Logical modelling of cell cycle control in eukaryotes: a comparative study.

Logical modelling of cell cycle control in eukaryotes: a comparative study. Adrien Fauré 1, Denis Thiery 1,2 April 8, 2009 1 Aix-Marseille University & INSERM U928 - TAGC, Marseille, France. 2 CONTRAINTES Project, INRIA-Rocquencourt, Le Chesnay, France. Contact: {faure,thiery}tagc.univ-mrs.fr Abstract Dynamical modelling is at the core of the systems biology paradigm. However, the daunting complexity of regulatory networks controlling crucial biological processes such a cell division, the paucity of currently available quantitative data, as well as the limited reproducibility of large-scale experiments complicate the development of comprehensive quantitative models. In this context, qualitative modelling approaches oer a useful alternative or complementary framework to build and analyse simplied, but still rigorous dynamical models. This point is illustrated here by analysing recent logical models of the molecular networks controlling mitosis in different organisms, from yeasts to mammals. After a short introduction to the cell cycle and logical modelling, we compare the modelling strategies and dynamical properties of these dierent models. Next, leaning on their transposition into a common logical framework, we compare their functional structure in terms of feedback circuits. Finally, we conclude this article by discussing assets and prospects of qualitative approaches for the modelling of the cell cycle. 1 Introduction 1.1 Overview of the cell cycle 1.1.1 The organisation of cell cycle in eukaryotes The cell cycle - the process by which a cell reproduces itself, dividing into two daughter cells - occupies a central place in the life in both unicellular and multicellular organisms [1, 2, 3, 4]. In animals, cell cycle is connected with dierentiation [5], apoptosis [6], organ size [7], tissue maintenance and regeneration [8], ageing [9], immune response [10]... Unrestricted proliferation is a hallmark of cancer, and cell division plays a role in the development of various diseases. Indeed, therapeutic approaches [11] that target the cell cycle [12] are considered promising, and related anti-cancer [13, 6, 5], antiviral [14] and anti-fungal/bacterial drugs (antibiotics) [15] have been, or are being developed. During the cell cycle, a eukaryotic cell goes through a series of well dened phases, in the course of which it grows and replicates all its components, before eventually dividing into two usually roughly identical daughter cells. While cell growth and replication of most cellular components is a continuous process, DNA replication occurs during the S phase (S for synthesis), along with synthesis of the histones necessary for the packaging of the new DNA. At the end of this process, chromosomes are duplicated in two sister chromatids held together by cohesin. Specialised structures known as kinetochores or cell pole bodies are also replicated early in the cycle. Separation of the replicated material occurs during M phase (M for mitosis), itself subdivided into several subphases. Chromosomes are condensed in prophase. In the course of metaphase, they are attached at the level of their centromeres to the mitotic spindle, a microtubular structure that stems from the kinetochores located at the opposite poles of the cell, and align at the spindle equatorial plane. The transition to the next phase occurs only when all chromosomes are properly attached to each pole and aligned. Chromosomes separate in anaphase, and are decondensed during telophase. M phase ends up with proper cell division, or cytokinesis. S and M phases are usually separated by two gap phases, G1 (between M and S) and G2 (between S and M). A fth phase called G0 can be reached from G1, that corresponds to a quiescence state of the cell. Gap phases enable the cell to monitor its environment and internal state before committing into S or M phase. The cell cycle is highly regulated. Indeed, external and internal signals may halt the cycle at particular checkpoints. An important checkpoint called Start in yeast, or the restriction point in mammalian cells, controls the G1/S transition. This checkpoint integrates signals depending on the presence of nutrients, cell size, or contact with other cells, thereby coordinating cell pro- 1

liferation with cell growth and the needs of the organism. In the course of the metaphase to anaphase transition, the spindle checkpoints monitors chromosome attachment to the microtubules, and their alignment on the metaphase plate. Additional checkpoints monitor DNA damage at dierent points of the cycle. The picture just drawn describes the canonical cell cycle. Specialised variants exist, that present signicant dierences with the classical G1-S-G2-M scheme. In the early stages of development of frog embryos, for example, the rst divisions involve fast and synchronous successions of S and M phases, with no gap phases between them [16]. During Drosophila development, early divisions are also fast and synchronous, but further limited to nuclei within a large syncytium, until gap phases appear along with true cellularisation around cell cycle 13 [17, 18]. Various specialised cell types in animals and plants undergo partial or complete endoreduplication cycles enabling various rounds of replication of (portions of) chromosomes without intervening nuclear division [19, 20]. Finally, meiosis can also be considered a specialised variant of the cell cycle that produces haploid germ cells, in two rounds of division, from diploid precursors [21]. All these events involve a complex machinery of enzymatic complexes, molecular motors and cytoskeleton. Here, we focus on the delineation of the regulatory network controlling cell division. 