19174
J. Phys. Chem. 1996, 100, 19174-19181
Thresholds and Oscillations in Enzymatic Cascades† Albert Goldbeter* and Jean-Michel Guilmot Faculte´ des Sciences, UniVersite´ Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Brussels, Belgium ReceiVed: July 30, 1996X
We determine the conditions in which sustained oscillations develop in a model for a bicyclic enzyme cascade regulated by negative feedback. The model, based on a cascade of two phosphorylation-dephosphorylation cycles, was previously proposed (Goldbeter, A. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 9107) as a minimal cascade model for the mitotic oscillator driving the early cell division cycles in amphibian embryos. We analyze the role of thresholds in the mechanism of oscillatory behavior by constructing stability diagrams as a function of the main parameters of the model. The thresholds arise from the phenomenon of zero-order ultrasensitivity naturally associated with the kinetics of phosphorylation-dephosphorylation cycles (Goldbeter, A.; Koshland, D. E., Jr. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 6840). The analysis shows that if the existence of a threshold in each of the two cycles markedly favors the periodic operation of the cascade, a single threshold suffices for sustained oscillatory behavior. Oscillations may even arise in the absence of any threshold in a small region of parameter space, but their amplitude is very much reduced. The model provides an example of biochemical oscillator based on negative feedback in which nonlinear amplification, instead of being due to allosteric cooperativity, results from the ultrasensitivity that arises from the kinetics of phosphorylation-dephosphorylation cycles.
Introduction How sustained oscillations occur in biochemical systems has been investigated theoretically both in a general manner and on specific examples. Such oscillations, which have been observed in a number of experimental systems, often result from positive feedback, as exemplified by the cases of glycolytic oscillations in yeast and muscle and cyclic AMP oscillations in Dictyostelium amoebae.1-4 That negative feedback can also give rise to sustained biochemical oscillations was shown by a number of theoretical studies, the first of which pertained to the conditions in which sustained oscillations arise from the repression of gene expression.5,6 Such models have recently regained interest7,8 in regard to the generation of circadian rhythms in organisms such as Drosophila. The question of how oscillations in gene expression arise from negative feedback was soon extended to the question of how end-product inhibition of enzyme activity can produce sustained oscillatory behavior.9,10 The key result of these studies is that, in a chain of enzyme reactions in which the first enzyme of the sequence is inhibited in a cooperative manner by the product of the last enzyme, oscillations may occur when the number of enzymatic steps and the degree of cooperativity of inhibition are sufficiently large; the degree of cooperativity required for sustained oscillations diminishes as the length of the chain increases.9-13 In these studies, the cooperativity of inhibition originates from the allosteric nature of the regulated enzyme. In the past two decades it has become clear that regulation of enzyme activity through covalent modification, and in particular through phosphorylation-dephosphorylation, has become as important and widespread, if not more, than allosteric regulation.14 Such mode of regulation often occurs in cascades in which the product of one phosphorylation-dephosphorylation cycle acts as catalyst in a second cycle, and so on. Multicyclic * To whom correspondence should be addressed. Tel (32-2) 650 5772; Fax (32-2) 650 5767; E-mail
[email protected]. † This paper is dedicated to Prof. John Ross on the occasion of his 70th birthday. X Abstract published in AdVance ACS Abstracts, November 15, 1996.
