BioSystems, 24(1990)85- 90 Elsevier Scientific Publishers Ireland Ltd.
85
MPF and cyclin: modelling of the cell cycle minimum oscillator Claude
Hyver a and
Herv~
Le
Guyader b
~Service de Biophysique, De'partement de Biologie, Centre d'Etudes Nucl~aires de Saclay, 91191 Gif-sur-Yvette (France) and bUniversit$ de Paris-Sud, Laboratoire de Biologie Cellulaire 4, Batiment, 444-91405 Orsay Cedex IFrance)
(Received January 22nd, 1990} (Revision received May 2nd, 1990) The cell cycle appears to be controlled by the interplay between two protein complexes, MPF and cyclin. Their interactions play an essential role in the structure of the oscillator governing the cell cycle. There seems to be no general agreemeat on this latter crucial question. Two different mechanisms are proposed: (i) cyclin and p34 kinase combine to form an oligomer with MPF activity; (ii) cyclin enzymatically activates the passage from inactive pre-MPF to active MPF, with the postulate that MPF initiates cyclin degradation. We have modelled these two hypotheses to see whether both actually lead to oscillatory behaviour. The p34-cyclin oligomerization does so without any difficulty. With the second mechanism, however, the strict hypothesis that cyclin degradation is activated by MPF must be re-examined: the system only oscillates if, in disappearing, the MPF and the cyclin react with each other stoichiometrically. The model also demonstrates that it is useless to seek cyclic control of cyclin proteolysis. Keywords: Cell cycle; MPF; Cyclin; Oscillator.
Introduction
Various simplified e x p e r i m e n t a l s y s t e m s for producing cell cycles have r e c e n t l y been designed both in vivo (Labb~ et al., 1989; Russell et al., 1989; Simanis and Nurse, 1986) and in vitro (Cyert and Kirschner, 1988; Lohka and Masui, 1984; Minshull et al., !989; M u r r a y and Kirschner, 1989a; M u r r a y et al., 1989). The essential basic c o m p o n e n t s of the cell oscillator e v i d e n t l y consist of at least one molecule with M P F activity, formed wholly (Labb~ et al., 1989) or p a r t l y (Draetta et al., 1989) of p34, and one cyclin (Murray and Kirschner, 1989a,b; Minshull et al., 1989). The whole question of their interactions, e x c e p t for the autocatalytic activation of the M P F (Cyert and Kirschner, 1988; Labb~ et al., 1989; M u r r a y and Kirschner, 1989b) often a dephosphorylation (Dunphy and N e w p o r t , Correspondence to: C. Hyver.
1989; Gautier et al., 1989; Morla et al., 1989) -- is still open to debate. Broadly speaking, t h e r e are two opposing schools of t h o u g h t (Luca and R u d e r m a n , 1989; M u r r a y and Kirschner, 1989b). For some (Booher et al., 1989; Brizuela et al., 1989; Dunphy and N e w p o r t , 1988; D r a e t t a and Beach, 1988; D r a e t t a et al., 1989; Pines and H u n t e r , 1989; W e s t e n d o r f et al., 1989), p34 and cyclin combine to form an oligomer with M P F activity when mitosis occurs. Before oligomerization, p34 u n d e r g o e s activation t h a t is p r e d o m i n a n t l y autocatalytic. Cyclin slowly accumulates during interphase, is oligomerized and t h e n u n d e r g o e s a sudden proteolysis at the end of mitosis, causing the M P F activity to disappear. F o r others (Cyert and Kirschner, 1988; Labb~ et al., 1989; M u r r a y and Kirschner, 1989a,b; M u r r a y et al., 1989), the cyclin is not oligomerized but enzymatically controls the kinase activity of p34, in o t h e r s words, the
0303-2647/90/$03.50 © 1990Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
86
K
aB
A (p34inac.)
B ( p 34 a c t .
)
+
-X(MPF)
C ( cyclin.
