Nat Comput (2011) 10:633–638 DOI 10.1007/s11047-011-9253-1
Preface: Petri nets for Systems and Synthetic Biology Monika Heiner
Published online: 19 April 2011 ! 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 current ‘‘Part1: Bridging Gaps’’ and the forthcoming ‘‘Part2: 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 at Cottbus, Cottbus, Germany e-mail:
[email protected] URL: http://www-dssz.informatik.tu-cottbus.de
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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 is 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), which 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 this 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 this 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 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.
Fig. 1 Framework unifying the qualitative, stochastic and continuous paradigms
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We briefly characterize here the most important ones contributed by the Petri net community. – 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 will be recalled in the paper by Chen et al. in the second part of this 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 will be demonstrated in the paper by Popova-Zeugmann in the second part of this special issue. – Generalised Stochastic Petri Nets (GSPN) extend SPN by immediate transitions (Ajmone Marsan et al. 1995). The paper by Lamprecht et al., which will appear in the 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 will also explain in detail—for the first time for HFPN—how the underlaying ODEs are generated and analysed. Both papers will appear in the second part of this special issue. – Generalised Hybrid Petri Nets combine all features of XSPN and CPN (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, will be introduced in this first part of the 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. We present selected contributions divided into two parts. Part1
Bridging Gaps
is devoted to network design techniques: Where and how do the models come from?
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Part2
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Unifying Diversity
will focus on network analysis and simulation techniques: What can be done with a model?
2 Outline of part 1: Bridging Gaps This first part of the special issue on Petri Nets in Systems and Synthetic Biology comprises six papers, which deploy a variety of different mathematical formalisms. The papers bridge quite a number of different gaps in various ways, each paper in its own way. (1) We begin with the paper ‘‘Petri nets as a framework for the reconstruction and analysis of signal transduction pathways and regulatory networks’’ by Wolfgang Marwan et al. It provides two bridges in its gentle introduction into the field. At first, it demonstrates that one and the same quantitative (kinetic) model can be read in two different ways: stochastically or continuously. In the stochastic case—in our modelling framework represented as stochastic Petri nets—the rate functions specify the state-dependent rates k of the exponentially distributed waiting time of the stochastic transitions (reactions), while in the continuous case—in our modelling framework represented as continuous Petri nets—the very same rate functions (up to some scaling adaptations) are taken to specify the strength of the continuous flow. The continuous perspective results in a deterministic behaviour, abstracting from the underlying stochasticity by just considering the averaged case. Secondly, this paper illustrates for a number of typical modelling problems in Systems Biology, how to represent the inherent mechanisms by means of (qualitative) Petri nets, among them competitive and allosteric inhibition, cooperative binding (in a continuous setting often summed up by one reaction following the Hill kinetics), negative feed-back loops (inhibition), or regulatory mechanisms in gene expression. Having agreed upon the model structure, e.g. by playing the token game and following the token flow and thus the causality among the elementary reactions, the Petri net may be enriched by the reaction kinetics (i.e., the precise rate functions), which then allows to read the model as a stochastic or continuous Petri net, opening the door to standard stochastic and continuous analysis techniques; some of them will be covered in the second part of this special issue. (2) In the opening introductory paper, modelling and specifically structure identification rely on the profound insights and the skills of the modeller in getting the right abstraction of the incredible diversity of bio-molecular phenomena. The next paper ‘‘The combinatorics of modeling and analyzing biological systems’’ by Annegret K. Wagler and Robert Weismantel takes another perspective and looks for a systematic and mathematically sound method to formally reconstruct the structure from experimental data, i.e., to detect the local mechanisms of species’ interactions in an algorithmic manner. A strength of the proposed combinatorial approach is that it reveals all structures in conformance with the observed data, and if there are none, the algorithm indicates new species (places) which could resolve the puzzle. All required background material from discrete mathematics and polyhedral theory is recalled. En passant the paper elucidates the relation between different representations of the continuous members of a cone (known in the Systems Biology community as elementary flux modes, extreme pathways, and minimal metabolic behaviour) and its integer counterpart (known in the Petri net community as minimal T-invariants). The paper bridges combinatorics and Systems Biology; specifically it shows that different concepts from
