Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

https://downloads.hindawi.com/archive/2013/897658.pdf

Biochemical Systems Theory: A Review

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.

/pdf/identifying-parameter-regions-for-multistationarity-2oq5lwasuy.pdf

Identifying parameter regions for multistationarity.

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.

/pdf/a-unique-transformation-from-ordinary-differential-equations-ks8bbp8q9s.pdf

A unique transformation from ordinary differential equations to reaction networks.

Water eutrophication evaluation based on rough set and petri nets: A case study in Xiangxi-River, Three Gorges Reservoir

https://hal.inria.fr/hal-01103692/document

Inferring reaction systems from ordinary differential equations

Petri nets are a simple formalism for modeling concurrent computation. They are also an interesting tool for modeling and analysing biochemical reaction systems, bridging the gap between purely qualitative and quantitative models. Biological networks can indeed be complex, large, and with many unknown kinetic parameters, which makes the development of quantitative models difficult. In this paper, we focus on the Petri net representation of biochemical reactions and on two structural properties of Petri nets, siphons and traps, that bring us information about the persistence of some molecular species, independently of the kinetics. We first study the theoretical time complexity of minimal siphon decision problems in general Petri nets, and present three new complexity results: first, we show that the existence of a siphon of a given cardinality is NP-complete; second, we prove that deciding the Siphon-Trap property is co-NP-complete; third, we prove that deciding the existence of a minimal siphon containing a given set of places, deciding the existence of a siphon of a given cardinality and deciding the Siphon-Trap property can be done in linear time in Petri nets of bounded tree-width. Then, we present a Boolean model of siphons and traps, and two method for enumerating all minimal siphons and traps of a Petri net, by using a SAT solver and a Constraint Logic Program (CLP) respectively. On a benchmark of 345 Petri nets of hundreds of places and transitions, extracted from biological models from the BioModels repository, as well as on a benchmark composed of 80 Petri nets from the Petriweb database of industrial processes, we show that both the SAT and CLP methods are overall faster by one or two orders of magnitude compared to the state-of-the-art algorithm from the Petri net community, and are in fact able to solve all the enumeration problems of our practical benchmarks. We investigate why these programs perform so well in practice, and provide some elements of explanation related to our theoretical complexity results.

/pdf/on-enumerating-minimal-siphons-in-petri-nets-using-clp-and-4d7d0rfk89.pdf

On enumerating minimal siphons in Petri nets using CLP and SAT solvers: theoretical and practical complexity

In recent years Systems Biology has become a rich field of study, trying to encompass all the information that has become available thanks to the new high-throughput techniques of biologists, in order to build detailed models of complex systems. Some models have been growing bigger and bigger, but lacking most of precise kinetic data. Other models remain of reasonable size, but have an even larger uncertainty about parameter values. Unfortunately, very few analyses allow to extract information about the dynamics of these models when pure symbolic computations fails. This article presents a way to generalize well-known results about the steady-state analysis of some symbolic Ordinary Differential Equations systems by taking into account the structure of the reaction network. The structural study of the underlying Petri net, usually used mostly for metabolic flux analysis, will provide classes where the computation of some steady states of the system is possible, even though the original symbolic model did not form an S-system and was not solvable by state-of-the-art symbolic computation software. This new method is then illustrated on some models of the Biomodels repository and is followed by a brief discussion.

Steady-state solution of biochemical systems, beyond S-systems via T-invariants

Petri nets are a simple formalism for modeling concurrent computation. Recently, they have emerged as a promising tool for modeling and analyzing biochemical interaction networks, bridging the gap between purely qualitative and quantitative models. Biological networks can indeed be large and complex, which makes their study difficult and computationally challenging. In this paper, we focus on two structural properties of Petri nets, siphons and traps, that bring us information about the persistence of some molecular species. We present two methods for enumerating all minimal siphons and traps of a Petri net by iterating the resolution of Boolean satisfiability problems executed with either a SAT solver or a CLP(B) program. We compare the performances of these methods with respect to a state-of-the-art algorithm from the Petri net community. On a benchmark with 80 Petri nets from the Petriweb database and 403 Petri nets from curated biological models of the Biomodels database, we show that miniSAT and CLP(B) solvers are overall both faster by two orders of magnitude with respect to the dedicated algorithm. Furthermore, we analyse why these programs perform so well on even very large biological models and show the existence of hard instances in Petri nets with unbounded degrees.

/pdf/a-boolean-model-for-enumerating-minimal-siphons-and-traps-in-48fuj78l6l.pdf

A boolean model for enumerating minimal siphons and traps in petri nets

Petri-nets are a simple formalism for modeling concurrent computation. Recently, they have emerged as a powerful tool for the modeling and analysis of biochemical reaction networks, bridging the gap between purely qualitative and quantitative models. These networks can be large and complex, which makes their study difficult and computationally challenging. In this paper, we focus on two structural properties of Petri-nets, siphons and traps, that bring us information about the persistence of some molecular species. We present two methods for enumerating all minimal siphons and traps of a Petri-net by iterating the resolution of a boolean model interpreted as either a SAT or a CLP(B) program. We compare the performance of these methods with a state-of-the-art dedicated algorithm of the Petri-net community. We show that the SAT and CLP(B) programs are both faster. We analyze why these programs perform so well on the models of the repository of biological models biomodels.net, and propose some hard instances for the problem of minimal siphons enumeration.

/pdf/un-modele-booleen-pour-l-enumeration-des-siphons-et-des-56eovxsk0k.pdf

Faten Nabli

Papers

On enumerating minimal siphons in Petri nets using CLP and SAT solvers: theoretical and practical complexity

Steady-state solution of biochemical systems, beyond S-systems via T-invariants

A boolean model for enumerating minimal siphons and traps in petri nets

Un modèle booléen pour l'énumération des siphons et des pièges minimaux dans les réseaux de Petri

Un mod\`ele bool\'een pour l'\'enum\'eration des siphons et des pi\`eges minimaux dans les r\'eseaux de Petri