Colored Petri net modeling and simulation of signal transduction pathways

Presented herein is a methodology for quantitatively analyzing the complex signaling network by resorting to colored Petri nets (CPN). The mathematical as well as Petri net models for two basic reaction types were established, followed by the extension to a large signal transduction system stimulated by epidermal growth factor (EGF) in an application study. The CPN models based on the Petri net representation and the conservation and kinetic equations were used to examine the dynamic behavior of the EGF signaling pathway. The usefulness of Petri nets is demonstrated for the quantitative analysis of the signal transduction pathway. Moreover, the trade-offs between modeling capability and simulation efficiency of this pathway are explored, suggesting that the Petri net model can be invaluable in the initial stage of building a dynamic model.     

Model validation of biological pathways using Petri nets--demonstrated for apoptosis

This paper demonstrates the first steps of a new integrating methodology to develop and analyse models of biological pathways in a systematic manner using well established Petri net technologies. The whole approach comprises step-wise modelling, animation, model validation as well as qualitative and quantitative analysis for behaviour prediction. In this paper, the first phase is addressed how to develop and validate a qualitative model, which might be extended afterwards to a quantitative model. The example used in this paper is devoted to apoptosis, the genetically programmed cell death. Apoptosis is an essential part of normal physiology for most metazoan species. Disturbances in the apoptotic process could lead to several diseases. The signal transduction pathway of apoptosis includes highly complex mechanisms to control and execute programmed cell death. This paper explains how to model and validate this pathway using qualitative Petri nets. The results provide a mathematically unique and valid model enabling the confirmation of known properties as well as new insights in this pathway.     

Learning Petri net models of non-linear gene interactions

Understanding how an individual's genetic make-up influences their risk of disease is a problem of paramount importance. Although machine-learning techniques are able to uncover the relationships between genotype and disease, the problem of automatically building the best biochemical model or "explanation" of the relationship has received less attention. In this paper, I describe a method based on random hill climbing that automatically builds Petri net models of non-linear (or multi-factorial) disease-causing gene-gene interactions. Petri nets are a suitable formalism for this problem, because they are used to model concurrent, dynamic processes analogous to biochemical reaction networks. I show that this method is routinely able to identify perfect Petri net models for three disease-causing gene-gene interactions recently reported in the literature.     

Application of Petri net theory for modelling and validation of the sucrose breakdown pathway in the potato tuber

MOTIVATION: Because of the complexity of metabolic networks and their regulation, formal modelling is a useful method to improve the understanding of these systems. An essential step in network modelling is to validate the network model. Petri net theory provides algorithms and methods, which can be applied directly to metabolic network modelling and analysis in order to validate the model. The metabolism between sucrose and starch in the potato tuber is of great research interest. Even if the metabolism is one of the best studied in sink organs, it is not yet fully understood. RESULTS: We provide an approach for model validation of metabolic networks using Petri net theory, which we demonstrate for the sucrose breakdown pathway in the potato tuber. We start with hierarchical modelling of the metabolic network as a Petri net and continue with the analysis of qualitative properties of the network. The results characterize the net structure and give insights into the complex net behaviour.     

Algebraic properties of automata associated to Petri nets and applications to computation in biological systems

Biochemical and genetic regulatory networks are often modeled by Petri nets. We study the algebraic structure of the computations carried out by Petri nets from the viewpoint of algebraic automata theory. Petri nets comprise a formalized graphical modeling language, often used to describe computation occurring within biochemical and genetic regulatory networks, but the semantics may be interpreted in different ways in the realm of automata. Therefore, there are several different ways to turn a Petri net into a state-transition automaton. Here, we systematically investigate different conversion methods and describe cases where they may yield radically different algebraic structures. We focus on the existence of group components of the corresponding transformation semigroups, as these reflect symmetries of the computation occurring within the biological system under study. Results are illustrated by applications to the Petri net modelling of intermediary metabolism. Petri nets with inhibition are shown to be computationally rich, regardless of the particular interpretation method. Along these lines we provide a mathematical argument suggesting a reason for the apparent all-pervasiveness of inhibitory connections in living systems.     

Topological analysis of metabolic networks based on Petri net theory

Petri net concepts provide additional tools for the modelling of metabolic networks. Here, the similarities between the counterparts in traditional biochemical modelling and Petri net theory are discussed. For example the stoichiometry matrix of a metabolic network corresponds to the incidence matrix of the Petri net. The flux modes and conservation relations have the T-invariants, respectively, P-invariants as counterparts. We reveal the biological meaning of some notions specific to the Petri net framework (traps, siphons, deadlocks, liveness). We focus on the topological analysis rather than on the analysis of the dynamic behaviour. The treatment of external metabolites is discussed. Some simple theoretical examples are presented for illustration. Also the Petri nets corresponding to some biochemical networks are built to support our results. For example, the role of triose phosphate isomerase (TPI) in Trypanosoma brucei metabolism is evaluated by detecting siphons and traps. All Petri net properties treated in this contribution are exemplified on a system extracted from nucleotide metabolism.     

