Modeling the Cell Cycle: Why Do Certain Circuits Oscillate?
2011; Cell Press; Volume: 144; Issue: 6 Linguagem: Inglês
10.1016/j.cell.2011.03.006
ISSN1097-4172
AutoresJames E. Ferrell, Tony Tsai, Qiong Yang,
Tópico(s)Bacterial Genetics and Biotechnology
ResumoComputational modeling and the theory of nonlinear dynamical systems allow one to not simply describe the events of the cell cycle, but also to understand why these events occur, just as the theory of gravitation allows one to understand why cannonballs fly in parabolic arcs. The simplest examples of the eukaryotic cell cycle operate like autonomous oscillators. Here, we present the basic theory of oscillatory biochemical circuits in the context of the Xenopus embryonic cell cycle. We examine Boolean models, delay differential equation models, and especially ordinary differential equation (ODE) models. For ODE models, we explore what it takes to get oscillations out of two simple types of circuits (negative feedback loops and coupled positive and negative feedback loops). Finally, we review the procedures of linear stability analysis, which allow one to determine whether a given ODE model and a particular set of kinetic parameters will produce oscillations. Computational modeling and the theory of nonlinear dynamical systems allow one to not simply describe the events of the cell cycle, but also to understand why these events occur, just as the theory of gravitation allows one to understand why cannonballs fly in parabolic arcs. The simplest examples of the eukaryotic cell cycle operate like autonomous oscillators. Here, we present the basic theory of oscillatory biochemical circuits in the context of the Xenopus embryonic cell cycle. We examine Boolean models, delay differential equation models, and especially ordinary differential equation (ODE) models. For ODE models, we explore what it takes to get oscillations out of two simple types of circuits (negative feedback loops and coupled positive and negative feedback loops). Finally, we review the procedures of linear stability analysis, which allow one to determine whether a given ODE model and a particular set of kinetic parameters will produce oscillations. In many eukaryotic cells, the cell cycle proceeds as a sequence of contingent events. A new cell must first grow to a sufficient size before it can begin DNA replication. Then, the cell must complete DNA replication before it can begin mitosis. Finally, the cell must successfully organize a metaphase spindle before it can complete mitosis and begin the cycle again. If cell growth, DNA replication, or spindle assembly is slowed down, the entire cell cycle slows. Thus, this type of cell cycle is like an “assembly line” or “succession of dominoes” (Hartwell and Weinert, 1989Hartwell L.H. Weinert T.A. Checkpoints: controls that ensure the order of cell cycle events.Science. 1989; 246: 629-634Crossref PubMed Scopus (2341) Google Scholar, Murray and Kirschner, 1989bMurray A.W. Kirschner M.W. Dominoes and clocks: the union of two views of the cell cycle.Science. 1989; 246: 614-621Crossref PubMed Scopus (499) Google Scholar). However, some cell cycles are qualitatively different in terms of their dynamics. Most notable of these exceptions is the early embryonic cell cycle in the amphibian Xenopus laevis. DNA replication is not contingent upon cell growth, probably because the frog egg is so big to start with. Mitotic entry is not contingent upon completion of DNA replication, and mitotic exit is not contingent upon the successful assembly of a metaphase spindle because the relevant checkpoints are ineffective in the context of the embryo's high cytoplasm:nucleus ratio (Dasso and Newport, 1990Dasso M. Newport J.W. Completion of DNA replication is monitored by a feedback system that controls the initiation of mitosis in vitro: studies in Xenopus.Cell. 1990; 61: 811-823Abstract Full Text PDF PubMed Scopus (250) Google Scholar, Minshull et al., 1994Minshull J. Sun H. Tonks N.K. Murray A.W. A MAP kinase-dependent spindle assembly checkpoint in Xenopus egg extracts.Cell. 1994; 79: 475-486Abstract Full Text PDF PubMed Scopus (349) Google Scholar). Lacking these contingencies, the early embryo simply pulses once every 25 min, irrespective of whether the endpoints of the cell cycle (DNA replication and mitosis) have been completed (Hara et al., 1980Hara K. Tydeman P. Kirschner M. A cytoplasmic clock with the same period as the division cycle in Xenopus eggs.Proc. Natl. Acad. Sci. USA. 1980; 77: 462-466Crossref PubMed Scopus (241) Google Scholar). Thus, this cell cycle is clock-like (Murray and Kirschner, 1989bMurray A.W. Kirschner M.W. Dominoes and clocks: the union of two views of the cell cycle.Science. 