S0022-3654(96)02298-8 CCC: $12.00
enzymatic cascades are characterized by an increased sensitivity to external signals: they can indeed lead to rate acceleration and to the amplification of small changes in effector concentrations.15-17 Not only can they generate a number of molecules of end product much larger than the magnitude of the initial signal, but they can also display a large change in response following a much smaller variation in stimulus.16,17 The latter phenomenon results from the existence of sharp thresholds associated with the kinetics of phosphorylationdephosphorylation cycles.18 Because these thresholds occur when the converting enzymes in the cascade (i.e., kinases and phosphatases in the case of phosphorylation-dephosphorylation cycles) operate in the domain of zero-order kinetics in which the converting enzymes are saturated by their substrates, such a phenomenon was called “zero-order ultrasensitivity”.18 The threshold-generating properties of cyclically organized enzymatic reactions18,19 were exploited by Ross and colleagues, who showed20 that these properties are analogous to logic functions that may be put to use in the control of metabolic pathways. A second line of research long pursued by Ross is the theoretical analysis of oscillations in chemical and biochemical systems.21,22 It seems fitting, therefore, in an article dedicated to him, to combine these two lines of research and to address the link between thresholds and oscillations in enzymatic cascades. We wish to investigate here the conditions in which sustained oscillations occur in enzymatic cascades regulated by negative feedback. The model considered23 was initially proposed as a simple description of the phosphorylation-dephosphorylation cascade underlying the periodic recurrence of mitosis in amphibian embryonic cells. More detailed models have since been analyzed for this phenomenon, which take into account the contribution of positive feedback to the mechanism of oscillations.24,25 Thus, the present paper should not be viewed so much as a theoretical study of the mitotic oscillator but rather as an investigation of the conditions in which phosphorylationdephosphorylation thresholds can contribute to oscillatory behavior in enzymatic cascades. © 1996 American Chemical Society
Thresholds and Oscillations in Enzymatic Cascades
Figure 1. Minimal model for oscillations in enzymatic cascades regulated by negative feedback. Although it is analyzed here in the general context of oscillations in enzymatic cascades, the model was initially proposed as a minimal model based on negative feedback for the mitotic oscillator in embryonic cells. In that particular context, an effector, C (cyclin), is synthesized at a constant rate (Vi) and triggers the transformation of inactive (M+) into active (M) cdc2 kinase, by enhancing the rate of phosphatase E1; kinase E2 reverts this modification. In the second cycle of the phosphorylation-dephosphorylation cascade, cdc2 kinase (identical to E3) elicits the transition from the inactive (X+) into the active (X) form of a protease that degrades cyclin; the activation of cyclin protease is reverted by phosphatase E4. Vi (i ) 1, ..., 4) denotes the effective, maximum rate of each of the four converter enzymes; Vd denotes the maximum rate of cyclin degradation by protease X. As shown in Figure 3, this minimal cascade model is capable of autonomous oscillatory behavior. More detailed models for the mitotic oscillator have been proposed, which take into account negative as well as positive feedback (see text).
The analysis indicates that thresholds in each phosphorylation-dephosphorylation cycle of the cascade favor the occurrence of oscillations. Although a single threshold in any of the cycles is needed to produce periodic behavior of significant amplitude, oscillations of minute amplitude can nevertheless occur in a tiny domain of parameter space in the absence of any threshold. The model investigated provides an example of a biochemical oscillator based on negative feedback in which oscillations do not rely on the cooperativity of allosteric proteins.26 In generating the thresholds that favor periodic behavior, the allosteric cooperativity traditionally considered5-13 in models for biochemical oscillations based on negative feedback is indeed replaced by the ultrasensitivity arising from phosphorylation-dephosphorylation kinetics. Minimal Cascade Model for Sustained Oscillations The model considered23 is that of a bicyclic cascade driven by the continuous synthesis of an effector protein. Although it is analyzed here in a more general context, we shall keep the notations and names of the variables used in its application to the case of the mitotic oscillator in embryonic cells. There a phosphorylation-dephosphorylation cascade leads to the periodic activation of an enzyme called cdc2 kinase, which triggers the onset of mitosis.27 In the first cycle of the cascade model (see Figure 1), the effector protein, called cyclin, promotes the activation of cdc2 kinase, through dephosphorylation by the phosphatase cdc25,28 while in the second cycle cdc2 kinase activates a cyclin protease, X, through phosphorylation.29 The fact that cyclin promotes the activation of cdc2 kinase and that the latter enzyme promotes the degradation of cyclin by a protease had suggested that such a negative feedback loop may give rise to sustained oscillatory behavior.29-31 This conjecture was tested by the study of the theoretical model schematized in Figure 1, which is based on a minimal cascade containing two phosphorylation-dephosphorylation cycles.23
