Modelling biological systems depends on the availability of data and components of the system at hand. As our understanding of these systems evolves, the ability to gradually refine models by adding new components of different formalisms covering stochastic, discrete, deterministic, and uncertainty without starting from scratch becomes essential. However, there remains a significant gap in the availability of methodologies and tool support for incrementally modelling and analysing complex biological systems in a flexible and intuitive manner. In this paper, we employ fuzzy hybrid Petri nets as a powerful expressive tool for presenting an incremental modelling and analysis protocol of biological systems. We demonstrate the utility of our protocol through a case study on cholesterol and lipoprotein metabolism and hypercholesterolemia therapy. Our model not only captures the underlying biochemical processes, but also quantitatively analyses how cholesterol levels are regulated, offering insights into potential therapeutic strategies for diseases associated with elevated cholesterol levels. The results confirm the validity and flexibility of our approach in representing complex biological processes and therapeutic interventions.

Computational biological models have proven to be an invaluable tool for understanding and predicting the behaviour of many biological systems. While it may not be too challenging for experienced researchers to construct such models from scratch, it is not a straightforward task for early stage researchers. Design patterns are well-known techniques widely applied in software engineering as they provide a set of typical solutions to common problems in software design. In this paper, we collect and discuss common patterns that are usually used during the construction and execution of computational biological models. We adopt Petri nets as a modelling language to provide a visual illustration of each pattern; however, the ideas presented in this paper can also be implemented using other modelling formalisms. We provide two case studies for illustration purposes and show how these models can be built up from the presented smaller modules. We hope that the ideas discussed in this paper will help many researchers in building their own future models.

Integrated modelling of biological systems is becoming a necessity for constructing models containing the major biochemical processes of such systems in order to obtain a holistic understanding of their dynamics and to elucidate emergent behaviours. Hybrid modelling methods are crucial to achieve integrated modelling of biological systems. This paper reviews currently popular hybrid modelling methods, developed for systems biology, mainly revealing why they are proposed, how they are formed from single modelling formalisms and how to simulate them. By doing this, we identify future research requirements regarding hybrid approaches for further promoting integrated modelling of biological systems.

Integrated modelling of biological systems is challenged by composing components with sufficient kinetic data and components with insufficient kinetic data or components built only using experts' experience and knowledge. Fuzzy continuous Petri nets (FCPNs) combine continuous Petri nets with fuzzy inference systems, and thus offer an hybrid uncertain/certain approach to integrated modelling of such biological systems with uncertainties. In this paper, we give a formal definition and a corresponding simulation algorithm of FCPNs, and briefly introduce the FCPN tool that we have developed for implementing FCPNs. We then present a methodology and workflow utilizing FCPNs to achieve hybrid (uncertain/certain) modelling of biological systems illustrated with a case study of the Mercaptopurine metabolic pathway. We hope this research will promote the wider application of FCPNs and address the uncertain/certain integrated modelling challenge in the systems biology area.

The modelling of biological systems is accompanied with epistemic uncertainties that range from structural uncertainty to parametric uncertainty due to such limitations as insufficient understanding of the underlying mechanism and incomplete measurement data of a system. Fuzzy logic approaches such as fuzzy Petri nets (FPNs) are effective in addressing these issues. In this paper, we review FPNs that have been used for modelling uncertain biological systems, which we classify in three categories: basic fuzzy Petri nets, fuzzy quantitative Petri nets and Petri nets with fuzzy kinetic parameters. For each category of these FPNs, we summarize its modelling capabilities and current applications, discuss its merits and drawbacks and give suggestions for further research. This understanding on how to use FPNs for modelling uncertain biological systems will assist readers in selecting appropriate FPN classes for specific modelling circumstances. This review may also promote the extensive research and application of FPNs in the systems biology area.

