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Oscillations in coupled enzymic reactions at high concentration of enzyme
118
Biochimica et Biophysica Acta, ! ! 19 (1992) ! ! 8- i 22 © 1992 Elsevier Science Publishers B.V. All rights reserved 0167-4838/92/$05.00
BBAPRO 34103
Oscillations in coupled enzymic reactions at high concentration of enzyme Ulf Ryde-Pettersson Al'delningen fi~r Biokemi, Kemicentrum, Lunds Unirersitet, Lund (Sweden) (Received 6 March lC)91) (Revised manuscript received 16 July 1991)
Key words: End-product inhibition: Metabolic oscillation; High enzyme concentration; Ribulose 1,5-bisphosphate carboxylase; Steady-state approximation
The transient-state kinetic consequences of the coupling of a single-enzyme reaction to other metabolic reactions have been examined in generalized terms. Analytical data are presented which specify under what conditions such coupling may lead to an oscillatory transient rate behaviour of a reaction system involving an enzyme operating by a Michaelian mechanism. The results indicate that the presence of enzyme in concentrations comparable to those of substrate and product may represent a hitherto unforeseen possible source of weakly damped, or even sustained, oscillations in metabolic networks. This observation has some important implications with regard to the occurrence of oscillations in biochemical reaction systems and to transient-state kinetic modelling of metabolic systems.
Introduction
A recent theoretical study showed that single-enzyme reactions may exhibit oscillations under standard conditions of transient-state kinetic experiments performed, for instance, by stopped-flow techniques [1]. Such oscillations were found to be strongly damped, however, and would hence be expected to readily escape experimental detection. This means that experimentally observed weakly damped or sustained metabolic oscillations are unlikely to be attributable to a single-enzyme source and hence can be anticipated to derive from the kinetic coupling of enzymic reactions. The present investigation provides a generalized treatment of the kinetic consequences of the coupling of an enzymatic reaction step to other metabolic reactions. A mathematical analysis is presented which establishes under what conditions such coupling may lead to the appearance of oscillatory transients in a reaction system involving an enzyme operating by a standard Michaelian mechanism. This analysis has been performed without introduction of the standard simplifying assumption that substrate and product are present in large excess to enzyme so that a steady-state
approximation may be applied to the enzymic reaction steps. The results provide evidence that the presence of enzyme at concentrations comparable to those of non-enzymic reactants may allow for the appearance of weakly damped, or even sustained, metabolic oscillations in systems that cannot exhibit any oscillations when the enzyme concentration is much lower than those of the non-enzymic reactants. Results
Basic kinetic properties o f the examined reaction system Consider the reaction system in Eqn. 1 t._~S + E ~ E S ~ E + P - ~ k_ l k -2
where S, P and E denote substrate, product and enzyme, respectively, v mrepresents the rate of substrate input into the system and r' 2 the rate of product output. The kinetics of this reaction sequence are governed by the differential equations
d[s] dt = t't + k_ t I E S ] - kllS]lE ]
(2)
d[ES] dt
Correspondence: U. Ryde-Pettersson, Avdelningen f6r Biokemi, Kemicentrum, Lunds Universitet, Box 124. S-22100 Lund, Sweden.
(1)
= k l [ S I [ E ] - k_ , [ E S ] - k2IES]+ k_z[Pl[E ]
(3)
diP] dt -- k2IES]- k - 2 [ P l l E ] - v2
(4)
119 and the mass-conservation constraint cr = IEI + IES]
¢5)
where c E is the total concentration of enzyme. The transient-sta~e kinetics of the reaction in Eqn. ! near a steady state may be described by application of a linear approximation [2] to Eqns. 2-4. This gives a linear system with the Jacobian matrix ( - kIll-I= + rill •~ IIE|~
kl[S[= + k - I
-(l~l[S]~+&-i+k2+k-2|Pl~)
- dz~
k2 + / , -z[P]=
d12 ~._,|E|= - I, -..['El= -
I d._.!
(6)
where [EL, [SL and [PL denote steady-state concentrations and where
ously shown [3,6], this system may give rise to damped oscillations if l,~ is feedback inhibited or r 2 is feed-forward activated, while sustained oscillations may arise if product activation or substrate inhibition is combined with, respectively, feedback inhibition or feed-forward activation.
Oscillatory properties of the exambled system When d~, =d2~ = 0, a similarity transformation of the Jacobian matrix in Eqn. 6 using the transforming matrix
i!
