Olivier F. Roux

Hybrid Modelling and Dynamical Analysis of Gene Regulatory Networks with Delays

René Thomas’ discrete modelling of gene regulatory networks (GRN) is a well-known approach to study the dynamics resulting from a set of interacting genes. It deals with some parameters which reflect the possible targets of trajectories. Those parameters are a priori unknown, but they may generally be deduced from a well-chosen set of biologically observed trajectories. Besides, it neglects the time delays for a gene to pass from one level of expression to another one. The purpose of this paper is to show that we can account for time delays of increasing or decreasing expression levels of genes in a GRN, while preserving powerful enough computer-aided reasoning capabilities. We designed a more accurate abstraction of GRN where delays are now supposed to be non-null unknown new parameters. We show that such models, together with hybrid model-checking algorithms, make it possible to obtain some results about the behaviour of a network of interacting genes, since dynamics depend on the respective values of the parameters. The characteristic of our approach is that, among possible execution trajectories in the model, we can automatically find out both viability cycles and absorption in capture basins. As a running example, we show that we are able to discriminate between various possible dynamics of mucus production in the bacterium Pseudomonas aeruginosa.

Using Zone Graph Method for Computing the State Space of a Time Petri Net

Presently, the method to verify quantitative time properties on Time Petri Nets is the use of observers. The state space is then computed to test the reachability of a given marking. The main method to compute the state space of a Time Petri Net has been introduced by Berthomieu and Diaz. It is known as the “state class method”. We present in this paper a new efficient method to compute the state space of a bounded Time Petri Net as a marking graph, based on the region graph method used for Timed Automaton. The algorithm is proved to be exact with respect to the reachability of a marking and it computes a graph which nodes are exactly the reachable markings of the Time Petri Net. The tool implemented computes faster than Tina, a tool for constructing the state space using classes, and allows to test on-the-fly the reachability of a given marking.

Semantics of Biological Regulatory Networks

The aim of the paper is to revisit the model of Biological Regulatory Networks (BRN) which was proposed by Rene Thomas to model the interactions between a set of genes. We give a formal semantics for BRN in terms of transition systems which formalizes the evolution rules given by Rene Thomas. Then we show how to use this model to find interesting properties of a BRN like the set of stable states, cycles etc using tools for analyzing transition systems.

Static analysis of biological regulatory networks dynamics using abstract interpretation

The analysis of the dynamics of Biological Regulatory Networks (BRNs) requires innovative methods to cope with the state-space explosion. This paper settles an original approach for deciding reachability properties based on Process Hitting, which is a framework suitable for modelling dynamical complex systems. In particular, Process Hitting has been shown to be of interest in providing compact models of the dynamics of BRNs with discrete values. Process Hitting splits a finite number of processes into so-called sorts and describes the way each process is able to act upon (that is, to hit) another one (or itself) in order to bounce it as another process of the same sort with further actions. n nBy using complementary abstract interpretations of the succession of actions in Process Hitting, we build a very efficient static analysis to over-and under-approximate reachability properties, which avoids the need to build the underlying states graph. The analysis is proved to have a low theoretical complexity, in particular when the number of processes per sorts is limited, while a very large number of sorts can be managed. n nThis makes such an approach very promising for the scalable analysis of abstract complex systems. We illustrate this through the analysis of a large BRN of 94 components. Our method replies quasi-instantaneously to reachability questions, while standard model-checking techniques regularly fail because of the combinatoric explosion of behaviours.

Refining dynamics of gene regulatory networks in a stochastic π-calculus framework

In this paper, we introduce a framework allowing to model and analyse efficiently Gene Regulatory Networks (GRNs) in their temporal and stochastic aspects. The analysis of stable states and inference of Rene Thomas discrete parameters derives from this logical formalism. We offer a compositional approach which comes with a natural translation to the Stochastic π-Calculus. The method we propose consists in successive refinements of generalised dynamics of GRNs. We illustrate the merits and scalability of our framework on the control of the differentiation in a GRN generalising metazoan segmentation processes, and on the analysis of stable states within a large GRN studied in the scope of breast cancer researches.

Compilation of the ELECTRE reactive language into finite transition systems

Abstract We present in this paper an operational semantics for the ELECTRE reactive language (Roux et al., 1992). This language is based on an asynchronous approach to real-time systems. First basic concepts and intuitive semantics are introduced. Then we give rules to model dynamic semantics of ELECTRE programs: this constitutes an operational semantics for the ELECTRE language. This operational semantics is used to define a model of execution for ELECTRE programs: transition system. In addition, we prove, using structural induction on the operational semantics, that this transition system is a finite state transition system. Eventually, we extend the previous transition system so as to handle multiple-storage events: it is important since the asynchronous ELECTRE language deals with multiple memorized occurrences of the events. This result gives a means of compiling the ELECTRE language into a finite-state machine.

Analysing formal models of genetic regulatory networks with delays

In this paper, we propose a refinement of the modelling of biological regulatory networks based on the discrete approach of Rene Thomas. We refine and automatise the use of delays of activation/inhibition in order to specify which variable is more quickly affected by a change of its regulators. The formalism of linear hybrid automata is well suited to allow such refinement. We then use HyTech for two purposes: to find automatically all paths from a specified initial state to another one; to synthesise constraints on the delay parameters in order to follow any specific path.

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 Thomass 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.

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

BackgroundModeling 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).ResultsWe 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.ConclusionsA 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.

Concretizing the process hitting into biological regulatory networks

The Process Hitting (PH) is a recently introduced framework to model concurrent processes. Its major originality lies in a specific restriction on the causality of actions, which makes the formal analysis of very large systems tractable. PH is suitable to model Biological Regulatory Networks (BRNs) with complete or partial knowledge of cooperations between regulators by defining the most permissive dynamics with respect to these constraints. n nOn the other hand, the qualitative modeling of BRNs has been widely addressed using Rene Thomas formalism, leading to numerous theoretical work and practical tools to understand emerging behaviors. n nGiven a PH model of a BRN, we first tackle the inference of the underlying Interaction Graph between components. Then the inference of corresponding Thomas models is provided using Answer Set Programming, which allows notably an efficient enumeration of (possibly numerous) compatible parametrizations. n nIn addition to giving a formal link between different approaches for qualitative BRNs modeling, this work emphasizes the ability of PH to deal with large BRNs with incomplete knowledge on cooperations, where Thomas approach fails because of the combinatorics of parameters.