J. L. Snoep, Triple J Group for Molecular Cell Physiology, Department of Biochemistry, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa Fax: +272 1808 5863 Tel: +272 1808 5844 E-mail: email@example.com
Numerous top-down kinetic models have been constructed to describe the cell cycle. These models have typically been constructed, validated and analyzed using model species (molecular intermediates and proteins) and phenotypic observations, and therefore do not focus on the individual model processes (reaction steps). We have developed a method to: (a) quantify the importance of each of the reaction steps in a kinetic model for the positioning of a switch point [i.e. the restriction point (RP)]; (b) relate this control of reaction steps to their effects on molecular species, using sensitivity and co-control analysis; and thereby (c) go beyond a correlation towards a causal relationship between molecular species and effects. The method is generic and can be applied to responses of any type, but is most useful for the analysis of dynamic and emergent responses such as switch points in the cell cycle. The strength of the analysis is illustrated for an existing mammalian cell cycle model focusing on the RP [Novak B, Tyson J (2004) J Theor Biol230, 563–579]. The reactions in the model with the highest RP control were those involved in: (a) the interplay between retinoblastoma protein and E2F transcription factor; (b) those synthesizing the delayed response genes and cyclin D/Cdk4 in response to growth signals; (c) the E2F-dependent cyclin E/Cdk2 synthesis reaction; as well as (d) p27 formation reactions. Nine of the 23 intermediates were shown to have a good correlation between their concentration control and RP control. Sensitivity and co-control analysis indicated that the strongest control of the RP is mediated via the cyclin E/Cdk2:p27 complex concentration. Any perturbation of the RP could be related to a change in the concentration of this complex; apparent effects of other molecular species were indirect and always worked through cyclin E/Cdk2:p27, indicating a causal relationship between this complex and the positioning of the RP.
Arthur Pardee  defined the restriction point (RP) as the apparent switch point in the late G1-phase, beyond which normal cells would only progress when supplied with a sufficient, mitogen containing culture medium. Once the cell has transgressed the RP, the necessity of a mitogen (i.e. growth factor) containing medium is relieved and cells commit to replicate their DNA (S-phase) and complete the remainder of the division cycle autonomously [2,3]. Cells in the G1-phase that have not yet passed RP, monitor their environment and own size to determine whether they are ready to commit to S-phase entry and division cycle completion or, in contrast, enter the resting G0-phase , where most non-cancerous somatic mammalian cells spend their lifetime. Disregulation of the RP has been linked to several disease states, most notoriously cancer [1,4], and quantification of the contribution of the different reaction steps in the cell cycle to the control of the RP would be important for drug target identification studies and for understanding the action mechanism of existing, RP affecting drugs.
Abrupt cell cycle transitions (e.g. G1/S) are properties of the complete underlying control system, which can be described without the need to introduce hypothetical regulatory proteins or hard code a decision event [5,6]. Numerous kinetic models that incorporate the existing knowledge of the molecular mechanism of the RP have been constructed [5,7–9]. Novak and Tyson  constructed their kinetic model on the RP of the mammalian cell cycle using a yeast-like core model of the cyclin–Cdk interactions (emphasizing the deep similarities of the Cdk-regulatory systems of yeast and mammalian cells). They extended this core model with kinetic modules for the growth factor sensing machinery of mammalian cells; the retinoblastoma and E2F transcription factor interactions and the antagonism between the cyclin-dependent kinase inhibitor (p27Kip1) and cyclin A/Cdk2 and cyclin E/Cdk2. With their kinetic model, Novak and Tyson could account for the findings obtained in the experiments by Zetterberg and Larsson , which demonstrated the existence of RP and positioned it somewhere in the G1-phase (a) cells pulse treated (1 h) with cycloheximide early in the cell cycle (before RP) immediately suffer a long delay in the cell division cycle; (b) cells treated late in the division cycle finish the current cycle similar to untreated cells but are significantly delayed in the subsequent division cycle; and (c) cells treated directly after the RP do not suffer any delays in the current or subsequent division cycles.
Kinetic models that are constructed on the basis of known biochemical information about the system and its components are routinely used to integrate this knowledge and compare the resulting simulations with experimental observations. More extensive analysis methods are necessary to quantify the importance of each of the reaction steps for the systemic behavior. Metabolic control analysis (MCA) [11,12] is a rigorous framework that enables the assessment of how a biological function is controlled by the various molecular processes in the cell sustaining that function. MCA has been used predominantly to analyse the control distribution in steady states, although the theory has been generalized for systems with dynamic behavior [13–17]. In this contribution, we develop and implement an extension to MCA that can be used not only to quantify the control of system variables, but also to infer by which molecular species this control was elicited.