0
kdE]~ ~[S]~ + k _ ~
,3
(12)
,/k~(k, + k_ ~[P]=)
dl'j dji = d i S ] ;
0
] = 1,2
(7)
j = 1.2
(8)
dFj d,, = diel :
represent the derivatives at the steady state of the reaction fluxes ~,t and r z with respect to the substrate and product concentrations. AS discussed previously [3], the sign of these derivatives may be interpreted in terms of inhibition (dig < 0) or activation (dig > 0) of the corresponding reaction flux rj by S or P. Changes of concentrations of reactants R) in Eqn. 1 near the steady state are characterized by three exponential transients 3
[r,I=IR,L+ Ea,,~'"
(9)
where [RjL is the concentration of the reactant at steady state and where the rate parameters Jt~ represent the eigenvalues of the Jacobian matrix [2]. If substrate and product are present in large excess to enzyme (c e << [S], [P]), the theory of time hierarchy separation [4,51 can be applied such that the enzymic reaction steps in Eqn. 1 may be assumed to be at steady state: diE] dl
dIES] - = dl
0
(10)
Concentration changes in the examined reaction system will then occur over two widely separated timescales which may be studied separately; [E] and [ES] change rapidly, while IS] and [P] vary slowly in a manner that may be described by the reduced system:
yields a symmetric matrix which implies that the eigenvalues of the Jacobian matrix are real and correspond to non-oscillatory transients [7]. Hence it may be concluded that the reaction in Eqn. 1 cannot exhibit any oscillations in the absence of feedback or feed-forward effects of S or P on the input ( r i) or output (v,) rates. As soon as such effects are present (i.e. when either d~2 #: 0 or d2~ ~ 0), however, oscillations may arise in the examined reaction system. This means that the general reaction in Eqn. 1 may exhibit an oscillatory behaviour in a number of cases where the reduced system in Eqn. 11 cannot support any oscillations. The following treatment focuses on these cases, which represent previously not recognized potential sources of oscillations attributable to a failure of the steady-state approximation in Eqn. 10.
Feed-forward inhibition and feedback actiration When the steady-state approximation fails, feed-forward inh~ition (d21 < 0) may trigger oscillations. This is illustrated by the typical results in Fig. 1. Numerical analysis of a large number of systems involving a variety of realistic rate equations indicated that oscillations triggered by feed-forward inhibition are generally so strongly damped that they may be difficult to distinguish experimentally from non-oscillatory transients. Furthermore, oscillations are obtained only when c E is so high that either IS] or [P] are equal or less than [ES] or [E]; those in Fig. 1, for example, disappear when c E < 0.4. Similar results were obtained with feedback activation (dl2 > 0).
Feedback inhibition -~ s - ~ p L~
(tt)
where v 3 represents a steady-state rate-equation of standard Michaelis-Menten type. As has been previ-
In systems involving feedback inhibition (d~, < 0) damped oscillations may occur both when the steadystate approximation holds and when it fails. In the latter case, however, sustained oscillations of the limit
120 TABLE I
0.12 ~ / . . . . . . . . . . . ~ -
Validity of the steady-staw approximation defined by Eqn. 10 0.1 r
Non-real transient-state rate p a r a m e t e r s for the system in Eqn. I calculated from Eqns. 2 - 5 using t'j=33/(l+lP]/O.O031, t,z-0.5[P]/([P] + 0.0001), k i = k 2 = ! a n d k _ i = k _ 2 = 0 in arbitrary units with (case A) a n d without (case B) application of the steady-state approximation in Eqn. IlL i = ifL-T.
L
0.08 0.06
[
P
0.o4 )
CE
c E/[P]~
0.50 0.51 0.53 0.6 0.8 1.0
2.6 2.6 2.7 3.1 4.1 5.1
0.02 ~
oI 0
5
l0
15 time
20
25
30
Fig. 1. Oscillatory relaxation of the system in Eqn. 1 in a case involving feedback activation, showing time-course for r e a t t a i n m e n t of steady-state concentration o f [S] following an initial perturbation of [P] to 10% of its steady-state value as calculated from Eqns. 2 - 5 using vl=O.2[P]/([P]+O.l), V z = [ P ] / ( [ P ] + I ) , C E = I , k l = k 2 = i a n d k _ t = k _ 2 = 0 in arbitrary units. T h e transient rate p a r a m e t e r s are - 1.32_+0.542 ~ and - 0 . 1 7 4 .
cycle type may also be obtained; a typical example is given in Fig. 2. These oscillations were obtained with v 1 = V,,2/(K i + [P]), v 2 = Vm2[PI/([P] + KM). In this particular case, sustained oscillations are most readily obtained when k_~ ---k_ 2 = 0 and K M is small. In the limiting case when KM << [P], the Routh-Hurwitz criterion for instability [8] reads
km(l+kttl-Vm2)2+
Vmzz(Vm2 - 11
Vrn,
)<0
(13,
(in units of c E and k2), which shows that Vm2 < 1, i'ml + Vm2 < 1 and V,,t < V,~ are necessary criteria for sustained oscillations and that small k~ favours their appearance. Similar results were obtained with other 0.08
0.06
m" 0.04
!
