Virtual mitochondrion: towards an integrated model of oxidative phosphorylation complexes and beyond.

The modelling of OXPHOS (oxidative phosphorylation) in order to integrate all kinetic and thermodynamic aspects of chemiosmotic theory has a long history. We briefly review this history and show how new ways of modelling are required to integrate a local model of the individual respiratory complexes into a global model of OXPHOS and, beyond that, into a reliable overall model of central metabolism.

Hybrid modeling of biological networks: mixing temporal and qualitative biological properties.

BACKGROUND: Modeling a dynamical biological system is often a difficult task since the a priori unknown parameters of such models are not always directly given by the experiments. Despite the lack of experimental quantitative knowledge, one can see a dynamical biological system as (i) the combined evolution tendencies (increase or decrease) of the biological compound concentrations, and: (ii) the temporal features, such as delays between two concentration peaks (i.e. the times when one of the components completes an increase (resp. decrease) phase and starts a decrease (resp. increase) phase). RESULTS: We propose herein a new hybrid modeling framework that follows such biological assumptions. This hybrid approach deals with both a qualitative structure of the system and a quantitative structure. From a theoretical viewpoint, temporal specifications are expressed as equality or inequality constraints between delay parameters, while the qualitative specifications are expressed as an ordered pattern of the concentrations peaks of the components. Using this new hybrid framework, the temporal specifications of a biological system can be obtained from incomplete experimental data. The model may be processed by a hybrid model-checker (e.g. Phaver) which is able to give some new constraints on the delay parameters (e.g. the delay for a given transition is exactly 5 hours after the later peak of a gene product concentration). Furthermore, by using a constraint solver on the previous results, it becomes possible to get the set of parameters settings which are consistent with given specifications. Such a modeling approach is particularly accurate for modeling oscillatory biological behaviors like those observed in the Drosophila circadian cycles. The achieved results concerning the parameters of this oscillatory system formally confirm the several previous studies made by numerical simulations. Moreover, our analysis makes it possible to propose an automatic investigation of the respective impact of per and tim on the circadian cycle. CONCLUSIONS: A new hybrid technique for an automatic formal analysis of biological systems is developed with a special emphasis on their oscillatory behaviors. It allows the use of incomplete and empirical biological data.

Temporal constraints of a gene regulatory network: Refining a qualitative simulation.

The modelling of gene regulatory networks (GRNs) has classically been addressed through very different approaches. Among others, extensions of Thomas's asynchronous Boolean approach have been proposed, to better fit the dynamics of biological systems: genes may reach different discrete expression levels, depending on the states of other genes, called the regulators: thus, activations and inhibitions are triggered conditionally on the proper expression levels of these regulators. In contrast, some fine-grained propositions have focused on the molecular level as modelling the evolution of biological compound concentrations through differential equation systems. Both approaches are limited. The first one leads to an oversimplification of the system, whereas the second is incapable to tackle large GRNs. In this context, hybrid paradigms, that mix discrete and continuous features underlying distinct biological properties, achieve significant advances for investigating biological properties. One of these hybrid formalisms proposes to focus, within a GRN abstraction, on the time delay to pass from a gene expression level to the next. Until now, no research work has been carried out, which attempts to benefit from the modelling of a GRN by differential equations, converting it into a multi-valued logical formalism of Thomas, with the aim of performing biological applications. This paper fills this gap by describing a whole pipelined process which orchestrates the following stages: (i) model conversion from a piece-wise affine differential equation (PADE) modelization scheme into a discrete model with focal points, (ii) characterization of subgraphs through a graph simplification phase which is based on probabilistic criteria, (iii) conversion of the subgraphs into parametric linear hybrid automata, (iv) analysis of dynamical properties (e.g. cyclic behaviours) using hybrid model-checking techniques. The present work is the outcome of a methodological investigation launched to cope with the GRN responsible for the reaction of Escherichia coli bacterium to carbon starvation. As expected, we retrieve a remarkable cycle already exhibited by a previous analysis of the PADE model. Above all, hybrid model-checking enables us to infer temporal properties, whose biological signification is then discussed.

Analysing gene regulatory networks by both constraint programming and model-checking.

In this article, we propose a formal method to analyse gene regulatory networks (GRN). The dynamics of such systems is often described by an ordinary differential equation system, but has also been abstracted into a discrete transition system. This modeling depends on parameters for which different values are possible. Each instantiation of these parameters defines a possible dynamics and verification tools can be used to select the tuples of values which lead to dynamics consistent with known behaviours. GRN are so complex that their discrete modeling gives a number of possible dynamics exponential in function of the GRN's size (number of genes and interactions). In this paper, we propose to use constraint programming and CTL formal language to determine the set of all dynamics consistent with the known behavioral properties without enumerating all of them. This approach allows us therefore to minimize the computation time necessary for the research of these parameters.