Nat Comput DOI 10.1007/s11047-011-9263-z Preface: Petri nets for Systems and Synthetic Biology Monika Heiner ! Springer Science+Business Media B.V. 2011 1 Introduction This special issue Petri Nets for Systems and Synthetic Biology presents selected highlights of a challenging and highly active research field. It consists of two parts, the former ‘‘Part 1: Bridging Gaps’’ and the current ‘‘Part 2: Unifying Diversity’’. Systems Biology is the biology-based interdisciplinary research area that focuses on complex interactions between the components of biological systems, and how these interactions give rise to function and behavior of these systems. One of the ambitions of Systems Biology is to discover the outcome of organic evolution and to describe this acquired knowledge in models, which are explanatory of the biological mechanisms as well as suitable for reliable prediction of behaviour when the system is perturbed by, e.g., mutations, chemical interventions or changes in the environment. In the emerging discipline Synthetic Biology, the very same kind of models are taken as design templates for novel synthetic biological systems, i.e., to design and construct new biological functions and systems not found in nature. Here, model verification and validation turn out to be crucial for reliable system design as models may serve as blueprints. One of the core issues in Systems and Synthetic Biology is the construction of biomolecular networks; either the reconstruction of networks which have been designed—as opposed to technical systems—by the organic evolution of living organisms, or the variation and/or design of novel networks, respectively. This kind of networks are most naturally described by bipartite graphs, e.g., Petri nets (PN), to distinguish between passive system components (such as chemical compounds, proteins, genes, etc.) and active system components (such as chemical reactions, complexation/decomplexation, activation/deactivation, etc.). Petri Nets have a well-defined semantics, which can either be an interleaving semantics captured in Labeled Transition Systems (LTS) to describe all possible behaviour by all interleaving sequences in the style of transition-labelled automata, or a partial order M. Heiner (&) Brandenburg University of Technology, Cottbus, Germany e-mail: monika.heiner@brunel.ac.uk URL: http://www-dssz.informatik.tu-cottbus.de 123 M. Heiner Fig. 1 Framework unifying the qualitative, stochastic and continuous paradigms semantics, usually given as finite prefix of the maximal branching process (PO prefix). Both descriptions of behaviour can be analysed for the purpose of model verification. Having agreed upon the structure, the next modelling step usually consists in getting the time-dependent behaviour right, which imposes specific timing constraints for the overall behaviour possible under any timing. This can basically be done in two different ways— stochastically or continuously. In summary, we get a family of related models with high analytical power. An overall framework unifying these three paradigms has been introduced in Gilbert et al. (2007) and is further discussed in Heiner et al. (2008); see also Fig. 1. Stochastic Petri Net (SPN) seem to be a natural choice as the behaviour of biochemical networks is inherently governed by stochastic laws. SPNs preserve the discrete state description, but in addition associate a probabilistically distributed firing rate (waiting time) with each transition (reaction). The underlying semantics defines a Continuous-Time Markov Chain (CTMC), which is isomorphic with the LTS if there are no parallel transitions. If molecules are in high numbers, and stochastic effects can be neglected, one can easily take a deterministic approach and read the given net structure and its attached kinetics as a Continuous Petri Net (CPN). A CPN uniquely defines a system of ordinary differential equations (ODEs), however generally not vice versa (Soliman and Heiner 2010). A CPN can immediately be analysed with all the standard ODEs’ analytical or numerical methods to explore how the averaged concentrations of species evolve over time. The paper by Marwan et al. in the first part of the special issue shows that one and the same quantitative (kinetic) model can be read either stochastically or continuously, with no changes required (to be precise: higher order reactions might be subjected to some scaling adaptations). The move from the discrete to the continuous world comes along with counter-intuitive effects; see the final paper in the first part of special issue by Angeli for an illustrative sample of such counter-intuitive behaviour. Besides the base case techniques, adequate and very efficient modelling and computational techniques are required, supporting such aspects as various abstraction levels, hierarchical thinking, and an holistic perspective on inherently multi-scale objects. As a consequence, quite a variety of different