Transient rate p a r a m e t e r s case A
case B
- 0.000658 + 0.00274i - 0.000759 + 0.0228i -0.0151 +0.0660i - 0 . 0 0 9 0 3 +0.205i -0.0570 +0.530i -0.126 + 0.785i
- 0.000658 + 0.00274i - 0.000754 + 0.0228i - 0 . 0 1 3 9 +0.0660i -0.00548 +0.204i -0.00801 +0.507/ +0.000261 +0.71 l i
realistic rate equations involving saturation kinetics for L"2 ,
Validity o f tile r,~:ady-state approximation The above :results show that the transient-state kinetic properties of the reaction system in Eqn. 1 may differ depending on whether or not the steady-state approximation expressed by Eqn. 10 holds. This means that erroneous analytical results may be obtained if the steady-state approximation is applied under conditions where it does not hold. This is illustrated by the numerical example considered in Table I, where the non-real transient rate parameters for a system involving feedback inhibition have been calculated as a function of CE, with and without introduction of the steady-state approximation. When c E < [P]~ ([SL > [PL in this example), application of the steady-state approximation provides an excellent description of the actual transient rate behaviour of the system in the slow time-scale of changes in concentrations of substrate and product. When CE/[P L increases above 2, the steady-state approximation breaks down and fails to account satisfactorily for the magnitude of the transient rate parameters. Application of the steady-state approximation under conditions where 5 < CE/[P L < 8 even leads to the qualitatively erroneous prediction that damped oscillations will occur in a system showing sustained oscillations.
0.02
0
0
20
40
60
80
time
Fig. 2. Sustained oscillations in the system in Eqn. 1 in a case involving feedback inhibition.The time-course for [S] followingan initial perturbation of [P] fo 10% of its steady-statevalue as calculated from Eqns. 2-5 using vI = 0.1/(1 +[Pl/0.01), v2 = 0.05[Pl/([P] +0.0001), c E = 1, k t = k 2 = 1 a n d k _ , = k _ 2 = 0 in arbitrary units. T h e transient rate parameters are + 0.02 + 1.07 ~ - T a n d - 2 . 0 9 .
Physiological example Ribulose 1,5-bisphosphate carboxylase (EC 4.1.1.39) is involved in kinetic interactions that are largely identical with those described by Eqn. 1 with feedback inhibition (S = ribulose 1,5-bisphosphate and P = 3phosphoglycerate). The enzyme is further known to have a high in vivo concentration (about 4 mM [9]) and therefore constitues a physiological example to which the present theory applies. Using experimentally supported [10-17] rate equations and rate constant esti-
121 T A B L E It
Transient-state kinetic behaviour o f thc ribulose 1,5-bisphosphate carboxylase system for different ralues o f the enzyme concentration Transient-sta~e rate p a r a m e t e r s for the system in Eqn. ! calculated from Eqns. 2 - 5 using
2500[RuSPItATP]
/'!
{[ATeI +0.05)([Ru5P] + 0.05)11 +
[PGA]
I and 2500 [ P G A ] [ A T P ] 0.5 0.I l, 2 = [PGA] [ATP] [PGA] [ATP l [ADP] 1+ + + - - - - + - 0.5 0.1 0.5 0.1 0.2 with [ R u 5 P ] = 0.01 m M , [ A T P ] = 0.1 m M and [ A D P ] = 0 A m M [ I 0 15]. k t a n d k 2 were chosen such that Vm~ = k2 cE = 1000/zmol h - l ( m g chlorophyll}- i and Ks1 = (k _ i + k z ) / k i = 0.05 m M for n b u l o s e 1,5-bisphosphate carboxylase with k _ l = I s - I [! 1,16A7]. T r a n s i e n t rate p a r a m e t e r s calculated u s i n g t h e steady-state approximation are -11800 and -671. i=~/-zT. cE
T r a n s i e n t rate p a r a m e t e r s
0.001 0.01 0.1 1.0 4.0
- | 300000 - 133000 - 2 1 100 - 15800 -- 15600
- 11800 - 1151"10 -7100
- 67! -672 -683 -740__+318i - 378__+ 148i
mates to express v~ and v 2, transient rate parameters aj of the ribulose 1,5-bisphosphate carboxylase reaction were calculated for different values of c E. The results in Table II show that the steady-state approximation is applicable only for c E < 0.1 mM. Damped oscillations appear when 0 . 6 < c w < 6 mM, i.e. at physiological values of the enzyme concentration. The system may exh~it also sustained oscillations, but only obtained for unphysiologically low values of K u in the expression for v 2. These observations illustrate the importance of verifying the applicability of the steadystate approximation when used for the modelling of metabolic systems and shows that the presence of high enzyme concentration may be an important factor triggering oscillations. Discussion The present investigation may be regarded as a generalization of the available transient-state kinetic theory for the appearance of oscillations in isolated single-enzyme reactions [1], in coupled two-reactant systems [3,6] and in end-product inhibition systems [4,18-21]. It expands upon the approaches previously chosen by considering the rate behaviour of the typical enzyme reaction in Eqn. 1 for arbitrary relative concentrations of enzyme and non-enzymic reactants, i.e.