J. Phys. Chem., Vol. 100, No. 49, 1996 19175 In regard to the mitotic oscillator in embryonic cells, the simple model of Figure 1 does not take into account the activation of the phosphatase cdc25 through phosphorylation by cdc2 kinase, which leads to the indirect self-activation of the latter enzyme.32 Also disregarded is the regulation by reversible phosphorylation of the kinase wee1 which reverses the action of the phosphatase cdc25.33 Moreover, there is evidence that cyclin degradation, which involves the ubiquitin pathway,34 itself occurs in a multicyclic phosphorylation cascade, the first step of which would be controlled by cdc2 kinase.29 Adding positive feedback or phosphorylation-dephosphorylation cycles to the minimal cascade model favors the occurrence of sustained oscillations.35,36 The present study focuses on the conditions in which negative feedback and zeroorder ultrasensitivity thresholds can produce sustained oscillatory behavior in the minimal, bicyclic system which can be viewed as a core mechanism for oscillations in enzymatic cascades. The dynamics of the bicyclic enzyme cascade is governed by the following system of three kinetic equations:23
dC C ) Vi - VdX - kdC dt Kd + C
(1a)
1-M M dM ) V1 - V2 dt K2 + M K1 + (1 - M)
(1b)
1-X X dX - V4 ) V3 dt K4 + X K3 + (1 - X)
(1c)
with
V1 )
C V , V3 ) MVM3 Kc + C M1
(2a,b)
In the above equations, C denotes the effector (cyclin) concentration while M and X represent the fraction of active cdc2 kinase and the fraction of active cyclin protease; 1 - M thus represents the fraction of inactive (i.e., phosphorylated) cdc2 kinase, while 1 - X represents the fraction of inactive (i.e., dephosphorylated) cyclin protease. Parameters Vi and Vd denote respectively the constant rate of cyclin synthesis and the maximum rate of cyclin degradation by protease X reached for X ) 1; Kd and Kc denote the Michaelis constants for cyclin degradation and for cyclin activation of the phosphatase acting on the phosphorylated form of cdc2 kinase; kd represents an apparent first-order rate constant related to nonspecific degradation of cyclin. (This reaction is introduced only to prevent the boundless increase of cyclin in conditions where the specific protease is inhibited.) The remaining parameters Vi and Ki (i ) 1, ..., 4) denote the effective maximum rate and the Michaelis constant for each of the enzymes Ei (i ) 1, ..., 4) involved in the two phosphorylation-dephosphorylation cycles, namely, on one hand, the phosphatase E1 and the kinase E2 acting on the cdc2 molecule and, on the other hand, the kinase cdc2 (E3) and the phosphatase E4 acting on the cyclin protease (see Figure 1). For each converter enzyme, parameters Vi and Ki have been normalized by dividing them by the total amount of target protein, i.e., MT (total amount of cdc2 kinase) for E1 and E2 and XT (total amount of cyclin protease) for E3 and E4; both MT and XT are considered as constant. The expressions for the effective, maximum rates V1 and V3 are given by eqs 2. Equation 2a reflects the assumption that cyclin activates phosphatase E1 in a Michaelian manner; VM1 is the maximum rate of that enzyme at saturating cyclin levels. On the other hand, eq 2b expresses the proportionality of the
19176 J. Phys. Chem., Vol. 100, No. 49, 1996
Goldbeter and Guilmot
Figure 3. Sustained oscillations in the minimal cascade model of Figure 1. The time evolution of the cyclin concentration (C), the fraction of active cdc2 kinase (M), and the fraction of active cyclin protease (X) is obtained by numerical integration of eqs 1 in conditions where a threshold exists in the two cycles of the cascade, namely, for the parameter values corresponding to curves a in Figures 2A,B, with Vi ) 0.05 µM min-1, Vd ) 0.25 µM min-1, Kd ) 0.02 µM, and kd ) 0.01 min-1. Initial conditions are C ) 0.01 µM and M ) X ) 0.01.
Figure 2. Dependence of the fraction of active cdc2 kinase (M) on cyclin (part A), and of the fraction of active cyclin protease (X) on M (part B). The curves in (A) are generated by means of an equation17,18,35 yielding the steady-state value of M as a function of the ratio of maximum modification rates V1/V2, which is itself a function of cyclin concentration (see eq 2a); the curves in part B are generated with a similar equation yielding the steady-state value of X as a function of the ratio V3/V4, which is proportional to M (see eq 2b). C* and M* refer to the thresholds apparent in curves a of parts A and B, respectively; the location of these thresholds is given by expressions 3b and 4b. The curves are established for the following parameter values (in min-1): VM1 ) 3, V2 ) 1.5, VM3 ) 1, V4 ) 0.5; moreover, Kc ) 0.5 µM. In parts A and B, curves a are obtained for Ki ) 0.01 (i ) 1, ..., 4) while curves b are obtained for Ki ) 100.