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discrete mathematics and polyhedral theory are crucial for the understanding and exploration of bio-molecular systems. (3) The paper ‘‘A Petri net representation of Bayesian message flows: importance of Bayesian networks for biological applications’’ by Kurt Lautenbach and Alexander Pinl follows another approach to reconstruct bio-molecular models. It favours Bayesian networks to express the knowledge which has been statistically extracted from array data, and to use them as executable and analyzable models of the data source. To support this kind of reasoning, the paper pioneers a Petri net representation for the propagation of probabilities and likelihoods in Bayesian networks, yielding a novel Petri net class called probability propagation nets. Bridging the gaps helps here to increase the transparency of propagation processes in the Bayesian world by exploiting structural and dynamic properties of Petri nets. (4) Yet another new Petri net class is launched out in the paper ‘‘Error-correcting Petri nets’’ by Anastasia Pagnoni. It bridges two apparently unrelated fields: reachability of states (markings) in Petri nets and Hamming codes. The mathematics of linear errorcorrecting codes (specifically modulo-p Hamming codes) guides the structural extension of a given Petri net in a way that allows for the algebraic detection and modulo-p correction of non-reachable Petri net states. Most importantly, in doing so the state space is never constructed. All notions required form coding theory are recalled. The method is demonstrated in full details through a didactical example. This result may gain importance in Synthetic Biology, e.g., to design biological systems with monitoring capabilities. The observation of states which are supposed to be nonreachable in the designed system will indicate some behavioural deviation from the original design. The method is able to pinpoint the source of the trouble to a single place (several places) for a single error (multiple errors) depending on the amount of structural redundancy spent. (5) The paper ‘‘Petri net representation of multi-valued logical regulatory graphs’’ by Claudine Chaouiya et al. considers the relation between two modelling techniques, which are typically applied on different abstraction levels. Petri nets are a natural choice to represent elementary chemical reactions following consumption/production semantics. The representation of gene regulatory mechanisms by means of standard Petri nets is less straightforward, because the regulation of gene expression is often characterised as just activating or inhibiting a target gene depending on the states of some other genes. Logical graphs provide a convenient abstraction level for such gene regulatory networks. This paper bridges the gap and proposes a systematic translation of logical graphs into elementary and coloured Petri nets to take advantage of the Petri net simulation and analysis tools, and to open the door to quantitative Petri net extensions. Vice versa, the circuit analysis—an established technique in logical graph analysis—complements existing Petri net methods by illuminating relationships between the occurrence of regulatory circuits and specific dynamical properties such as multi-stationarity or sustained oscillations. (6) We close this first part of the special issue with the paper ‘‘Boundedness analysis for open Chemical Reaction Networks with mass-action kinetics’’ by David Angeli. It is devoted to a challenging research field and requires discrete as well as continuous reasoning. The paper bridges the gap between structure (topology) analysis of a continuous Petri net and the dynamic behaviour of the underlaying ordinary differential equation system. With other words, it looks for structure-induced constraints on the qualitative properties of the continuous behaviour. Networks are considered with all reactions
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following the popular mass-action kinetics and independently of specific values for the kinetic parameters or specific initial concentrations. Starting from some rather mild assumptions, the paper performs a qualitative study of input/output stability for open networks, i.e., networks with input/output flows (input/ output transitions), to rule out dynamic behavior which is not compatible with the given network structure. The proposed algorithm classifies each variable (continuous place, species) either as converging to 0 (extinction), bounded, or diverging to ? (i.e., , unbounded), and returns all extinction/boundedness/divergence scenarios compatible with the network structure. The introduced analysis approach is demonstrated for a network of 9 chemical species, with a potential for 39 = 19,683 possible scenarios. P-invariants rule out unboundedness for six species, thus we are left with 33 9 62 = 972 potentially possible scenarios. However, in the end only four scenarios get approved by the proposed simple, but efficient analysis algorithm. Most remarkably, the scenario in which three species simultaneously diverge to infinity (as one would expect based on the x-state in the coverability graph) is discarded by the algorithm and is therefore not compatible with mass-action kinetics. The most intuitive scenario, in which all species are bounded and do not approach zero, appears only for relatively small inflow rates. 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, Newyork Baldan P, Cocco N, Marin A, Simeoni M (2010) Petri nets for modelling metabolic pathways: a survey. J Nat Comput 9:955–989 Chaouiya C (2007) Petri net modelling of biological networks. Brief in Bioinform 8(4):210–219 David R, Alla H (2005) Discrete, continuous, and hybrid Petri nets. Springer, Berlin Doi A, Drath R, Nagaska M, Matsuno H, Miyano S (1999) Protein dynamics observations of Lambda-Phage by hybrid Petri net. Genome Inform 10:217–218 Heiner M, Gilbert D, Donaldson R (2008) Petri nets in systems and synthetic biology. In: Schools on formal methods (SFM), LNCS, vol 5016. Springer, Heidelberg, pp 215–264 Heiner M, Lehrack S, Gilbert D, Marwan W (2009a) Extended stochastic Petri nets for model-based design of Wetlab experiments. transactions on computational systems biology XI. Springer, Heidelberg, pp 138–163 Heiner M, Schwarick M, Tovchigrechko A (2009b) DSSZ-MC—a tool for symbolic analysis of extended Petri nets. In: Proceedings of the PETRI NETS 2009. LNCS, vol 5606. Springer, pp 323–332, June Herajy M, Heiner M (2010) Hybrid Petri Nets for modelling of hybrid biochemical interactions. In: Proceedings of the 17th German workshop on algorithms and tools for Petri nets (AWPN 2010). CEUR workshop proceedings, vol 643, pp 66–79. CEUR-WS.org, October Merlin PM (1974) A study of the recoverability of computing systems. PhD Thesis, University of California, Irvine 1974. Available from University Microfilms, Ann Arbor, No. 75–11026 Valk R (1978) Self-modifying nets, a natural extension of Petri nets. In: LNCS, automata, languages and programming, vol 62, pp 464–476
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