without introducing the simplifying standard assumption that a steady-state approximation (Eqn. 10) can be applied to the concentrations of enzymic reactants. Such an extension of the available kinetic theory would seem well motivated. A recent literature survey of 40 metabolically central enzymes showed that a third of them act upon substrates being present at concentration of the same order of magnitude as the active-site concentration of the enzyme [22]. This means that analyses of the kinetics and dynamics of metabolic pathways cannot be generally based on the assumption that non-enzymic reactants are present in large excess to et~yme, such that the above steady-state approximation applies. The consequences of this for the mathematical modelling of metabolic pathways have been previously discussed, but mainly in terms of the binding to enzymes of stoichiometrically significant amounts of reactants in the steady-state reaction phase [17,22-24]. The present investigation draws attention to the effects of a high enzyme concentration on the transient-state phase of enzymic reactions. Data in Table I and 1I illustrate that inadequate application of the steady-state approximation does lead to incorrect analytical predictions. Furthermore, it follows that enzyme reactions may exhibit oscillations under hitherto unforeseen circumstances when the steady-state approximation does not apply, e.g., damped oscillations in reaction systems involving feedback activation or feedtorward inhibition (Fig. 1) and sustained oscillations in systems involving feedback inhibition (Fig. 2). From a mathematical point of view, oscillations represent a two-variable phenomenon; they may be generally described by models involving two independent variables. This has led to the widely held view that oscillations should originate basically from a specific two-reactant source. Theoretical analyses have been published [3,6] which characterizes the mechanistic cases in which the kinetic interplay of two reactants may give rise to metabolic oscillations. When examined in view of the latter results, the data in Fig. 1 and 2 provide the inference that the reaction in Eqn. 1 may exhibit oscillations which cannot be attributed to any two-reactant source but which must derive from the kinetic interplay of three independent reactants. Hence it may be concluded that the mechanistic origin of metabolic oscillations cannot be un:,mbiguously established merely be a systematic search for, and examination of, possible two-reactant sources. Feedback inhibition represents a prominent mechanism for the regulation of metabolic pathways and is known to be a possible source of metabolic oscillations [18]. Previous theoretical studies of linear reaction sequences involving end-product inhibition of the first reacton step [4,18-21] have led to the conclusion that such systems may exhibit sustained oscillations only under quite restrictive conditions, e.g., if the number of
122 reactants is high ( > 8) or if the feedback inhibition exhibits kinetic cooperativity with a Hill coefficient exceeding unity. Results now presented (Fig. 2) establish that non-cooperative feedback inhibition may lead to sust~'ned oscillation already in systems involving only thr¢~e independent variables, provided that the system includes an enzyme being present at so high a concentr~,tion that the steady-state approximation in Eqn. 10 does not hold. Considering the evidence indicating that physiological enzyme and substrate concentrations often are of the same order of magnitude [22,23-25], it may be concluded from the present results that the participation of such systems in feedbark inhibition may represent an important and hitherto unforeseen source of weakly damped or even sustained metabolic oscillations. Acknowledgement This investigation was supported by grants from the Swedish Natural Science Research Council. References I Ryde-Pettersson, U. (1989) Eur. J. Biochem. 186, 145-148. 2 Andronov, A.A., Vitt, A.A. and Khaikin, S.E. (1966) Theory of Oscillations, Pergamon Press, Oxford. 3 Ryde-Pettersson, U. (1990) Eur. J. Biochcm. 194, 431-436.