Our approach is well suited for systems biology studies where the primary goal is to come to a quantitative understanding of systemic properties (such as the emergent RP in the mammalian cell cycle) in terms of the multitude of reaction steps and interactions between the cell's molecular parts (e.g. cell cycle proteins) . Numerous studies have recognized the need for theoretical approaches in molecular biology [19–24]. Our novel framework would be useful for molecular and systems biologists to determine which reactions exert important control on a high-level system property, and explain by which molecular species these reaction steps exert their control.
Applying our approach to the RP, as modeled in the kinetic model developed by Novak and Tyson , revealed that the control of RP was distributed over the different reaction steps in the network. The highest control was exerted by the reactions responsible for: (a) the interaction between retinoblastoma protein (Rb) and E2F transcription factor; (b) synthesis of delayed response genes (DRGs) and cyclin D/Cdk4 in response to growth signals; (c) the E2F-dependent cyclin E/Cdk2 synthesis reaction; and (d) the p27 formation reactions. In addition, our analysis revealed that RP control was exerted via the cyclin E/Cdk2:p27 complex. Independent of which reaction step or which molecular species was perturbed, all of the effects on the RP could be explained via an effect of the perturbation on cyclin E/Cdk2:p27.
The model presented by Novak and Tyson  consisted of 18 ordinary differential equations (ODEs) and several algebraic equations. Each of these ODEs were used, along with the wiring diagram of the mammalian cell cycle network, to distil 52 reaction steps from the network, such that each step formed a functional unit within the network. The retinoblastoma and E2F transcription factor interactions which were coded as steady-state algebraic equations in the original model were coded explicitly as ODEs, increasing the number of ODEs to 23 (see Table 1). The reaction network of the cell cycle model is schematically represented in Fig. 1.
v(29) = E2F-Rb(t)(K20(CYCDT · LD + LA · CYCA(t) + LB · CYCB(t) + LE · CYCE(t)))
v(30) = P-E2F-Rb(t)(K20(CYCDT · LD + LA · CYCA(t) + LB · CYCB(t) + LE · CYCE(t)))
v(32) = K28 · GM(t)
v(33) = eps(t) · MU · GM(t)
v(35) = eps(t)(K11a + K11 · CYCB(t))
v(36) = eps(t) · K29 · E2F(t) · MASS(t)
v(37) = eps(t) · K33
v(38) = eps(t)(K7a+K7 · E2F(t))
v(39) = eps(t) · K9 · DRG(t)
v(40) = eps(t) · K5
v(43) = Rb(t)(K20(CYCDT · LD + LA · CYCA(t) + LB · CYCB(t) + LE · CYCE(t)))
v(44) = PP-Rb(t)(K19a(PP1T − PP1A) + K19 · PP1A)
v(45) = E2F-Rb(t) · K26R
v(46) = E2F(t)(K23a + K23(CYCA(t) + CYCB(t)))
v(47) = P-E2F(t) · K22
v(48) = E2F(t) · Rb(t) · K26
v(49) = P-E2F-Rb(t) · K26R
v(50) = Rb(t) · P-E2F(t) · K26
v(51) = P-E2F-Rb(t) · K22
v(52) = E2F-Rb(t)(K23a + K23(CYCA(t) + CYCB(t)))
mathematica, version 6.0 (Wolfram Research, Champaign, IL, USA, http://www.wolfram.com/) was used for all simulations. The ODEs were solved using the NDSolve function in combination with the EventLocator method to precisely locate events during the simulation. A simulation result in which the RP and the different phases of the cell cycle are indicated is shown in Fig. 2. In the Novak and Tyson model, the variable cyclin E represents the cyclin E/Cdk2 complex (i.e. cyclin E in its active form). It is therefore not possible to distinguish between active and inactive forms of cyclin E in the Novak and Tyson model.
MCA is a theoretical framework that can be used to calculate the importance of each of the steps in a reaction network for the systemic behavior, using so-called control coefficients [11,12]. We used a perturbation method to calculate control coefficients by adding a multiplier to each of the 52 steps (α1 to α52) and perturbing each reaction individually (0.1 per million perturbation up and down). After each perturbation, the model was simulated until a new limit cycle was reached and then control coefficients for stationary behavior (e.g. position of RP or the length of the G1-phase) were calculated and expressed as a fraction of the cell division cycle. For nonstationary behavior (i.e. time-dependent variables such as concentrations of metabolites or flux values for reaction steps), an extension of MCA was used that enables the determination of control coefficients as the system progresses through an arbitrary fixed interval of time (e.g. the cell division cycle); a detailed description of the method is provided elsewhere .