effective maximum rate of cdc2 kinase to the fraction M of active enzyme; VM3 is the maximum velocity of the kinase reached for M ) 1. All nonlinearities in the model are of the Michaelian type, so that no allosteric cooperativity is assumed, neither in the synthesis or proteolysis of cyclin nor in any of the reactions of phosphorylation or dephosphorylation. Zero-Order Ultrasensitivity Thresholds The analysis of the minimal cascade model shows that sustained oscillations in cdc2 kinase activity readily arise when the dependence at steady state of active cdc2 kinase on cyclin and of active cyclin protease on cdc2 kinase both exhibit a sharp threshold. Such thresholds originate naturally in the model from phosphorylation-dephosphorylation kinetics as a result of “zeroorder ultrasensitivity” when the kinase and/or phosphatase become progressively saturated by their respective substrates.17,18 To address the role of thresholds in the mechanism of oscillations, it is necessary to first characterize the dependence at steady state of the fraction of active cdc2 kinase on cyclin and of the fraction of active cyclin protease on active cdc2 kinase. Shown in Figure 2A is the steady-state dependence of M on C, for different values of the reduced Michaelis constants K1 and K2 of the phosphatase E1 and kinase E2 that act on cdc2 kinase. When these constants are much smaller than unity (see curve a established for K1 ) K2 ) 0.01), the activation curve
for cdc2 kinase is characterized by a sharp threshold.18 In contrast, the activation curve possesses a quasi-Michaelian form at values of the reduced Michaelis constants much higher than unity (see curve b established for K1 ) K2 ) 100). A similar difference exists in the steepness of the activation of cyclin protease by cdc2 kinase (Figure 2B), depending on the values of the reduced Michaelis constants of enzymes E3 and E4. Thus, the sharp threshold visible when K3 ) K4 ) 0.01 (curve a) disappears when K3 ) K4 ) 100 (curve b). The precise location of the thresholds in curves a of Figure 2A,B can be obtained as follows (see ref 35 for further details). In the first cycle, the critical value of the ratio V1/V2 corresponding to the half-maximum fraction of phosphorylated cdc2 kinase, i.e., M ) 0.5, is given by eq 3a, while the corresponding value of the threshold C* is obtained from eq 3b:
( ) ( )( ) ( )/[ V1 * VM1 ) V2 V2
C* ) Kc
1 + 2K1 C* ) Kc + C* 1 + 2K2
(3a)
]
(3b)
1 + 2K1 1 + 2K2
VM1 1 + 2K1 V2 1 + 2K2
Similarly, for the second cycle, the threshold values of V3/V4 and M in which X ) 0.5 are given by expressions 4a and 4b, respectively:
() (
V3 * 1 + 2K3 ) V4 1 + 2K4
M* )
)/( )
1 + 2K3 1 + 2K4
VM3 V4
(4a)
(4b)
Role of Phosphorylation-Dephosphorylation Thresholds in the Mechanism of Oscillations The importance of thresholds for oscillatory behavior is readily illustrated by the fact that sustained oscillations occur in the conditions of curves a in Figure 2, as demonstrated in Figure 3, whereas such oscillations disappear when the thresholds are suppressed, in the conditions of curves b. The periodic behavior shown in Figure 3 corresponds to sustained oscillations of the limit cycle type.23
Thresholds and Oscillations in Enzymatic Cascades To better grasp the role of thresholds, it is useful to recall the succession of events in the course of one period of oscillations in the model of Figure 1.23 When starting from a low level, cyclin accumulates at a constant rate until the threshold value C* is reached (see Figure 3). Then, as C increases beyond C*, the fraction of active cdc2 kinase abruptly increases according to the activation curve a in Figure 2A. If M goes then beyond the value M* characterizing the threshold of the activation curve a for cyclin protease in Figure 2B, X will abruptly increase. As soon as the rate of specific degradation of cyclin by protease X exceeds the rate of cyclin synthesis, Vi, C will begin to decrease. When C falls below the threshold value C*, M will abruptly diminish; then, as M goes below M*, X will also decrease. A new cycle of the oscillations begins as soon as C resumes its accumulation, once the rate of cyclin synthesis is again larger than the rate of its specific degradation by X. Thus, periodic behavior originates from the negative feedback loop, the phosphorylation-dephosphorylation thresholds, and the associated time delays which result from that the passage through the thresholds is followed only after a lag by a significant effect on the level of the three variables in the cascade. To determine in a detailed, systematic manner the role played by phosphorylation-dephosphorylation thresholds in the occurrence of oscillations in the enzymatic cascade, it is useful to perform a linear stability analysis as a function of the parameters which govern both the steepness of the thresholds and their location in Figure 2A,B. Solving eqs 1 