Ebook: Biological Petri Nets
It was suggested some years ago that Petri nets might be well suited to modeling metabolic networks, overcoming some of the limitations encountered by the use of systems employing ODEs (ordinary differential equations). Much work has been done since then which confirms this and demonstrates the usefulness of this concept for systems biology. Petri net technology is not only intuitively understood by scientists trained in the life sciences, it also has a robust mathematical foundation and provides the required degree of flexibility. As a result it appears to be a very promising approach to modeling biological systems. This unique compilation of articles is the first book from the journal In Silico Biology (ISB) on Biological Petri Nets. Subjects covered include a summary of essential topics, basic requirements and constraints; quantitative modeling of metabolic networks, gene regulatory and cell-cell communication processes; the concept of hybrid Petri nets; modeling signal transduction pathways; net models of metabolic steady states as well as a number of further applications of Petri net based modeling. The book helps to illustrate the value of Petri net approach to modern life sciences. It is dedicated to the memory of Prof. Dr. Carl Adam Petri.
Department of Bioinformatics, University Medical Center Göttingen, Göttingen, Germany, E-mail: email@example.com
As “classical” bioinformatics developed further towards modern systems biology, the idea of a holistic view of a biological system was not completely new: the aim to provide a comprehensive picture of genes and the regulatory features encoded in a genome had been inherent in bioinformatics research from the very beginning, as was the attempt to come up with an integrative view across the different levels of organisation, which was at least conceptually implicit in the numerous approaches to integrating the rapidly growing information about biological objects into comprehensive knowledge bases. However, transcendence of the research focus on static objects and progress towards the computer-aided investigation of biological processes was significantly advanced by the emerging field of systems biology. The new paradigm for formally representing the processes that make up a biological system is now the “network”.
The term “process” itself implies dynamic events, changes, that we may wish to simulate with the aid of a computer in order to predict the behavior of a biological system under certain circumstances. Biochemistry provides the formal instruments to do so for defined (bio)chemical reactions, usually resulting in a set of ordinary differential equations (ODEs). Solving the large number of ODEs that are required to describe the behavior of a complex biological system exactly may be cumbersome, but computationally feasible as soon as we have at hand all the necessary parameters, such as the corresponding kinetic constants for all reactions involved. Even in those cases where these kinetics have been studied in vitro, it is still questionable whether the insights gained from these experiments are applicable to specific in vivo conditions. Nevertheless, this approach has been proven to work for (parts of) the metabolic network of living cells, but regulatory events that depend on a very low number of individual molecules per cell may require different approaches. Moreover, applying ODEs to a large complex system may be mere overkill, and, presumably, a less exact approach might be of even more appropriate granularity, at least for the larger part of the network under consideration.
Several years ago, Petri nets were suggested as being well suited to modeling metabolic networks, overcoming some of the limitations outlined above [Reddy et al., 1993]. Since then, a great deal of further conceptual work, technical tool implementations and applications to biological problems have been reported which have demonstrated the usefulness of this concept for what we know today as systems biology. Being intuitively understandable to scientists trained in the life sciences, they also have a robust mathematical foundation and provide the required flexibility with regard to the models' granularity. As a result, Petri net technology appears to be a very promising approach to modeling biological systems.
A significant part of the progress in this field has been published by In Silico Biology since it began in 1998. Four articles constituted the first Petri Net Special in 2003 (“Petri Nets for Metabolic Networks”; http://www.bioinfo.de/isb/toc_vol_03.html). R. Hofestädt's introduction to that Special Issue summarized the most essential topics as well as some of the basic requirements and constraints relevant for applying Petri net technology, and it therefore follows this preface [Hofestädt, 2003].