1.1.2 The cell cycle molecular engine The cell cycle is controlled by a complex network of interacting proteins known as the cell cycle engine [22]. Regulatory components contributing to this molecular machinery control each other as well as a range of downstream processes necessary for cell duplication. These processes feed back on the engine, forming checkpoints able to halt the progression of the cycle and to ensure enough time to complete each crucial step. At the core of the cell cycle engine lies the MPF (Maturation or Mitosis Promoting Factor). Discovered in 1971 for its role in meiotic maturation of frog oocytes [23], MPF was later found to display oscillating activity, with a period coincident with that of the cell cycle [24]. In the course of the 1980s, MPF has been resolved as a heterodimer of cyclin and cdk (for cyclin-dependent kinase) [25, 26, 27]. Oscillations of the regulatory cyclin subunit, driven by an alternation of synthesis and degradation phases, control the activity of the enzymatic cdk subunit. A combination of positive and negative feedback circuits is responsible for these oscillations. Early work had already shown that the cell cycle can be blocked in stable states of high or low MPF activity [24]. The underlying multistable behaviour is ensured by various positive feedback mechanisms. On the one hand, MPF self-activates through a positive eect on cyclin synthesis, as well as via post-transcriptional modications controlled by the homologs of the Wee1 kinase and the Cdc25 phosphatase. On the other hand, MPF inhibits its own inhibitors, sometimes called the G1 stabilisers: the Cdk inhibitors (CKI) and Cdh1, an activator of the APC (Anaphase Promoting Complex). CKI sequester Cyclin-Cdk complexes, thereby inactivating them. Cdh1 activates the degradation of the cyclin subunit through the APC, a ubiquitinating enzymatic complex. Thus, in the course of cell proliferation, states with low MPF and high CKI and Cdh1 alternate with states with high MPF and low CKI and Cdh1 activity. How does the cell switch from a state of low cyclin activity to a state of high cyclin activity, and vice versa? The cyclin protein identied by Evans in 1983 has later been related to a larger family of cyclins, whose members peak at dierent time points in the cycle: G1 cyclins are active in late G1 and play a key role in the Start transition. These cyclins are represented by members of the cyclin E family. Homologous members of the Cyclin A family are activated at the G1 transition and trigger DNA synthesis; their expressions last until mitotic entry. B-type cyclins are mitotic cyclins, that promote entry into mitosis and the formation of the mitotic spindle, and whose degradation triggers mitotic exit and cytokinesis. The cyclin responsible for MPF activity belongs to this family. G1 and S cyclins play a major role in the transition from low to high MPF activity. Indeed, G1 cyclins are not inhibited by the G1 stabilisers [28]. In budding yeast, G1 cyclins Cln3 and Cln2 rst inhibit CKIs, allowing the activation of the S cyclins Clb5 and Clb6 [29, 30]. Together with the G1 cyclins, they inhibit Cdh1, allowing the accumulation of Clb1 and Clb2, the mitotic cyclins of budding yeast. Clb1 and Clb2 are sucient to maintain their own activity by triggering their own synthesis and inhibiting the G1 stabilisers. They further inhibit the G1 and S cyclins. The transition from high to low MPF state, which corresponds to mitotic exit, is regulated by another negative feedback circuit enabling mitotic cyclins to trigger their own degradation. Given the role of proteolysis in the inactivation of MPF activity, a factor triggering Cyclin degradation, under the control of the mitotic spindle, has been suspected early on [31]. It was not until the late 1990s that this factor has been identied as Cdc20, and its regulator as the checkpoint protein Mad2 [32, 33]. The activation of Cdc20 by Cyclin B [34, 35] completes the negative feedback circuit, by which mitotic cyclins trigger their own degradation. This circuit had already been postulated and integrated in mathematical models of the cell cycle [36] prior to the discovery of its molecular components. The progression of cell cycle is further constrained by checkpoint mechanisms that condition the activation and inactivation of key regulatory components to the 2

Cell cycle phase G0/G1 S/G2 M molecular denition low CycA and CycB activity high CycA, low CycB high CycB Table 1: Cell cycle phases. Components' names based on mammalian nomenclature. completion of specic events. Activation of Cdc20 by Cyclin B is controlled by the spindle checkpoint to ensure that sisters chromatids are not separated before chromosomes are properly attached to the spindle and aligned on the metaphase plate. Cdc20 promotes mitosis by triggering the degradation of cyclins as well as the separation of sister chromatids. Additional checkpoint mechanisms condition the completion of mitotic exit to the separation of sister chromatids by regulating the activation of Cdh1 and CKI by Cdc14. A G2/M checkpoint monitors both DNA damage and unreplicated DNA, thereby ensuring that replication is complete before entering M phase. In budding yeast, the morphogenesis checkpoint conditions the activation of Clb2 to the formation of a bud. Consistent with the crucial importance of cell division, cell cycle engine components are highly conserved among eukaryotes [37]. Table 2 presents the homology relationships existing between key regulatory components of the cell cycle control network in budding yeast, ssion yeast, arabidopsis, drosophila and mammals. However, substantial dierences exist between organisms in terms of precise wiring of the network as well as of timing of expression and activity pattern of regulatory components [38]. In this respect, Jensen et al. recently showed that timing of expression of key players, in particular, cyclins and Cdc20, is relatively consistent between dierent organisms, but that the timing of expression of many other cell cycle-regulated proteins is poorly conserved [38]. Moreover, components that are cyclically expressed or posttranscriptionally modied in one organism do not appear to be regulated in others. However, Jensen et al. showed that such components often take part in molecular complexes involving other cycle regulated subunit(s) (a principle called just-in-time assembly, [39]). In brief, although the molecular details may dier, the general organisation of the regulatory network may still be conserved. 