The analysis of the dynamic behaviour of genome-scale models of metabolism (GEMs) currently presents considerable challenges because of the difficulties of simulating such large and complex networks. Bacterial GEMs can comprise about 5000 reactions and metabolites, and encode a huge variety of growth conditions; such models cannot be used without sophisticated tool support. This article is intended to aid modellers, both specialist and non-specialist in computerized methods, to identify and apply a suitable combination of tools for the dynamic behaviour analysis of large-scale metabolic designs. We describe a methodology and related workflow based on publicly available tools to profile and analyse whole-genome-scale biochemical models. We use an efficient approximative stochastic simulation method to overcome problems associated with the dynamic simulation of GEMs. In addition, we apply simulative model checking using temporal logic property libraries, clustering and data analysis, over time series of reaction rates and metabolite concentrations. We extend this to consider the evolution of reaction-oriented properties of subnets over time, including dead subnets and functional subsystems. This enables the generation of abstract views of the behaviour of these models, which can be large-up to whole genome in size-and therefore impractical to analyse informally by eye. We demonstrate our methodology by applying it to a reduced model of the whole-genome metabolism of Escherichia coli K-12 under different growth conditions. The overall context of our work is in the area of model-based design methods for metabolic engineering and synthetic biology.

Owing to the availability of data of one biological phenomenon at different levels/scales, modelling of biological systems is moving from single level/scale to multiple levels/scales, which introduces a number of challenges. Coloured Petri nets (ColPNs) have been successfully applied to multilevel, multiscale and multidimensional modelling of some biological systems, addressing many of these challenges. In this article, we first review the basics of ColPNs and some popular extensions, and then their applications for multilevel, multiscale and multidimensional modelling of biological systems. This understanding of how to use ColPNs for modelling biological systems will assist readers in selecting appropriate ColPN classes for specific modelling circumstances.

BACKGROUND: Quorum sensing drives biofilm formation in bacteria in order to ensure that biofilm formation only occurs when colonies are of a sufficient size and density. This spatial behaviour is achieved by the broadcast communication of an autoinducer in a diffusion scenario. This is of interest, for example, when considering the role of gut microbiota in gut health. This behaviour occurs within the context of the four phases of bacterial growth, specifically in the exponential stage (phase 2) for autoinducer production and the stationary stage (phase 3) for biofilm formation. RESULTS: We have used coloured hybrid Petri nets to step-wise develop a flexible computational model for E.coli biofilm formation driven by Autoinducer 2 (AI-2) which is easy to configure for different notions of space. The model describes the essential components of gene transcription, signal transduction, extra and intra cellular transport, as well as the two-phase nature of the system. We build on a previously published non-spatial stochastic Petri net model of AI-2 production, keeping the assumptions of a limited nutritional environment, and our spatial hybrid Petri net model of biofilm formation, first presented at the NETTAB 2017 workshop. First we consider the two models separately without space, and then combined, and finally we add space. We describe in detail our step-wise model development and validation. Our simulation results support the expected behaviour that biofilm formation is increased in areas of higher bacterial colony size and density. Our analysis techniques include behaviour checking based on linear time temporal logic. CONCLUSIONS: The advantages of our modelling and analysis approach are the description of quorum sensing and associated biofilm formation over two phases of bacterial growth, taking into account bacterial spatial distribution using a flexible and easy to maintain computational model. All computational results are reproducible.

BACKGROUND: Determining the value of kinetic constants for a metabolic system in the exact physiological conditions is an extremely hard task. However, this kind of information is of pivotal relevance to effectively simulate a biological phenomenon as complex as metabolism. RESULTS: To overcome this issue, we propose to investigate emerging properties of ensembles of sets of kinetic constants leading to the biological readout observed in different experimental conditions. To this aim, we exploit information retrievable from constraint-based analyses (i.e. metabolic flux distributions at steady state) with the goal to generate feasible values for kinetic constants exploiting the mass action law. The sets retrieved from the previous step will be used to parametrize a mechanistic model whose simulation will be performed to reconstruct the dynamics of the system (until reaching the metabolic steady state) for each experimental condition. Every parametrization that is in accordance with the expected metabolic phenotype is collected in an ensemble whose features are analyzed to determine the emergence of properties of a phenotype. In this work we apply the proposed approach to identify ensembles of kinetic parameters for five metabolic phenotypes of E. Coli, by analyzing five different experimental conditions associated with the ECC2comp model recently published by Hädicke and collaborators. CONCLUSIONS: Our results suggest that the parameter values of just few reactions are responsible for the emergence of a metabolic phenotype. Notably, in contrast with constraint-based approaches such as Flux Balance Analysis, the methodology used in this paper does not require to assume that metabolism is optimizing towards a specific goal.