The early publication of Hofestädt and Thelen, 1998, demonstrated how the Petri net concept can be extended to a more quantitative modeling of metabolic networks, and also expanded these ideas to gene regulatory and cell-cell communication processes. In a subsequent work,Chen and Hofestädt were able to demonstrate how the Petri net approach can deal with an integrated process comprising gene regulatory and metabolic events, making use of the concept of hybrid Petri nets (HPN) [Chen and Hofestädt, 2003]. Having analyzed the requirements of the biochemical particularities of metabolic networks, Zevedei-Oancea and Schuster, 2003, studied the resulting topological properties of the corresponding Petri net models. Takai-Igarashi suggested a consistent definition of the Petri net units that are required for modeling signal transduction pathways; this was based on a specific ontology, the Cell Signaling Networks Ontology (CSNO) [Takai-Igarashi, 2005]. Voss et al., 2003, complemented these efforts by studying the Petri net models of metabolic steady states including all relevant reverse reactions. Similarly, Gambin et al. made an attempt to model the stationary state of a gene regulatory network in a Petri net [Gambin et al., 2006]. A big step forward towards dynamic simulation with the aid of Petri net models was undertaken by Matsuno et al., 2003, by extending the underlying approach further to the concept of hybrid functional Petri nets (HFPN). It was proved that this could be applied to simulating the dynamics of two regulatory pathways, and it was demonstrated how such a Petri net model could be constructed [Matsuno et al., 2003; Doi et al., 2004]. More recently, the concept of firing delay times was introduced into the HFPN approach and applied to a signaling process [Miwa et al., 2010].
The conceptualization phase was also accompanied by the active development of a number of tools assisting in the creation and manipulation of biological Petri nets. While the early efforts used a platform that was originally developed for technical purposes (Visual Object Net ++, VON++) [Chen and Hofestädt, 2003], a tool specifically developed soon afterwards for modeling biological processes with Petri nets was published: Genome Object Net, GON [Matsuno et al., 2003; Doi et al., 2004]. GON evolved subsequently into Cell Illustrator (CI), a platform that is suitable for easy modeling and simulating the dynamics of cellular processes [Nagasaki et al., 2010]. A specialized tool (STEPP) for searching in a Petri net for those paths that connect, e.g., two metabolites and thus generating hypotheses about the interconversion of these two compounds was introduced by Koch et al., 2004; the program is still available for download (http://www.bioinformatik.uni-frankfurt.de/tools/stepp/stepp.html). Janowski et al. developed a powerful network editor, VANESA, that is able to work with Cell Illustrator as a simulation engine [Janowski et al., 2010].
A number of applications of Petri net-based modeling and simulation have been published in recent years. While early work usually focused on certain parts of the metabolic network such as glycolysis and pentose phosphate pathway [Voss et al., 2003; Zevedei-Oancea and Schuster, 2003; Doi et al., 2004], nucleotide metabolism [Zevedei-Oancea and Schuster, 2003], urea cycle [Chen and Hofestädt, 2003], or the conversion of sucrose into starch in potato [Koch et al., 2004], different kinds of regulatory networks have attracted more attention of late. Thus, several signal transduction networks have been studied in great detail: the Fas ligand-induced cascade leading to apoptosis [Matsuno et al., 2003], the TGF-beta pathway [Takai-Igarashi, 2005], and the p53 network [Doi et al., 2006]. This year, the IL-1 pathway was added [Miwa et al., 2010].
Gene regulatory events have also been studied, such as the control of circadian rhythm in Drosophila [Matsuno et al., 2003], the regulation of glycolysis by the lac operon [Doi et al., 2004], the flower morphogenesis of Arabidopsis [Gambin et al., 2006]. The latter topic has now been resumed by Kaufmann et al., with the aid of Cell Illustrator [Kaufmann et al., 2010]. The mRNA turnover for a number of components relevant for cell-cycle regulation was studied with a stochastic Petri net by Csikász-Nagy and Mura, 2010. The assembly of the splicesosome has been modeled by Bortfeldt et al., 2010, analyzing the modular nature of this regulatory network.
How intercellular communication processes can be linked with intracellular regulatory events and simulated has been demonstrated in the contributions of Janowski et al., taking the bacterial quorum sensing as an example [Janowski et al., 2010]. The impact of delays and noise on the dopamine signal transmission was investigated by E. Voit and colleagues [Wu et al., 2010], and both studies made use of the HFPN-base simulation engine of Cell Illustrator [Nagasaki et al., 2010].
This recent overview was published as a Special Issue on Petri Net Applications in Molecular Biology of ISB volume 10, and the entire collection now constitutes this First ISB Book on Biological Petri Nets. We are confident that the reader will benefit from this unique compilation of articles, and hope that it helps to illustrate the value of the Petri Net approach to modern life sciences.
