylate energy charge of the influx and the fructose-1,6bisphosphatase maximal activity, are potential candidates to provoke such irreversible transitions from one steady state to the other: (a) when varying the maximal activity of fructose-l,6-bisphosphatase, the system can jump irreversibly from a low to a high stable steady state, and (b) when the adenykdte energy charge of the influx is the changing parameter, irreversible transitions occur from a high stable steady state to a stable oscillatory state (limit cycle motion). This behavior can be predicted by constructing the loci of limit points and Hopf bifurcation points.
An important feature of the cellular metabolism is the existence of moiety-conserved cycles [l, 21. The two best known groups of metabolites participating in such operations are ATP/ADP and NAD(P)/NAD(P)H. The role played by cycles in the regulation of metabolism is still a matter of controversy. However, from experimental data and theoretical considerations, it is likely that substrate cycling is involved in thermogenesis [3,4], in the orientation of fluxes, in the control of concentrations [5], and in the amplification of sensitivity [6 - 91. When dealing with cycles subjected to a destabilising factor (product activation, substrate inhibition, etc.), more sophisticated dynamics may be observed, such as multiple steady states, damped and sustained oscillations or even chaotic motions [2, lo]. When dealing with bistability, the transition between the two steady states might be irreversible. By irreversible transitions, we mean that, contrary to what occurs in a classical bistable system, the switching from one branch of steady states to the other is, at the most, possible only once. These irreversibility features are in no way due to a bounded domain of variation of one parameter (experimental constraints) but are an intrinsic property of the highly nonlinear system considered. As a minimum system, a simple substrate cycle involving two antagonist enzymes, one of them being inhibited by excess Correspondence to J. F. Hervagault, U.R.A. no. 523 duC.N.R.S., UniversitC de Compiegne Boite Postale no. 649, F-60206 Compiegne, France Abbreviations. Fru6P, fructose-6-phosphate; Glc6P, glucose 6phosphate; Fru(l,6)P2, fructose 1,6-bisphosphate; PPrv, phosphoenolpyruvate; [AECIIN,influx adenylate energy charge; Fru(l,6)P,ase, fructose-l,6-bisphosphatase;Glc6Pase, glucose-6-phosphate isomerase. Enzymes. 6-Phosphofructokinase (EC 2.7.1.11); pyruvate kinase (EC 2.7.1.40); fructose-1,6-bisphosphatase(EC 3.1.3.11); glucose-6phosphate isomerase (EC 5.3.1.9); adenylate kinase (EC 2.7.4.3).
of its substrate was studied theoretically [Ill. In addition to classical bistability (with respect to both the maximal activity of one of the enzymes and the sum of the metabolite concentrations), irreversible transitions were also found to occur, but only with respect to the sum of the metabolite concentrations. The model was extended in order to investigate the changes in the dynamic properties which may arise when such a cycle is open to mass transfers [12]. In particular, the coexistence of both reversible and irreversible transitions could be observed when, for instance, the level of exchanges with the surroundings are altered. Recently, it was shown experimentally [13] that irreversible transitions could actually be observed in such a model minimum substrate cycle (ATP/ADP, with enzymes phosphofructokinase and pyruvate kinase from rabbit muscle). Yeast glycolysis is a prototype of a nonlinear dynamical system. Damped and sustained oscillations, as well as chaotic behaviour, were observed experimentally in yeast cells and cell-free yeast extracts [14, 151. There is general concern that, in yeast glycolysis, the source of nonlinear dynamic phenomena is the 6-phosphofructokinase/fructose-l,6-bisphosphatase cycle [14-171. There is evidence for a regulation of 6phosphofructokinase and fructose-l,6-bisphosphatase by allosteric and epigenetic mechanisms [15]. With respect to the allosteric regulation, the effects of AMP and fructose 2,6bisphosphate are regarded as most important. These metabolites, acting reciprocally, have effects on the activities of 6phosphofructokinase and fructose-l,6-bisphosphatase:both are efficient activators of 6-phosphofructokinase and inhibitors of fructose-l,6-bisphosphatase[16, 171. While AMP is directly involved in the cellular energy metabolism, synthesis and degradation of fructose 2,6-bisphosphate is accomplished by a cyclic bypass of glycolysis, catalyzed by phosphofructokinase-2 and several fructose-2,6-bisphosphate-degradingenzymes [17].