at steady state and expressing C and M as a function of X yields a ninth-degree equation for X. The coefficients of that polynomial equation were obtained by means of the MAPLE symbolic computation program. Solving this ninth-degree equation always yielded a single physically acceptable solution. The stable or unstable nature of this unique steady state was determined by solving the third-degree characteristic equation obtained from linear stability analysis of eqs 1. This procedure allowed us to construct stability diagrams as a function of the main parameters of the model. The role of thresholds is best illustrated by the stability diagram established as a function of the Michaelis constants of the converter enzymes in the two cycles of the cascade. Shown in Figure 4A is such a diagram established as a function of the pairs of reduced Michaelis constants K1 ) K2 versus K3 ) K4. Corroborating the mechanism outlined above, the diagram confirms that oscillations preferentially occur in a domain where the reduced Michaelis constants are much smaller than unity.23 As shown in Figure 2, such conditions indeed favor the occurrence of sharp thresholds in the two cycles of the cascade. The diagram nevertheless indicates that sustained oscillations can occur when only one of the two cycles presents a sharp threshold. The region of instability indeed extends upward along the vertical axis at sufficiently low values of K1 ) K2, in which case a threshold is observed only in the first cycle, and to the right along the horizontal axis, at low values of K3 ) K4, when a threshold occurs only in the second cycle. The effect of unequal Michaelis constants for the converter enzymes in each cycle is illustrated in Figure 4B. That the domain of oscillations is enlarged in these conditions stems from the fact that the threshold values (V1/V2)* and (V3/V4)* in the two cycles depend on these Michaelis constants according to eqs 3a and 4a. Taking K2 ) 10K1 and K3 ) 10K4 as in Figure 4B results in favoring a large rise in M following a change in C, since the threshold for the activation of cdc2 kinase is lowered while the threshold for the activation of cyclin protease
J. Phys. Chem., Vol. 100, No. 49, 1996 19177
Figure 4. Stability diagram showing the behavior of the minimal cascade model as a function of the reduced Michaelis constants of the converter enzymes in the two phosphorylation-dephosphorylation cycles. The results are obtained by means of linear stability analysis of the steady-state solution admitted by eqs 1 (see text). Shown in each diagram are the regions where the system evolves toward a stable steady state or to sustained oscillations. Part A is established for equal values of the Michaelis constants in each cycle (K1 ) K2 and K3 ) K4), while in part B, K2 ) 10 K1 and K3 ) 10 K4. Parameter values are those of Figure 3, with Vi ) 0.025 µM min-1. Thresholds in the first and second cycles of the cascade are respectively associated with values of K1, K2 and K3, K4 smaller than unity (see Figure 2).
increases. Such conditions appear to favor the occurrence of oscillatory behavior. If the cascade can oscillate in the presence of a single threshold, the form of the oscillations markedly changes depending on the cycle in which the threshold occurs. When the threshold is in the second cycle only, the variation in cdc2 kinase, M, remains reduced (Figure 5A). In contrast, the oscillations in M possess a much larger amplitude when the threshold is in the first cycle which governs the M vs C dependence (Figure 5B); then the variation in X remains small. The fact that X varies only slightly but keeps a significant level throughout the oscillations results in a lengthening of the period as cyclin takes more time to accumulate up to the threshold value C*. A comparison of Figure 5A,B with Figure 3 shows that the amplitude of oscillations in M and X is larger when the two cycles of the cascade possess a sharp threshold. The period of the oscillations in that case is also shorter, all other parameters remaining fixed. Effect of Maximum Rates of Phosphorylation-Dephosphorylation Besides the Michaelis constants, the other main parameters affecting the dynamics of the cascade are the maximum rates of the converter enzymes E1, E2 and E3, E4. These parameters govern the location of the thresholds, as shown by expressions
19178 J. Phys. Chem., Vol. 100, No. 49, 1996
Figure 5. Sustained oscillations in the minimal cascade model, in conditions where a single threshold exists in the second (part A) or first (part B) cycle of the cascade. Figure 3, in contrast, was established for the case where a threshold occurs in both cycles. Parameter values are those of Figure 4. Initial conditions are C (in µM) ) M ) X ) 0.5 in (A) and 0.2 in (B). Moreover, in part A, K1 ) K2 ) 100 and K3 ) K4 ) 0.001, while in part B, K1 ) K2 ) 0.001 and K3 ) K4 ) 100.