Sadly, on the 2nd of July, 2010, while this book was in the last stages of editing, Prof. Dr. Carl Adam Petri passed away. We mourn the loss of a great scientist. His work has inspired researchers from a broad range of disciplines, which clearly indicates his perspicacious mindset. We would like to honor his outstanding scientific merits by dedicating this book to his memory.
• Bortfeldt, R. H., Schuster, S. and Koch, I. (2010). Exhaustive analysis of the modular structure of the spliceosomal assembly network - a Petri net approach. In Silico Biology 10, 0007.
• Chen, M. and Hofestaedt, R. (2003). Quantitative Petri net model of gene regulated metabolic networks in the cell. In Silico Biology 3, 0030.
• Csikász-Nagy, A. and Mura, I. (2010). Role of mRNA gestation and senescence in noise reduction during the cell cycle. In Silico Biology 10, 0003.
• Doi, A., Fujita, S., Matsuno, H., Nagasaki, M. and Miyano, S. (2004). Constructing biological pathway models with hybrid functional Petri net. In Silico Biology 4, 0023.
• Doi, A., Nagasaki, M., Matsuno, H. and Miyano, S. (2006). Simulation based validation of the p53 transcriptional activity with hybrid functional Petri net. In Silico Biology 6, 0001.
• Gambin, A., Lasota, S. and Rutkowski, M. (2006). Analyzing stationary states of gene regulatory network using Petri nets. In Silico Biology 6, 0010.
• Hofestädt, R. (2003). Petri nets and the simulation of metabolic networks. In Silico Biology 3, 0028.
• Hofestädt, R. and Thelen, S. (1998). Quantitative modeling of biochemical networks. In Silico Biology 1, 0006.
• Janowski, S., Kormeier, B., Töpel, T., Hippe, K., Hofestädt, R., Willassen, N., Friesen, R., Rubert, S., Borck, D., Haugen, P. and Chen, M. (2010). Modeling of cell-cell communication processes with Petri nets using the example of quorum sensing. In Silico Biology 10, 0003.
• Kaufmann, K., Nagasaki, M. and Jáuregui, R. (2010). Modelling the molecular interactions in the flower developmental network of Arabidopsis thaliana. In Silico Biology 10, 0008.
• Koch, I., Schüler, M. and Heiner, M. (2004). STEPP – Search Tool for Exploration of Petri net Paths: A new tool for Petri net-based path analysis in biochemical networks. In Silico Biology 5, 0014.
• Matsuno, H., Tanaka, Y., Aoshima, H., Doi, A., Matsui, M. and Miyano, S. (2003). Biopathways representation and simulation on hybrid functional Petri net. In Silico Biology 3, 0032.
• Miwa, Y., Li, C., Ge, Q.-W., Matsuno, H. and Miyano, S. (2010). On determining firing delay time of transitions for Petri net based signaling pathways by introducing stochastic decision rules. In Silico Biology 10, 0004.
• Nagasaki, M., Saito, A., Jeong, E., Li, C., Kojima, K., Ikeda, E. and Miyano, S. (2010). Cell Illustrator 4.0: A computational platform for systems biology. In Silico Biology 10, 0002.
• Reddy, V. N., Mavrovouniotis, M. L. and Liebman, M. N. (1993). Petri net representation in metabolic pathways. Proc. Int. Conf. Intell. Syst. Mol. Biol. 1, 328-336.
• Takai-Igarashi, T. (2005). Ontology based standardization of Petri net modeling for signaling pathways. In Silico Biology 5, 0047.
• Voss, K., Heiner, M. and Koch, I. (2003). Steady state analysis of metabolic pathways using Petri nets. In Silico Biology 3, 0031.
• Wu, J., Qi, Z. and Voit, E. O. (2010). Impacts of delays and noise on dopamine signal transduction. In Silico Biology 10, 0005.
• Zevedei-Oancea, I. and Schuster, S. (2003). Topological analysis of metabolic networks based on Petri net theory. In Silico Biology 3, 0029.
Today different database systems for molecular structures (genes and proteins) and metabolic pathways are available. All these systems are characterized by the static data representation. For progress in biotechnology, the dynamic representation of this data is important. The metabolism can be characterized as a complex biochemical network. Different models for the quantitative simulation of biochemical networks are discussed, but no useful formalization is available. This paper shows that the theory of Petrinets is useful for the quantitative modeling of biochemical networks.
