Petri Net Based Model Validation in Systems Biology

Qualitative Modeling, Analysis and Control of Synthetic Regulatory Circuits

Qualitative modeling approaches are promising and still underexploited tools for the analysis and design of synthetic circuits. They can make predictions of circuit behavior in the absence of precise, quantitative information. Moreover, they provide direct insight into the relation between the feedback structure and the dynamical properties of a network. We review qualitative modeling approaches by focusing on two specific formalisms, Boolean networks and piecewise-linear differential equations, and illustrate their application by means of three well-known synthetic circuits. We describe various methods for the analysis of state transition graphs, discrete representations of the network dynamics that are generated in both modeling frameworks. We also briefly present the problem of controlling synthetic circuits, an emerging topic that could profit from the capacity of qualitative modeling approaches to rapidly scan a space of design alternatives.

Modeling the Metabolism of Arabidopsis thaliana: Application of Network Decomposition and Network Reduction in the Context of Petri Nets

Motivation:Arabidopsis thaliana is a well-established model system for the analysis of the basic physiological and metabolic pathways of plants. Nevertheless, the system is not yet fully understood, although many mechanisms are described, and information for many processes exists. However, the combination and interpretation of the large amount of biological data remain a big challenge, not only because data sets for metabolic paths are still incomplete. Moreover, they are often inconsistent, because they are coming from different experiments of various scales, regarding, for example, accuracy and/or significance. Here, theoretical modeling is powerful to formulate hypotheses for pathways and the dynamics of the metabolism, even if the biological data are incomplete. To develop reliable mathematical models they have to be proven for consistency. This is still a challenging task because many verification techniques fail already for middle-sized models. Consequently, new methods, like decomposition methods or reduction approaches, are developed to circumvent this problem. Methods: We present a new semi-quantitative mathematical model of the metabolism of Arabidopsis thaliana. We used the Petri net formalism to express the complex reaction system in a mathematically unique manner. To verify the model for correctness and consistency we applied concepts of network decomposition and network reduction such as transition invariants, common transition pairs, and invariant transition pairs. Results: We formulated the core metabolism of Arabidopsis thaliana based on recent knowledge from literature, including the Calvin cycle, glycolysis and citric acid cycle, glyoxylate cycle, urea cycle, sucrose synthesis, and the starch metabolism. By applying network decomposition and reduction techniques at steady-state conditions, we suggest a straightforward mathematical modeling process. We demonstrate that potential steady-state pathways exist, which provide the fixed carbon to nearly all parts of the network, especially to the citric acid cycle. There is a close cooperation of important metabolic pathways, e.g., the de novo synthesis of uridine-5-monophosphate, the γ-aminobutyric acid shunt, and the urea cycle. The presented approach extends the established methods for a feasible interpretation of biological network models, in particular of large and complex models.

In Silico Knockout Studies of Xenophagic Capturing of Salmonella

The degradation of cytosol-invading pathogens by autophagy, a process known as xenophagy, is an important mechanism of the innate immune system. Inside the host, Salmonella Typhimurium invades epithelial cells and resides within a specialized intracellular compartment, the Salmonella-containing vacuole. A fraction of these bacteria does not persist inside the vacuole and enters the host cytosol. Salmonella Typhimurium that invades the host cytosol becomes a target of the autophagy machinery for degradation. The xenophagy pathway has recently been discovered, and the exact molecular processes are not entirely characterized. Complete kinetic data for each molecular process is not available, so far. We developed a mathematical model of the xenophagy pathway to investigate this key defense mechanism. In this paper, we present a Petri net model of Salmonella xenophagy in epithelial cells. The model is based on functional information derived from literature data. It comprises the molecular mechanism of galectin-8-dependent and ubiquitin-dependent autophagy, including regulatory processes, like nutrient-dependent regulation of autophagy and TBK1-dependent activation of the autophagy receptor, OPTN. To model the activation of TBK1, we proposed a new mechanism of TBK1 activation, suggesting a spatial and temporal regulation of this process. Using standard Petri net analysis techniques, we found basic functional modules, which describe different pathways of the autophagic capture of Salmonella and reflect the basic dynamics of the system. To verify the model, we performed in silico knockout experiments. We introduced a new concept of knockout analysis to systematically compute and visualize the results, using an in silico knockout matrix. The results of the in silico knockout analyses were consistent with published experimental results and provide a basis for future investigations of the Salmonella xenophagy pathway.