1.2 Dynamical modeling of the cell cycle regulatory network 1.2.1 Quantitative modeling Rapid progress in the elucidation of the molecular nature of the cell cycle engine has triggered the development of mathematical models to explain how the observed oscillatory behaviour stems from molecular interactions. As far as cell cycle is concerned, mathematical modelling with systems of Ordinary Dierential Equations (ODE) is the most common approach [51, 52, 53]. Each equation gives the rate of change of the concentration or activity of a given component as a function of the concentrations of its regulators (see [54] for a recent review on ODE modelling). Several articles co-authored by Béla Novák and John Tyson during the last decades arguably embody the state of the art of dierential modelling of cell cycle regulation [36, 55, 56, 30, 57, 58]. Of particular interest is their comprehensive model of the budding yeast cell cycle, which enabled consistent simulations of the eect of over one hundred reported single or multiple mutations, for dierent growth conditions [30]. More recently, these authors have supervised the development of a generic model enabling consistent simulations of cell cycle in four dierent eukaryotes (xenopus, budding and ssion yeasts, and mammalian cells [58]). Numerical integration and analysis of dierential models have already led to specic predictions, some already experimentally validated (see for example [29] and [59], or [36] and [60]). In most cases, however, the results obtained should be considered as qualitative, despite the quantitative potential of the modelling method. This is due to two main issues. On the one hand, the precise nature of the mathematical relationships and the corresponding parameters remain dicult to estimate on the basis of available experimental data. On the other hand, the use of non linear functions complicates the analysis and forces the recourse to intrinsically partial numerical approaches (simulations, one or two-dimensional parameter bifurcation, etc.) [58, 61]. In addition, it is dicult to further extend large dierential models, as numerical instabilities arise when the number of variables and the complexity of control terms increases. For the same reasons, stochastic modelling is dicult to apply to large regulatory networks. However, a signicant step in this direction is made in [62], which presents a stochastic Petri net model of budding yeast cell cycle engine. 1.2.2 Qualitative modeling To cope with complex networks and match qualitative experimental data, one can rely on qualitative representations of molecular interactions in terms of graphs [63], Boolean models and their multilevel extensions [64, 65], standard Petri nets ([66] and references therein), or yet piecewise linear equations [67, 68]. Among these approaches, logical modelling is increasingly used to model cell cycle [69, 70, 71, 72, 73, 74]. Graph-based representations are certainly the most intuitive for biologists, as they arguably formalise their practice of drawing regulatory diagrams. Leaning on standardised regulatory graphs, logical models rely on 3

Budding Yeast Fission Yeast Drosophila Arabidopsis Mammals Function Cln3 Puc1 Cyclin D Cyclin D1 to D7 Cyclin D G1 progression [17, 40, 41] Cln1/2 - DmCycE Nicta;CYCA3;2 Cyclin E G1/S transition [42, 43, 41] Clb5/6 cig2 Cyclin A Cyclin A1, A2 and A3 CyclinA1 and A2 S phase progression, G2/M transition [42, 44, 40, 41] Clb1/2 Cdc13 cyclin B and B3 Cyclin B1, B2 and B3 Cyclin B1/2/3 G2/M transition and intra-m control [44, 40, 41] Sic1 Rum1 Rux, Dacapo KRP1 to KRP7 p21, p27kip1 Cdk inhibitors [17, 40, 45] Swe1 Mik1 Wee1 WEE1 WEE1/Myt1 Kinase, inhibits Cyclin B [40, 46] Mih1 Cdc25 Cdc25String - Cdc25B Phosphatase, activates Cyclin B [17, 40] Cdc20 Slp1 Fizzy Cdc20, Ccd52B Cdc20 Activator of the APC, active in mitosis [32, 47, 46] Cdh1 (HCT1) Ste9 Fizzy-related Ccs52A1, Ccs52A2 Cdh1 Activator of the APC, active in late mitosis and G1 [32, 17, 47] SBF, MBF MBF DmE2F-1 E2Fa, b, c E2F-1, -2, -3 Transcription factor, controls the G1/S transition [42, 48, 17, 40, 46] Whi5 - Rb Rb Rb binds and inactivates E2F [42, 49, 50] Table 2: Homology relationships between cell cycle regulatory components. Most of the components presented in this table are sequence homologs. However in some cases, even when sequence homologs have not been found, unrelated components may fulll homologous functions. This is the case for example for Nicta;CYCA3;2 in Plants (related to the Cyclin A family but functional homolog of Cyclin E), or Rb and E2F on the one hand, and Whi5 and SBF and MBF on the other. Not shown in the table, whereas Yeast usually have only one member of each family of component, plants and, to a lesser extent, mammals display plethora of paralogs (cf [41] regarding cyclins in plants for example). 4

Boolean rules to dene the eects of dierent combinations of interactions on their targets, thereby enabling qualitative simulations. In many cases, regulatory components can be simply considered as present or absent (or yet as active or inactive), thereby matching qualitative reasoning pervasive among biologists. Logical modelling has been successfully applied to various biological regulatory networks, from the bacteriophage lambda lysis/lysogeny switch ([75] and references therein), to the polarisation of the immune response [76, 77, 78, 79], the specication of arabidopsis ower organs [80, 81, 82], the delineation of the segmentation pattern in drosophila embryo [83, 84, 85, 86, 87, 88], the development of sense organs in drosophila [89], the dierentiation of keratinocytes [90]... All these applications deal with decision-making systems enabling the selection of specic cell fates, and can thus be analysed in terms of alternative stable states. In contrast, in the case of the cell cycle, what matters is the actual sequence of transitions, the succession of changes in the levels of activity of components controlling the progression of the cell through the dierent phases of the cycle. This does not impede many regulatory components to adopt a switch-like behaviour [91], which can be approximated by logical variables. Proper logical modelling of cell cycle engine is still in its infancy. However, a series of Boolean models have been recently proposed, from [69] presenting a very simplied logical model of the mammalian cell cycle engine, to increasingly sophisticated models for budding yeast [70, 74], ssion yeast [72, 73], and mammals [71]. Before turning to the description and comparison of specic models in more details, the following section encapsulates the main aspects and variants of the Boolean modelling framework. 