approaches have been proposed over the last 123 Preface years, and new ones are constantly emerging. At the same time, many modelling techniques, well-known from Computer Science and well-equipped with elaborated analysis methods, have been applied to this exciting and challenging research field. We briefly characterize here the most important ones contributed by the Petri net community; see Heiner and Gilbert (2011) for more details. – Extended Petri Nets (XPN) support read and inhibitor arcs; see, e.g., Chaouiya (2007) and Heiner et al. (2009b). – Functional Petri Nets (FPN) pick up the idea of self-modifying nets (Valk 1978) and use state-dependent functions to dynamically adjust arc multiplicities. This net class is recalled in the paper by Chen et al. in this second part of the special issue. – Time Petri Nets (TPN) equip transitions with a deterministic firing delay, typically characterized by a continuous interval (Merlin 1974). The particular capabilities of this net class for Systems and Synthetic Biology is demonstrated in the paper by PopovaZeugmann in this second part of the special issue. – Generalised Stochastic Petri Nets (GSPN) extend SPN by immediate transitions (Ajmone Marsan et al. 1995). The paper by Lamprecht et al. in this second part of this special issue discusses scalability issues that evolve in the use of GSPNs for stochastic models of Ca2? signaling complexes. – Extended Stochastic Petri Nets (XSPN) enrich GSPN with transitions having deterministic firing delay, typically characterized by a constant; see, e.g., Heiner et al. (2009a). – Autonomous Continuous Petri Nets (ACPN) let transitions fire continuously, but timefree; we get a continuous state space (David and Alla 2005). – Hybrid Petri Nets (HPN) enrich CPN with discrete places and discrete transitions having deterministic firing delay, see, e.g., Doi et al. (1999). – Hybrid Functional Petri Nets (HFPN) are a popular extension of HPN combining them with the self-modifying arcs of FPN. This net class is deployed in the paper by Chen et al. to discuss the semi-automatic construction of large scale biological networks and in the paper by Matsuno et al. for the simulation of biochemical pathways. The latter paper goes even one step further by introducing Hybrid Functional Petri Nets with extensions (HFPNe). Both papers appear in this second part of the special issue. – Generalised Hybrid Petri Nets combine all features of XSPN and CPN. They are specifically powerful if dynamic partitioning into the discrete and continuous net elements is supported (Herajy and Heiner 2010). This variety clearly demonstrates one of the big advantages of using Petri nets as a kind of umbrella formalism—the models may share the network structure, but vary in their kinetic details (quantitative information). For a recent survey of case studies applying various Petri net classes in the context of Systems and Synthetic Biology see Baldan et al. (2010). Two further novel Petri net classes, which may turn out to be useful in Systems and Synthetic Biology, were introduced in the first part of this special issue: the Probability Propagation Nets (see third paper by Lautenbach and Pinl), and the Error-correcting Petri Nets (see fourth paper by Pagnoni). In summary, Systems and Synthetic Biology are obviously full of challenges and open issues, with adequate modelling and analysis techniques being one of them; see Heiner and Gilbert (2011) for a recent review paper. In this special issue, we present selected contributions divided into two parts. PART 1—Bridging Gaps 123 M. Heiner Is devoted to network design techniques: where and how do the models come from? PART 2—Unifying Diversity Focusses on network analysis and simulation techniques: what can be done with a model? 2 Outline of part 2: Unifying Diversity This second part of the special issue on Petri Nets in Systems and Synthetic Biology comprises five papers, which deploy a variety of different mathematical formalisms to gain new insights into the modelled subject. Each paper takes its own perspective on formal models, which all share the same structuring principles—being bipartite. Together the papers contribute to a unifying framework applying various analysis techniques which look at the very first glance rather diverse. We present the papers (and thus their respective analysis and simulation techniques) in an order which might be appropriate for a step-wise model evaluation procedure. (1) (2) (3) We begin with the paper ‘‘A discrete Petri net model for cephalostatin-induced apoptosis in leukemic cells’’ by Eva Rodriguez et al., which introduces the first discrete model for cephalostatin 1-induced apoptosis. The authors explain the development of a qualitative Petri net by combining experimental data and results from a search through the literature. Thus, the medium-sized Petri net itself represents new knowledge. To validate the model, standard Petri net analysis techniques