Modeling and analysis of innate immune responses induced by the host cells against hepatitis C virus infection

An in-depth understanding of complex systems such as hepatitis C virus (HCV) infection and host immunomodulatory response is an open challenge for biologists. In order to understand the mechanisms involved in immune evasion by HCV, we present a simplified formalization of the highly dynamic system consisting of HCV, its replication cycle and host immune responses at the cellular level using hybrid Petri net (HPN). The approach followed in this study comprises of step wise simulation, model validation and analysis of host immune response. This study was performed with an objective of making correlations among viral RNA levels, interferon (IFN) production and interferon stimulated genes (ISGs) induction. The results correlate with the biological data verifying that the model is very useful in predicting the dynamic behavior of the signaling proteins in response to a stimulus. This study implicates that HCV infection is dependent upon several key factors of the host immune response. The effect of host proteins on limiting viral infection is effectively overruled by the viral pathogen. This study also analyzes activity levels of RNase L, miR-122, IFN, ISGs and PKR induction and inhibition of TLR3/RIG1 mediated pathways in response to targeted manipulation in the presence of HCV. The results are in complete agreement at the time of writing with the published expression studies and western blot experiments. Our model also provides some biological insights regarding the role of PKR in the acute infection of HCV. It might help to explain why many patients fail to clear acute HCV infection while others, with low ISG basal levels, clear HCV spontaneously. The described methodology can easily be reproduced, which suitably supports the study of other viral infections in a formal, automated and expressive manner. The Petri net-based modeling approach applied here may provide valuable insights for study design and analyses to evaluate other disease associated integrated pathways in biological systems.

Modeling integrated cellular machinery using hybrid Petri-Boolean networks

The behavior and phenotypic changes of cells are governed by a cellular circuitry that represents a set of biochemical reactions. Based on biological functions, this circuitry is divided into three types of networks, each encoding for a major biological process: signal transduction, transcription regulation, and metabolism. This division has generally enabled taming computational complexity dealing with the entire system, allowed for using modeling techniques that are specific to each of the components, and achieved separation of the different time scales at which reactions in each of the three networks occur. Nonetheless, with this division comes loss of information and power needed to elucidate certain cellular phenomena. Within the cell, these three types of networks work in tandem, and each produces signals and/or substances that are used by the others to process information and operate normally. Therefore, computational techniques for modeling integrated cellular machinery are needed. In this work, we propose an integrated hybrid model (IHM) that combines Petri nets and Boolean networks to model integrated cellular networks. Coupled with a stochastic simulation mechanism, the model simulates the dynamics of the integrated network, and can be perturbed to generate testable hypotheses. Our model is qualitative and is mostly built upon knowledge from the literature and requires fine-tuning of very few parameters. We validated our model on two systems: the transcriptional regulation of glucose metabolism in human cells, and cellular osmoregulation in S. cerevisiae. The model produced results that are in very good agreement with experimental data, and produces valid hypotheses. The abstract nature of our model and the ease of its construction makes it a very good candidate for modeling integrated networks from qualitative data. The results it produces can guide the practitioner to zoom into components and interconnections and investigate them using such more detailed mathematical models.

Reconstruction of extended Petri nets from time-series data by using logical control functions

The aim of this work is to extend a previously presented algorithm (Durzinsky et al. 2008b in Computational methods in systems biology, LNCS, vol 5307. Springer, Heidelberg, pp 328–346; Marwan et al. 2008 in Math Methods Oper Res 67:117–132) for the reconstruction of standard place/transition Petri nets from time-series of experimental data sets. This previously reported method finds provably all networks capable to reproduce the experimental observations. In this paper we enhance this approach to generate extended Petri nets involving mechanisms formally corresponding to catalytic or inhibitory dependencies that mediate the involved reactions. The new algorithm delivers the set of all extended Petri nets being consistent with the time-series data used for reconstruction. It is illustrated using the phosphate regulatory network of enterobacteria as a case study.