U
=
)
\
b
dA dt
= K- EA-aAB
dB = ~A + aAB-fB-~BC dt dC = ~dt
O~BC-~PC
dX = O~BC - uX dt
20
//
10
/ / /
/
JX \
/
tB
/
\
/
\
II I
\ \
i
% x \
I
\
/ /
/ 1
/
/ /
X I l/I
,,, i i > ~
1 11
.., ,~
10
Fig. 1. Oscillating s y s t e m modelling a p34-cyclin interaction by oligomerization. (a) Graph; (b) e q u a t i o n s of t h e s y s t e m . A r e p r e s e n t s t h e inactive p34, B t h e active p34, C t h e cyclin and X t h e B-C oligomer with an M P F activity. T h e A ~ B reaction t a k e s place either s p o n t a n e o u s l y (eA) or by autocatalysis (aAB). T h e active p34 and t h e cyclin interact (aBC) to form X, an oligomer with M P F activity. K and I~ are t h e c o n s t a n t inputs of A and C. B, C and X h a v e linear losses - f B , - (pC and - ~X. (c) C u r v e s of t h e t i m e variations in t h e s y s t e m , s h o w i n g in particular t h e oscillations of cyclin and MPF. In this special case, we h a v e t a k e n ¢p = 0 in order to show t h a t t h e s y s t e m oscillates e v e n w h e n t h e cyclin is d e g r a d e d only after oligomerization. Values of t h e p a r a m e t e r s are ~ = 0.12, K = 12, a = 1, f = 1.15, a = 2.9, ~ = 9.6, ¢p = 0, u = 2. T h e s e t of dimensional e q u a t i o n s allows c h a n g e s in t h e time and concentration scales to be made.
87 MPF activity. In this way, MPF becomes active firstly by autocatalysis and secondly under the influence of the cyclin. As before, the accumulation of cyclin occurs in interphase. However, such a system without feedback can a priori only oscillate with difficulty. For this reason, Murray and Kirschner (1989a,b) have recently postulated that the proteolysis of cyclin is activated by MPF. It seemed to us important that a theoretical test of the oscillatory behaviour of an MPF-cyclin system according to these two hypotheses should be carried out, a matter to be resolved before considering the inclusion of any possible additional regulators in the model, regulators such as genes cdc25, sucl, wee1, nirn 1, etc. Because of the simplicity required in the constructed models, a large number of forms and feedback mechanisms have been deliberately ignored, for example the number of degrees of phosphorylation implicated in the MPF activity regulation, which is not actually well known (Draetta and Beach, 1988; Dunphy and Newport, 1989; Labb~ et al., 1989). Moreover, we did not consider it wise to take into account, at all costs, the relative stability of the p34 pool for two main reasons: (i) only a small fraction of the p34 molecule participates in the formation of high molecular weight complexes with other proteins including cyclin -- and exhibits any MPF activity (Gautier et al., 1989; Wittenberg and Reed, 1988). (ii) p34 is suspected of being differently localized during the cell cycle; in the absence of cyclin, it is dispersed throughout the cell. In the presence of cyclin, it is preferentially detected in the nucleus (Booher et al., 1989) and perhaps around the centrosome (Riabowol et al., 1989). Thus it is suggested that cyclin acts to regulate both the catalytic properties and the localization of the small fraction of the p34 pool which is implicated in the MPF activity. It is essentially this active fraction which is of interest for our model. By analogy with the experimental proce-
dure, we wished, using a differential system, to model a corresponding minimal structure having the ability to oscillate. Such a "theoretical experiment" seemed to us desirable in this case, since it is well known that it is difficult, if not impossible, to determine intuitively and without calculation, the oscillatory capabilities of a given system (Hyver, 1985; Tyson, 1977). Figure 1 sums up the structure of the model incorporating the oligomerization hypothesis, p34 becomes active through spontaneous activity and by autocatalysis. There is a continuous supply of inactive p34 (molecule A), e.g. from a stored pool (not modelled). Cyclin synthesis occurs at a constant rate (Luca and Ruderman, 1989). The MPF is formed by oligomerization of active p34 and cyclin. At the start, we laid down that cyclin proteolysis occurs either directly on the free form of the molecule or on the oligomerized form (Richardson et al., 1989). An analytical study like the simulation of the model (Fig. lc) shows that this system oscillates in a way which agrees well with the available experimental data both in vivo (Westendorf et al., 1989) and in vitro (Murray and Kirschner, 1989a; Murray et al., 1989); for example a 4 : 1 ratio between the times for the accumulation and degradation of cyclin, and the collapse of the cyclin before the MPF peak. Note that in these curves the MPF is given as a molecular concentration and not as an activity, which makes it difficult to make a comparison between the amplitude of the modelled variations and experiment. Finally, the system oscillates whether or not cyclin degradation occurs only after oligomerization, i.e. whether or not the parameter ~o is equal to zero. Figure 2 corresponds to the hypothesis of enzymatic activation of the MPF by cyclin. We have kept most points the same as in the first system (inputs of A and C, losses of X and C, autocatalysis). However, the cyclin now enzymatically activates the passage from the pre-MPF to the MPF. First of all, we wished to carry out a rigorous modelling of the
88 a
1)
g'= 0 K
.J
aX
A (pre- MPF)
m
X (MPF)
_._
~C
/~
=
C
9X
~-
2) g ' = g
K
.x
_- A ( p r e - M P F )
/
X (MPF)
'~
#
-
"C
b dA = Kdt
(~CA-aAX
dX = ECA + aAX-fX-g'CX dt dC_ dt
g'=O
or g ' = g
/~ - g C X - ~ C
A
C
/"
i/i-\
\
/
/ /
/
/
/
/
/
\
\
\
,
\
/
/
/ /
I
I
5
10
t
89
I
I
2l
i /
i
~ C
/ l / /
2
l
///~A ill/
15
1[ /I
t2
10
• ,,
If
5
'fl
'If: I
i
10
20
30
t
Fig. 3. Simulation of the blockage of cyclin degradation by the cytostatic factor (CSF). Equations as in Fig. 2b with g' = g. Initial values of the parameters as in Fig. 2c. A t time t 1, g is put equal to 0; the cyclin degradation induced by MPF stops. At time t2, g takes on its initial value (g = 2.9). The stoppage of the degradation leads to high values of cyclin and MPF in good agreement with experimental results.