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.
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.
Computer assisted analysis and simulation of biochemical pathways can improve the understanding of the structure and the dynamics of cell processes considerably. The construction and quantitative analysis of kinetic models is often impeded by the lack of reliable data. However, as the topological structure of biochemical systems can be regarded to remain constant in time, a qualitative analysis of a pathway model was shown to be quite promising as it can render a lot of useful knowledge, e. g., about its structural invariants. The topic of this paper are pathways whose substances have reached a dynamic concentration equilibrium (steady state). It is argued that appreciated tools from biochemistry and also low-level Petri nets can yield only part of the desired results, whereas executable high-level net models lead to a number of valuable additional insights by combining symbolic analysis and simulation.
The following two matters should be resolved in order for biosimulation tools to be accepted by users in biology/medicine: (1) remove issues which are irrelevant to biological importance, and (2) allow users to represent biopathways intuitively and understand/manage easily the details of representation and simulation mechanism. From these criteria, we firstly define a novel notion of Petri net called Hybrid Functional Petri Net (HFPN). Then, we introduce a software tool, Genomic Object Net, for representing and simulating biopathways, which we have developed by employing the architecture of HFPN.
In order to show the usefulness of Genomic Object Net for representing and simulating biopathways, we show two HFPN representations of gene regulation mechanisms of Drosophila melanogaster (fruit fly) circadian rhythm and apoptosis induced by Fas ligand. The simulation results of these biopathways are also correlated with biological observations. The software is available to academic users from http://www.GenomicObject.Net/.
In many research projects on modeling and analyzing biological pathways, the Petri net has been recognized as a promising method for representing biological pathways. From the pioneering works by Reddy et al., 1993, and Hofestädt, 1994, that model metabolic pathways by traditional Petri net, several enhanced Petri nets such as colored Petri net, stochastic Petri net, and hybrid Petri net have been used for modeling biological phenomena. Recently, Matsuno et al., 2003b, introduced the hybrid functional Petri net (HFPN) in order to give a more intuitive and natural modeling method for biological pathways than these existing Petri nets. Although the paper demonstrates the effectiveness of HFPN with two examples of gene regulation mechanism for circadian rhythms and apoptosis signaling pathway, there has been no detailed explanation about the method of HFPN construction for these examples. The purpose of this paper is to describe method to construct biological pathways with the HFPN step-by-step. The method is demonstrated by the well-known glycolytic pathway controlled by the lac operon gene regulatory mechanism.
To understand biochemical processes caused by, e.g., mutations or deletions in the genome, the knowledge of possible alternative paths between two arbitrary chemical compounds is of increasing interest for biotechnology, pharmacology, medicine, and drug design. With the steadily increasing amount of data from high-throughput experiments new biochemical networks can be constructed and existing ones can be extended, which results in many large metabolic, signal transduction, and gene regulatory networks. The search for alternative paths within these complex and large networks can provide a huge amount of solutions, which can not be handled manually. Moreover, not all of the alternative paths are generally of interest. Therefore, we have developed and implemented a method, which allows us to define constraints to reduce the set of all structurally possible paths to the truly interesting path set. The paper describes the search algorithm and the constraints definition language. We give examples for path searches using this dedicated special language for a Petri net model of the sucrose-to-starch breakdown in the potato tuber.
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.
MDM2 and p19ARF are essential proteins in cancer pathways forming a complex with protein p53 to control the transcriptional activity of protein p53. It is confirmed that protein p53 loses its transcriptional activity by forming the functional dimer with protein MDM2. However, it is still unclear that protein p53 keeps its transcriptional activity when it forms the trimer with proteins MDM2 and p19ARF. We have observed mutual behaviors among genes p53, MDM2, p19ARF and their products on a computational model with hybrid functional Petri net (HFPN) which is constructed based on information described in the literature. The simulation results suggested that protein p53 should have the transcriptional activity in the forms of the trimer of proteins p53, MDM2, and p19ARF. This paper also discusses the advantages of HFPN based modeling method in terms of pathway description for simulations.
We introduce and formally define the notion of a stationary state for Petri nets. We also propose a fully automatic method for finding such states. The procedure makes use of the Presburger arithmetic to describe all the stationary states. Finally we apply this novel approach to find stationary states of a gene regulatory network describing the flower morphogenesis of A. thaliana. This shows that the proposed method can be successfully applied in the study of biological systems.