110 Influx of metabolites and enzymes
Glc6P
Fru6P
ATP
Fru (1,6)
-*
.. . ADP
uvate kinase for the regeneration of ATP from phosphoenolpyruvate. Quasi-equilibria of the adenine nucleotides and the hexose monophosphates are maintained by high concentrations of adenylate kinase and glucose-6-phosphate isomerase, respectively. The reaction network operates under conditions far from equilibrium by using a stirred flowthrough reactor. The system is maintained with adenine nucleotides, fructose 6-phosphate and phosphoenolpyruvate, as well as with the enzymes by different channels. In the reaction chamber the metabolites are converted by the enzymes involved. The efflux contains the metabolites according to their actual concentration in the reaction chamber. The cooperation of 6-phosphofructokinase, Fru( 1,6)P,ase, pyruvate kinase, adenylate kinase and Glc6P isomerase is governed by a set of coupled differential equations for the metabolite concentrations. These equations take into account the influx and efflux processes as well as the enzymatic conversions. According to the stoichiometric structure of the system, conservation equations hold for the total concentrations of the metabolite pools: [FrubP]
Efflux
Fig. 1. Cooperation of 6-plzosph~f~uctokinase (PFK), Fru(l,6)P2ase, pyruvate kinasse ( P K l , adenylate kirzase I A K ) and Glc6P isomerase in an open real system alimented with adenine nucleotides { ([ATP] + [ADP] + [AMP])IN},hexosemonophosphates ([Fru6PjlNj and phosphoenolpyruvute ( [ P P ~ v ] ~Substrates ~). and enzymes are pumped into the reaction chamber (5 = time of residence). The metabolite concentrations are measured in the efflux
In the open systems reconstituted from purified enzymes containing yeast phosphofructokinase as the central allosteric enzyme, multiple stationary states and oscillations could be observed. The experimental data were found to be predictable by mathematical models based on the kinetic properties of the enzymes involved [18]. Recently, the dynamics of a partial glycolytic reaction sequence, which converts glucose 6-phosphate to triose phosphates, was studied in cell-free extracts of yeast. Multiple stationary states and a hysteretic cycle formed by reversible transitions between unique and stable states could be established. When the activity concentrations of the enzymes involved were taken into account, the dynamics of the reaction sequence was predictable by a variant of the models used for investigating reconstituted enzyme systems ~91. In this paper the dynamics of the reconstituted enzyme system are investigated with respect to the occurrence of irreversible transitions between alternative stationary solutions. For the analysis, dynamic structures were investigated with respect to the maximum activity of fructose-1,6-bisphosphatase and the influx adenylate energy charge. The influx energy charge characterizes the energy supply of the system. Yeast fructose-I ,6-bisphosphatase is a strong candidate for epigenetic regulation in that it changes significantly during the cell cycle and growth of yeast [20,21].
[ATP]
+ [Glc6P] + [Fru(l,6)P2]= ( [ F ~ U ~ P ] )(1)~ ~
+ [ADP] + [AMP] = ([ATPI + [ADP] + [AMP])m
(2)
+
[PPrv] [Prv] = [PPrv],, (3) where the subscript IN indicates influx solution. The reversible and fast reactions catalyzed by adenylate kinase and glucose-6-phosphate isomerase give rise to substrate equilibration. Therefore it is suficient to consider the time evolution of the hexose monophosphates, ATP and phosphoenolpyruvate (Eqns 4- 6). d((Fru6PI
{[ F ~ u ~ P I I([FrufjPI N
+ [Glc6P])/dt =
+ [Glc6Pl)}l~ ~ ~ P F+K I ~ F H P (4) ~ ~ ~ -
+
d(2[ATP] [ADP])/dt = {(2[ATPl + [ADPI)IN-(~[ATP] [ADPl))/T-Upm
+
+ UPK
(5)
d[PPrv]/dt = ([PPrv],,-[PPrv])/z -upK. (6) These equations contain expressions for the activities of 6-phosphofructokinase (upFK), Fru( 1,6)P2ase (oFBpasc)and pyruvate kinase (up& which were taken from the kinetic investigation of the individual enzymes [19,20]. The concentrations of metabolites in the influx, the adenylate energy charge of the influx solution, the time of residence (7) and the maximum activities of the enzymes can be adjusted experimentally and are the control parameters of this system. The steady-state curves as a function of one control parameter (bifurcation diagrams) and the continuation of limit points and Hopf bifurcation points (parametric planes) were computed by using the software package AUTO [22]. The time evolution of the metabolite concentrations was obtained by a semi-implicite Gear algorithm [23]. For the sake of presentation and simplification, instead of the metabolite concentrations ([ATPI, [PPrv] and [Fru6P] + [Glc6P]) the norm of the concentration vector (Eqn 7) was used: Norm = [ATPI’ ([Fru6P] [ G ~ c ~ P ] [PPrvI2 )~ ([ATPI [ADP] [AMPI)& [Fru6P]k [PPrv]?,
+
+
+
r2.