Petri net modelling of biological networks

Mathematical modelling is increasingly used to get insights into the functioning of complex biological networks. In this context, Petri nets (PNs) have recently emerged as a promising tool among the various methods employed for the modelling and analysis of molecular networks. PNs come with a series of extensions, which allow different abstraction levels, from purely qualitative to more complex quantitative models. Noteworthily, each of these models preserves the underlying graph, which depicts the interactions between the biological components. This article intends to present the basics of the approach and to foster the potential role PNs could play in the development of the computational systems biology.     

基于Petri网T不变量的PHB新陈代谢建模分析(英文)

从拓扑结构的角度分析生化反应网络是生物信息学研究中的一个热点问题。通过将两种传统的途径分析方法(基元模式和极端途径)与Petri网的T不变量分析进行了比较,结果表明:它们本质上是一致的,但是采用Petri网的T不变量分析更便捷。然后,利用Petri网技术构建了PHB代谢模型。对该模型作了结构分析,将计算得到的23个T不变量进行了分组:I组表示简单的可逆反应,II组表示循环的反应,III组可用于调控ATP/ADP比率,IV组是与PHB生产直接相关的反应,可用于代谢工程以提高PHB的产率。最后讨论了Petri网的T不变量分析在这个领域中的应用。     

Network modeling of signal transduction: establishing the global view

Embryonic development and adult tissue homeostasis are controlled through activation of intracellular signal transduction pathways by extracellular growth factors. In the past, signal transduction has largely been regarded as a linear process. However, more recent data from large-scale and high-throughput experiments indicate that there is extensive cross-talk between individual signaling cascades leading to the notion of a signaling network. The behavior of such complex networks cannot be predicted by simple intuitive approaches but requires sophisticated models and computational simulations. The purpose of such models is to generate experimentally testable hypotheses and to find explanations for unexpected experimental results. Here, we discuss the need for, and the future impact of, mathematical models for exploring signal transduction in different biological contexts such as for example development.     

Petri net modeling of high-order genetic systems using grammatical evolution

Understanding how DNA sequence variations impact human health through a hierarchy of biochemical and physiological systems is expected to improve the diagnosis, prevention, and treatment of common, complex human diseases. We have previously developed a hierarchical dynamic systems approach based on Petri nets for generating biochemical network models that are consistent with genetic models of disease susceptibility. This modeling approach uses an evolutionary computation approach called grammatical evolution as a search strategy for optimal Petri net models. We have previously demonstrated that this approach routinely identifies biochemical network models that are consistent with a variety of genetic models in which disease susceptibility is determined by nonlinear interactions between two DNA sequence variations. In the present study, we evaluate whether the Petri net approach is capable of identifying biochemical networks that are consistent with disease susceptibility due to higher order nonlinear interactions between three DNA sequence variations. The results indicate that our model-building approach is capable of routinely identifying good, but not perfect, Petri net models. Ideas for improving the algorithm for this high-dimensional problem are presented.     

Dynamic modelling and analysis of biochemical networks: mechanism-based models and model-based experiments

Systems biology applies quantitative, mechanistic modelling to study genetic networks, signal transduction pathways and metabolic networks. Mathematical models of biochemical networks can look very different. An important reason is that the purpose and application of a model are essential for the selection of the best mathematical framework. Fundamental aspects of selecting an appropriate modelling framework and a strategy for model building are discussed. Concepts and methods from system and control theory provide a sound basis for the further development of improved and dedicated computational tools for systems biology. Identification of the network components and rate constants that are most critical to the output behaviour of the system is one of the major problems raised in systems biology. Current approaches and methods of parameter sensitivity analysis and parameter estimation are reviewed. It is shown how these methods can be applied in the design of model-based experiments which iteratively yield models that are decreasingly wrong and increasingly gain predictive power.     

An algebraic-combinatorial model for the identification and mapping of biochemical pathways

We develop the mathematical machinery for the construction of an algebraic-combinatorial model using Petri nets to construct an oriented matroid representation of biochemical pathways. For demonstration purposes, we use a model metabolic pathway example from the literature to derive a general biochemical reaction network model. The biomolecular networks define a connectivity matrix that identifies a linear representation of a Petri net. The sub-circuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariants, that are called S-(state) and T-(transition) invariants. Each invariant has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. We prove that the family of signed sub-circuits has an implicit order that defines an oriented matroid. The oriented matroid is then used to identify the feasible sub-circuit pathways that span the biochemical network as the positive cycles in a hyper-digraph.     