An integrated pathway system modeling of Saccharomyces cerevisiae HOG pathway: a Petri net based approach

Biochemical networks comprise many diverse components and interactions between them. It has intracellular signaling, metabolic and gene regulatory pathways which are highly integrated and whose responses are elicited by extracellular actions. Previous modeling techniques mostly consider each pathway independently without focusing on the interrelation of these which actually functions as a single system. In this paper, we propose an approach of modeling an integrated pathway using an event-driven modeling tool, i.e., Petri nets (PNs). PNs have the ability to simulate the dynamics of the system with high levels of accuracy. The integrated set of signaling, regulatory and metabolic reactions involved in Saccharomyces cerevisiae's HOG pathway has been collected from the literature. The kinetic parameter values have been used for transition firings. The dynamics of the system has been simulated and the concentrations of major biological species over time have been observed. The phenotypic characteristics of the integrated system have been investigated under two conditions, viz., under the absence and presence of osmotic pressure. The results have been validated favorably with the existing experimental results. We have also compared our study with the study of idFBA (Lee et al., PLoS Comput Biol 4:e1000-e1086, 2008) and pointed out the differences between both studies. We have simulated and monitored concentrations of multiple biological entities over time and also incorporated feedback inhibition by Ptp2 which has not been included in the idFBA study. We have concluded that our study is the first to the best of our knowledge to model signaling, metabolic and regulatory events in an integrated form through PN model framework. This study is useful in computational simulation of system dynamics for integrated pathways as there are growing evidences that the malfunctioning of the interplay among these pathways is associated with disease.

Colored petri nets to model gene mutation and amino acids classification

The genetic code is the triplet code based on the three-letter codons, which determines the specific amino acid sequences in proteins synthesis. Choosing an appropriate model for processing these codons is a useful method to study genetic processes in Molecular Biology. As an effective modeling tool of discrete event dynamic systems (DEDS), colored petri net (CPN) has been used for modeling several biological systems, such as metabolic pathways and genetic regulatory networks. According to the genetic code table, CPN is employed to model the process of genetic information transmission. In this paper, we propose a CPN model of amino acids classification, and further present the improved CPN model. Based on the model mentioned above, we give another CPN model to classify the type of gene mutations via contrasting the bases of DNA strands and the codons of amino acids along the polypeptide chain. This model is helpful in determining whether a certain gene mutation will cause the changes of the structures and functions of protein molecules. The effectiveness and accuracy of the presented model are illustrated by the examples in this paper.

Implicit methods for qualitative modeling of gene regulatory networks

Advancements in high-throughput technologies to measure increasingly complex biological phenomena at the genomic level are rapidly changing the face of biological research from the single-gene single-protein experimental approach to studying the behavior of a gene in the context of the entire genome (and proteome). This shift in research methodologies has resulted in a new field of network biology that deals with modeling cellular behavior in terms of network structures such as signaling pathways and gene regulatory networks. In these networks, different biological entities such as genes, proteins, and metabolites interact with each other, giving rise to a dynamical system. Even though there exists a mature field of dynamical systems theory to model such network structures, some technical challenges are unique to biology such as the inability to measure precise kinetic information on gene-gene or gene-protein interactions and the need to model increasingly large networks comprising thousands of nodes. These challenges have renewed interest in developing new computational techniques for modeling complex biological systems. This chapter presents a modeling framework based on Boolean algebra and finite-state machines that are reminiscent of the approach used for digital circuit synthesis and simulation in the field of very-large-scale integration (VLSI). The proposed formalism enables a common mathematical framework to develop computational techniques for modeling different aspects of the regulatory networks such as steady-state behavior, stochasticity, and gene perturbation experiments.

Petri nets in Snoopy: a unifying framework for the graphical display, computational modelling, and simulation of bacterial regulatory networks

Using the example of phosphate regulation in enteric bacteria, we demonstrate the particular suitability of stochastic Petri nets to model biochemical phenomena and their simulative exploration by various features of the software tool Snoopy.

New insights into the human body iron metabolism analyzed by a Petri net based approach

Iron homeostasis is one of the most important biochemical processes in the human body. Despite this fact, the process is not fully understood and until recently only rough descriptions of parts of the process could be found in the literature. Here, an extension of the recently published formal model of the main part of the process is presented. This extension consists in including all known mechanisms of hepcidin regulation. Hepcidin is a hormone synthesized in the liver which is mainly responsible for an inhibition of iron absorption in the small intestine during an inflammatory process. The model is expressed in the language of Petri net theory which allows for its relatively easy analysis and simulation.