Results and Discussion
Control of the RP
We used a perturbation method to quantify the control of the respective reaction steps in a mammalian cell cycle model on the time of occurrence of the RP, . The analysis revealed (Fig. 3) that the control is distributed over the reaction network, with the majority of the reactions steps exerting a small to moderate control (). Eight reaction steps exerted a high control () on the RP; four of which caused a delay (positive control), whereas the others advanced the RP (negative control). The for all reaction steps sums up to 0; this is to be expected on the basis of our definition of RP as a fraction of the cell division cycle. The control coefficient of reaction step x on RP is the percentage change in the fraction of the cell cycle length at which RP occurs upon a 1% change in the activity of x. If we had defined the RP as an absolute time point, then the sum of control coefficients would add up to −1 (as they do for the control on the cell division time).
The four steps delaying the RP the most (i.e. with the highest positive control values) are: (a) the breakdown reaction of the DRG (Reaction 2, ); (b) the step involved in proteolysis of cyclin D in the cyclin D:p27 complex, releasing the cyclin dependent kinase inhibitor (p27), (Reaction 3, ); (c) the phosphatase reaction responsible for the dephosphorylation of retinoblastoma (Reaction 44, ); and (d) the reaction step responsible for the synthesis of p27 (Reaction 40, ).
The steps advancing the RP the most (i.e. with the highest negative control values) are: (a) the synthesis reactions of the DRGs and cyclin D (Reactions 41 and 39, and ); (b) the cyclin mediated phosphorylation of retinoblastoma to release it from the retinoblastoma:E2F complex yielding free E2F (Reaction 29, ); and (c) the reaction responsible for the production of cyclin E (Reaction 38, ).
To identify the mechanism underlying the high control coefficients of these steps for RP, we first analyzed which molecular species showed the same control pattern as that observed for the RP. A strong correlation between the control on RP and the concentration control coefficients of species directly involved in the molecular machinery governing the RP is to be expected. To test this, we plotted the concentration control coefficients for the model variables against the RP control for each of the reaction steps (Fig. 4). For nine of the 23 molecular species, a strong correlation was observed, with some species showing a positive correlation, whereas others have a negative correlation and all species have a different slope in the correlation plots (note that the y-axes between the plots differ). The nine variables that showed a strong correlation were cyclin A, cyclin D, cyclin E, cyclin E:p27 complex, p27, hyperphosphorylated retinoblastoma, E2F transcription factor, unphosphorylated retinoblastoma and unphosphorylated retinoblastoma bound to E2F transcription factor. All of these species either reside in the early and DRG module, in the cyclin–cyclin-dependent kinase inhibitor module or the retinoblastoma–E2F transcription factor module (Fig. 1, modules B, C and D). These are the same three modules that also contain the aforementioned eight steps with the highest absolute values.
A sensitivity analysis for the system variables was made at the RP. For this, we initialized all variables in the model at their concentrations at RP, except for one variable to which we made a small perturbation. A subsequent simulation was analyzed with respect to cell cycle completion, (i.e. testing whether the perturbation made to the one variable could prevent RP). The perturbation that we made to the variable was to change its value to a value it had just before RP (i.e. followed its integration path back in time). In this way, we determined the minimal time period that a variable must be regressed from RP along its time integration in order to stall the cell cycle progression. From this analysis, it was evident that the RP was most sensitive for changes made to the cyclin E/Cdk2:p27 complex; changing its value to the value it had 0.04 s before the RP resulted in the cells entering quiescence. The species with values that needed to be regressed 0.5 s or less to prevent RP are listed in Table 2; each of these variables also showed a good correlation in Fig. 4.
Table 2. Species-dependent regression time necessary for cells to enter quiescence. The six molecular intermediates for which the RP showed the highest sensitivity were each regressed in time (at the same time as initializing all other intermediates at their RP concentrations) until, upon subsequent simulation, the cells enter quiescence. This minimal time that a species must be regressed to prevent RP is listed as the Regression time in the table. The RP concentration and the percentage change in concentration of the intermediate concentration at the regression time point compared to the RP are also given. AU, arbitrary units.