METHODS
[--
The Fru6P/Fru(l,6)P2 cycle operating in an open and homogeneous system reconstituted from purified enzymes is depicted in Fig. 1. In addition to 6-phosphofructokinase and fructose-1,6-bisphosphatase[Fru(l ,6)Pzase], it contains pyr-
RESULTS In Fig. 2 is shown a bifurcation diagram of the enzyme system with respect to the influx adenylate energy charge,
+
+
+
+
(7)
111 5 4
$ 3 m
P
2 1
0 0.5
1.o
2.0
1.5
2.5
3.0
[A=],,
1
2
[Adenylate energy charge],,
0.4
0.3
f
0.2
P 0.1
0
0.4
1
1
I
1
I
0.5
0.6
0.7
0.8
0.9
1.0
[AECI,,
0
1
2
3
Fig. 2. Bifurcation diagrams in the [ A ECjIN/vFBPose parametric plane. Qualitative properties of stationary solutions obtained by bifurcation analysis for [AECIlN = 0.5-3.0 (A) and 0.4-1.0 (B). Control parameters: ([ATP] [ADP] [AMP])IN= 3 mM, [PPrv],, = 6.3 mM, ([Fru6P]),, = 6 mM, upFK = 1 U/ml, upK = 10 U/ml and z = 40 min. The light and the heavy curves mark limit points and Hopf bifurcations, respectively. Domains: (A, E) Unique stable states (S, and SH); (B) unique unstable states (UL);(C) bistability (SH, saddle point and UL);(D, F) bistability (SH, saddle point, SL);(F) complex dynamic behavior
Fig. 3. Dependence of stationary states on the influx energy charge. vFBPase = 2 U/ml and the other parameters correspond to Fig. 2. (a) Concentration of metabolites; (b) scaled norm of the concentrations. (0)Limit points; (m) Hopf bifurcation points. The broken parts of the curves refer to unstable stationary solutions. The full curves correlate to stable stationary states. The limit points cover a domain of input adenylate energy charges in which three alternative stationary solutions exist
[AECIIN,and the maximum activity of Fru(l,6)P2ase. The other parameters fit to the experimental conditions used in [18] for which the occurrence of multistability and of sustained oscillations was proved experimentally and shown to be in good agreement with model calculations. The light solid curves in the parameter domain connect limit points of the system and hence limit a domain in which multiple steady states exist. The heavy solid curve indicates Hopf bifurcations. Accordingly, seven parameter domains can be distinguished. In A and E, unique and stable stationary states occur, respectively, characterized by low and high, respectively, levels of the adenylate energy charge and phosphoenolpyruvate and are denoted as low, SL, and high, SH, energy states respectively. In the domain B, the single stationary solutions are unstable and correspond to the emergence of stable oscillations (checked by numerical integration). In C, three stationary solutions coexist, a stable attractor (S"), a saddle point and an unstable low-energy state (U,). In the domain D, the low-energy state becomes stable (S,) and relates to stable or unstable states. Domain F too is a region of classical bistability. In a rather narrow region (domain G) the dynamic behavior is much more complex (i.e. SH,UL, fivefold
stationarities). Which of the low or high energy attractor applying in domains C, D, F (and G) is actually reached, depends on the initial conditions (history of the system). In this way, two basins of attraction are defined, from which the trajectories lead to the states SHor SL (U,) respectively. Fig. 3 shows steady-state curves in dependence on the influx adenylate energy charge, [AECIIN.The control parameter was selected so that the graphs relate to a horizontal line in Fig. 2 (uFBPase= 2 Ujml), which crosses the domains A, B and C. Fig. 3 a shows the concentrations of the hexose monophosphates, ATP and phosphoenolpyruvate and in Fig. 3 b the scaled norm of the concentration vector is indicated. The graphs give the steady-state curves in a region of [AECIIN between 0-3, although values above unity are without any physical meaning. At [AECIIN= 1, two stationary solutions coexist (SH,U,). By decreasing [AECIIN,a limit point is passed at which the branch of high-energy states disappears. Hence, decreasing [AECIINbelow the limit point will induce a transition to the low-energy branch of stationary states. The other limit point however, corresponds to [AECIIN> 1 and can not be reached by changes of the adenylate energy charge. Hence, the transitions SH--t SL induced by decreasing [AECIINare irreversible.
