Multiclock discrete models of the eukaryotic cell cycle.
Abstract
Among the various formalisms proposed to model biological system, the discrete models are especially appealing because the modeled events can easily reflect biological observations. It also benefits from a wealth of work done in circuit and program verification with well-established concepts and languages such as temporal logic, and efficient data structures and decision procedures like Binary Decision Diagrams (BDD) and satisfiability solvers (SAT solvers). In such models each entity (gene, protein, ...) is represented by a finite-state variable and the values reflect their different biological properties. For instance, in discrete boolean models, the value indicates whether a molecule is active or not. The evolution equation between two states of a molecule is then conditioned by the values of the other variables of the system. At each state of the system, some entities of the model can evolve. However this model does not contain any specification on the order of the transitions. This is deferred to the interpretation of the model in a simulator. The commonly used approaches are the synchronous and asynchronous interpretations. In synchronous mode, all the possible changes occur in one evolution step whereas in asynchronous mode, only one change is allowed at each step. The asynchronous interpretation is considered more interesting since it explores all the potential evolutions of the model. Nevertheless, it is often unsatisfactory, i.e. giving too many solutions not always biologically relevant or with questionable trajectories. In an attempt to get finer description of time, some authors introduced the notion of priorities on transitions [1]. However, in all cases, these assumptions are still not part of the model but directives to the simulator. When using model checking, specification of time outside the model becomes a source of problems. The different interpretations of the models must be implemented in the model-checker. With more complex directives on the sequencing of transitions, it is much more difficult to use formal verification methods. Moreover, it is rather difficult, even impossible, to perform computations for control purposes or model fitting. Here, we propose a new formalism to include timing specifications in the models and use a unique interpretation for all models. This formalism is inspired by the formal models underlying real time programming languages such as Esterel, Lustre and Signal [2,3]. Time is the logical time used in computer science: it does not correspond to the duration of events but to their relative sequencing. This allows the description of several biological signals with different clocks, i.e., multiclock systems. To illustrate the power of the formalism, we modeled the core network controlling the mammalian cell cycle. We showed that the standard asynchronous and synchronous interpretations of discrete models are equivalent to particular timing specifications in our formalism. One main improvement of our formalism is its capacity to support model-checking technique for properties involving biological entities and reaction time. This computational power can also be used for enforcing properties paving the way to model fitting and experiment planning. Finally, our example shows that pure timing constraints may not be sufficient to correctly model some biological phenomena. Priorities on reactions (transitions) have been used to finely tune timing conditions. In a forthcoming work we will investigate and implement model-checking approaches to explore our models. [1] A. Faure et al. Bioinformatics, 14:e124-131, 2006. [2] A. Benveniste et al. 40th IEEE Conference on Decision and Control, 2001. [3] A. Benveniste et al. Proceedings of the IEEE, Special issue on Modeling and Design of Embedded Systems, 91:64-83, 2003.