A Unique Transformation from Ordinary Differential Equations to Reaction Networks

Many models in Systems Biology are described as a system of Ordinary Differential Equations, which allows for transient, steady-state or bifurcation analysis when kinetic information is available. Complementary structure-related qualitative analysis techniques have become increasingly popular in recent years, like qualitative model checking or pathway analysis (elementary modes, invariants, flux balance analysis, graph-based analyses, chemical organization theory, etc.). They do not rely on kinetic information but require a well-defined structure as stochastic analysis techniques equally do. In this article, we look into the structure inference problem for a model described by a system of Ordinary Differential Equations and provide conditions for the uniqueness of its solution. We describe a method to extract a structured reaction network model, represented as a bipartite multigraph, for example, a continuous Petri net (CPN), from a system of Ordinary Differential Equations (ODEs). A CPN uniquely defines an ODE, and each ODE can be transformed into a CPN. However, it is not obvious under which conditions the transformation of an ODE into a CPN is unique, that is, when a given ODE defines exactly one CPN. We provide biochemically relevant sufficient conditions under which the derived structure is unique and counterexamples showing the necessity of each condition. Our method is implemented and available; we illustrate it on some signal transduction models from the BioModels database. A prototype implementation of the method is made available to modellers at http://contraintes.inria.fr/~soliman/ode2pn.html, and the data mentioned in the “Results” section at http://contraintes.inria.fr/~soliman/ode2pn_data/. Our results yield a new recommendation for the import/export feature of tools supporting the SBML exchange format

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dualrail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations in PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks

Graphical Conditions for Rate Independence in Chemical Reaction Networks

Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real function $f : R+$\rightarrow$R+$, if for any initial concentration x of the input species, the concentration of the output molecular species stabilizes at concentration f (x). The Turing-completeness of that notion of chemical analog computation has been established by proving that any computable real function can be computed by a CRN over a finite set of molecular species. Rate-independent CRNs form a restricted class of CRNs of high practical value since they enjoy a form of absolute robustness in the sense that the result is completely independent of the reaction rates and depends solely on the input concentrations. The functions computed by rate-independent CRNs have been characterized mathematically as the set of piecewise linear functions from input species. However, this does not provide a mean to decide whether a given CRN is rate-independent. In this paper, we provide graphical conditions on the Petri Net structure of a CRN which entail the rate-independence property either for all species or for some output species. We show that in the curated part of the Biomodels repository, among the 590 reaction models tested, 2 reaction graphs were found to satisfy our rate-independence conditions for all species, 94 for some output species, among which 29 for some non-trivial output species. Our graphical conditions are based on a non-standard use of the Petri net notions of place-invariants and siphons which are computed by constraint programming techniques for efficiency reasons

SBGN support in BIOCHAM

BIOCHAM - BioChemical Abstract Machine is an environment for modeling biological systems and formalizing experimental knowledge. Mainly, it is composed of:
* a rule-based language for modeling biochemical systems (compatible with SBML)
* several simulators (boolean, differential, stochastic)
* a temporal logic based language to formalize the temporal properties of a biological system and validate models with respect to such specifications,
* unique features for developing/correcting/completing/reducing/coupling models, including the inference of kinetic parameters in high dimension from temporal logic constraints.

BIOCHAM is presented with a user friendly graphical interface, that is easy to use and gives the user richer experience throughout his work. The current release of its graphical user interface has a SBGN compliant Reaction Graph Editor for drawing and editing biochemical networks. It also offers export features to SBML and other formats.

It is implemented by the CONTRAINTES team in INRIA ( National Institute for Research in Computer Science and Control), in Paris - Rocquencourt, France.
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Pi-calculus and LCC, a Space Odyssey

We present a translation of the asynchronous pi-calculus into linear concurrent constraint languages (LCC), and use that translation and the recent advances in the logical semantics of LCC to give an account of the restriction operator in Intuitionistic Linear Logic. This allows us to express as a Linear Logic theory, a notion of space similar to what has recently been introduced by Gabbay and Pitts in modal logics. It also permits to relate more closely the pi-calculus and CC paradigms that people have wanted to compare for a long time

Coupling the Cell cycle and the Circadian Cycle

Cancer treatments based on the administration of medicines at different times of the day have been shown to be more efficient against malign cells and less damaging towards healthy ones. These results might be related to the recent discovery of links between the circadian clock, (controlled by the light/dark cycle of a day), and the cell cycle. However, if many models have been developed to describe both of these cycles, to our knowledge none has described a real interaction between them. We will try to establish a relation at a molecular level and we will then study the conditions of entrainment in period of these cycles with the modeling environment BIOCHAM. In other words, we will search how and where in the parameter space of our model the two cycles get synchronized. This technical report will insist on the conditions of entrainment of the cell cycle by the circadian cycle via a common protein kinase WEE1