+
+
[Adenylate energy charge],,
112 I
I
1.0 [
I
1.0 I
I
I
I
I I I
I
I I I I I I 0 '
I
I
2
I
,
4
I
I
6
I
I
8
I
2
I
I
6
8
1
4
Time (rnin x l o 3 )
Time (rnin x l o 3 )
Fig. 4. Time e ~ d t r t i o nof the norm of the concentration vector. Parameter values as in Fig. 3. The time phases differ in the adenylate energy charge ([AEC],,) applied. Phases I, 111: [AEC],, = 1 and phase TI: [AECIIN= 0.5
1.0 I
0
I I
I
I
1
Fig. 5. Stutionary .stutes in dependence on the maximum activity qf Fru(l,6)P2ase, vFHPart,.[AECIIN= 1 and the other parameters as in Fig. 2. (n)Limit point; (0)critical uFBPase value below which the limit cycle surrounding the states U L becomes unstable. Broken and solid curves have the same meaning as in Fig. 3
In Fig. 4 is shown the respective computer simulation. In phase 1 ([AEC]" = 1) the metabolites evolve to a stable attractor (SH). In 11, [AEC]INis decreased down to 0.5. In accordance with Figs 2 and 3, the system evolves to a lowenergy stationary state. In phase 111, the adenylate energy charge is increased again to its upper positive level ([AEC],, = 1). This causes the appearence of sustained oscillations, the original state (phase I) cannot be reached again by any change of [AECIIN. In Fig. 5 a steady-state curve is shown, referring to a vertical intersection of the parameter plane shown in Fig. 2 ([AECIIN= 1). The curve crosses the regions B, C (and D) of the bifurcation diagram. For maximum activities of Fru(l,6)P2ase above a critical value (uFBPase = 0.26 Ujml), three stationary solutions exist (SH, saddle point, U,). In the plotted parameter region, the low-energy branch of the stationary solutions contains only unstable points and the HoDf bifurcation point, at which the low-energy _ _ branch becomes stable ( I I ~ = ~ 6.17 ~ ~ ~Ujml) , ~ is out of the frame of this
Fig. 6. Time evolution of' the norm qf the concentration vector. Parameter values as in Fig. 5. The time phases differ in the maximum activity of fructose-I ,6-bisphosphatase (uFBPase) applied. Phases I, IV, uFBPase = 20 U/ml; phase TI, uFBPase = 2 Uirnl; phase 111. tiFHPase = 1.75 Uiml
diagram. The high-energy branch of stationary solutions exists for all values of uFBPase,while the low-energy branch disappears at the limit point. Hence, when the stationary state correlates to the low-energy branch, decreases of u~~~~~~ cause a transition UL+ SH,which can not be reversed by increasing i)FRPasc again. A corresponding computer simulation is shown in Fig. 6. The computer experiment starts with a high value of uFBpase (phase I) : the system evolves to a stable low-energy stationary state (S,). Then in phase 11, the maximum activity of Fru(l,6)P2ase is decreased. After this, a Hopf bifurcation point is passed and sustained oscillations around a state U1 emerge. On further decreases of uFBPase,the limit cycle becomes unstable and the trajectories come into the domain of attraction of a state SH, which is approached in phase 111. Interestingly, the transition to the high-energy branch of the stationary solutions arises at levels of u F B which ~ ~ ~ are ~ significantly above the respective limit point ( u ~ ~= ~1.8 U/ ~ ~ ml). The critical instability of the limit cycles, however, is not obtained by a bifurcation analysis of the stationary solutions and the respective critical curve is not involved in the bifurcation diagram shown in Fig. 2. The states S,, exist for all possible values of uFBPasc. Hence, further increases of the maximum activity of Fru( 1,6)P2ase (phase IV) will not drive the system to switch to the branch of low-energy states: the transition SL+ S, is thus irreversible. DISCUSSION AND CONCLUSION The raison d'ctre of a bistable system is to allow dramatic reversible transitions between two different steady states. These transitions are observed when a parameter is varied back and forth, that is, the reversible switching from one steady-state concentration to another occurs at two different threshold values of the control parameter (limit points). The main consequences of this are (a) if the parameters cross the domain of bistability, a jump between two states and the reverse transition will proceed by different routes forming a dynamic hysteretic loop; (b) between the limit points, some of the variables of the system are insensitive to any fluctuation. a property which may be used by the cell to improve