hypothesis of Murray and Kirschner (1989a,b) by writing in an enzymatic activation of the process of cyclin proteolysis by MPF (see Fig. 2, case when g' = 0). However, it can easily be shown analytically, by the first method of Liapunov, that this system cannot oscillate. From a strictly mathematical point of view, therefore, the system must be modified for it to oscillate. A simple way of doing this is to add a term to the equation for X (see Fig. 2, case when g' = g). Biochemically, this means that the
cyclin and the MPF disappear from the system stoichiometrically at the same time. For example, it is possible to postulate a chemical type of interaction facilitating the action of proteases on the cyclin and entailing an inactivation or sequestration of the MPF. The addition of this term renders the system capable of oscillation (Fig. 2c). Note that the equations and curves obtained for this system are very similar to those of the first system, the curve for X in Fig. 2c corresponding to that for B in Fig. lc.
Fig. 2. Oscillating system modelling an enzymatic pre-MPF-cyclin interaction. (a) Graphs; (b) equations of the two possible systems (see text). A represents the pre-MPF (often inactive p34), X the MPF (often active p34) and C the cyclin. The A -~ X reaction takes place firstly by autocatalysis (aAX) and secondly by enzymatic catalysis due to the cyclin (~CA). K, ~, -
85
MPF and cyclin: modelling of the cell cycle minimum oscillator Claude
Hyver a and
Herv~
Le
Guyader b
~Service de Biophysique, De'partement de Biologie, Centre d'Etudes Nucl~aires de Saclay, 91191 Gif-sur-Yvette (France) and bUniversit$ de Paris-Sud, Laboratoire de Biologie Cellulaire 4, Batiment, 444-91405 Orsay Cedex IFrance)
(Received January 22nd, 1990} (Revision received May 2nd, 1990) The cell cycle appears to be controlled by the interplay between two protein complexes, MPF and cyclin. Their interactions play an essential role in the structure of the oscillator governing the cell cycle. There seems to be no general agreemeat on this latter crucial question. Two different mechanisms are proposed: (i) cyclin and p34 kinase combine to form an oligomer with MPF activity; (ii) cyclin enzymatically activates the passage from inactive pre-MPF to active MPF, with the postulate that MPF initiates cyclin degradation. We have modelled these two hypotheses to see whether both actually lead to oscillatory behaviour. The p34-cyclin oligomerization does so without any difficulty. With the second mechanism, however, the strict hypothesis that cyclin degradation is activated by MPF must be re-examined: the system only oscillates if, in disappearing, the MPF and the cyclin react with each other stoichiometrically. The model also demonstrates that it is useless to seek cyclic control of cyclin proteolysis. Keywords: Cell cycle; MPF; Cyclin; Oscillator.
Introduction
Various simplified e x p e r i m e n t a l s y s t e m s for producing cell cycles have r e c e n t l y been designed both in vivo (Labb~ et al., 1989; Russell et al., 1989; Simanis and Nurse, 1986) and in vitro (Cyert and Kirschner, 1988; Lohka and Masui, 1984; Minshull et al., !989; M u r r a y and Kirschner, 1989a; M u r r a y et al., 1989). The essential basic c o m p o n e n t s of the cell oscillator e v i d e n t l y consist of at least one molecule with M P F activity, formed wholly (Labb~ et al., 1989) or p a r t l y (Draetta et al., 1989) of p34, and one cyclin (Murray and Kirschner, 1989a,b; Minshull et al., 1989). The whole question of their interactions, e x c e p t for the autocatalytic activation of the M P F (Cyert and Kirschner, 1988; Labb~ et al., 1989; M u r r a y and Kirschner, 1989b) often a dephosphorylation (Dunphy and N e w p o r t , Correspondence to: C. Hyver.
