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Over the last twenty years, Petri nets have been increasingly adopted for modelling and simulating biological systems, as they offer an intuitive and graphical approach for this purpose. Their usability convenience comes from the fact that they offer many types of elements to describe systems in a qualitative and quantitative way. Coloured Petri nets are particularly useful to model systems with repeated components in a compact fashion. Our tool Snoopy for modelling and simulating Petri nets is one of the most well-known tools supporting a family of related Petri net classes comprising stochastic, continuous and hybrid Petri nets, and covering uncoloured and coloured Petri nets alike. However, kinetic information of a biological system, i.e. kinetic parameters may be uncertain, due to many reasons, e.g. environmental factors. Besides, coloured Petri nets as they were previously supported in Snoopy suffered from some inconsistencies. Due to these inconsistencies, exploring the model behaviour using different sizes (scaleability) was not feasible. Both challenges call for a new and more powerful approach integrating the modelling of uncertainties together with modelling features supporting repeated structures in a compact and scalable way.
This thesis comprises two major contributions: Firstly, we introduce the definition and present the simulation algorithm for both uncoloured and coloured fuzzy Petri nets, by extending the existing quantitative uncoloured and coloured Petri nets in Snoopy. This includes discretising the uncertain kinetic parameters to crisp values by using sampling strategies. Secondly, we harmonise coloured Petri nets in Snoopy with their uncoloured counterparts and we extend the Snoopy’s coloured Petri nets by all the features, which are supported by the coloured abstract net description language - an exchange format of coloured Petri nets in our PetriNuts tool family.
By performing fuzzy simulation, one can obtain two kinds of output: fuzzy bands of each output variable and their corresponding timed-membership functions. Each fuzzy band describes the uncertainties associated with the input, whereas membership functions give more accurate information about the associated uncertainties. The most important features that we obtain by harmonising coloured Petri nets are to develop scaleable models, by defining scaling factors as constants and unifying the usage of coloured Petri nets with the other tools in our PetriNuts tool family.
Computational steering is an interactive remote control of a long running application. The user can adopt it to adjust the simulation parameters on the fly. Correspondingly, simulation of large scale biochemical networks is computationally expensive, particularly stochastic and hybrid simulation. Such extremely intensive computations necessitate an interactive mechanism to permit users to try different paths and ask simultaneously "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 (called also stiff systems). In this context, Petri nets are of considerable importance in the modelling and analysis of biochemical networks, since they provide an intuitive visual representation of reaction networks. The contributions of this thesis are twofold: firstly, we introduce the definition and present simulation algorithms of Generalised Hybrid Petri Nets (GHPNbio) to represent and simulate stiff biochemical networks where fast reactions are represented and simulated continuously, while slow reactions are carried out stochastically. GHPNbio provide rich modelling and simulation functionalities by combining all features of Continuous Petri Nets (CPN) and Extended Stochastic Petri Nets (XSPN), including three types of deterministic transitions. Moreover, the partitioning of the reaction networks can either be done off-line before the simulation starts or on-line while the simulation is in progress. Secondly, we introduce a novel framework which combines Petri nets and computational steering for the representation and interactive simulation of biochemical networks. The main merits of the framework proposed in this thesis are: the tight coupling of simulation and visualisation, distributed; collaborative; and interactive simulation, and intuitive representation of biochemical networks by means of Petri nets. Generalised hybrid Petri nets and computational steering will together provide an invaluable tool for systems biologists to help them to obtain a deeper system level understanding. GHPNbio speed up the simulation and simultaneously preserve accuracy, while computational steering enables users of different background to share, collaborate and interactively simulate biochemical models. Finally, the implementation of the proposed framework is given as part of Snoopy - a tool to design and animate/simulate hierarchical graphs, among them qualitative, stochastic, continuous and hybrid Petri nets.
Modeling plays a crucial role in Systems Biology in order to provide a system-level understanding of biological systems. With the rapid development of systems biology, modeling of biological systems has shifted from single scales to multiple scales. This introduces a series of challenges that should be addressed, e.g. repetition of components such as genes and cells, variation of components, or hierarchical organization of components. Traditional modeling approaches, e.g. Petri nets, cannot afford to cope with these challenges, which, however, can be tackled using colored Petri nets. This thesis aims to present a technology based on colored Petri nets and associated techniques to address challenges introduced by multiscale modeling in systems biology and to implement them in our modeling tool, Snoopy. To this aim, we present a colored Petri net framework for systems biology, which relates three modeling paradigms: colored qualitative Petri net (QPNC), colored stochastic Petri net (SPNC) and colored continuous Petri net (CPNC). Using this framework, we can model and analyze a biological system from three different perspectives: qualitative, stochastic and continuous by converting them into each other. We implement this framework in our modeling tool, Snoopy, and therefore in this thesis we explore three key problems concerning the implementation of colored Petri nets. For animating/simulating colored Petri nets, we present an efficient algorithm for the computation of enabled transition instances. In order to utilize the analysis techniques of Petri nets we present an efficient unfolding algorithm for large-scale colored Petri nets. In addition, we discuss three special cases for automatic folding (colorizing): colorizing T-invariants, master nets and twin nets in order to reduce the amount of work for folding Petri nets. Petri nets offer a large variety of analysis techniques ranging from informal techniques, e.g. animation/simulation to formal techniques, e.g. model checking. We summarize those analysis techniques that can be used for colored Petri nets, e.g. structural analysis, numerical and simulative model checking from the application point of view. We discuss some scenarios to illustrate the potential capability of colored Petri nets to cope with challenges in systems biology. Moreover, we apply our colored Petri net technology and techniques to three case studies, C. elegans vulval development, coupled Ca2+ channels and membrane systems. These case studies not only demonstrate how to use the colored Petri net framework and related analysis techniques for modeling and analyzing biological systems, but also show how to address the challenges of systems biology.