Cell Illustrator is a software platform for Systems Biology that uses the concept of Petri net for modeling and simulating biopathways. It is intended for biological scientists working at bench. The latest version of Cell Illustrator 4.0 uses Java Web Start technology and is enhanced with new capabilities, including: automatic graph grid layout algorithms using ontology information; tools using Cell System Markup Language (CSML) 3.0 and Cell System Ontology 3.0; parameter search module; high-performance simulation module; CSML database management system; conversion from CSML model to programming languages (FORTRAN, C, C++, Java, Python and Perl); import from SBML, CellML, and BioPAX; and, export to SVG and HTML. Cell Illustrator employs an extension of hybrid Petri net in an object-oriented style so that biopathway models can include objects such as DNA sequence, molecular density, 3D localization information, transcription with frame-shift, translation with codon table, as well as biochemical reactions.
The understanding of the molecular mechanism of cell-to-cell communication is fundamental for system biology. Up to now, the main objectives of bioinformatics have been reconstruction, modeling and analysis of metabolic, regulatory and signaling processes, based on data generated from high-throughput technologies. Cell-to-cell communication or quorum sensing (QS), the use of small molecule signals to coordinate complex patterns of behavior in bacteria, has been the focus of many reports over the past decade. Based on the quorum sensing process of the organism Aliivibrio salmonicida, we aim at developing a functional Petri net, which will allow modeling and simulating cell-to-cell communication processes. Using a new editor-controlled information system called VANESA (http://vanesa.sf.net), we present how to combine different fields of studies such as life-science, database consulting, modeling, visualization and simulation for a semi-automatic reconstruction of the complex signaling quorum sensing network. We show how cell-to-cell communication processes and information-flow within a cell and across cell colonies can be modeled using VANESA and how those models can be simulated with Petri net network structures in a sophisticated way.
Parameter determination is important in modeling and simulating biological pathways including signaling pathways. Parameters are determined according to biological facts obtained from biological experiments and scientific publications. However, such reliable data describing detailed reactions are not reported in most cases. This prompted us to develop a general methodology of determining the parameters of a model in the case of that no information of the underlying biological facts is provided. In this study, we use the Petri net approach for modeling signaling pathways, and propose a method to determine firing delay times of transitions for Petri net models of signaling pathways by introducing stochastic decision rules. Petri net technology provides a powerful approach to modeling and simulating various concurrent systems, and recently have been widely accepted as a description method for biological pathways. Our method enables to determine the range of firing delay time which realizes smooth token flows in the Petri net model of a signaling pathway. The availability of this method has been confirmed by the results of an application to the interleukin-1 induced signaling pathway.
Dopamine is a critical neurotransmitter for the normal functioning of the central nervous system. Abnormal dopamine signal transmission in the brain has been implicated in diseases such as Parkinson's disease (PD) and schizophrenia, as well as in various types of drug addition. It is therefore important to understand the dopamine signaling dynamics in the presynaptic neuron of the striatum and the synaptic cleft, where dopamine synthesis, degradation, compartmentalization, release, reuptake, and numerous regulatory processes occur. The biochemical and biological processes governing this dynamics consist of interacting discrete and continuous components, operate at different time scales, and must function effectively in spite of intrinsic stochasticity and external perturbations. Not fitting into the realm of purely deterministic phenomena, the hybrid nature of the system requires special means of mathematical modeling, simulation and analysis. We show here how hybrid functional Petri-nets (HFPNs) and the software Cell Illustrator® facilitate computational analyses of systems that simultaneously contain deterministic, stochastic, and delay components. We evaluate the robustness of dopamine signaling in the presence of delays and noise and discuss implications for normal and abnormal states of the system.