4 Heinrich, R.. Rapoport, S.M. and Rapoporl, T.A. (1977) Prog. Biophys. Molec. Biol. 32, 1-82. 5 KIonowski, W. (1983) Biophys. Chem. 18, 73-87. 6 Higgins, J. (1967) Ind. Eng. Chem. 59, 19-62. 7 Hearon, J.Z. (1953) Bull. Math. Biophys. 15, 121-141. 8 Hahn, W. (1967) Stability Of Motion, 19-22. 9 Jensen, R.G. and Bahr, J.T. (1977) Annu. Rev. Plant Physiol. 28, 379-400. 10 Bassham, J.A. and Krause, G.H. (1969) Biochim. Biophys. Acta 189, 207-221. 11 Walker, D.A. (1975) Encycl. Plant Physiol. (new Series) 3, 85-136. 12 Portis, A.R., Chon, CJ., Mosbach, A.M. and Held, H.W. (1977) Biochim. Biophys. Acta 461,313-325. 13 Wirtz, W., Stitt, M. and Heldt, H.W. (1980) Plant Physiol. 66, 187-193. 14 Gardemann, A., Stin, M. and Held, H.W. (1983) Biochim. Biophys. Acta 722, 51-60. 15 Leegood, R.C., Walker, D.A. and Foyer, C.H. (1985)Top. Photosynth. Res. 6, 190-258, 16 Robinson, S.P. and Walker, D.A. (1981) in Biochemistry of Plants, Voi. 8, pp. 193-236, Academic Press. 17 Laisk, A., Eichelmann, H., Oja, V., Eatherall, A. and Walker, D.A. (1989) Proc. R. Soc. Lond. B 237, 389-415. 18 Goodwin, B.C. (1963) Temporal organization in cells, Academic Press, London. 19 Walter, C.F. (1970) J. Theor. Biol. 27, 259-272. 20 Tyson, J.J. (1975) J. Math. Biol. 1, 311-315. 21 Palsson, B.O. (1984)J. Theor. Biol. 111, 273-302. 22 AIbe, K.R., Butler, M.H. and Wright, B.E. (1990) J. Theor. Biol. 143, 163-195. 23 Srere, P.A. (1967) Science 158, 936-937. 24 Sols, A. and Marco, R. (1970) Curt. Top. Cell. Regul. 2, 227-273. 25 Srivastava, D.K. and Bernhard S.A. (t980) Curr. Top. Cell. Regul. 28, 1-68.
Biochimica et Biophysica Acta, ! ! 19 (1992) ! ! 8- i 22 © 1992 Elsevier Science Publishers B.V. All rights reserved 0167-4838/92/$05.00
BBAPRO 34103
Oscillations in coupled enzymic reactions at high concentration of enzyme Ulf Ryde-Pettersson Al'delningen fi~r Biokemi, Kemicentrum, Lunds Unirersitet, Lund (Sweden) (Received 6 March lC)91) (Revised manuscript received 16 July 1991)
Key words: End-product inhibition: Metabolic oscillation; High enzyme concentration; Ribulose 1,5-bisphosphate carboxylase; Steady-state approximation
The transient-state kinetic consequences of the coupling of a single-enzyme reaction to other metabolic reactions have been examined in generalized terms. Analytical data are presented which specify under what conditions such coupling may lead to an oscillatory transient rate behaviour of a reaction system involving an enzyme operating by a Michaelian mechanism. The results indicate that the presence of enzyme in concentrations comparable to those of substrate and product may represent a hitherto unforeseen possible source of weakly damped, or even sustained, oscillations in metabolic networks. This observation has some important implications with regard to the occurrence of oscillations in biochemical reaction systems and to transient-state kinetic modelling of metabolic systems.
Introduction
A recent theoretical study showed that single-enzyme reactions may exhibit oscillations under standard conditions of transient-state kinetic experiments performed, for instance, by stopped-flow techniques [1]. Such oscillations were found to be strongly damped, however, and would hence be expected to readily escape experimental detection. This means that experimentally observed weakly damped or sustained metabolic oscillations are unlikely to be attributable to a single-enzyme source and hence can be anticipated to derive from the kinetic coupling of enzymic reactions. The present investigation provides a generalized treatment of the kinetic consequences of the coupling of an enzymatic reaction step to other metabolic reactions. A mathematical analysis is presented which establishes under what conditions such coupling may lead to the appearance of oscillatory transients in a reaction system involving an enzyme operating by a standard Michaelian mechanism. This analysis has been performed without introduction of the standard simplifying assumption that substrate and product are present in large excess to enzyme so that a steady-state
approximation may be applied to the enzymic reaction steps. The results provide evidence that the presence of enzyme at concentrations comparable to those of non-enzymic reactants may allow for the appearance of weakly damped, or even sustained, metabolic oscillations in systems that cannot exhibit any oscillations when the enzyme concentration is much lower than those of the non-enzymic reactants. Results
Basic kinetic properties o f the examined reaction system Consider the reaction system in Eqn. 1 t._~S + E ~ E S ~ E + P - ~ k_ l k -2
where S, P and E denote substrate, product and enzyme, respectively, v mrepresents the rate of substrate input into the system and r' 2 the rate of product output. The kinetics of this reaction sequence are governed by the differential equations
d[s] dt = t't + k_ t I E S ] - kllS]lE ]
(2)
d[ES] dt
Correspondence: U. Ryde-Pettersson, Avdelningen f6r Biokemi, Kemicentrum, Lunds Universitet, Box 124. S-22100 Lund, Sweden.
(1)
= k l [ S I [ E ] - k_ , [ E S ] - k2IES]+ k_z[Pl[E ]
(3)
diP] dt -- k2IES]- k - 2 [ P l l E ] - v2
(4)
119 and the mass-conservation constraint cr = IEI + IES]
¢5)
where c E is the total concentration of enzyme. The transient-sta~e kinetics of the reaction in Eqn. ! near a steady state may be described by application of a linear approximation [2] to Eqns. 2-4. This gives a linear system with the Jacobian matrix ( - kIll-I= + rill •~ IIE|~
kl[S[= + k - I
-(l~l[S]~+&-i+k2+k-2|Pl~)
- dz~
k2 + / , -z[P]=
d12 ~._,|E|= - I, -..['El= -
I d._.!