1.3 Logical modeling 1.3.1 Regulatory graph and Boolean rules The skeleton of a logical model is often represented in terms of an oriented graph, where the nodes denote regulatory components, and the arcs denote cross-regulatory interactions. A logical variable is associated with each node, which can represent dierent types of biological elements, from molecular compounds (genes, mrna, proteins, protein complexes...) to phenomenological variables (e.g., cellular mass, DNA synthesis, or cell division). These variables take integer values within an interval ranging from 0 (absence of a component or undetectable activity) to a maximum activity level. In most cases, this maximum value is equal to one (binary variables). If all variables are binary (i.e., if they take their values within the [0;1] interval), we face a Boolean model. However, it is often necessary to consider intermediary activity levels for some nodes. One possibility is to use multiple binary variables to represent the dierent activity levels of a single component [72, 73]; alternatively, one may use multilevel logical variables [64]. Similarly, arcs can be dened for a particular interval ranging from i (i>0) to the maximum level of the source node. They can represent dierent types of inuences, from transcriptional regulation (activation or repression of the transcription of a particular gene), to biochemical reactions (de/phosphorylation, degradation...), to subtler situations (modulation of the activity of another component within a complex, regulation through implicit pathways...). The regulatory graph is a useful tool to get a rst glance of a logical model, but a logical model fundamentally consists in a set of logical rules, each directing the evolution of one component. In our context, one can distinguish two main approaches to dene these logical rules. On the one hand, logical rules can be dened as an arithmetic sum of (weighted) positive and negative inuences received by each node. Depending on whether the sum lies above, below or at a particular threshold, the value of the node considered will tend toward 1, 0, or remain unchanged [70, 90, 72, 73]. Alternatively, Boolean formulae combining Boolean variables, NOT, AND and OR operators can be used to dene the target value of a component depending on the values of its regulators (see, e.g., [71] or [74]). 1.3.2 State updating, stable states, and mutant simulations To study the dynamical behaviour of a logical model, we have to further select an updating schedule. Indeed, at a given logical state (dened as a specic combinations of values for the network components), the logical rules may imply updating calls for several components (there is an updating call on a component whenever the target value is dierent from the current one). The simplest assumption considers that all current updating calls are updated simultaneously at each step (synchronous schedule). The resulting behaviour is deterministic since each state can have at most one successor state (although several states may lead to the same state). Alternatively, one may consider that one component at most can be updated at each time step (asynchronous schedul e), either considering all possible single changes, or using a deterministic transition order, or yet using probabilistic selection rules. For a given schedule, the dynamical behaviour of a logical model can be represented in the form of a state transition graph, where the nodes denote dierent states of the system and the arcs denote (allowed) transitions from one state to another [70, 71, 72, 73]. Alternatively, a particular trajectory corresponding to a linear sequence of transitions may be represented in a table, where the columns correspond to the dierent components and successive rows to successive dynamical steps 5

[74]. Within a logical framework, it is relatively straightforward to dene and simulate mutants and other kinds of perturbations, provided that they correspond to clearcut qualitative outcome [71, 74]. Various solutions have been proposed to rene logical simulations while avoiding the complexity of fully asynchronous schemes, for example by distinguishing between fast and slow subnetworks, simulated sequentially [92, 93, 86], or by grouping transitions into ranked synchronous or asynchronous classes [71], or yet by introducing delaying nodes [74]. Whatever the updating schedule, a logical model conserves the same stable states (states for which each component has a target value identical to its current one, and thus from which no transition is possible). In this respect, powerful algorithms have been recently proposed to compute stable states in large regulatory networks [90, 94, 95]. This is particularly useful to exclude the occurrence of stable state for a viable cell cycle model, or yet to identify specic mutant arrest states [71, 74]. As we shall see, in all the models considered here, the underlying network wiring appears tight and robust enough to enable the generation of fully synchronous state transitions remarkably consistent with available biological data. 1.3.3 Regulatory circuits and their dynamical roles A regulatory circuit is formally dened as a simple circular path in the regulatory graph dened above. The sign of a circuit is given by the product of the signs of its constitutive interactions and reect the (indirect) eect of each component of the circuit on itself [64, 96, 97]. It is now well established that positive regulatory circuits are necessary for multiple stable states or attractors, whereas negative circuits are needed to generate sustained oscillations [98]. As mentioned earlier, negative and positive regulatory circuits have been regularly involved at the core of the cell cycle engine, but rigorous delineation of circuit roles is complicated by the increasing number of crossregulations in recent models, implying an exponential increase of the number of regulatory circuits. In this respect, within the logical framework, it is possible to identify all feedback circuits embedded into a complex regulatory networks and determine the role of each of these circuits depending on the levels of external regulators [94]. 