Regression time (s)
% Change in concentration
Concentration at RP (AU)
The six metabolites for which RP showed a high sensitivity each had a different gradient in the correlation plots (Fig. 4); for some intermediates, the correlation was positive, whereas, for others, it was negative, and each had a different slope. The correlations observed in Fig. 4 suggest that several of the intermediates also correlate with one another (independent of which reaction is perturbed). Such covariances between intermediates make it difficult to asses a causal relationship. The RP showed the highest sensitivity for changes in the concentration of cyclin E/Cdk2:p27 (Table 2); all of the other species needed to be perturbed more, either in terms of time regression or in terms of the percentage change in their concentrations. This high sensitivity indicates that this intermediate is important for the regulation of RP, and we tested whether the difference in slopes in the correlation plots could be related to co-correlations between the intermediates and cyclin E/Cdk2:p27.
For this, we first determined the co-control coefficients of cyclin E/Cdk2:p27 with the nine intermediates that showed a good correlation in Fig. 4. A co-control coefficient is defined as the ratio of two concentration control coefficients and quantifies the correlation between the two intermediates upon a small perturbation of a reaction step . Interestingly, the values for the co-control coefficients were largely independent of the reaction step that was perturbed, (i.e. two intermediates would co-vary, independent of which step was perturbed). Thus, the co-control coefficients of the model species with the cyclin E/Cdk2:p27 complex quantify the extent that these species co-vary with the complex. For example, cyclin E/Cdk2 (CYCE) and cyclin E/Cdk2:p27 (CE) showed a positive co-variation, with a co-control coefficient of 0.53, indicating that changes in the concentration of CE were correlated with approximately half the concentration change in CYCE. By contrast, CE concentration is negatively correlated with p27Kip1 concentration (i.e. if one goes up, the other goes down) quantitatively expressed in a co-control coefficient, of −1.2.
Subsequently, we calculated the ratio of the gradients in the correlation plots (Fig. 4) for the intermediates, and for cyclin E/Cdk2:p27. For each of the intermediates, the ratio of its gradient in the correlation plot and the gradient of cyclin E/Cdk2:p27 was close to the value of the co-control coefficient of the intermediate with cyclin E/Cdk2:p27. For example, the gradient ratio for cyclin E/Cdk2 and cyclin E/Cdk2:p27 equals 0.49 and, for p27Kip1 and cyclin E/Cdk2:p27, the ratio equals −1.1 (these values are close to the respective co-control coefficients 0.53 and −1.2 as calculated above). This result is in agreement with the hypothesis that the correlations observed in Fig. 4, between changes in system intermediate concentrations and RP, can be accounted for by co-correlations of those intermediates with cyclin E/Cdk2:p27.
The observation that cyclin E/Cdk2:p27 needed to be regressed for the shortest time period (Table 2) indicated that this molecular species on its own could prevent RP. Regressing any of the other species further back in time could potentially effect RP indirectly via cyclin E/Cdk2:p27. We further tested this by analyzing the effect on the cyclin E/Cdk2:p27 concentration upon perturbing each of the other variables to which the RP showed a high sensitivity. Indeed, as can be seen in Fig. 5A, changing any of these variables to such an extent that they interfered with the RP did lead to a change in cyclin E/Cdk2:p27. Each of the curves in Fig. 5A is the result of a separate time simulation for the cyclin E/Cdk2:p27 concentration upon a perturbation of the indicated (red circle) model variable. The wt curve is the reference curve where no perturbation was made. The time point (x-axis value) corresponding with the red symbols indicates RP. From Fig. 5, it can be seen that the shift brought about to the RP was in very good agreement with the extent that the concentration of cyclin E/Cdk2:p27 was affected by the perturbation, for most metabolite perturbations, RP was shifted to the time point where the cyclin E/Cdk2:p27 complex reached a critical concentration. Because a causal relationship between the cyclin E/Cdk2:p27 concentration and the RP would necessitate that, for each perturbation of the complex, the RP must shift accordingly, we perturbed all reaction steps with a high control on RP and plotted the occurrence of RP together with the trajectory for cyclin E/Cdk2:p27 (Fig. 5B). The reactions that were perturbed for each of the curves in Fig. 5 are indicated by number (corresponding to the numbers in Fig. 1). Again, a very good correlation, in excellent agreement with the shift in RP upon metabolite perturbations, was observed for the reaction perturbations. The correlation plots along with the sensitivity analysis are indicative of a causal relationship between the cyclin E/Cdk2:p27 concentration and RP; independent of the type of perturbation made, a critical cyclin E/Cdk2:p27 concentration must be reached for RP transition.
