Modelling the cell cycle regulatory network

Transcription

1 Chapter 3 Modelling the cell cycle regulatory network 3.1 Dynamical modelling Dynamical properties of a system arise from the interaction of its components. In the case of the cell division cycle, cellular behaviour depends on various types of components (genes, mrna, proteins, protein complexes, up to the cell itself) that all influence each other, forming multiple, intricate regulatory circuits. Mathematical models are necessary tools to handle such complex systems. In a first approach, a model can consist in a static regulatory network. Several sets of rules have been proposed to formalise and encode knowledge on regulatory relationships between molecules and genes implicated in the control of biological processes (Kohn, 1999; Pirson et al., 2000; Le Novere et al., 2008; Longabaugh et al., 2008). Beyond oriented edges, these formalisms often offer a complex semantics that associates biological meaning to the shape and colour of arcs and nodes of the diagram, adapted to different degrees of abstraction, depending on the processes to be described, the biological questions to be addressed, and the available data. This type of approach has been applied to the mammalian cell cycle (Kohn, 1999, cf. Figure 3.1), the Rb/E2F pathway (Calzone et al., 2008, cf. Figure 3.2), and is central to databases such as KEGG (Kanehisa and Goto, 2000) or data integration softwares like Biotapestry (Longabaugh et al., 2008) or CellDesigner (Funahashi et al., 2003). Graph-based representations are certainly the most intuitive for biologists, as they arguably formalise their practice of drawing regulatory diagrams. Not only are these diagrams extremely useful in recapitulating in a non-ambiguous way the precise relationships between regulatory components, they can also be used to gain fruitful insights in terms of network topology. An approach of modularity can be made based on topology to identify functional subsystems (Kohn, 1999; Calzone et al., 2008). Homology relationships can be used to identify conserved pathway motifs (Kanehisa and Goto, 2000). However these representations remain static by nature even though some authors have proposed to sequentially represent network dynamics through this type of diagrams (Longabaugh et al., 2008). The dynamical behaviour that emerges from regulatory networks may be intuitively apprehended from the static representation of simple systems. However, in most cases dynamical behaviour can not be 37

2 38 CHAPTER 3. MODELLING THE CELL CYCLE REGULATORY NETWORK Figure 3.1: Map of the mammalian cell cycle control network. Published in 1999, this diagram synthetised the data on the molecular interactions between cell cycle regulatory components. Unambiguous graphical conventions on the shape and colours of the symbols distinguish different types of regulation (transcriptional activation or repression, post-transcriptional modifications, molecular complex formation...). Parts of the network that fulfill a specific function are grouped into functional subsystems. From Kohn (1999) and

3 3.1. DYNAMICAL MODELLING 39 Figure 3.2: Rb-E2F transcription (top) and protein-protein interaction (bottom) map. Blue denotes tumor suppressors, red denotes oncogenes. From Calzone et al. (2008) and

