Oncogene (1997) 15, 317 ± 325 1997 Stockton Press All rights reserved 0950 ± 9232/97 $12.00 Bistable biochemical switching and the control of the events of the cell cycle* CD Thron 5 Barrymore Road, Hanover, New Hampshire 03755, USA Some of the events of the cell cycle appear to be development (Edelstein, 1971; Seelig and Denzel, 1972; triggered by a bistable mechanism. A bistable biochem- Meinhart, 1982; Murray, 1989). Biochemical switches ical system can respond to a small, slow signal and is of the simple types considered here have been described carried by positive feedback from one stable steady state and discussed by Edelstein-Keshet (1988) and Murray directly to another, in an all-or-none manner. Slow or (1989). Lewis et al. (1977) have particularly stressed subthreshold stimuli do not cause accommodation or loss that other types of biochemical switch do not show of excitability. Switching is not readily reversible by true thresholds of discontinuity, and do not provide removing the stimulus, i.e. there is hysteresis: reversal either the sharpness or the irrevocability of bistable generally requires a stronger, opposite stimulus. Bio- switches. Bistable switching is typically caused by a chemically, bistable biochemical switching requires change in a rate constant or other parameter which positive feedback, and mechanisms for stabilizing the changes the biochemical dynamics and destabilizes the system against premature activation and for destabiliza- current stable state, and thereby forces the system to tion in response to a biological signal. Three bistable bio- go to the alternate stable state. A change of steady chemical models, all suggested by reported experimental state may alternatively be caused by addition of observations, are described and analysed. These models substrate, as in ampli®cation of MPF; but mitosis is suggest that a titratable inhibitor may play an important not normally initiated by addition of MPF, and we will part in bistable switching, because the end-point of focus here on switching by destabilization of a steady titration can form a natural threshold for enhancement state. of positive feedback. Some cell cycle events other than mitosis are controlled by cyclin-dependent kinases (Cdks) other Keywords: biochemical switch; bistability; cell cycle; than Cdc2, and some of these Cdks appear to be cyclin-dependent kinase inhibitors; positive feedback regulated by Cdk inhibitors (CKIs) (Hunter, 1993; Pines, 1994; Elledge and Harper, 1994; Peter and Herskowitz, 1994b). Nasmyth and Hunt (1993) have suggested that Cdks `build up like water behind a CKI Introduction dam,' and that over¯ow of Cdk from the dam triggers The onset of mitosis is thought to be caused by proteolysis or inactivation of the CKI, releasing a ¯ood maturation promoting factor (MPF), a complex of the of Cdks that irrevocably commit the cell to the next protein kinase Cdc2 with cyclin B (Murray and Hunt, phase of the cell cycle. Several CKIs appear to be 1993). MPF shows the phenomenon of `ampli®cation': regulated by proteolysis (McKinney et al., 1993; a small added amount of MPF causes activation of a Schwob et al., 1994; Pagano et al., 1995), and while much larger amount from an inactive precursor (Cyert mechanisms for rapidly activating the proteolysis have and Kirschner, 1988; Dunphy and Newport, 1988). not been generally demonstrated, Sherr (1996) cites a Ampli®cation is apparently due to the fact that Cdc2/ report that Cdk2/cyclin E accelerates turnover of cyclin B activates its own activating kinase Cdc25 p27Kip1. If a Cdk activates CKI proteolysis this releases (Homan et al., 1993), giving rise to a positive more Cdk, creating positive feedback and possibly feedback loop that makes the activation of MPF leading to complete destruction of CKI and a stable eectively autocatalytic. Evidently a cell can exist in a state with CKI obliterated and Cdk freed. To prevent stable steady state with low MPF activity; but on activation, CKI must inhibit Cdk-activated CKI displacement of the cell from this steady state, by proteolysis enough to protect itself and stabilize a addition of a small amount of MPF, the autocatalytic steady state with low Cdk activity. The system is then eect of MPF causes the cell to shift toward a dierent bistable and has the potential for bistable switching. stable state, one with high MPF activity. Ampli®cation The important characteristics of a bistable biochem- therefore indicates the existence of two potential stable ical switch are (1) it always commits itself to one of steady states, i.e. bistability. two stable steady states, and does not linger in any Bistable switching is well known in chemical intermediate or transitional states; (2) it can be engineering, and a clear and fairly simple explanation switched rapidly from one stable steady state to is given by Gray and Scott (1990). In biology it has another, and this may require only a