Modeling of the U1 snRNP assembly pathway in alternative splicing in human cells using Petri nets

The investigation of spliceosomal processes is currently a topic of intense research in molecular biology. In the molecular mechanism of alternative splicing, a multi-protein-RNA complex - the spliceosome - plays a crucial role. To understand the biological processes of alternative splicing, it is essential to comprehend the biogenesis of the spliceosome. In this paper, we propose the first abstract model of the regulatory assembly pathway of the human spliceosomal subunit U1. Using Petri nets, we describe its highly ordered assembly that takes place in a stepwise manner. Petri net theory represents a mathematical formalism to model and analyze systems with concurrent processes at different abstraction levels with the possibility to combine them into a uniform description language. There exist many approaches to determine static and dynamic properties of Petri nets, which can be applied to analyze biochemical systems. In addition, Petri net tools usually provide intuitively understandable graphical network representations, which facilitate the dialog between experimentalists and theoreticians. Our Petri net model covers binding, transport, signaling, and covalent modification processes. Through the computation of structural and behavioral Petri net properties and their interpretation in biological terms, we validate our model and use it to get a better understanding of the complex processes of the assembly pathway. We can explain the basic network behavior, using minimal T-invariants which represent special pathways through the network. We find linear as well as cyclic pathways. We determine the P-invariants that represent conserved moieties in a network. The simulation of the net demonstrates the importance of the stability of complexes during the maturation pathway. We can show that complexes that dissociate too fast, hinder the formation of the complete U1 snRNP.

Automatic reconstruction of molecular and genetic networks from discrete time series data

We apply a mathematical algorithm which processes discrete time series data to generate a complete list of Petri net structures containing the minimal number of nodes required to reproduce the data set. The completeness of the list as guaranteed by a mathematical proof allows to define a minimal set of experiments required to discriminate between alternative network structures. This in principle allows to prove all possible minimal network structures by disproving all alternative candidate structures. The dynamic behaviour of the networks in terms of a switching rule for the transitions of the Petri net is part of the result. In addition to network reconstruction, the algorithm can be used to determine how many yet undetected components at least must be involved in a certain process. The algorithm also reveals all alternative structural modifications of a network that are required to generate a predefined behaviour.

Causal mapping as a tool to mechanistically interpret phenomena in cell motility: application to cortical oscillations in spreading cells

Biological processes that occur at the cellular level and consist of large numbers of interacting elements are highly nonlinear and generally involve multiple time and spatial scales. The quantitative description of these complex systems is of great importance but presents large challenges. We outline a new systems biology approach, causal mapping (CMAP), which is a coarse-grained biological network tool that permits description of causal interactions between the elements of the network and overall system dynamics. On one hand, the CMAP is an intermediate between experiments and physical modeling, describing major requisite elements, their interactions and paths of causality propagation. On the other hand, the CMAP is an independent tool to explore the hierarchical organization of cell and the role of uncertainties in the system. It appears to be a promising easy-to-use technique for cell biologists to systematically probe verbally formulated qualitative hypotheses. We apply the CMAP to study the phenomenon of contractility oscillations in spreading cells in which microtubules have been depolymerized. The precise mechanism by which these oscillations are governed by a complex mechano-chemical system is not known but the data observed in experiments can be described by a CMAP. The CMAP suggests that the source of the oscillations results from the opposing effects of Rho activation leading to a decreased level of myosin light chain phosphatase and a cyclic calcium influx caused by increased membrane tension and leading to a periodically enhanced activation of myosin light chain kinase.

Reconstructing the regulatory network controlling commitment and sporulation in Physarum polycephalum based on hierarchical Petri Net modelling and simulation

We reconstruct the regulatory network controlling commitment and sporulation of Physarum polycephalum from experimental results using a hierarchical Petri Net-based modelling and simulation framework. The stochastic Petri Net consistently describes the structure and simulates the dynamics of the molecular network as analysed by genetic, biochemical and physiological experiments within a single coherent model. The Petri Net then is extended to simulate time-resolved somatic complementation experiments performed by mixing the cytoplasms of mutants altered in the sporulation response, to systematically explore the network structure and to probe its dynamics. This reverse engineering approach presumably can be employed to explore other molecular or genetic signalling systems where the activity of genes or their products can be experimentally controlled in a time-resolved manner.