4 40 CHAPTER 3. MODELLING THE CELL CYCLE REGULATORY NETWORK inferred from the wiring (Ingram et al., 2006), and the complexity of molecular regulatory networks and particularly of the cell cycle engine is beyond the scope of our limited intuition. Moreover, the state of a biological system may depend on its own history, such that it is not possible to predict the state of a system based only on the inputs it receives. For example, a cell that differentiates upon receiving a signal will remain differentiated even after the signal is removed (Thomas and D Ari, 1990; Xiong and Ferrell, 2003). Dynamical, mathematical modelling is thus a necessary tool to help us decipher the complexity of the cell cycle engine. For example, current descriptions heavily rely on concepts of positive and negative feedback loops, switches and oscillations. Bistability has been particularly investigated (Pomerening et al., 2003; Pomerening, 2008). Although these concepts have been considered out of the mathematical framework (Murray and Kirschner, 1989b), they are central to mathematical modelling approaches of the cell cycle (Tyson et al., 2003; Pomerening et al., 2003; Novák et al., 2007). Mathematical modelling is indeed well suited to study the emergence of dynamical behaviour from a regulatory network. In particular, the emergence of multistability and oscillations from simple positive and negative feedback loops is well known, both in the ODE and logical formalisms (Thomas and D Ari, 1990). Applied to the cell cycle, mathematical approaches can be used to validate a regulatory network by demonstrating that it can produce the expected behaviour, to suggest experiments to discriminate between two hypotheses, to predict unknown interactions, and to evaluate the robustness of the dynamical behaviour (Ingolia and Murray, 2004). Out of the many modelling approaches (de Jong, 2002; Schlitt and Brazma, 2007), two bear a particular relevance to my work on the cell cycle: ordinary differential equations (ODE), arguably the most widespread formalism for cell cycle modelling, and the logical formalism. Both approaches have their assets and drawbacks, and complement rather than oppose each other. ODE models usually focus on the biochemical scale, providing quantitative, mechanistic description of biological interactions: not only who interacts with who, but what are the affinities, the rates of synthesis and degradation, how components diffuse through the cell... Typically, ODE models rely on mass action law or Michaelis-Menten kinetics to describe the temporal evolution of the concentrations of the components of a system (see Sible and Tyson (2007) for a recent review on ODE modelling). This approach is extremely powerful, but focuses on a very local scale and may be difficult to implement on a large scale. It is not by chance if classical metaphors for the phenomenons modelled with this quantitative approach come from Newtonian physics: switches and springs, balls and seesaws... (cf. Figure 3.3). Although extremely useful to understand the behaviour of regulatory interactions at the molecular scale, this approach may be difficult to apply to larger networks, due to the overwhelming problem of parameter estimation. In contrast, the logical formalism has its focus on a higher scale, closer to genetic data than to biochemical data, and rely on simply knowing that knocking the gene X out will cause such particular defect, or that Y is epistatic to Z. It is a much more coarse-grained, qualitative approach, but it can offer a more global view on the dynamics of larger systems. Logical modelling is introduced in further details in Part 4.

5 3.2. MODELS OF THE CELL CYCLE NETWORK 41 Figure 3.3: Examples of physical metaphors for cell cycle control mechanisms. Left: a seesaw, a metaphor for the positive feedbacks controlling the activity of cyclins in the cell (Tyson et al., 2002); the seesaw has two stable states: one with the left end up and the ball on the right, the other with the right end up and the ball on the left. Right: irreversible switch; the switch from the upper to the lower steady state depends on the strength of a stimulus (water); if the stimulus is strong enough, the system switches to the lower steady state; a positive feedback (the ball) makes this transition irreversible, by maintaining the lower steady state even if the stimulus is removed (Novák et al., 2007). 3.2 Models of the cell cycle network Quantitative models Rapid progress in the elucidation of the molecular nature of the mechanisms controlling cell division 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 ODE systems is the most common approach (Norel and Agur, 1991; Goldbeter, 1991; Tyson, 1991). Several articles co-authored by Béla Novák and John Tyson during the last decades arguably embody the state of the art of differential modelling of cell cycle regulation (Novák and Tyson, 1993; Ciliberto and Tyson, 2000; Novák and Tyson, 2004; Chen et al., 2004; Sveiczer et al., 2004; Csikász-Nagy et al., 2006). Of particular interest is their comprehensive model of the budding yeast cell cycle (Chen et al., 2004), which enabled consistent simulations of the effect of over one hundred reported single or multiple mutations, for different growth conditions (see Figure 3.4 and Part 6 below). More recently, these authors have supervised the development of a generic model enabling consistent simulations of cell cycle in four different eukaryotes (xenopus, budding and fission yeasts, and mammalian cells) (Csikász-Nagy et al., 2006). Numerical integration and analysis of differential models have already led to specific predictions, some already experimentally validated (see for example (Chen et al., 2000) and (Cross et al., 2002), or (Novák and Tyson, 1993) and (Sha et al., 2003)). 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, even though high throughput quantitative measurement techniques are rising (Kittler et al., 2008), the precise nature of the mathematical relationships between variables, and the corresponding parameters remain difficult 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 ap-