1989; 246: 614-621Crossref PubMed Scopus (499) Google Scholar); it behaves as if it is being driven by an autonomous biochemical oscillator. Although many biological processes seem almost unfathomably complex and incomprehensible, oscillators and clocks are the types of processes that we might have a good chance of not just describing, but also understanding. Accordingly, much effort has gone into understanding how simple cell cycles work in model systems like Xenopus embryos and the fungi S. pombe and S. cerevisiae. This requires the identification of the proteins and genes needed for the embryonic cell cycle and the elucidation of the regulatory processes that connect these proteins and genes. Over the past three decades, enormous progress has been made toward these ends. In each case, the cell cycle is driven by a protein circuit centered on the cyclin-dependent protein kinase CDK1 and the anaphase-promoting complex (APC) (Figure 1A ). The activation of CDK1 drives the cell into mitosis, whereas the activation of APC, which generally lags behind CDK1, drives the cell back out (Figure 1B). There are still some missing components and poorly understood connections, but overall, the cell-cycle network is fairly well mapped out. But a satisfying understanding of why the CDK1/APC system oscillates requires more than a description of components and connections; it requires an understanding of why any regulatory circuit would oscillate instead of simply settling down into a stable steady state. What types of biochemical circuits can oscillate, and what is required of the individual components of the circuit to permit oscillations? Such insights are provided by the theory of nonlinear dynamics and by computational modeling. Indeed, cell-cycle modeling has become a very popular pursuit. Hundreds of models have been published (Table 1), beginning with Kauffman, Wille, and Tyson's prescient proposal that the cell cycle of the yellow slime mold Physarum polycephalum is driven by a relaxation oscillator (Kauffman and Wille, 1975Kauffman S. Wille J.J. 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Biol. 2010; 264: 771-781Crossref PubMed Scopus (11) Google Scholar) Open table in a new tab Through time, many of the models have become more complicated and more like chemical engineering models, consisting of dozens of variables and regulatory processes. The purpose of this type of modeling is to account for and test our understanding of specific details of the system that, because of the complexity of the system, cannot always be understood through intuition. This type of detailed model has successfully accounted for the phenotypes of dozens of budding yeast mutants (Chen et al., 2004Chen K.C. Calzone L. Csikasz-Nagy A. Cross F.R. Novak B. Tyson J.J. Integrative analysis of cell cycle control in budding yeast.Mol. Biol. Cell. 2004; 15: 3841-3862Crossref PubMed Scopus (470) Google Scholar). Both types of modeling have their place in understanding cell-cycle regulation, and both have their adherents. Modeling approaches range from simple Boolean modeling to stochastic modeling and partial differential equation modeling. However, to date, the majority of effort has focused on ordinary differential equation (ODE) modeling (Table 1), which gets at the basic solution phase biochemistry of cell-cycle regulation. Here, we address the question of what it takes to make a simple protein circuit like the CDK1/APC system oscillate. We will start with Boolean modeling, which provides intuition into the logic of biochemical oscillators. We then move on to ODE models, which translate this logic into chemical terms. The basic methods for analyzing ODE models of oscillators are well known in the field of nonlinear dynamics but are not so well known among biologists. We believe that it is high time that they were; after all, we biologists are studying what are probably the world's most interesting nonlinear dynamical systems. We will emphasize the basic concepts of oscillator function and, to the extent possible, keep the algebra to a minimum. For further information, the reader is directed to lucid reviews by Goldbeter (Goldbeter, 2002Goldbeter A. Computational approaches to cellular rhythms.Nature. 2002; 420: 238-245Crossref PubMed Scopus (444) Google Scholar) and Novák and Tyson (Novák and Tyson, 2008Novák B. Tyson J.J. Design principles of biochemical oscillators.Nat. Rev. Mol. Cell Biol. 2008; 9: 981-991Crossref PubMed Scopus (684) Google Scholar, Tyson et al., 2003Tyson J.J. Chen K.C. Novak B. Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell.Curr. Opin. Cell Biol. 2003; 15: 221-231Crossref PubMed Scopus (1081) Google Scholar), as well as Strogatz's outstanding textbook (Strogatz, 1994Strogatz S.H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Cambridge, MA1994Google Scholar). We begin by paring the cell cycle down to a simple two-component model in which CDK1 activates APC and APC inactivates CDK1 (Figure 2B ). This is the essential negative feedback loop upon which the cell-cycle oscillator is built (Murray et al., 1989Murray A.W. Solomon M.J. Kirschner M.W. The role of cyclin synthesis and degradation in the control of maturation promoting factor activity.Nature. 