The method presented here, combining control analysis with sensitivity analysis, and showing that all perturbations work via the same mechanism, comprises a generic method to test for causal relationships. We illustrate the method for RP control in the Novak and Tyson model and the results obtained clearly indicate that the control of RP runs mostly via cyclin E/Cdk2:p27. Newer models have been constructed for the RP [4,9] but such models do not describe the complete cell cycle. Using a core model and experimental data of individual gene expression levels, Yao et al.  demonstrated that the interplay between retinoblastoma and E2F creates a bistable switch that probably regulates the RP. Although p27 was not modeled explicitly in this core model (its importance was stated in the text), there is still a strong similarity between the reactions that control RP, together with the feedback loops from cyclin D and cyclin E, in our model and the reactions involved in the bistable switch in the Yao model. Thus, according to our analysis of the model of Novak and Tyson, taken together with the summation theorem, the interplay between retinoblastoma and E2F creates a bistable switch that has a strong effect on the RP, but with a net positive control (i.e. advancing the RP), which is countered by the reactions affecting p27 (i.e. delaying the RP).
RP control in cancer cells
Deregulation of the RP is implicated in almost all tumor cells and it has been suggested that regulation of the RP is essential to prevent cells from becoming cancerous . We thus investigated whether the species shown by our analysis to alter the RP were indeed implicated in malignancies in mammals. Comparing phenotypic observations of tumor cells with the results of our analysis, we indeed observed that changes to cells lines leading to an increased concentration of cyclin E/Cdk2:p27 concentration, such as cyclin E overexpression [26–29] and p27Kip1 downregulation [28–32], yielded uncontrolled tumour proliferation in a wide range of human tissues. Furthermore, it was observed that, in aggressive stomach, prostate, breast, lung and pituitary cancers, p27Kip1 levels were low, either as a result of degradation or translocation to the cytoplasm [33,34].
These observations are in agreement with our co-control analysis; the co-control coefficients, and quantify a strong positive correlation between cyclin E and cyclin E/Cdk2:p27 and a strong negative correlation between p27 and cyclin E/Cdk2:p27, where these co-control coefficients are largely independent of which step is perturbed. These observations and model analyses lead us to hypothesize that RP is advanced in cancer cell lines as a result of an increased concentration of cyclin E/Cdk2:p27.
Even though the Novak and Tyson model does not explicitly model the localization of p27 from the nucleus to the cytoplasm (i.e. the transport of p27 from the nucleus is implicit in the reaction that degrades p27; Reaction 15, ), our results are consistent with the observation that the aggressiveness of cancer cells can be linked to a shift in the localization of p27 from the nucleus to the cytoplasm . As we have shown in our analysis, a decrease in the concentration of p27 is strongly correlated with an increase of the cyclin E/Cdk2:p27 concentration, which will advance the RP. This is in agreement with the apparent absence of RP in cancer cells, which might be translated as advancing RP to such an extent that it never prevents G1/S transition. The relatively high negative control of the consumption reaction of p27 (Reaction 15, ) is also in agreement with experiments showing that a reduction in p27 levels promotes the passage of the RP .
In cancer cells, RP control appears to be completely absent. On the basis of an extensive analysis of a detailed kinetic model, we hypothesize that such a shift of the RP to a much earlier point in the cell cycle might be caused by an increased concentration in the cyclin E/Cdk2:p27 complex concentration. We showed that a perturbation of cyclin E/Cdk2:p27 on its own was sufficient to shift the RP. In addition, we showed that the effect on RP upon perturbating any of the reaction steps could always be explained via a change in the cyclin E/Cdk2:p27 concentration, leading to the proposal that RP control by the reaction steps works via the cyclin E/Cdk2:p27 complex. The method that we have used to quantify the control of a reaction step on any of the systems variables can be used to identify the reaction steps that would have the biggest effect on the cyclin E/Cdk2:p27 concentration and thereby shift RP most strongly. An important implication is that it should be possible to shift RP in cancer cells back to its normal point via perturbations of steps that effect the cyclin E/Cdk2:p27 concentration.
R.C. and J.L.S. acknowledge the National Bioinformatics Network (South Africa) for funding and support for the JWS Online project. F.J.B. thanks the Netherlands Institute for Systems Biology and NWO for funding. A.C. acknowledges support from the Italian research fund FIRB (project RBPR0523C3). B.N. thanks UNICELLSYS, YSBN and BBSRC for funding. H.V.W. thanks EC-FPs BioSim, EC-MOAN, NucSys, UNICELLSYS, YSBN, ESF-FuncDyn, NWO-FALW and the BBSRC and EPSRC for support.