1989; Gautier et al., 1989; Morla et al., 1989) -- is still open to debate. Broadly speaking, t h e r e are two opposing schools of t h o u g h t (Luca and R u d e r m a n , 1989; M u r r a y and Kirschner, 1989b). For some (Booher et al., 1989; Brizuela et al., 1989; Dunphy and N e w p o r t , 1988; D r a e t t a and Beach, 1988; D r a e t t a et al., 1989; Pines and H u n t e r , 1989; W e s t e n d o r f et al., 1989), p34 and cyclin combine to form an oligomer with M P F activity when mitosis occurs. Before oligomerization, p34 u n d e r g o e s activation t h a t is p r e d o m i n a n t l y autocatalytic. Cyclin slowly accumulates during interphase, is oligomerized and t h e n u n d e r g o e s a sudden proteolysis at the end of mitosis, causing the M P F activity to disappear. F o r others (Cyert and Kirschner, 1988; Labb~ et al., 1989; M u r r a y and Kirschner, 1989a,b; M u r r a y et al., 1989), the cyclin is not oligomerized but enzymatically controls the kinase activity of p34, in o t h e r s words, the
0303-2647/90/$03.50 © 1990Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ireland
86
K
aB
A (p34inac.)
B ( p 34 a c t .
)
+
-X(MPF)
C ( cyclin.
U
=
)
\
b
dA dt
= K- EA-aAB
dB = ~A + aAB-fB-~BC dt dC = ~dt
O~BC-~PC
dX = O~BC - uX dt
20
//
10
/ / /
/
JX \
/
tB
/
\
/
\
II I
\ \
i
% x \
I
\
/ /
/ 1
/
/ /
X I l/I
,,, i i > ~
1 11
.., ,~
10
Fig. 1. Oscillating s y s t e m modelling a p34-cyclin interaction by oligomerization. (a) Graph; (b) e q u a t i o n s of t h e s y s t e m . A r e p r e s e n t s t h e inactive p34, B t h e active p34, C t h e cyclin and X t h e B-C oligomer with an M P F activity. T h e A ~ B reaction t a k e s place either s p o n t a n e o u s l y (eA) or by autocatalysis (aAB). T h e active p34 and t h e cyclin interact (aBC) to form X, an oligomer with M P F activity. K and I~ are t h e c o n s t a n t inputs of A and C. B, C and X h a v e linear losses - f B , - (pC and - ~X. (c) C u r v e s of t h e t i m e variations in t h e s y s t e m , s h o w i n g in particular t h e oscillations of cyclin and MPF. In this special case, we h a v e t a k e n ¢p = 0 in order to show t h a t t h e s y s t e m oscillates e v e n w h e n t h e cyclin is d e g r a d e d only after oligomerization. Values of t h e p a r a m e t e r s are ~ = 0.12, K = 12, a = 1, f = 1.15, a = 2.9, ~ = 9.6, ¢p = 0, u = 2. T h e s e t of dimensional e q u a t i o n s allows c h a n g e s in t h e time and concentration scales to be made.