1989; 339: 280-286Crossref PubMed Scopus (802) Google Scholar). Perhaps the simplest way to think about the dynamics of a system like this is through Boolean or logical analysis (Glass and Kauffman, 1973Glass L. Kauffman S.A. The logical analysis of continuous, non-linear biochemical control networks.J. Theor. Biol. 1973; 39: 103-129Crossref PubMed Scopus (719) Google Scholar). Suppose that both CDK1 and APC are perfectly switch-like in their regulation; that is, they are either completely on or completely off. Then, together, the system of CDK1 plus APC has four possible discrete states (APCon/CDK1on, APCon/CDK1off, APCoff/CDK1on, and APCoff/CDK1off) (Figure 2E). Now suppose the system starts in an interphase-like state, with APCoff/CDK1off. In the first increment of time, what will happen? If the APC is off, then CDK1 turns on. Thus, we define a rule: state 1, with APCoff/CDK1off, goes to state 2 with APCoff/CDK1on. Next, the active CDK1 activates APC; thus, state 2 goes to 3. The active APC then inactivates CDK1, and state 3 goes to state 4. Finally, in the absence of active CDK1, the APC becomes inactive, and state 4 goes to state 1. This completes the cycle. We can depict the dynamics of this oscillator as a diagram in “state space” (Figure 2E). The model goes through a never-ending cycle, and all of the possible states of the system are visited during each run through the cycle. If we add one more component to the system—for example, a protein like Polo-like kinase 1 (Plk1), which here we assume is activated by CDK1 and, in turn, contributes to the activation of APC (Figure 2C)—then there are eight (2 × 2 × 2) possible states for the system. If we start with all of the proteins off and assume six biologically reasonable rules (active CDK1 activates Plk1, active Plk1 activates APC, active APC inactivates CDK1…), once again we get a never-ending cycle of states (Figure 2F). But this time, only some of the possible states (states 1–6 in Figure 2F) lie on the cycle. The other two states (7 and 8) feed into the cycle in a manner determined by the rules we assume. Thus, no matter where the system starts, it will converge to the cycle sooner or later. The behavior of this Boolean model is analogous to “limit cycle oscillations,” which we will encounter again in the next section. With Boolean models, it is easy to obtain oscillations. Indeed, one can even get oscillations from a model with a single species (CDK1) that flips on when it is off and flips off when it is on (Figures 2A and 2D), a discrete representation of a protein that negatively regulates itself. Although Boolean analysis is simple and appealing, it is not completely realistic. First, all three Boolean models with negative feedback loops (Figures 2A–2C) yielded oscillations even though we know that real negative feedback loops do not always oscillate. The problem is the simplifying assumptions that underpin Boolean analysis: the discrete activity states and time steps. Even if individual CDK1 and APC molecules actually flip between discrete on/off states, a cell contains a number of CDK1 and APC molecules, and they would not be expected to all flip simultaneously. The framework for describing the dynamics of such a system is chemical kinetic theory, and, assuming that the numbers of CDK1 and APC molecules are large, the activation and inactivation of CDK1 and APC can be described by a set of differential equations. Here, we will build up an ODE model of the system, starting with a one-ODE model, which fails to produce oscillations. We then add additional complexity to the ODEs until the model succeeds in producing sustained, limit cycle oscillations. By definition, the rate of change of active CDK1 (denoted CDK1∗) is the rate of CDK1 activation minus the rate of CDK1 inactivation. For simplicity, we will assume that CDK1 is activated by the rapid, high-affinity binding of cyclin, which is being synthesized at a constant rate of α1 (Equation 1, blue). For CDK1 inactivation, we will assume mass action kinetics (Equation 1, pink). This gives us the first-order differential equation:There are two time-dependent variables, CDK1∗ and APC∗. To allow the system to be described by an ODE with a single time-dependent variable (Figure 3A ), we assume that the activity of APC is regulated rapidly enough by CDK1∗ so that it can be considered an instantaneous function of CDK1∗. What functional form should we use for APC's response function? Here, we will assume that APC's response to CDK1∗ is ultrasensitive—sigmoidal in shape, like the response of a cooperative enzyme—and that the response is described by a Hill function. This assumption is reasonable because APC activation is a multistep process; multistep processes often yield ultrasensitive, sigmoidal responses; and, for our purposes, the Hill equation with a Hill coefficient (n) greater than 1 can be thought of as a generic sigmoidal function. Substituting a Hill function for APC∗ in Equation 1, we get a one-ODE model of a negative feedback loop:dCDK1∗dt=α1−β1CDK1∗CDK1∗n1K1n1+CDK1∗n1[Equation 2] We now choose, somewhat arbitrarily, values for the model's parameters (α1 = 0.1, β1 = 1, K1 = 0.5, n1 = 8) and initial conditions (CDK1∗[0] = 0). We can then numerically integrate Equation 2 over time and see how the concentration of activated CDK1∗ evolves.Figure 3A Model of CDK1 Regulation with One Differential EquationShow full caption(A) Schematic of the model. The parameters chosen for the model were α1 = 0.1, β1 = 1, K1 = 0.5, and n1 = 8.