Cell cycle

Leaning on the differential model published by Chen et al. in 2004 [1], we have delineated a discrete, logical model that reproduces the main qualitative results reported in this study, in terms of cycle viability or arrest in a particular stable state, for the wild type as well as over one hundred mutant conditions [2]. In a first step, we have defined a regulatory graph encompassing the main documented interactions between core regulators of the cell cycle (Cdk/cyclins, APC, Cdk inhibitors).
For proper logical rules, in the wild type situation, our model accounts for the following sequence of events: firing of the origins of replication (ORI goes up), spindle alignment (SPN goes up), inhibition of the securin (Pds1 goes down), division (MITOSIS goes up to level 2) after the formation of a bud (BUD must reach at least transiently the level 1), and origin relicensing (ORI goes down). This sequence serves as a criterion to evaluate the viability of the cell. Based on the levels of activity of key variables (Clb5, Clb2, Pds1), we can divide the cycle into three different phases: G0/G1 (low Clb5 and Clb2 activity, either OFF or sequestered by Sic1/Cdc6), S/G2 (high Clb5 activity, i.e. not sequestered by the CKI, and low Clb2), and M (high Clb2 activity). The M phase is itself subdivided into prophase/metaphase (high Pds1, low Esp1, sister chromatids not separated) and anaphase/telophase (low Pds1, high Esp1). These rules are used to characterised cell arrests in various mutants. Although the logical formalism is particularly well suited to represent information fluxes (activation, inhibition), it is not adapted to the representation of mass flow, reactant consumption, or complex formation. Consequently, we represent the complexes implicitly: a complex is considered present if all its components are present; all components, whether regulatory or enzymatic subunits, regulate the targets of the enzymatic member, with a logical AND rule (or AND NOT in the case of sequestration). For example, in the case of the inhibition of the Clbs by the CKI, we have represented inhibition of the Clbs by introducing arrows from the CKI towards the Clbs targets, so that inhibition of the Clbs is represented by an inhibition of the components they activate and by an activation of the components they inhibit. In Chen's model, cytokinesis was triggered by a fall of clb2's activity below a certain threshold. To formulate this rule in the logical formalism, we introduced a multilevel CYTOKINESIS variable. This variable is fully activated (level 2) only when CycB is inactivated, either by a decrease of CycB level from level 2 to level 1, or by an increase of the CKI activity when Clb2 is still high. To differentiate between a rise and a fall of CycB activity, CYTOKINESIS has to be pre-activated (up to level 1) by high Clb2 activity before being allowed to reach level 2. Our model simulations qualitatively agree with the behaviour reported for the wild-type Yeast, as well as for over one hundred mutant conditions. This led us to further develop this model by defining additional regulatory modules such as the morphogenetic checkpoint. We also developed a model for mitotic exit based on more recent evidence. The resulting logical models are presented as separate entries of our model repository. In conclusion, our core cell cycle model served as a benchmark to assess the power of logical modelling applied to a complex oscillatory system, as well as the first step towards the development of a more comprehensive model of the budding yeast cell cycle network.

References

On the basis of a previous modelling study by Novak and Tyson [1] , we have recently proposed a generic Boolean model of the core network controlling the restriction point of the mammalian cell cycle [2].

For proper logical parameter values, the simulation of this Boolean model leads to dynamical behaviours (sequences of activations and inactivations of key regulatory products) in qualitative agreement with current experimental data.

However, as kinetic details are still lacking, many different (in)activation pathways are compatible with existing data, including fully synchronous transition pathways. To further evaluate these different possibilities, we have analysed the asymptotical behaviour of this network under synchronous versus asynchronous updating assumptions. Furthermore, we consider intermediate updating strategies to improve the computation of asymptotical properties depending on available kinetic data. This approach has been implemented through user-defined priority classes in the logical modelling and simulation software GINsim.

Regulatory Graph

The Figure 1 presents the regulatory graph corresponding to our Boolean model of the restriction point control for the mammalian cell cycle. In this graph, each node represents the activity of a key regulatory element, whereas the edges represent functional interactions between these elements. Blunt arrows stand for inhibitory effects, whereas normal arrows stand for activations. Details and justifications can be found in [2].

Figure 1: Regulatory graph

Simulations

Depending on CycD activity at the initial state, our model can lead to two different asymptotical behaviour:

• In the absence of CycD (representing a lack of growth factors), the system is quickly trapped in a stable state with (only) Rb, p27 and Cdh1 stably activated.
• In the presence of CycD, the system reaches a unique (multi-)cycle attractor, whose complexity depends on the updating assumption selected.

To illustrate the impact of the updating assumption, we present here the cyclic attractors (terminal, maximal, strongly connected components) corresponding to three different updating assumptions:

• Synchronous updating: all updating calls are applied simultaneously; as shown in Figure 2-A, the terminal cycle contains 7 states.
• Asynchronous updating: competing updating calls lead to alternative transitions; as shown in Figure 2-B, the terminal attractor contains 112 states.
• Mixed a/synchronous updating: the regulatory components are distributed into four different priority classes; as shown in Figure 2-C, the attractor contains 18 states. We have first built two priority classes, which arguably group faster versus slower biochemical processes. In the highest ranked transition priority class, we have included the degradations of E2F, CycE, CycA, Cdc20, UbcH10, CycB, as well as all transitions (in both directions) for CycD, Rb, p27 (Kip1) and Ccdh1. The remaining transitions correspond to synthesis rates (of E2F, CycE, CycA, Cdc20, UbcH10, and CycB) and are grouped in a lower priority class. Using these two priority classes, both considered under the asynchronous assumption, we still obtain a single terminal strongly connected component (not shown) involving 34 states (to compare with 7 in the standard synchronous treatment, versus 112 in the fully asynchronous case without priority). The analysis of this component reveals that some pathways are clearly unrealistic, as they skip the activation of some specific cyclin, for example. To eliminate these spurious pathways, one can further refine the priority classes, taking into account additional information. Here, we can exploit the fact that several transitions are controlled by similar regulatory mechanisms to group them in synchronous classes.

(A) synchronous	(B) asynchronous	(C) mixed

Figure 2: attractors, depending on the updating policy (Node order in the state labelling: CycD Rb E2F CycE CycA p27Kip1 Cdc20 Cdh1 UbcH10 CycB)

Mutant simulations

References