3b and 4b for C* and M*. Shown in panels A and B of Figure 6 are the stability diagrams established as a function of VM1 or V2, on one hand, and K1 ) K2 on the other. For these diagrams, K3 ) K4 are fixed at the value of 5 × 10-3 so that a threshold always exists in the second cycle of the cascade. The results confirm the conclusions reached in Figure 4A as to the possibility of oscillations at large values of K1 ) K2 such that no threshold exists in the first cycle. Then, however, the range of VM1 or V2 values yielding oscillations is reduced as compared to the situation where thresholds occur in both cycles, when K1 and K2 are also much smaller than unity. When taking K1 ) K2 ) 5 × 10-3, that is, when a sharp threshold exists in the first cycle, diagrams similar to those of Figure 6A,B can be established when VM3 is varied as a function of K3 ) K4, at a fixed value of V4. Two separated regions of instability in the VM3-K3 ()K4) parameter plane can sometimes be observed, as shown by the diagrams of Figure 7 established for three increasing values of V4. The reasons for the breaking up of the instability domain into two regions are unclear; the tendency to such break is already apparent in the diagram of Figure 6A. Oscillations in the Absence of Thresholds In relation with parameter kd, which measures the nonspecific degradation of cyclin, the stability analysis of the model has revealed a somewhat surprising possibility, illustrated in Figure 8. Shown in part A is the stability diagram established as a function of the maximum rate VM1 of the phosphatase E1 versus the reduced Michaelis constants of the four converter enzymes,
Goldbeter and Guilmot
Figure 6. Stability diagrams showing the behavior of the minimal cascade as a function of the reduced Michaelis constants of the converter enzymes in the first phosphorylation-dephosphorylation cycle and of the maximum rate of enzymes E1 (part A) and E2 (part B). The diagrams in parts A and B are established for K3 ) K4 ) 5 × 10-3; other parameter values are those of Figure 4.
all taken as equal. The main difference with Figure 6A is that, there, a threshold always exists in the second phosphorylationdephosphorylation cycle, as K3 and K4 are fixed at the value 5 × 10-3. In the diagram of Figure 8A, the region of instability is smaller as no threshold exists at the larger values of the four Michaelis constants. Whereas the data in Figure 8A were obtained for kd ) 10-2 min-1 (a value used in all our previous simulations), the corresponding diagram in Figure 8B was established for the value kd ) 10-4 min-1. This diagram looks very similar to that of panel A, except for an additional, minute region of instability which appears at low values of the rate VM1 when the reduced Michaelis constants become extremely large, i.e., in the absence of any phosphorylation-dephosphorylation threshold. (This additional region might correspond to the second instability domain seen in the diagram of Figure 7, although there a threshold existed in the first cycle of the cascade.) The dynamic behavior of the system in that small region of instability was checked by numerical integration of the kinetic equations. These simulations indicate (Figure 8C) that if the oscillations possess a significant amplitude in C, they are accompanied by only minute oscillations in the other variables. The period of these oscillations is much larger than in previous figures, because taking such huge values for the Michaelis constants amounts to slowing down the dynamics of the cascade by dividing by these constants the maximum rates of the
Thresholds and Oscillations in Enzymatic Cascades
J. Phys. Chem., Vol. 100, No. 49, 1996 19179
Figure 7. Stability diagrams showing the behavior of the cascade model as a function of the reduced Michaelis constants of the converter enzymes in the second phosphorylation-dephosphorylation cycle and of the maximum rate of kinase E3, at three increasing values of the maximum rate of the phosphatase E4 that inactivates the cyclin protease. As the value of V4 increases, the single domain of oscillations is seen to break up into two separate oscillatory domains. Parameter values are the same as in Figure 4, with K1 ) K2 ) 5 × 10-3.