are applied, such as structural and invariant analyses, complemented by a recently developed modularisation approach using abstract dependent transition sets (ADT sets). The submodules represented by the ADT sets were compared with the functional modules of apoptosis identified by another modularisation approach. All results of these analyses are consistent with known biological behavior. This kind of model validation is an important first step in gaining confidence in the structure of newly developed models. The next paper ‘‘Quantitative evaluation of time-dependent Petri nets and applications to biochemical networks’’ by Louchka Popova-Zeugmann explores the potential of time-dependent Petri nets for analysing biochemical networks. It specifically considers a well-known extension of standard Petri nets, where each transition gets a continuous time interval, specifying the range of the transitions’ reaction time; this net class is commonly known as Time Petri Net (TPN). A crucial point in applying TPN is that they permit to be analysed in a discrete manner, despite the continuous time intervals associated to transitions. For this purpose, the paper exploits basically two ideas: parametric state description and discretization of the state space. Altogether, eight problems are introduced, characterised by their input/output relation. A sketch of the solution idea as well as possible application scenarios to evaluate biochemical systems are given as well. The approach is illustrated using a classical 6-variable model of cell division cycle. The paper ‘‘Stochastic Petri net models of Ca2? signaling complexes and their analysis’’ by Ruth Lamprecht et al. illustrates how Stochastic Petri Nets (SPN) and in particular Stochastic Activity Networks (SAN) can contribute to gain new insights into dynamical phenomena in cell biology. The paper emphasizes scalability issues that evolve in the use of stochastic models. It investigates how state-of-the-art techniques for the numerical and simulative analysis of Markov chains scale with 123 Preface (4) (5) increasing model size, while taking advantage of the structural knowledge if Markov chains are induced by stochastic Petri nets. The paper considers as case study Ca2? release site models, which are composed of a number of individual channel models whose dynamic behavior depends on the local Ca2? concentration which in turn is influenced by the state of all channels. This application area highlights two characteristics of complex (stochastic) models in biology: models are structured by their compositional description, and models may have a spatial aspect that a modeling formalism should be able to take into account. The paper ‘‘Petri net models for the semi-automatic construction of large scale biological networks’’ by Ming Chen et al. favours Hybrid Functional Petri Nets (HFPN) as an adequate method to model and simulate complex biological networks. It recalls the historical background of hybrid Petri nets, which concludes with an overview of some contemporary biological simulation tools, among them some Petri net simulators. The paper also addresses the timely issue of model construction and data integration; currently there are more than 1000 relevant molecular databases comprising fundamental molecular data. The authors propose an integration process to support semi-automatic generation of hybrid Petri net models and present a tool which has been developed with these objectives in mind. A cardio-disease related generegulated network serves as case study. We close this second part of the special issue with the paper ‘‘Hybrid Petri net based modeling for biological pathway simulation’’ by Matsuno et al., which starts with hybrid Petri nets and then introduces a new net class, the Hybrid Functional Petri Nets with extensions (HFPNe), which may serve as a highly abstract language for biological process modelling and simulation. The paper illustrates the approach taken with two case studies: the k phage genetic switch system including induction and retro-regulation mechanisms, and the cyanobacterial circadian gene clock system. We wish all readers of this special issue an enjoyable journey through some selected highlights of the challenging field of Systems and Synthetic Biology. Adequate modelling techniques and dedicated analysis approaches will certainly have their share in getting the ongoing puzzles right, sometimes. References Ajmone Marsan M, Balbo G, Conte G, Donatelli S, Franceschinis G (1995) Modelling with generalized stochastic Petri nets. Wiley series in parallel computing, 2nd edn. Wiley, New York Baldan P, Cocco N, Marin A, Simeoni M (2010) Petri nets for modelling metabolic pathways: a survey. J Nat Comput 9(4):955–989 Chaouiya C (2007) Petri net modelling of biological networks. Briefings Bioinform 8(4):210–219 David R, Alla H (2005) Discrete, continuous, and hybrid Petri nets. 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