Invariants and Other Structural Properties of Biochemical Models as a Constraint Satisfaction Problem

Background

We present a way to compute the minimal semi-positive invariants of a Petri net representing a biological reaction system, as resolution of a Constraint Satisfaction Problem. The use of Petri nets to manipulate Systems Biology models and make available a variety of tools is quite old, and recently analyses based on invariant computation for biological models have become more and more frequent, for instance in the context of module decomposition.

Results

In our case, this analysis brings both qualitative and quantitative information on the models, in the form of conservation laws, consistency checking, etc. thanks to finite domain constraint programming. It is noticeable that some of the most recent optimizations of standard invariant computation techniques in Petri nets correspond to well-known techniques in constraint solving, like symmetry-breaking. Moreover, we show that the simple and natural encoding proposed is not only efficient but also flexible enough to encompass sub/sur-invariants, siphons/traps, etc., i.e., other Petri net structural properties that lead to supplementary insight on the dynamics of the biochemical system under study.

Conclusions

A simple implementation based on GNU-Prolog's finite domain solver, and including symmetry detection and breaking, was incorporated into the BIOCHAM modelling environment and in the independent tool Nicotine. Some illustrative examples and benchmarks are provided.     

Bio-Logic Builder: A Non-Technical Tool for Building Dynamical,Qualitative Models

Computational modeling of biological processes is a promising tool in biomedical research. While a large part of its potential lies in the ability to integrate it with laboratory research, modeling currently generally requires a high degree of training in mathematics and/or computer science. To help address this issue, we have developed a web-based tool, Bio-Logic Builder, that enables laboratory scientists to define mathematical representations (based on a discrete formalism) of biological regulatory mechanisms in a modular and non-technical fashion. As part of the user interface, generalized “bio-logic” modules have been defined to provide users with the building blocks for many biological processes. To build/modify computational models, experimentalists provide purely qualitative information about a particular regulatory mechanisms as is generally found in the laboratory. The Bio-Logic Builder subsequently converts the provided information into a mathematical representation described with Boolean expressions/rules. We used this tool to build a number of dynamical models, including a 130-protein large-scale model of signal transduction with over 800 interactions, influenza A replication cycle with 127 species and 200+ interactions, and mammalian and budding yeast cell cycles. We also show that any and all qualitative regulatory mechanisms can be built using this tool.     

Quantitative Petri net model of gene regulated metabolic networks in the cell

A method to exploit hybrid Petri nets (HPN) for quantitatively modeling and simulating gene regulated metabolic networks is demonstrated. A global kinetic modeling strategy and Petri net modeling algorithm are applied to perform the bioprocess functioning and model analysis. With the model, the interrelations between pathway analysis and metabolic control mechanism are outlined. Diagrammatical results of the dynamics of metabolites are simulated and observed by implementing a HPN tool, Visual Object Net ++. An explanation of the observed behavior of the urea cycle is proposed to indicate possibilities for metabolic engineering and medical care. Finally, the perspective of Petri nets on modeling and simulation of metabolic networks is discussed.     

Ontology based standardization of Petri net modeling for signaling pathways

Taking account of the great availability of Petri nets in modeling and analyzing large complicated signaling networks, semantics of Petri nets is in need of systematization for the purpose of consistency and reusability of the models. This paper reports on standardization of units of Petri nets on the basis of an ontology that gives an intrinsic definition to the process of signaling in signaling pathways.     

Hierarchical Cluster-based Partial Least Squares Regression (HC-PLSR) is an efficient tool for metamodelling of nonlinear dynamic models

Background

Network inference methods reconstruct mathematical models of molecular or genetic networks directly from experimental data sets. We have previously reported a mathematical method which is exclusively data-driven, does not involve any heuristic decisions within the reconstruction process, and deliveres all possible alternative minimal networks in terms of simple place/transition Petri nets that are consistent with a given discrete time series data set.

Results

We fundamentally extended the previously published algorithm to consider catalysis and inhibition of the reactions that occur in the underlying network. The results of the reconstruction algorithm are encoded in the form of an extended Petri net involving control arcs. This allows the consideration of processes involving mass flow and/or regulatory interactions. As a non-trivial test case, the phosphate regulatory network of enterobacteria was reconstructed using in silico-generated time-series data sets on wild-type and in silico mutants.

Conclusions

The new exact algorithm reconstructs extended Petri nets from time series data sets by finding all alternative minimal networks that are consistent with the data. It suggested alternative molecular mechanisms for certain reactions in the network. The algorithm is useful to combine data from wild-type and mutant cells and may potentially integrate physiological, biochemical, pharmacological, and genetic data in the form of a single model.