Reifying Global Constraints

Global constraints were introduced two decades ago as a means to model some core aspects of combinatorial problems with one single constraint for which an efficient domain filtering algorithm can be provided, possibly using a complete change of representation. However, global constraints are just constraint schemas on which one would like to apply usual constraint operations such as reification, i.e. checking entailment, disentailment and negating the constraint. This is currently not the case in state-of-the-art tools and was not considered in the global constraint catalog until recently. In this paper, we propose a general framework for reifying global constraints and apply it to some important constraints of the catalog, such as the cumulative constraint for instance. We show that several global constraints that were believed to be hard to negate can in fact be efficiently negated, and that entailment and disentailment can be efficiently tested. We also point out some new global constraints that are worth studying from this point of view and provide some performance figures obtained with an implementation in Choco.Les contraintes globales ont été introduites il y a une vingtaine d'années afin de modéliser certains aspects centraux des problèmes combinatoires avec une seule contrainte dotée d'un algorithme de filtrage efficace, au besoin via un changement complet de représentation. Cependant, les contraintes globales ne sont que des schémas de contraintes sur lesquelles on souhaiterait pouvoir appliquer les opérations usuelles des contraintes comme la réification, ce qui suppose de tester l'implication et de nier la contrainte. Ceci n'est pas le cas dans les outils de l'état de l'art et n'a été considéré que récemment dans le catalogue des contraintes globales. Dans cet article nous proposons un cadre général pour réifier les contraintes globales, et l'appliquons aux principales contraintes du catalogue, comme par exemple la contrainte cumulative. Nous montrons que plusieurs contraintes réputées difficiles à nier peuvent l'être efficacement, et que l'implication peuvent être testée efficacement. Nous montrons aussi que de nouvelles contraintes globales vaudraient la peine d'être étudiées de ce point de vue, et fournissons une évaluation préliminaire des performances obtenues avec une implémentation en Choco

Reactmine: a search algorithm for inferring chemical reaction networks from time series data

Inferring chemical reaction networks (CRN) from time series data is a challenge encouraged by the growing availability of quantitative temporal data at the cellular level. This motivates the design of algorithms to infer the preponderant reactions between the molecular species observed in a given biochemical process, and help to build CRN model structure and kinetics. Existing ODE-based inference methods such as SINDy resort to least square regression combined with sparsity-enforcing penalization, such as Lasso. However, when the input time series are only available in wild type conditions in which all reactions are present, we observe that current methods fail to learn sparse models. Results: We present Reactmine, a CRN learning algorithm which enforces sparsity by inferring reactions in a sequential fashion within a search tree of bounded depth, ranking the inferred reaction candidates according to the variance of their kinetics, and re-optimizing the CRN kinetic parameters on the whole trace in a final pass to rank the inferred CRN candidates. We first evaluate its performance on simulation data from a benchmark of hidden CRNs, together with algorithmic hyperparameter sensitivity analyses, and then on two sets of real experimental data: one from protein fluorescence videomicroscopy of cell cycle and circadian clock markers, and one from biomedical measurements of systemic circadian biomarkers possibly acting on clock gene expression in peripheral organs. We show that Reactmine succeeds both on simulation data by retrieving hidden CRNs where SINDy fails, and on the two real datasets by inferring reactions in agreement with previous studies

On the Complexity of Quadratization for Polynomial Differential Equations

International audienceChemical reaction networks (CRNs) are a standard formalism used in chemistry and biology to reason about the dynamics of molecular interaction networks. In their interpretation by ordinary differential equations, CRNs provide a Turing-complete model of analog computattion, in the sense that any computable function over the reals can be computed by a finite number of molecular species with a continuous CRN which approximates the result of that function in one of its components in arbitrary precision. The proof of that result is based on a previous result of Bournez et al. on the Turing-completeness of polyno-mial ordinary differential equations with polynomial initial conditions (PIVP). It uses an encoding of real variables by two non-negative variables for concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with degrees at most 2) for restricting ourselves to at most bimolecular reactions. In this paper, we study the theoretical and practical complexities of the quadratic transformation. We show that both problems of minimizing either the number of variables (i.e., molecular species) or the number of monomials (i.e. elementary reactions) in a quadratic transformation of a PIVP are NP-hard. We present an encoding of those problems in MAX-SAT and show the practical complexity of this algorithm on a benchmark of quadratization problems inspired from CRN design problems

A Constraint Solving Approach to Tropical Equilibration and Model Reduction

International audienceModel reduction is a central topic in systems biology and dynamical systems theory, for reducing the complexity of detailed models, finding important parameters, and developing multi-scale models for instance. While perturbation theory is a standard mathematical tool to analyze the different time scales of a dynamical system, and decompose the system accordingly, tropical methods provide a simple algebraic framework to perform these analyses systematically in polynomial systems. The crux of these tropicalization methods is in the computation of tropical equilibrations. In this paper we show that constraint-based methods, using reified constraints for expressing the equilibration conditions, make it possible to numerically solve non-linear tropical equilibration problems, out of reach of standard computation methods. We illustrate this approach first with the reduction of simple biochemical mechanisms such as the Michaelis-Menten and Goldbeter-Koshland models, and second, with performance figures obtained on a large scale on the model repository biomodels.net