We consider trophic networks, a kind of networks used in ecology to represent feeding interactions (what-eats-what) in an ecosystem. Starting from the observation that trophic networks can be naturally modelled as Petri nets, we explore the possibility of using Petri nets for the analysis and simulation of trophic networks. We define and discuss different continuous Petri net models, whose level of accuracy depends on the information available for the modelled trophic network. The simplest Petri net model we construct just relies on the topology of the network. We also propose a technique for deriving a more refined model that embeds into the Petri net the known constraints on the transition rates that represent the knowledge on metabolism and diet of the species in the network. Finally, if the information of the biomass amounts for each species at steady state is available, we discuss a way of further refining the Petri net model in order to represent dynamic behaviour. We apply our Petri net technology to a case study of the Venice lagoon and analyse the results.

Continuous Petri nets (CPN) provide a graphical tool to model and analyse the deterministic dynamic behaviour of biological reaction networks. They can be considered as an alternative to the traditional ODE representation of biological models, enjoying a visual depiction of reaction networks. A mod el constructed as CPN can take advantages of quantitative (e.g., transient and steady state analysis) as well as qualitative (e.g., structural analysis) techniques. However, there are different semantics of CPN due to varying interpretations of transition rates. Choosing an appropriate semantics and corresponding simulator is not a straightforward procedure for the modelling of certain biological systems. In this paper, we compare two widely used semantics of CPN: adaptive semantics and bio-semantics. In the adaptive case, the enabling of continuous transitions may vary and the ODEs are correspondingly adjusted during model execution in order to avoid negative markings, while continuous transitions are always enabled in the bio-semantics and ODEs are never altered during the whole simulation period. We discuss the implementation complexity of both approaches in the context of systems biology and present two case studies to illustrate the best utilisation and individual strength of the two interpretations.

We consider localised DNA computation, where a DNA strand walks along a binary decision graph to compute a binary function. One of the challenges for the design of reliable walker circuits consists in leakage transitions, which occur when a walker jumps into another branch of the decision graph. We automatically identify leakage transitions, which allows for a detailed qualitative and quantitative assessment of circuit designs, design comparison, and design optimisation. The ability to identify leakage transitions is an important step in the process of optimising DNA circuit layouts where the aim is to minimise the computational error inherent in a circuit while minimising the area of the circuit. Our 2D modelling approach of DNA walker circuits relies on coloured stochastic Petri nets which enable functionality, topology and dimensionality all to be integrated in one two-dimensional model. Our modelling and analysis approach can be easily extended to 3-dimensional walker systems.

Both Petri nets and differential equations are important modeling tools for biological processes. In this paper we demonstrate how these two modeling techniques can be combined to describe biological gradient formation. Parameters derived from partial differential equation describing the process of gradient formation are incorporated in an abstract Petri net model. The quantitative aspects of the resulting model are validated through a case study of gradient formation in the fruit fly.

Computational steering is an interactive remote control of a long running application. The user can adopt it, e.g., to adjust simulation parameters on the fly. Simulation of large-scale biochemical networks is often computationally expensive, particularly stochastic and hybrid simulation. Such extremely time-consuming computations necessitate an interactive mechanism to permit users to try different paths and ask “what-if-questions” while the simulation is in progress. Furthermore, with the progress of computational modelling and the simulation of biochemical networks, there is a need to manage multi-scale models, which may contain species or reactions at different scales. In this context, Petri nets are of special importance, since they provide an intuitive visual representation of reaction networks. In this paper, we introduce a framework and its implementation for combining Petri nets and computational steering for the representation and interactive simulation of biochemical networks. The main merits of the developed framework are: intuitive representation of biochemical networks by means of Petri nets, distributed collaborative and interactive simulation, and tight coupling of simulation and visualisation.