1.3.4 Tools Dierent tools are available to simulate and analyse Boolean networks. For example, DDLab enables the computation of the attractors of large synchronous Boolean networks [99]. REVEAL oers means to reverse-engineer Boolean networks from dynamical data [100]. CellNetAnalyser implements several methods to analyse uxes in metabolic or signal transduction networks represented in terms of bipartite graphs [92]. Finally, GINsim supports the denition, simulation and analysis of Boolean and multilevel logical models for regulatory networks [101]. All the models reviewed here have been encoded or adapted using GINsim and are made available in the companion model repository (www.gin.univ-mrs.fr). 2 Logical models of the cell cycle engine in eukaryotes 2.1 Budding and ssion yeast cell cycle models Yeast, and most particularly budding yeast, has for long been a reference model system for the study of the cell cycle control mechanism. Saccharomyces cerevisiae is a simple, unicellular organism, used from time immemorial for brewing and bakery - hence its common name of brewer's or baker's yeast. During the 1970s, extensive mutation screens have been carried on, facilitated by the possibility to grow haploid cells, thereby avoiding the pitfall of recessive mutations. Many key regulators of cell division have been discovered in this context [102]. Consequently, yeast cell cycle also constitutes a system of choice to develop or assess computational methods dedicated to the dynamical modeling of biological regulatory networks. The model of the budding yeast cell cycle presented in [70] gathers all the usual suspects of cell cycle regulation: four groups of cyclins with dierent functions (Cln3 activates Cln1 and Cln2, which inhibits Cdh1 and Sic1, allowing on the one hand Clb5 and Clb6 to activate S phase, and on the other hand, Clb1 and Clb2 to activate M phase); the Cdk inhibitor Sic1; activators of the APC Cdc20 and Cdh1; plus several transcription factors (SBF and MBF for Cln1,2 and Clb5,6, Mcm1/SFF for Clb1,2, and Swi5 for Sic1). A few other key players are abstracted (Pds1) or lumped together with other components (Cdc14 with Cdc20). This scheme matches current textbook descriptions of the core engine of the budding yeast cell cycle (see, e.g., [103] Figure 3-34). In their model, Li et al represent all components by Boolean variables and use generic rules (arithmetic sum of positive or negative inuences, with weight 1 or -1, respectively, and a threshold xed at 0) to compute the response of by each component to multiple regulatory inputs. The logical rules can thus be directly derived from the topology of the network. However, in the case of components that are not negatively regulated (namely Cln3, Cln1,2, Mcm1/SFF, Swi5 and Cdc20&Cdc14), the authors have introduced self-degradation loops to inactivate them when all their activators are OFF. Simula- 6

tions are carried out synchronously. The authors simulate the behaviour of their model from all possible initial states, showing that most trajectories converge towards a main attractor, a stable state corresponding to the G1 phase. Other stable states attract minor portions of the state space. Cell cycle simulations are performed starting from the main stable state, and switching Cln3 on: the resulting sequence of synchronous transitions matching cell cycle evolution up to returning to G1 resting state. The authors further analysed the eect of perturbation by introducing slight changes in the network topology (adding, suppressing or changing the sign of one arrow). However they did not systematically analyse the eect of reported mutations. We have transcribed the logical rules used by the authors into our formalism. The resulting model is shown on Figure 1, top left. A colour code highlights protein homology relationships with other organisms. Although the global topology of the network is preserved, our transcription highlights the presence of positive regulatory circuits on several components, namely SBF, MBF, Clb5/6 and Clb1/2, Cdh1 and Sic1 (i.e., all variables that have both positive and negative regulators). In contrast, the negative loops introduced by the authors to match available kinetic data do not appear in our regulatory graph, as they do not correspond to true selfdegradation or inhibitory mechanisms; in our transposition, the corresponding components are assigned zero basal value to implement spontaneous decays. Analysis of the functionality of the feedback circuits of this model conrms that the positive self-activating feedback loops play an important role in its dynamics, in particular regarding the maintenance of alternative, artefactual stable states. This observation led us to simplify Li's model by eliminating all ad hoc positive loops (Figure 1, top right). For proper logical rules, the resulting model converges towards a single stable state corresponding to G1. However, in the absence of SBF and MBF autocatalytic feedback loops, the start signal is not maintained long enough to ensure a proper sequence of state transitions towards the G1 stable state under the fully synchronous updating mode. This problem can be compensated by the introduction of delays for SBF/ MBF switching OFF (using the priority system introduced in [71]). Another possibility would be to include additional biological data showing that Cln1/2 and Clb5/6 may regulate their own transcription factors, and that there is some redundancy between SBF and MBF [104, 105, 106]. Such feedbacks would ensure that Cln2 and Clb5 signals are maintained long enough for Clb2 to take up the repression of Sic1 and Cdh1. However, integration of these data into the original modelling framework would largely perturb the behaviour of the system, due to the rigidity of the dynamical rules. Using this revised model, we could simulate various documented mutants. In particular, overexpression of Cln3 yields a cyclic attractor whose sequence corresponds to the cell cycle (even under the synchronous updating mode), consistent with biological data. As already hinted by [74], this was not the case in the original model, where overexpression of Cln3 yielded a M-phase arrest. This cyclic behaviour is consistent with the result of a feedback circuit analysis, leading to the identication of functional positive and negative circuits in the adapted model (see Table 3). In [72], Davidich and Bornholdt proposed a Boolean model of the ssion yeast cell cycle, dened in