small, slow been proposed as a mechanism of memory, of signal; and (3) it shows hysteresis, in that removing the dierentiation, or of conversion of continuous chemi- signal does not undo the response, and restoring the cal gradients into discontinuous patterns of embryonic original steady state requires a stronger opposite signal. These characteristics, in a switch controlling the onset of mitosis or other cell cycle events, can prevent an *Most of this work was done while the author was in the department incomplete or vacillating commitment, which might of Pharmacology and Toxicology, Darmouth Medical School, Hanover, NH 03755, U.S.A. well be catastrophic. This paper attempts to give a Received 24 January 1997; revised 2 April 1997; accepted 3 April qualitative understanding of the dynamics and 1997 biochemical kinetics underlying several types of Bistable biochemical switching in the cell cycle CD Thron 318 bistable switching which may be involved in controlling Newport, 1989), and this mechanism could explain events of the cell cycle. The models used are based on several of the puzzling eects of Suc1 (Thron, 1994). the experimental literature but are simpli®ed in order However this role of Suc1 now seems questionable. to show the dynamical principles. Recent ®ndings (Patra and Dunphy, 1996; Pines, 1996) indicate that the action of Suc1 is more complicated. Moreover, the quantity of Suc1 present vastly exceeds Autocatalytic activation the quantity of cyclin B (Kobayashi et al., 1991a, b; We use the terms Cdc2 and Cdc2P to denote the active Patra and Dunphy, 1996). The amount of Cdc2/cyclin (tyrosine 15-dephosphorylated) and inactive (tyrosine B is therefore much too small to titrate Suc1 15-phosphorylated) forms of the Cdc2/cyclin B stoichiometrically. Moreover, if Cdc2/Suc1 binding is heterodimer, respectively. partially reversible, the large excess of Suc1 would With any autocatalytic process, a key problem is inhibit all concentrations of Cdc2/cyclin B approxi- how to prevent premature activation. In the generally mately equally, and there would not be much accepted mechanism for Cdc2 activation and inactiva- preferential inhibition of lower concentrations. How- tion (Figure 1a) the relative strength of autocatalysis ever other Cdc2-binding substances have been reported must be relatively weak at low Cdc2 concentrations, in (Kumagai and Dunphy, 1995; Lee and Kirschner, order to stabilize the steady state with predominantly 1996) which may act as the inhibitor X in Figure 1b. the inactive phosphorylated form Cdc2P. Binding of In this model, there is continuous metabolic turn- Cdc2 to an inhibitor (X, Figure 1b), preventing over between the active and inactive forms, Cdc2 and autocatalysis, would weaken the autocatalysis prefer- Cdc2P respectively. To analyse the mechanism we entially at low Cdc2 concentrations because, by the consider the case where the total of Cdc2 and Cdc2P is laws of chemical equilibrium, binding of Cdc2 to a constant. Cdc2 is assumed to bind to the inhibitor X, ®xed amount of inhibitor would be proportionately but Cdc2P binding to X is assumed to be relatively greater at lower [Cdc2]. If the binding is very strong, much less strong (Thron, 1994, 1996). Unbound Cdc2 the inhibitor would almost completely inhibit Cdc2 up eectively (through activation of Cdc25) accelerates its to a stoichiometric amount, but excess Cdc2 would be own formation, i.e. the conversion of Cdc2P to Cdc2. uninhibited. Since the total of Cdc2P and Cdc2 is assumed to be It has been proposed (Thron, 1994, 1996), that the constant, the net rate of formation of Cdc2 equals the protein Suc1 acts as the inhibitor X in this model. Suc1 net rate of decrease of Cdc2P. This rate is equal to the binds to Cdc2 and inhibits activation (Dunphy and dierence between the rate of activation F and the rate of deactivation R: dgdP dgdP p ÿ I a dt dt where [Cdc2] is the total concentration of free and X- bound Cdc2, and F and R are mathematical functions of the concentration of Cdc2. For present purposes these functions need only be described qualitatively, and the mathematical details can be disregarded. (F and R are also functions of the concentrations of free Cdc2 and Cdc2P, but these in turn are functions of [Cdc2]: the proportion of free Cdc2 is determined by the equilibrium of X binding, and [Cdc2P] is equal to the ®xed total of [Cdc2] and [Cdc2P] minus [Cdc2]). b The key to the biochemical switching dynamics is in how the rates of activation and inactivation depend on [Cdc2]. In Figure 2a, the rate of Cdc2 deactivation R is shown as directly proportional to [Cdc2] (free and X- bound Cdc2 are assumed to be deactivated at equal rates). The rate of Cdc2 formation F might be expected to be a similar straight line sloping in the opposite direction, because its substrate Cdc2P increases as [Cdc2] decreases (because the total [Cdc2]+[Cdc2P] is con- stant). However, curve F is more complicated.