87 MPF activity. In this way, MPF becomes active firstly by autocatalysis and secondly under the influence of the cyclin. As before, the accumulation of cyclin occurs in interphase. However, such a system without feedback can a priori only oscillate with difficulty. For this reason, Murray and Kirschner (1989a,b) have recently postulated that the proteolysis of cyclin is activated by MPF. It seemed to us important that a theoretical test of the oscillatory behaviour of an MPF-cyclin system according to these two hypotheses should be carried out, a matter to be resolved before considering the inclusion of any possible additional regulators in the model, regulators such as genes cdc25, sucl, wee1, nirn 1, etc. Because of the simplicity required in the constructed models, a large number of forms and feedback mechanisms have been deliberately ignored, for example the number of degrees of phosphorylation implicated in the MPF activity regulation, which is not actually well known (Draetta and Beach, 1988; Dunphy and Newport, 1989; Labb~ et al., 1989). Moreover, we did not consider it wise to take into account, at all costs, the relative stability of the p34 pool for two main reasons: (i) only a small fraction of the p34 molecule participates in the formation of high molecular weight complexes with other proteins including cyclin -- and exhibits any MPF activity (Gautier et al., 1989; Wittenberg and Reed, 1988). (ii) p34 is suspected of being differently localized during the cell cycle; in the absence of cyclin, it is dispersed throughout the cell. In the presence of cyclin, it is preferentially detected in the nucleus (Booher et al., 1989) and perhaps around the centrosome (Riabowol et al., 1989). Thus it is suggested that cyclin acts to regulate both the catalytic properties and the localization of the small fraction of the p34 pool which is implicated in the MPF activity. It is essentially this active fraction which is of interest for our model. By analogy with the experimental proce-
dure, we wished, using a differential system, to model a corresponding minimal structure having the ability to oscillate. Such a "theoretical experiment" seemed to us desirable in this case, since it is well known that it is difficult, if not impossible, to determine intuitively and without calculation, the oscillatory capabilities of a given system (Hyver, 1985; Tyson, 1977). Figure 1 sums up the structure of the model incorporating the oligomerization hypothesis, p34 becomes active through spontaneous activity and by autocatalysis. There is a continuous supply of inactive p34 (molecule A), e.g. from a stored pool (not modelled). Cyclin synthesis occurs at a constant rate (Luca and Ruderman, 1989). The MPF is formed by oligomerization of active p34 and cyclin. At the start, we laid down that cyclin proteolysis occurs either directly on the free form of the molecule or on the oligomerized form (Richardson et al., 1989). An analytical study like the simulation of the model (Fig. lc) shows that this system oscillates in a way which agrees well with the available experimental data both in vivo (Westendorf et al., 1989) and in vitro (Murray and Kirschner, 1989a; Murray et al., 1989); for example a 4 : 1 ratio between the times for the accumulation and degradation of cyclin, and the collapse of the cyclin before the MPF peak. Note that in these curves the MPF is given as a molecular concentration and not as an activity, which makes it difficult to make a comparison between the amplitude of the modelled variations and experiment. Finally, the system oscillates whether or not cyclin degradation occurs only after oligomerization, i.e. whether or not the parameter ~o is equal to zero. Figure 2 corresponds to the hypothesis of enzymatic activation of the MPF by cyclin. We have kept most points the same as in the first system (inputs of A and C, losses of X and C, autocatalysis). However, the cyclin now enzymatically activates the passage from the pre-MPF to the MPF. First of all, we wished to carry out a rigorous modelling of the
88 a
1)
g'= 0 K
.J
aX
A (pre- MPF)
m
X (MPF)
_._
~C
/~
=
C
9X
~-
2) g ' = g
K
.x
_- A ( p r e - M P F )
/
X (MPF)
'~
#
-
"C
b dA = Kdt
(~CA-aAX
dX = ECA + aAX-fX-g'CX dt dC_ dt
g'=O
or g ' = g
/~ - g C X - ~ C
A
C
/"
i/i-\
\
/
/ /
/
/
/
/
/
\
\
\
,
\
/
/
/ /
I
I
5
10
t
89
I
I
2l
i /
i
~ C
/ l / /
2
l
///~A ill/
15
1[ /I
t2
10
• ,,
If
5
'fl
'If: I
i
10
20
30
t
Fig. 3. Simulation of the blockage of cyclin degradation by the cytostatic factor (CSF). Equations as in Fig. 2b with g' = g. Initial values of the parameters as in Fig. 2c. A t time t 1, g is put equal to 0; the cyclin degradation induced by MPF stops. At time t2, g takes on its initial value (g = 2.9). The stoppage of the degradation leads to high values of cyclin and MPF in good agreement with experimental results.
hypothesis of Murray and Kirschner (1989a,b) by writing in an enzymatic activation of the process of cyclin proteolysis by MPF (see Fig. 2, case when g' = 0). However, it can easily be shown analytically, by the first method of Liapunov, that this system cannot oscillate. From a strictly mathematical point of view, therefore, the system must be modified for it to oscillate. A simple way of doing this is to add a term to the equation for X (see Fig. 2, case when g' = g). Biochemically, this means that the
cyclin and the MPF disappear from the system stoichiometrically at the same time. For example, it is possible to postulate a chemical type of interaction facilitating the action of proteases on the cyclin and entailing an inactivation or sequestration of the MPF. The addition of this term renders the system capable of oscillation (Fig. 2c). Note that the equations and curves obtained for this system are very similar to those of the first system, the curve for X in Fig. 2c corresponding to that for B in Fig. lc.
Fig. 2. Oscillating system modelling an enzymatic pre-MPF-cyclin interaction. (a) Graphs; (b) equations of the two possible systems (see text). A represents the pre-MPF (often inactive p34), X the MPF (often active p34) and C the cyclin. The A -~ X reaction takes place firstly by autocatalysis (aAX) and secondly by enzymatic catalysis due to the cyclin (~CA). K, ~, -