terms of equations similar to those found in [70]. Consistent with biological data showing that a single B-type cyclin, Cdc13, is sucient for both S and M phase progression [107], and with the preponderance of the G2 phase in the ssion yeast cell cycle [108], this model puts the emphasis on the control of Cdc13 activity. The starter kinase SK is the only component representing G1 or S cyclins in this model. Consistent with the results of [107] and with the quantitative model published by [57], Cdc13 is assigned two activity levels, represented by two distinct Boolean variables. We have transcribed this model using GINsim (Figure 1, bottom left). Similar in its construction, the model yields results comparable to those reported in [70]: a main attractor corresponding to G1 gathers most trajectories; switching on the start signal from this attractor triggers a sequence of state transitions qualitatively matching cell cycle progression. However, this model also suers from similar drawbacks, as it generates several spurious stable states, due to the introduction of ad hoc positive feedback loops. In contrast with the budding yeast model presented by [70], however, circuit analysis of this ssion yeast model identies functional positive as well as negative feedback circuits. Accordingly, overexpressing the Starter kinase SK, which corresponds to both Cln3 and Cln1/2 in the budding yeast model, does yield a cyclic attractor. However, oscillations do not aect the G1 stabilisers Ste9 and Rum1. To get rid of the spurious stable states, we have adapted Davidich & Bornholdt's model. The resulting network is shown Figure1, bottom right. All ad hoc positive feedback loops have been removed and compensated by the introduction of priorities to account for the maintenance of the start signal and its eect on Rum1 and Ste9. Furthermore, we have replaced the two Boolean Cdc13 variables by a single, ternary one. The resulting model reproduces the cell cycle sequence, provided priorities are used to maintain the Start signal active long enough. It has three functional two-elements circuits: one negative and two positive (Table 3). Overexpression of SK yields a cyclic attractor, where all downstream components oscillate, and whose sequence corresponds to the cell cycle. Recently, a more detailed model for the budding yeast 7

Figure 1: Top: Budding yeast cell cycle. Left: model transcribed from [70]; dynamical rules are generated from the network structure as the sum of the positive and negative inuences exerted on each nodes by its regulators. Right: an adaptation of the same model using adjusted logical rules. See text for details. Bottom: Fission yeast cell cycle. Left: model transcribed from [72]. Dynamical rules are generated as in [70]. Note that two variables are used to represent dierent levels of activity for Cdc13 in a strictly Boolean formalism. Right: an adaptation of the same model using adjusted logical rules. Cdc13 is now represented by a single, ternary variable. Graphical conventions: ovals: Boolean variables; rectangles: multilevel variables; green arrows: activations; red blunt arrows: inhibitions; node colours emphasise homology relationships. 8

cell cycle control network has been published by Irons [74]. This model encompasses several checkpoint modules in addition to the core cycling engine, albeit in a very schematic form. The FEAR and MEN pathways are present, but reduced to single variables, and the morphogenesis checkpoint, that conditions entry into mitosis to the formation of a BUD, as well as the spindle checkpoint, that blocks exit from mitosis until chromosomes are properly aligned on the metaphase plate, are represented by direct eect from the variables representing the bud and M-phase onto the targets of these checkpoints. Several of these modules have been previously modelled in the dierential framework [109, 30, 110], but, to date, these modules have not yet been integrated into a single comprehensive model. Iron's model thus provides a simple, but relatively extended view of the budding yeast cell cycle. Irons' model is dened in terms of a series of logical rules, completed with a system of temporisation based on the introduction of additional nodes to delay the activation or degradation of particular components. Contrary to previous models ([70] and [72]), this one yields a single, cyclic attractor. Both fully synchronous and temporised dynamics have been considered, yielding results qualitatively consistent with available kinetic data. Finally, Irons further simulated the behaviour of several mutants in terms of viability or arrest in a particular phase of the cycle, as well as checkpoint-mediated arrest. We have transposed the logical rules using GINsim and simulated Iron's model under the synchronous mode. The corresponding regulatory graph is shown in Figure 2 and enables the recapitulation wild-type and mutant simulation results for the rules dened by Irons. Using our feedback circuit analysis tool, we could further identify the underlying functional circuits (3). 2.2 Logical cell cycle models for multicellular eukaryotes Proper dynamical modelling and simulation of cell cycle in multicellular eukaryotes still constitutes a daunting challenge, in particular regarding mammalian systems. Beyond the involvement of additional components such as Rb and E2F proteins, complexity stems from the high level of redundancy within each family of components. Indeed, databases [111, 112, 113] and publications point to innumerable cyclins [114, 115], cyclin-dependent kinases [116], cdk inhibitors [117], E2F transcription factors [118], pocket proteins [119], Cdc25 phosphatases [120], etc. Most of these components are in fact essential in particular cell types or growth conditions, but dispensable in others. The overwhelming complexity of the mammalian cell cycle network (see, e.g., [63, 121]) discourage the integration of all details into a comprehensive dynamical model. However, provisionally leaving aside redundancies and specicities, several groups are working on the delineation of generic models of the mammalian cell cycle, leaning in part on current knowledge about the yeast cell cycle engine (see, e.g., [69], [56] and [71]). In 2006, we have proposed a generic Boolean model for the mammalian cell cycle [71], based on