converter enzymes. Since the four enzymes operate in the domain of first-order kinetics, it is possible to approximate eqs 1 by polynomial equations; their integration yields results similar to those shown in Figure 8C. Discussion We have analyzed the dynamic properties of an enzymatic cascade model regulated by negative feedback and focused on the role played by phosphorylation-dephosphorylation thresholds in the mechanism of sustained oscillatory behavior. Key to the oscillatory mechanism is the observation that any cycle of phosphorylation-dephosphorylation may behave as a switch when the kinase and/or phosphatase become saturated by their respective substrates.17,18,37 This phenomenon of zero-order ultrasensitivity18 is associated with a sharp threshold in the curve yielding the steady-state level of phosphorylated target protein as a function of the ratio of maximum rates of the kinase and phosphatase active in a given cycle. Experimental evidence for such ultrasensitivity has been obtained for isocitrate dehydrogenase38 and glycogen phosphorylase.39 In the bicyclic cascade model considered here (Figure 1), two phosphorylation-dephosphorylation thresholds can arise. Previous analysis of the model already pointed to the role of thresholds and associated time delays in the mechanism of oscillations.23,35 That analysis solely rested on the steady-state activation curves for the target proteins in each cycle and on the numerical integration of the three kinetic equations 1. Here we have extended these results and given them firmer founda-
Figure 8. Oscillations in the absence of threshold in any of the two phosphorylation-dephosphorylation cycles of the cascade. The stability diagram established as a function of the maximum rate of phosphatase E1 and of the reduced Michaelis constants of the four converter enzymes (Ki, i ) 1, ..., 4) generally shows a single domain of oscillations at values of Ki smaller than unity, which correspond to the existence of thresholds in the two cycles of the cascade (part A). In the absence or at very low rates of nonspecific cyclin degradation, however, a small domain of instability appears at small values of VM1 and values of Ki much larger than unity, i.e., in the absence of any threshold in the two cycles of the cascade (see part B established for kd ) 10-4 min-1 instead of the value 10-2 min-1 used in part A). The oscillations which occur in that domain (part C) are characterized by a very low amplitude in both M and X; the curves are obtained by numerical integration of eqs 1 for Ki ) 300, VM1 ) 0.2 min-1, and kd ) 10-4 min-1. Other parameter values are as in Figure 4.
tions by carrying out a linear stability analysis of the minimal cascade model. Such an analysis has allowed us to construct
19180 J. Phys. Chem., Vol. 100, No. 49, 1996 stability diagrams as a function of the main parameters, namely, the Michaelis constants of the four converter enzymes of the cascade and their maximum rates. These diagrams give the domains in parameter space where sustained oscillations occur around a nonequilibrium unstable steady state. The main result of this analysis is that if thresholds in the two cycles clearly favor oscillations, a single threshold suffices for periodic behavior (Figure 4). The oscillations, however, differ by their amplitude, depending on whether the threshold is in the first or second cycle. Thresholds in the two cycles of the cascade definitely favor large-amplitude oscillations in all three variables of the model, as shown by a comparison of Figure 3 with Figure 5A,B. Nevertheless, periodic behavior cannot absolutely be ruled out in the absence of any threshold. Indeed, when the reduced Michaelis constants of the converter enzymes are all extremely large, a tiny domain of instability can sometimes be found (Figure 7B). Such oscillations, however, would not possess much physiological significance as the amplitude of oscillations in M and X is then markedly reduced (Figure 8). If the existence of at least one threshold is needed, it does not ensure per se the occurrence of periodic behavior of significant amplitude. The relative maximum rates of phosphorylation and dephosphorylation in each cycle are crucial in transforming the necessary condition of the threshold existence into a sufficient one for sustained oscillations. Stability diagrams clearly indicate that, for oscillations to occur, the system must be capable of going back and forth through each threshold in the course of one cycle of periodic behavior. For example, if the ratio VM1/V2 is too low, the system is unable to pass above the threshold in the first cycle; it will then be blocked in a stable steady state characterized by high levels of C and low levels of M and X. If, on the contrary, the ratio VM1/V2 is too large, the system will not be capable of falling below the threshold and will be trapped in a stable steady state characterized by high levels of M and X and low levels of C. Similar remarks hold with respect to the threshold in the second cycle. In particular, a stable steady state in which M and C are high and X is low will be established when the ratio VM3/V4 is too small; the system will then be