Stochastic Modeling of Complex Systems and Systems Biology: From Stochastic Transition Systems to Hybrid Systems

Rodrigo Assar 1, * Martín A Montecino 1 David James Sherman 2
* Auteur correspondant
2 MAGNOME - Models and Algorithms for the Genome
Inria Bordeaux - Sud-Ouest, UB - Université de Bordeaux, CNRS - Centre National de la Recherche Scientifique : UMR5800
Abstract : Modeling complex dynamic systems requires to reuse and to combine models in a non-ambiguous way, integrating processes with different information levels and temporal dependences. System behaviors are consequence of interacting processes, which are affected by external factors often not controlled. In particular, biological functions are the result of processes that connect different hierarchy levels, associated by physical and chemical relations (H. Kitano (2002)). Each process works in different way and it is common to observe that changes in the conditions, such as quantity of nutrients or environment, modify the behavior of the systems. The firstt approach we discuss is the use of Stochastic Transition Systems (STSs, L.D. Alfaro (1998)). It considers the dynamics of system variables given by transitions, changing their values, described by Markov processes with continuous time. Transitions are provoked by specific conditions of system variables, which can be ambiguous and generate non-determinism. Although STSs allow us to incorporate randomness and non-determinism, we do not capture the complexity of behaviors nor the continuity of the variables. To describe the behavior of complex systems over time, it is convenient to combine different types of models: continuous models for gradual changes, discrete models for instantaneous changes, deterministic models for completely predictable behaviors, and stochastic or non-deterministic models to describe behaviors with imprecise or incomplete information. To do that, we use the Hybrid Systems theory and the composition of models. System variables evolve according to continuous models, but deterministic, stochastic and non-deterministic transitions can change the definition of these models. Composition is the action of combining different models into an integrated model. We connect them by using input-output relations, and by processes synchronization (e.g. activation or repression signals). A very intuitive example of hybrid system is the motion of an automobile with a manual gearbox. The dynamics of the velocity and position evolve in a continuous specific way depending of the engaged gear. With an automatic gearbox the gear changes are deterministic, but if it is manual many factors influence the decisions of the driver, and the transitions are stochastic or non-deterministic. According to the form of the model changes, we consider hybrid systems with coefficient switches, or with strong switches. For coefficient switches, the transitions provoke changes in specific coefficients of the continuous model. For strong switches, transitions control the activation of models allowing radical changes. First type favors the interpretation of the effect of transitions on the continuous model, while strong switches are useful for reconciling models with different nature (R. Assar (2011)). This approach allows us to build more complete descriptions of complex biological systems. As application, we built a hybrid model of the osteo-adipo differentiation process. It combines known validated models to predict the bone and fat formation in response to activation of pathways such as the Wnt pathway, stochastic factors, and changes of conditions affecting these functions (R. Assar et al. (2012)). This model is our first phase to simulate physiological responses to treatments of bone mass disorders in silico, and to explore the efficiency of new medical strategies before testing them in vitro or in vivo.
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Soumis le : samedi 7 avril 2012 - 02:49:15
Dernière modification le : jeudi 11 janvier 2018 - 01:49:29

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Rodrigo Assar, Martín A Montecino, David James Sherman. Stochastic Modeling of Complex Systems and Systems Biology: From Stochastic Transition Systems to Hybrid Systems. XII Latin American Congress of Probability and Mathematical Statistics, Mar 2012, Viña del Mar, Chile. 2012. 〈hal-00686072〉

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