the quantitative model presented in [56] (Figure 3, left). This model encompasses the key players of the original model: four cyclins (D, E, A and B), the CKI p27kip1, the transcription factor E2F and its inhibitor Rb, as well as the activators of the APC Cdc20 and Cdh1. The eects of growth factors are represented by the activation of Cyclin D. More importantly, we introduced the recently discovered UbcH10 [122] to account for the fact that Cyclin A is a target of Cdh1 in late mitosis or early G1, but not in late G1, thereby allowing the rise of CycA and the concomitant inactivation of Cdh1. Compared to the models described above, an original aspect lies in the implicit representation of molecular complexes. For example, the sequestration of cyclin-cdk complexes by the CKI (p27 in the case of the mammalian cell cycle, shown in yellow in the Figures of this paper) is represented by direct inhibitions in [70], [72] and [74]. In contrast, in [71], we implicitly consider that such complexes are formed whenever all their components are activated. For example, sequestering and inactivation of cyclin-cdk complexes by CKI are modelled by arcs from CKI onto cyclin targets, along with logical rules restricting cyclins activities on their targets to the absence of p27 (see [71] for more details). This representation has two main advantages: (i) a lowering of the number of logical components to consider; (ii) immediate component activation following their release from a sequestering complex. Simulation of this model yields a cyclic attractor qualitatively matching reported cell cycle oscillations. In addition, several mutant phenotypes could be properly recapitulated (cf. [71]). However, this very simplied Boolean model has several drawbacks. In particular, Rb remains OFF in the wild-type cycle, while some mutant simulations disagree with reported phenotype (for example, p27- and CycEop cells should cycle in absence of CycD, whereas in our model they remain arrested in G1). The consideration of additional components or of multiple levels for some regulatory factors (such as Rb) could in principle resolve these discrepancies. Mammals display a large variety of cell types, some of which do not follow the canonical G1-S-G2-M pattern of cell division: asymmetric division, endocycles, meiosis... Such variations happen in various organisms. However they have been particularly well studied in the context of drosophila development [17, 123]. We are currently developing a model of the cell cycle control network in drosophila, with the aim of accounting for developmental variants of mitosis: namely, endocycles 9

Figure 2: Budding yeast cell cycle regulatory graph build on the basis of [74]. This model reproduce the simulations presented in Figure 3B of [74] for the same logical rules, under the synchronous updating mode (without delaying nodes). Graphical convention as in Figure 1. 10

Figure 3: Left: A generic Boolean model for the mammalian cell cycle (for details, see [71]). Right: Drosophila cell cycle model. Graphical conventions: thinner arrows indicate indirect regulation through complex formation, other graphical conventions as in Figure 1. and syncytial cycles. Endocycles (or endore(du)plication cycle) happen in particular drosophila cell lineages, including larval salivary glands and chorion (eggshell) cells [124]. They consist in nested S phases without intervening mitosis, which results in polyploid or polytene cells, characterised by gene amplications enabling very high production of specic proteins. Syncytial cycles occur during the early developmental stages of the drosophila egg, where S and M phases succeed at a very fast pace, in absence of growth, gap phases or cytokinesis. The nucleus divide synchronously until the thirteenth round of division, after which cellularisation occurs. We have derived a model using published data on drosophila cell cycle, completed when necessary with information transferred from other organisms, principally mammals, using orthology relationships between regulatory components. Shown in Figure 3, right, this model can be used to simulate the canonical cell cycle, syncytial cycles, as well as endocycles. Transition from canonical cycles to endocycles is regulated by Notch and Archipelago [125, 126], while the transition from syncytial cycles to canonical cycles originates from the titration of maternally expressed cyclins [127], along with de novo zygotic expression of canonical cell cycle components. 3 Discussion 3.1 Similarities and variations and in cell cycle models Beyond homology relationships between regulators, mathematical models of the cell cycle highlight the conservation of the regulatory circuits governing cell division. In 1998, using ordinary dierential equations, Novák et al. [128] proposed a prospective cell cycle model, emphasising the universality of the corresponding regulatory mechanisms. More recently, [58] published a generic, modular model of the eukaryotic cell cycle, enabling the recapitulation of the results obtained by Novák's and Tyson's groups for dierent organisms, by considering specic subsets of modules and tuning appropriate parameters. Here, we have focused on recent Boolean models of the eukaryotic cell cycle. These models have been published by dierent groups, and designed to reproduce the behaviour of the core cycle oscillator in dierent organisms. Still, their comparison reveals that, in spite of the specicity of the wiring of each model and some dierences in modelling assumptions (e.g. regarding the use of Boolean versus multilevel components, or the representation of protein complexes), model behaviour rely on similar sets of conserved functional circuits. Indeed, once transposed into a common logical framework, these dierent models can be compared in terms of functional circuits. The results of this comparison are summarised in Table 3. A general observation is that only a relatively low number of positive and negative circuits are found functional, even in the most complex 11