(6)
where [EL, [SL and [PL denote steady-state concentrations and where
ously shown [3,6], this system may give rise to damped oscillations if l,~ is feedback inhibited or r 2 is feed-forward activated, while sustained oscillations may arise if product activation or substrate inhibition is combined with, respectively, feedback inhibition or feed-forward activation.
Oscillatory properties of the exambled system When d~, =d2~ = 0, a similarity transformation of the Jacobian matrix in Eqn. 6 using the transforming matrix
i!
0
kdE]~ ~[S]~ + k _ ~
,3
(12)
,/k~(k, + k_ ~[P]=)
dl'j dji = d i S ] ;
0
] = 1,2
(7)
j = 1.2
(8)
dFj d,, = diel :
represent the derivatives at the steady state of the reaction fluxes ~,t and r z with respect to the substrate and product concentrations. AS discussed previously [3], the sign of these derivatives may be interpreted in terms of inhibition (dig < 0) or activation (dig > 0) of the corresponding reaction flux rj by S or P. Changes of concentrations of reactants R) in Eqn. 1 near the steady state are characterized by three exponential transients 3
[r,I=IR,L+ Ea,,~'"
(9)
where [RjL is the concentration of the reactant at steady state and where the rate parameters Jt~ represent the eigenvalues of the Jacobian matrix [2]. If substrate and product are present in large excess to enzyme (c e << [S], [P]), the theory of time hierarchy separation [4,51 can be applied such that the enzymic reaction steps in Eqn. 1 may be assumed to be at steady state: diE] dl
dIES] - = dl
0
(10)
Concentration changes in the examined reaction system will then occur over two widely separated timescales which may be studied separately; [E] and [ES] change rapidly, while IS] and [P] vary slowly in a manner that may be described by the reduced system:
yields a symmetric matrix which implies that the eigenvalues of the Jacobian matrix are real and correspond to non-oscillatory transients [7]. Hence it may be concluded that the reaction in Eqn. 1 cannot exhibit any oscillations in the absence of feedback or feed-forward effects of S or P on the input ( r i) or output (v,) rates. As soon as such effects are present (i.e. when either d~2 #: 0 or d2~ ~ 0), however, oscillations may arise in the examined reaction system. This means that the general reaction in Eqn. 1 may exhibit an oscillatory behaviour in a number of cases where the reduced system in Eqn. 11 cannot support any oscillations. The following treatment focuses on these cases, which represent previously not recognized potential sources of oscillations attributable to a failure of the steady-state approximation in Eqn. 10.
Feed-forward inhibition and feedback actiration When the steady-state approximation fails, feed-forward inh~ition (d21 < 0) may trigger oscillations. This is illustrated by the typical results in Fig. 1. Numerical analysis of a large number of systems involving a variety of realistic rate equations indicated that oscillations triggered by feed-forward inhibition are generally so strongly damped that they may be difficult to distinguish experimentally from non-oscillatory transients. Furthermore, oscillations are obtained only when c E is so high that either IS] or [P] are equal or less than [ES] or [E]; those in Fig. 1, for example, disappear when c E < 0.4. Similar results were obtained with feedback activation (dl2 > 0).
Feedback inhibition -~ s - ~ p L~
(tt)
where v 3 represents a steady-state rate-equation of standard Michaelis-Menten type. As has been previ-
In systems involving feedback inhibition (d~, < 0) damped oscillations may occur both when the steadystate approximation holds and when it fails. In the latter case, however, sustained oscillations of the limit
120 TABLE I
0.12 ~ / . . . . . . . . . . . ~ -
Validity of the steady-staw approximation defined by Eqn. 10 0.1 r
Non-real transient-state rate p a r a m e t e r s for the system in Eqn. I calculated from Eqns. 2 - 5 using t'j=33/(l+lP]/O.O031, t,z-0.5[P]/([P] + 0.0001), k i = k 2 = ! a n d k _ i = k _ 2 = 0 in arbitrary units with (case A) a n d without (case B) application of the steady-state approximation in Eqn. IlL i = ifL-T.