Recent innovations in experimental techniques on single molecule detection resulted in advances in the quantification of molecular noise in several systems, and provide suitable data for defining stochastic computational models of biological processes. Some of the latest stochastic models of cell cycle regulation analyzed the effect of noise on cell cycle variability. In their study, Kar et al. (Proc. Natl. Acad. Sci. USA 106, 6471–6476, 2009) found that the observed variances of cell cycle time and cell division size distributions cannot be matched with the measured long half-lives of mRNAs. Here, we investigate through modeling and simulation how the noise created by the transcription and degradation processes of a key cell cycle controller mRNA affect the statistics of cell cycle time and cell size at division. Our model consists of an encoding of the model of Kar et al. into a stochastic Petri net, with the extensions necessary to represent multiple synthesis (gestation) and degradation (senescence) steps in the regulation of mRNAs. We found that few steps of gestation and senescence of mRNA are enough to give a good match for both the measured half-lives and variability of cell cycle-statistics. This result suggests that the complex process of transcription can be more accurately approximated by multi-step linear processes.
Spliceosomes are macro-complexes involving hundreds of proteins with many functional interactions. Spliceosome assembly belongs to the key processes that enable splicing of mRNA and modulate alternative splicing. A detailed list of factors involved in spliceosomal reactions has been assorted over the past decade, but, their functional interplay is often unknown and most of the present biological models cover only parts of the complete assembly process. It is a challenging task to build a computational model that integrates dispersed knowledge and combines a multitude of reaction schemes proposed earlier.
Because for most reactions involved in spliceosome assembly kinetic parameters are not available, we propose a discrete modeling using Petri nets, through which we are enabled to get insights into the system's behavior via computation of structural and dynamic properties. In this paper, we compile and examine reactions from experimental reports that contribute to a functional spliceosome. All these reactions form a network, which describes the inventory and conditions necessary to perform the splicing process. The analysis is mainly based on system invariants. Transition invariants (T-invariants) can be interpreted as signaling routes through the network. Due to the huge number of T-invariants that arise with increasing network size and complexity, maximal common transition sets (MCTS) and T-clusters were used for further analysis. Additionally, we introduce a false color map representation, which allows a quick survey of network modules and the visual detection of single reactions or reaction sequences, which participate in more than one signaling route.
We designed a structured model of spliceosome assembly, which combines the demands on a platform that i) can display involved factors and concurrent processes, ii) offers the possibility to run computational methods for knowledge extraction, and iii) is successively extendable as new insights into spliceosome function are reported by experimental reports. The network consists of 161 transitions (reactions) and 140 places (reactants). All reactions are part of at least one of the 71 T-invariants. These T-invariants define pathways, which are in good agreement with the current knowledge and known hypotheses on reaction sequences during spliceosome assembly, hence contributing to a functional spliceosome. We demonstrate that present knowledge, in particular of the initial part of the assembly process, describes parallelism and interaction of signaling routes, which indicate functional redundancy and reflect the dependency of spliceosome assembly initiation on different cellular conditions. The complexity of the network is further increased by two switches, which introduce alternative routes during A-complex formation in early spliceosome assembly and upon transition from the B-complex to the C-complex. By compiling known reactions into a complete network, the combinatorial nature of invariant computation leads to pathways that have previously not been described as connected routes, although their constituents were known. T-clusters divide the network into modules, which we interpret as building blocks in spliceosome maturation. We conclude that Petri net representations of large biological networks and system invariants, are well-suited as a means for validating the integration of experimental knowledge into a consistent model. Based on this network model, the design of further experiments is facilitated.
We present a dynamical model of the gene network controlling flower development in Arabidopsis thaliana. The network is centered at the regulation of the floral organ identity genes (AP1, AP2, AP3, PI and AG) and ends with the transcription factor complexes responsible for differentiation of floral organs. We built and simulated the regulatory interactions that determine organ specificity using an extension of hybrid Petri nets as implemented in Cell Illustrator. The network topology is characterized by two main features: (1) the presence of multiple autoregulatory feedback loops requiring the formation of protein complexes, and (2) the role of spatial regulators determining floral patterning. The resulting network shows biologically coherent expression patterns for the involved genes, and simulated mutants produce experimentally validated changes in organ expression patterns. The requirement of heteromeric higher-order protein complex formation for positive autoregulatory feedback loops attenuates stochastic fluctuations in gene expression, enabling robust organ-specific gene expression patterns. If autoregulation is mediated by monomers or homodimers of proteins, small variations in initial protein levels can lead to biased production of homeotic proteins, ultimately resulting in homeosis. We also suggest regulatory feedback loops involving miRNA loci by which homeotic genes control the activity of their spatial regulators.