unable to pass above the second threshold even if it is capable of passing through the first one. The present results have been obtained in a minimal cascade model based on negative feedback. In the case of the mitotic oscillator, positive feedback due to indirect activation of phosphatase E1 by cdc2 kinase32 is known to also play a role in the onset of oscillations.24,25,35,40 Such positive feedback, as well as additional phosphorylation-dephosphorylation cycles on the path leading to activation of the cyclin protease, tend to increase the domain of oscillatory behavior and thereby make the requirement for steep thresholds less stringent.35,36 The present results suggest ways to arrest the oscillatory behavior of the cascade. Thus, oscillations can be suppressed if the Michaelis constant of one or several of the converter enzymes is sufficiently increased so that the threshold in the corresponding cycle is shifted, becomes less steep, or disappears. Such an effect could be brought about by a competitive inhibitor of one of the kinases or phosphatases. The stability diagrams indicate that each of the maximum rates of phosphorylation or dephosphorylation possesses a window in which oscillatory behavior occurs. Oscillations in the cascade can therefore be suppressed (or induced) by either increasing or decreasing the value of either one of the four converter enzyme activities. This result bears on the arrest of the mitotic oscillator in relation to the control of cell proliferation.41 Stability diagrams established with respect to Vi and Vd show that yet another way of arresting
Goldbeter and Guilmot the oscillations is to alter the balance between synthesis and degradation of the effector that drives the enzymatic cascade. Conclusion The minimal cascade considered here belongs to the class of models for biochemical oscillations based on negative feedback. Such models have previously been studied in relation to the possible occurrence of oscillations in biochemical systems regulated at the genetic level by repression5-8 or in metabolic chains controlled by end-product inhibition of enzyme activity.9-13 These studies showed that oscillations generally require cooperativity of the negative feedback. The degree of cooperativity needed for sustained oscillations was found to decrease as the number of enzymatic steps in the metabolic chain increases.9-13 While most of these studies relied on the assumption of a linear sink of end product, it was later found that sustained oscillations can be obtained in the absence of cooperativity when the sink is of a saturable, e.g., Michaelis-Menten nature;42,43 a similar result was also obtained in a model based on positive feedback.44 The results obtained here in Figure 8 can be related to these findings. As in the present model, however, it remains clear that the cooperativity of the feedback process favors oscillatory behavior. In all the previous studies of biochemical oscillations due to negative (or positive) feedback, the source of cooperativity was provided by allosteric regulation.26 Here, in contrast, the negative feedback does not rely on allosteric cooperativity to produced oscillatory behavior; the steep thresholds instead originate from zero-order ultrasensitivity.17,18 The model for the enzymatic cascade thus provides an example of a biochemical oscillator based on negative feedback in which the nonlinear amplification, rather than being due to allosteric interactions, arises from the phenomenon of zero-order ultrasensitivity associated with the kinetics of phosphorylation-dephosphorylation cycles. Acknowledgment. This work was supported by Grant 3.4588.93 from the Fonds de la Recherche Scientifique Me´dicale (FRSM, Belgium) and by the Programme “Actions de Recherche Concerte´e” (Convention 94/99-180) launched by the Division of Science and Higher Education, French Community of Belgium. References and Notes (1) Hess, B.; Boiteux, A. Annu. ReV. Biochem. 1971, 40, 237. (2) Gerisch, G.; Wick, U. Biochem. Biophys. Res. Commun. 1975, 65, 364. (3) Goldbeter, A.; Caplan, S. R. Annu. ReV. Biophys. Bioeng. 1976, 5, 449. (4) Goldbeter, A. Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic BehaViour; Cambridge University Press: Cambridge, UK, 1996. (5) Goodwin, B. C. AdV. Enzyme Regul. 1965, 3, 425. (6) Griffith, J. S. J. Theor. Biol. 1968, 20, 202. (7) Goldbeter, A. Proc. R. Soc. London B 1995, 261, 319. (8) Ruoff, P.; Rensing, L. J. Theor. Biol. 1996, 179, 275. (9) Morales, M.; McKay, D. Biophys. J. 1967, 7, 621. (10) Walter, C. J. Theor. Biol. 1970, 27, 259. (11) Hunding, A. Biophys. Struct. Mech. 1974, 1, 47. (12) Rapp, P. E. Math. Biosci. 1975, 25, 165. (13) Tyson, J. J.; Othmer, H. G. In Progress in Theoretical Biology; Snell, F., Rosen, R., Eds.; Academic Press: New York, 1978; Vol. 5, pp 2-62. (14) Boyer, P. D., Krebs, E. G., Eds. The Enzymes, 3rd ed.; Academic Press: Orlando, FL, 1986; Vol. XVII. (15) Stadtman, E. R.; Chock, P. B. Curr. Top. Cell. Regul. 1978, 13, 53. (16) Koshland, D. E., Jr.; Goldbeter, A.; Stock, J. B. Science 1982, 217, 1982. (17) Goldbeter, A.; Koshland, D. E., Jr. Q. ReV. Biophys. 1982, 15, 555.
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