L
0.08 0.06
[
P
0.o4 )
CE
c E/[P]~
0.50 0.51 0.53 0.6 0.8 1.0
2.6 2.6 2.7 3.1 4.1 5.1
0.02 ~
oI 0
5
l0
15 time
20
25
30
Fig. 1. Oscillatory relaxation of the system in Eqn. 1 in a case involving feedback activation, showing time-course for r e a t t a i n m e n t of steady-state concentration o f [S] following an initial perturbation of [P] to 10% of its steady-state value as calculated from Eqns. 2 - 5 using vl=O.2[P]/([P]+O.l), V z = [ P ] / ( [ P ] + I ) , C E = I , k l = k 2 = i a n d k _ t = k _ 2 = 0 in arbitrary units. T h e transient rate p a r a m e t e r s are - 1.32_+0.542 ~ and - 0 . 1 7 4 .
cycle type may also be obtained; a typical example is given in Fig. 2. These oscillations were obtained with v 1 = V,,2/(K i + [P]), v 2 = Vm2[PI/([P] + KM). In this particular case, sustained oscillations are most readily obtained when k_~ ---k_ 2 = 0 and K M is small. In the limiting case when KM << [P], the Routh-Hurwitz criterion for instability [8] reads
km(l+kttl-Vm2)2+
Vmzz(Vm2 - 11
Vrn,
)<0
(13,
(in units of c E and k2), which shows that Vm2 < 1, i'ml + Vm2 < 1 and V,,t < V,~ are necessary criteria for sustained oscillations and that small k~ favours their appearance. Similar results were obtained with other 0.08
0.06
m" 0.04
!
Transient rate p a r a m e t e r s case A
case B
- 0.000658 + 0.00274i - 0.000759 + 0.0228i -0.0151 +0.0660i - 0 . 0 0 9 0 3 +0.205i -0.0570 +0.530i -0.126 + 0.785i
- 0.000658 + 0.00274i - 0.000754 + 0.0228i - 0 . 0 1 3 9 +0.0660i -0.00548 +0.204i -0.00801 +0.507/ +0.000261 +0.71 l i
realistic rate equations involving saturation kinetics for L"2 ,
Validity o f tile r,~:ady-state approximation The above :results show that the transient-state kinetic properties of the reaction system in Eqn. 1 may differ depending on whether or not the steady-state approximation expressed by Eqn. 10 holds. This means that erroneous analytical results may be obtained if the steady-state approximation is applied under conditions where it does not hold. This is illustrated by the numerical example considered in Table I, where the non-real transient rate parameters for a system involving feedback inhibition have been calculated as a function of CE, with and without introduction of the steady-state approximation. When c E < [P]~ ([SL > [PL in this example), application of the steady-state approximation provides an excellent description of the actual transient rate behaviour of the system in the slow time-scale of changes in concentrations of substrate and product. When CE/[P L increases above 2, the steady-state approximation breaks down and fails to account satisfactorily for the magnitude of the transient rate parameters. Application of the steady-state approximation under conditions where 5 < CE/[P L < 8 even leads to the qualitatively erroneous prediction that damped oscillations will occur in a system showing sustained oscillations.
0.02
0
0
20
40
60
80
time
Fig. 2. Sustained oscillations in the system in Eqn. 1 in a case involving feedback inhibition.The time-course for [S] followingan initial perturbation of [P] fo 10% of its steady-statevalue as calculated from Eqns. 2-5 using vI = 0.1/(1 +[Pl/0.01), v2 = 0.05[Pl/([P] +0.0001), c E = 1, k t = k 2 = 1 a n d k _ , = k _ 2 = 0 in arbitrary units. T h e transient rate parameters are + 0.02 + 1.07 ~ - T a n d - 2 . 0 9 .
Physiological example Ribulose 1,5-bisphosphate carboxylase (EC 4.1.1.39) is involved in kinetic interactions that are largely identical with those described by Eqn. 1 with feedback inhibition (S = ribulose 1,5-bisphosphate and P = 3phosphoglycerate). The enzyme is further known to have a high in vivo concentration (about 4 mM [9]) and therefore constitues a physiological example to which the present theory applies. Using experimentally supported [10-17] rate equations and rate constant esti-
121 T A B L E It
Transient-state kinetic behaviour o f thc ribulose 1,5-bisphosphate carboxylase system for different ralues o f the enzyme concentration Transient-sta~e rate p a r a m e t e r s for the system in Eqn. ! calculated from Eqns. 2 - 5 using
2500[RuSPItATP]
/'!