models analysed. Strikingly, two circuits are conserved in all ve models: one negative circuit involving the homologs of Cyclin B and Cdc20, and one positive circuit involving the homologs of Cyclin B and Cdh1. Additional circuits are conserved in subsets of organisms. Following cyclin discovery in the 1980s[25], and the demonstration [26] that it corresponds to Masui and Markert's MPF [23], oscillations of cyclins, and most particularly of B-type cyclins, have been placed at the heart of the molecular control of the cell cycle. Cyclin activity oscillate between a stable low state, corresponding to interphase, to a stable high state, corresponding to mitosis (see [31] for an early review). The positive feedback between Cyclin B and Cdh1 (an activator of APC) has been considered at the basis of B cyclin bistable behaviour since long [129]. Other positive feedbacks have been identied, most particularly feedbacks involving the Cdk modiers Cdc25 (activator) and Wee1 (inhibitor) [36, 130, 131]. However, these two factors have not been considered in most of the models reviewed here, with the exception of the ssion yeast and drosophila models. The negative feedback circuit between Cdc20 and Cyclin B singled our by our analysis is also generally considered as crucial. This circuit had already been singled out [31], and integrated in mathematical models of the cell cycle [36] well before the deciphering of the corresponding molecular mechanisms [32, 34, 33, 35]. Interestingly, Cdc20 has been outlined by Jensen et al. [38] as one of the components for which periodicity is the most conserved among eukaryotes. Our comparative study conrms that this negative feedback circuit is conserved and functional in all four organisms considered. The roles of the connected positive and negative circuits highlighted by our study has been previously emphasized as crucial for cell cycle oscillations by Pomerening[132]. Cross et al. further proposed that this combination could be widespread among eukaryotes [133]. 3.2 Outlook A limitation or our comparative analysis stems from the limits of the models themselves. As discussed above, published logical models are relatively simple, focusing on the core engine of the cell cycle. Various components and interactions are missing. For example, the logical mammalian model [71] inspired by [56] overlooks the possibility that p27 also sequesters Cyclin B-Cdk1 complexes [134]. As previously mentioned, dierential modelling has been the most popular approach as far as cell cycle modelling is concerned. Recent eorts focus on the development of less comprehensive, but more detailed models, focusing on the G1 phase [135, 136, 137, 138, 139, 140], or the spindle assembly checkpoint and metaphase to anaphase transition [141, 142]. However, the complexity of the mammalian engine remains daunting: beyond the introduction of components specic of higher eukaryotes, such as Rb and E2F factors, complexity stems from the high level of redundancy within each component family. Databases [111, 112, 113] and publications point to innumerable cyclins [114, 115], cyclin-dependent kinases [116], cdk inhibitors [117], E2F transcription factors [118], pocket proteins [119], Cdc25 phosphatases [120]... Most of these redundant family members are in fact essential components in particular cell types, particular pathways, but dispensable in others. Due to the lack of knowledge on the specicity of these family members, and to the overwhelming complexity of a comprehensive map of the cell cycle network [63, 121], the integration of all these details into a comprehensive dynamical model is presently out of reach. In this respect, modular approaches may help to design comprehensive models or to simplify regulatory graph by selecting representative members of each family (cf. [121] for a rst step in this direction). To cope with such complexity, the logical framework has several assets. Logical models are relatively easy to dene, and do not necessitate detailed quantitative data. As mentioned above, various tools are available to study the dynamics of logical models, in particular for the identication of stable states, as well as for the analysis of the dynamical roles of regulatory circuits. Logical models can also be easily used to predict known or novel behaviour in response to various types of perturbations. Furthermore, logical modelling has interesting compositional properties, which could be exploited to develop incremental modelling strategies. Indeed, the core cell cycle engine can be divided in specic modules controlling each transition. Moreover, various pathways and checkpoints inuence the core engine to regulate cell cycle progression, including response to growth factors or DNA damage. However, logical model cannot generate truly quantitative predictions. It is thus important to build bridges between qualitative and quantitative approaches. For example,[71] transposes Novák & Tyson's generic model for the mammalian cell cycle in the logical framework, whereas [73] proposes a more formal translation method and applies it to Novák's ssion yeast cell cycle model [143]. Referencing to other biological applications, several software suites, including SQUAD [144] and Odefy [145], even automate the translation of logical models into generic dierential equations. In parallel, ODE models have also been adapted into probabilistic frameworks, such as stochastic Petri nets[62]. Beyond the core cycling network, numerous control modules could be integrated to the mammalian model, starting with the heavily studied and modeled the DNA damage checkpoint [146, 147, 148, 149, 150, 151, 152, 12

Organism Budding yeast (adapted from [70] Fission yeast (adapted from [72]) Total number of circuits Functional negative circuits Functional positive circuits Budding yeast [74] Drosophila Mammals [71] 64 9 701 199 132 Clb1_2/MBF/Clb5_6 Cdc20_Cdc14/Clb5_6/Mcm1_SFF Clb1_2/Cdc20_Cdc14 Sic1/Clb1_2 Clb1_2/Cdh1, Sic1/Clb5_6 Cdc2_Cdc13/Slp1 Cdc20/FEAR/Cdc14/CKI/Clb2 SFF/Swi5/CKI CKI/Clb2/MEN/Cdc14 Clb2/MEN/Cdc14/Cdh1 Cdc20/Clb2 Ste9/Cdc2_Cdc13 Cdc2_Cdc13/Rum1 Cdc2_Cdc13[1]/Cdc25 CKI/Clb2 Clb2/Cdh1 CycA/E2F Fzy/CycB CycE/Dap CycE/Rb Fzr/CycB CycA/Fzr CycB/Stg CycA/ Rb CycB/Wee1 Cdc20/CycB Rb/CycA CycA/Cdh1 Rb/CycE CycB/Cdh1 Table 3: Functional regulatory circuits identied in logical models for the core regulatory network controlling cell cycle in budding yeast, ssion yeast, drosophila, and mammals (all auto-regulatory loops are functional but not listed here). A color code emphasises homology relationships between key regulatory factors (cf. Table 2). 13