6 42 CHAPTER 3. MODELLING THE CELL CYCLE REGULATORY NETWORK Figure 3.4: Wiring diagram of the Chen model. This model of the budding yeast cell cycle has been an important inspiration for my work, and is presented in further detail in subsection 6 and in the corresponding article. Wiring diagram taken from Chen et al. (2004) and proaches (simulations, one- or two-dimensional parameter bifurcation analysis, etc.) (Csikász-Nagy et al., 2006; Ingram et al., 2006). In addition, it is difficult to further extend large differential models, as numerical instabilities arise when the number of variables and the complexity of control terms increases. For the same reasons, stochastic modelling is difficult to apply to large regulatory networks. However, a significant step in this direction is made by Mura and Csikász- Nagy (2008), who presents a stochastic Petri net model of budding yeast cell cycle engine Qualitative models To cope with complex networks and match qualitative experimental data, one can rely on qualitative representations of regulatory graphs in terms of Boolean models and their multilevel extensions (Thomas and D Ari, 1990; Kauffman, 1993), standard Petri nets (Chaouiya, 2007, and references therein), or yet piecewise linear equations (Glass and Kauffman, 1973; De Jong et al., 2004). Among these approaches, logical modelling is increasingly used to model the cell cycle (Huang and Ingber, 2000; Li et al., 2004; Fauré et al., 2006; Davidich and Bornholdt, 2008a,b; Irons, 2009). Leaning on standardised regulatory graphs, logical models rely on Boolean rules to define the effects of different 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 widespread among biologists.

7 3.2. MODELS OF THE CELL CYCLE NETWORK 43 Figure 3.5: Quantitative and qualitative simulations. Davidich and Bornholdt (2008a) present a method to adapt a model based on ODE into the logical formalism, applied to a model of the fission yeast cell cycle (Novák et al., 2001). This figure presents the temporal evolution of two components of the system, IEP (in red) and Ste9 (in black), in simulations of the original ODE model (left) and of the adapted Boolean model (right). Abscissa represents time in the left panel, and the number of successive steps of the synchronous simulation in the right panel. Logical modelling has been successfully applied to various biological regulatory networks, from the bacteriophage lambda lysis/lysogeny switch (Thieffry and Thomas, 1995, and references therein), to the polarisation of the immune response (Kaufman et al., 1985; Kaufman and Thomas, 1987; Muraille et al., 1996; Mendoza, 2006), the specification of arabidopsis flower organs (Mendoza and Alvarez-Buylla, 1998; Mendoza et al., 1999; Espinosa-Soto et al., 2004), the delineation of the segmentation pattern in drosophila embryo (Sánchez and Thieffry, 2001; Albert and Othmer, 2003; Sánchez and Thieffry, 2003; Chaves et al., 2006; Chaves and Albert, 2008; Sánchez et al., 2008), the development of sense organs in drosophila (Ghysen and Thomas, 2003), the differentiation of keratinocytes (Schaub et al., 2007), etc.. All these applications deal with decision-making systems enabling the selection of specific 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 different phases of the cycle. Nonetheless, many regulatory components adopt a switch-like behaviour (Tyson et al., 2003), which can be approximated by logical variables (Figure 3.5). Proper logical modelling of cell cycle engine is still in its infancy. However, a series of Boolean models have been recently proposed, from Huang and Ingber (2000), who presented a very simplified logical model of the mammalian cell cycle engine, to increasingly sophisticated models for budding yeast (Li et al., 2004; Irons, 2009), fission yeast (Davidich and Bornholdt, 2008a,b), and mammals (Fauré et al., 2006).

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