{[ATeI +0.05)([Ru5P] + 0.05)11 +
[PGA]
I and 2500 [ P G A ] [ A T P ] 0.5 0.I l, 2 = [PGA] [ATP] [PGA] [ATP l [ADP] 1+ + + - - - - + - 0.5 0.1 0.5 0.1 0.2 with [ R u 5 P ] = 0.01 m M , [ A T P ] = 0.1 m M and [ A D P ] = 0 A m M [ I 0 15]. k t a n d k 2 were chosen such that Vm~ = k2 cE = 1000/zmol h - l ( m g chlorophyll}- i and Ks1 = (k _ i + k z ) / k i = 0.05 m M for n b u l o s e 1,5-bisphosphate carboxylase with k _ l = I s - I [! 1,16A7]. T r a n s i e n t rate p a r a m e t e r s calculated u s i n g t h e steady-state approximation are -11800 and -671. i=~/-zT. cE
T r a n s i e n t rate p a r a m e t e r s
0.001 0.01 0.1 1.0 4.0
- | 300000 - 133000 - 2 1 100 - 15800 -- 15600
- 11800 - 1151"10 -7100
- 67! -672 -683 -740__+318i - 378__+ 148i
mates to express v~ and v 2, transient rate parameters aj of the ribulose 1,5-bisphosphate carboxylase reaction were calculated for different values of c E. The results in Table II show that the steady-state approximation is applicable only for c E < 0.1 mM. Damped oscillations appear when 0 . 6 < c w < 6 mM, i.e. at physiological values of the enzyme concentration. The system may exh~it also sustained oscillations, but only obtained for unphysiologically low values of K u in the expression for v 2. These observations illustrate the importance of verifying the applicability of the steadystate approximation when used for the modelling of metabolic systems and shows that the presence of high enzyme concentration may be an important factor triggering oscillations. Discussion The present investigation may be regarded as a generalization of the available transient-state kinetic theory for the appearance of oscillations in isolated single-enzyme reactions [1], in coupled two-reactant systems [3,6] and in end-product inhibition systems [4,18-21]. It expands upon the approaches previously chosen by considering the rate behaviour of the typical enzyme reaction in Eqn. 1 for arbitrary relative concentrations of enzyme and non-enzymic reactants, i.e.
without introducing the simplifying standard assumption that a steady-state approximation (Eqn. 10) can be applied to the concentrations of enzymic reactants. Such an extension of the available kinetic theory would seem well motivated. A recent literature survey of 40 metabolically central enzymes showed that a third of them act upon substrates being present at concentration of the same order of magnitude as the active-site concentration of the enzyme [22]. This means that analyses of the kinetics and dynamics of metabolic pathways cannot be generally based on the assumption that non-enzymic reactants are present in large excess to et~yme, such that the above steady-state approximation applies. The consequences of this for the mathematical modelling of metabolic pathways have been previously discussed, but mainly in terms of the binding to enzymes of stoichiometrically significant amounts of reactants in the steady-state reaction phase [17,22-24]. The present investigation draws attention to the effects of a high enzyme concentration on the transient-state phase of enzymic reactions. Data in Table I and 1I illustrate that inadequate application of the steady-state approximation does lead to incorrect analytical predictions. Furthermore, it follows that enzyme reactions may exhibit oscillations under hitherto unforeseen circumstances when the steady-state approximation does not apply, e.g., damped oscillations in reaction systems involving feedback activation or feedtorward inhibition (Fig. 1) and sustained oscillations in systems involving feedback inhibition (Fig. 2). From a mathematical point of view, oscillations represent a two-variable phenomenon; they may be generally described by models involving two independent variables. This has led to the widely held view that oscillations should originate basically from a specific two-reactant source. Theoretical analyses have been published [3,6] which characterizes the mechanistic cases in which the kinetic interplay of two reactants may give rise to metabolic oscillations. When examined in view of the latter results, the data in Fig. 1 and 2 provide the inference that the reaction in Eqn. 1 may exhibit oscillations which cannot be attributed to any two-reactant source but which must derive from the kinetic interplay of three independent reactants. Hence it may be concluded that the mechanistic origin of metabolic oscillations cannot be un:,mbiguously established merely be a systematic search for, and examination of, possible two-reactant sources. Feedback inhibition represents a prominent mechanism for the regulation of metabolic pathways and is known to be a possible source of metabolic oscillations [18]. Previous theoretical studies of linear reaction sequences involving end-product inhibition of the first reacton step [4,18-21] have led to the conclusion that such systems may exhibit sustained oscillations only under quite restrictive conditions, e.g., if the number of
122 reactants is high ( > 8) or if the feedback inhibition exhibits kinetic cooperativity with a Hill coefficient exceeding unity. Results now presented (Fig. 2) establish that non-cooperative feedback inhibition may lead to sust~'ned oscillation already in systems involving only thr¢~e independent variables, provided that the system includes an enzyme being present at so high a concentr~,tion that the steady-state approximation in Eqn. 10 does not hold. Considering the evidence indicating that physiological enzyme and substrate concentrations often are of the same order of magnitude [22,23-25], it may be concluded from the present results that the participation of such systems in feedbark inhibition may represent an important and hitherto unforeseen source of weakly damped or even sustained metabolic oscillations. Acknowledgement This investigation was supported by grants from the Swedish Natural Science Research Council. References I Ryde-Pettersson, U. (1989) Eur. J. Biochem. 186, 145-148. 2 Andronov, A.A., Vitt, A.A. and Khaikin, S.E. (1966) Theory of Oscillations, Pergamon Press, Oxford. 3 Ryde-Pettersson, U. (1990) Eur. J. Biochcm. 194, 431-436.
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