Modeling across multiple scales is a current challenge in Systems Biology, especially when applied to multicellular organisms. In this paper, we present an approach to model at different spatial scales, using the new concept of Hierarchically Colored Petri Nets (HCPN). We apply HCPN to model a tissue comprising multiple cells hexagonally packed in a honeycomb formation in order to describe the phenomenon of Planar Cell Polarity (PCP) signaling in Drosophila wing. We have constructed a family of related models, permitting different hypotheses to be explored regarding the mechanisms underlying PCP. In addition our models include the effect of well-studied genetic mutations. We have applied a set of analytical techniques including clustering and model checking over time series of primary and secondary data. Our models support the interpretation of biological observations reported in the literature.

SUMMARY: To investigate biomolecular networks, Snoopy provides a unifying Petri net framework comprising a family of related Petri net classes. Models can be hierarchically structured, allowing for the mastering of larger networks. To move easily between the qualitative, stochastic and continuous modelling paradigms, models can be converted into each other. We get models sharing structure, but specialized by their kinetic information. The analysis and iterative reverse engineering of biomolecular networks is supported by the simultaneous use of several Petri net classes, while the graphical user interface adapts dynamically to the active one. Built-in animation and simulation are complemented by exports to various analysis tools. Snoopy facilitates the addition of new Petri net classes thanks to its generic design. AVAILABILITY: Our tool with Petri net samples is available free of charge for non-commercial use at http://www-dssz.informatik.tu-cottbus.de/snoopy.html; supported operating systems: Mac OS X, Windows and Linux (selected distributions).

Quantitative models of biochemical networks (signal transduction cascades, metabolic pathways, gene regulatory circuits) are a central component of modern systems biology. Building and managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Here we provide a general introduction to the field of formal modelling, which emphasizes the intuitive biochemical basis of the modelling process, but is also accessible for an audience with a background in computing science and/or model engineering. We show how signal transduction cascades can be modelled in a modular fashion, using both a qualitative approach--qualitative Petri nets, and quantitative approaches--continuous Petri nets and ordinary differential equations (ODEs). We review the major elementary building blocks of a cellular signalling model, discuss which critical design decisions have to be made during model building, and present a number of novel computational tools that can help to explore alternative modular models in an easy and intuitive manner. These tools, which are based on Petri net theory, offer convenient ways of composing hierarchical ODE models, and permit a qualitative analysis of their behaviour. We illustrate the central concepts using signal transduction as our main example. The ultimate aim is to introduce a general approach that provides the foundations for a structured formal engineering of large-scale models of biochemical networks.