(B) Trajectories in one-dimensional phase space, approaching a stable steady state (designated by the filled circle) at CDK1∗≈0.43.(C) Time course of the system, starting with CDK1∗[0] = 0 and evolving toward the steady state.View Large Image Figure ViewerDownload Hi-res image Download (PPT) (A) Schematic of the model. The parameters chosen for the model were α1 = 0.1, β1 = 1, K1 = 0.5, and n1 = 8. (B) Trajectories in one-dimensional phase space, approaching a stable steady state (designated by the filled circle) at CDK1∗≈0.43. (C) Time course of the system, starting with CDK1∗[0] = 0 and evolving toward the steady state. As shown in Figure 3C, the system moves monotonically from its initial state toward a steady state; there is no hint of oscillation. This monotonic approach to steady state is observed no matter what we assume for the parameters and initial conditions. Thus, we have not yet built an oscillator model. Even though we were able to produce sustained oscillations with a one-variable Boolean model of a negative feedback loop (Figures 2A and 2D), translating the model into a differential equation eliminated the oscillations. Another way of representing the system's behavior is through a phase plot, which shows all possible activities of the system. This is similar to the state-space plots that we used for the Boolean analysis, but instead of having a few discrete states, the phase plot displays a continuum, showing how the system's transition between states occurs through a smooth continuum (as we would expect, given that the numerous CDK1 molecules do not all activate simultaneously but “smoothly” turn on.). The phase plot contains one dimension for each time-dependent variable. Therefore, in this one-variable model, the phase plot possesses one axis, representing the concentration of activated CDK1∗ (Figure 3B). In addition, the system's phase plot shows one stable steady state with CDK1∗≈0.43. If the system starts off with CDK1 activity less or greater than 0.43, the system will move along a trajectory back to 0.43. In other words, any initial condition to the left or right of the steady state yields a trajectory moving to the right or left, respectively. Why did the one variable Boolean model produce oscillations (Figures 2A and 2D), whereas the one-ODE model (Equation 2) did not (Figure 3)? The discrete time steps of the Boolean model help to segregate CDK1 activation from inactivation in time. Thus, perhaps adding another ODE (Figure 4A ), which acknowledges the fact that APC regulation is not instantaneous, might allow us to generate oscillations. First, we write an ODE for the activation and inactivation of CDK1 (Equation 3). We once again assume that CDK1 is activated by a constant rate of cyclin synthesis (α1). We assume that the multistep process through which APC∗ inactivates CDK1∗ is described by a Hill function. The inactivation rate is therefore proportional to the concentration of CDK1∗ (the substrate being inactivated) times a Hill function of APC∗. Now for APC (Equation 4), we assume that its rate of its activation by CDK1∗ is proportional to the concentration of inactive APC (which, assuming the total concentration of active and inactive APC to be constant, we take to be 1−APC∗) times a Hill function of CDK1∗, and the rate of inactivation of APC∗ is described by simple mass action kinetics. The resulting two-ODE model is:dCDK1∗dt=α1−β1CDK1∗APC∗n1K1n1+APC1∗n1[Equation 3] dAPC∗dt=α2(1−APC∗)CDK1∗n2K2n2+CDK1∗n2−β2APC∗[Equation 4] Again, we choose kinetic parameters and initial condition (as described in the caption to Figure 4) and integrate the ODEs numerically. The results are shown in Figures 4B and 4C. The CDK1 activity initially rises as the system moves from interphase (low CDK1 activity) toward M phase (high CDK1 activity) (Figure 4C). After a lag, the APC activity begins to rise too. Then, the rate of CDK1 inactivation (driven by APC activation) exceeds the rate of CDK1 activation (driven by cyclin synthesis), and the CDK1 activity starts to fall. After a few wiggles up and down, the system approaches a steady state with intermediate levels of both CDK1 and APC activities. Thus, we have generated damped oscillations, but not sustained oscillations. Figure 4B shows the phase space view of these damped oscillations. The phase space is now two dimensional because there are two time-dependent variables. There is a stable steady state that sits at the intersection of two curves called the nullclines (green and red curves, Figure 4B). These two nullclines can be thought of as stimulus-response curves for the two individual legs of the CDK1/APC system. The red nullcline (defined by the equation dCDK1∗/dt=0) represents what the steady-state response of CDK1∗ to constant levels of APC activity would be if there were no feedback from CDK1∗ to APC∗ (Figure 4B). The green nullcline (defined by dAPC∗/dt=0) represents what the steady-state response of APC∗ to CDK1∗ would be if there were no feedback from APC∗ to CDK1∗ (Figure 4B). For the whole system to be in s
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