~
113 homeostasis; and (c) the sensitivity of the response to effectors, modulators or fluctuations may also be increased through bistability [24, 251. If however, one of the two limit points disappears or lies outside of the domain of physically attractible parameter values, irreversible transitions may arise, which will not allow any more cyclic changes between the alternative branches of stationary solutions. It follows that the stationary branch of which the correlated limit point is out of the range of accessibility has a definite preference in that transitions may drive the system to this branch, while the reverse transitions, and hence the formation of a hysteretic loop, is almost impossible. The irreversibility of transitions between alternative stationary solutions may be of principal importance for physiological properties such as memory, evolution, or the maintenance of homeostasis in the cellular metabolism. Although the classical references to bistable dynamic systems still refer to reversible transitions between the alternative branches of stationary states and the occurrence of hysteretic loops, irreversible transitions are not an exclusive property of nonlinear systems, but are very likely to occur in systems exhibiting bistability. Surprisingly, little attention has been paid so far to the existence and/or the significance of these irreversible transitions. (a) Studying analytically a Monod/Jacob model for induction and repression of synthesis of proteins, Babloyantz and Nicolis [26] have shown that the system is able to switch between two states of largely different enzyme concentration and, once coupled with the environment, it may be unable to trace back its previous history even when the constraints are released. (b) The analysis of isothermal chemical reactions (e. g. Michaelis-Menten mechanism) occurring in a volume bounded by a membrane and immersed in a reservoir of reactants (substrates) and products at fixed concentrations was made by Hahn et al. [27]. If the permeability of the membrane to a given species is taken to be a function of the concentration of that species or of another one, then the coupling between reaction and permeation provides feedback and irreversible transitions between stable branches of steady states can be induced. (c) A mathematical model for the glycolysis of human erythrocytes which takes into account ATP synthesis and consumption was proposed by Rapoport and Heinrich [28]. In the frame of this model, below a critical load of ATP consumption three steady states of the system exist: the physiological ATPproducing state, a saddle point and a stable non-energized steady state ([ATP] = 0). Above the critical level of energy consumption (high ATPase and diphosphoglycerate mutase activities) the system breaks down irreversibly to the nonenergized state. The physiological state can not be restored by decreasing the rate of ATP consumption. In this paper, irreversible transitions were shown to exist in a reconstituted enzyme system containing the Fru6Pl Fru( 1,6)P2 cycle. The irreversible character of transitions between alternative stationary solutions was shown to exist with respect to the adenylate energy charge input and the maximum activity of fructose-l,6-bisphosphatase. Varying the adenylate energy charge input, two limit points were shown to exist, one of which was not accessible ([AEC],, > 1). The existence of alternative stationary branches, even for uFBPase which tends to infinity, result in the occurrence of irreversible transitions with respect to the maximum activity of Fru(l,6)P2ase. Both irreversible transitions with respect to uFBPaseand [AECIIN might be of interest for the function of the 6-
phosphofructokinase/Fru(l ,6)P2ase cycle in vivo : irreversibility with respect to [AECIINcould refer to critical properties pertaining to the recovery of cellular energy metabolism. The irreversibility of transitions with respect to uFBPasf could be of significance for metabolic changes occurring during the yeast cell cycle. Experimental demonstrations of the occurrence of irreversible transitions in yeast cell extracts are presently under progress. This work was made in the frame of project BIO/CHE-4 of the French-GDR Program for Scientific and Technical Research.
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