BACKGROUND: Structural analysis of biochemical networks is a growing field in bioinformatics and systems biology. The availability of an increasing amount of biological data from molecular biological networks promises a deeper understanding but confronts researchers with the problem of combinatorial explosion. The amount of qualitative network data is growing much faster than the amount of quantitative data, such as enzyme kinetics. In many cases it is even impossible to measure quantitative data because of limitations of experimental methods, or for ethical reasons. Thus, a huge amount of qualitative data, such as interaction data, is available, but it was not sufficiently used for modeling purposes, until now. New approaches have been developed, but the complexity of data often limits the application of many of the methods. Biochemical Petri nets make it possible to explore static and dynamic qualitative system properties. One Petri net approach is model validation based on the computation of the system's invariant properties, focusing on t-invariants. T-invariants correspond to subnetworks, which describe the basic system behavior.With increasing system complexity, the basic behavior can only be expressed by a huge number of t-invariants. According to our validation criteria for biochemical Petri nets, the necessary verification of the biological meaning, by interpreting each subnetwork (t-invariant) manually, is not possible anymore. Thus, an automated, biologically meaningful classification would be helpful in analyzing t-invariants, and supporting the understanding of the basic behavior of the considered biological system. METHODS: Here, we introduce a new approach to automatically classify t-invariants to cope with network complexity. We apply clustering techniques such as UPGMA, Complete Linkage, Single Linkage, and Neighbor Joining in combination with different distance measures to get biologically meaningful clusters (t-clusters), which can be interpreted as modules. To find the optimal number of t-clusters to consider for interpretation, the cluster validity measure, Silhouette Width, is applied. RESULTS: We considered two different case studies as examples: a small signal transduction pathway (pheromone response pathway in Saccharomyces cerevisiae) and a medium-sized gene regulatory network (gene regulation of Duchenne muscular dystrophy). We automatically classified the t-invariants into functionally distinct t-clusters, which could be interpreted biologically as functional modules in the network. We found differences in the suitability of the various distance measures as well as the clustering methods. In terms of a biologically meaningful classification of t-invariants, the best results are obtained using the Tanimoto distance measure. Considering clustering methods, the obtained results suggest that UPGMA and Complete Linkage are suitable for clustering t-invariants with respect to the biological interpretability. CONCLUSION: We propose a new approach for the biological classification of Petri net t-invariants based on cluster analysis. Due to the biologically meaningful data reduction and structuring of network processes, large sets of t-invariants can be evaluated, allowing for model validation of qualitative biochemical Petri nets. This approach can also be applied to elementary mode analysis.

BACKGROUND: Signal transduction pathways are usually modelled using classical quantitative methods, which are based on ordinary differential equations (ODEs). However, some difficulties are inherent in this approach. On the one hand, the kinetic parameters involved are often unknown and have to be estimated. With increasing size and complexity of signal transduction pathways, the estimation of missing kinetic data is not possible. On the other hand, ODEs based models do not support any explicit insights into possible (signal-) flows within the network. Moreover, a huge amount of qualitative data is available due to high-throughput techniques. In order to get information on the systems behaviour, qualitative analysis techniques have been developed. Applications of the known qualitative analysis methods concern mainly metabolic networks. Petri net theory provides a variety of established analysis techniques, which are also applicable to signal transduction models. In this context special properties have to be considered and new dedicated techniques have to be designed. METHODS: We apply Petri net theory to model and analyse signal transduction pathways first qualitatively before continuing with quantitative analyses. This paper demonstrates how to build systematically a discrete model, which reflects provably the qualitative biological behaviour without any knowledge of kinetic parameters. The mating pheromone response pathway in Saccharomyces cerevisiae serves as case study. RESULTS: We propose an approach for model validation of signal transduction pathways based on the network structure only. For this purpose, we introduce the new notion of feasible t-invariants, which represent minimal self-contained subnets being active under a given input situation. Each of these subnets stands for a signal flow in the system. We define maximal common transition sets (MCT-sets), which can be used for t-invariant examination and net decomposition into smallest biologically meaningful functional units. CONCLUSION: The paper demonstrates how Petri net analysis techniques can promote a deeper understanding of signal transduction pathways. The new concepts of feasible t-invariants and MCT-sets have been proven to be useful for model validation and the interpretation of the biological system behaviour. Whereas MCT-sets provide a decomposition of the net into disjunctive subnets, feasible t-invariants describe subnets, which generally overlap. This work contributes to qualitative modelling and to the analysis of large biological networks by their fully automatic decomposition into biologically meaningful modules.

Abstract 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. Availability: Free availability of the Petri net editor PED, the animator PedVisor via http://www-dssz.informatik.tu-cottbus.de/~wwwdssz, and the analysis tool Integrated Net Analyser (INA) via http://www.informatik.hu-berlin.de/~starke/ina.html Contact: [email protected]

In order to adapt a Petri net based verification framework to programmable logic controllers, a Petri net semantics is introduced formally for a subset of the standardized instruction list language [IEC 1131-3]. For that purpose, the subset's syntax as well as static and operational semantics are specified strictly. Having that, the operational reference semantics is substituted by an equivalent Petri net semantics. Due to this prudent practice, the equivalence proof of the substitution step is obvious.