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

Disentangling a complex response in cell reprogramming and probing the Waddington landscape by automatic construction of Petri nets

We analyzed the developmental switch to sporulation of a multinucleate Physarum polycephalum plasmodial cell, a complex response to phytochrome photoreceptor activation. Automatic construction of Petri nets representing finite state machines assembled from trajectories of differential gene expression in single cells revealed alternative, genotype-dependent interconnected developmental routes and identified reversible steps, metastable states, commitment points, and subsequent irreversible steps together with molecular signatures associated with cell fate decision and differentiation. Formation of cyclic transits identified by transition invariants in mutants that are locked in a proliferative state is remarkable considering the view that oncogenic alterations may cause the formation of cancer attractors. We conclude that the Petri net approach is useful to probe the Waddington landscape of cellular reprogramming, to disentangle developmental routes for the reconstruction of the gene regulatory network, and to understand how genetic alterations or physiological conditions reshape the landscape eventually creating new basins of attraction. Unraveling the complexity of pathogenesis, disease progression, drug response or the analysis of attractor landscapes in other complex systems of uncertain structure might be additional fields of application.

Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.

Cellular differentiation and individuality in the 'minor' multicellular taxa

Biology needs a concept of individuality in order to distinguish organisms from parts of organisms and from groups of organisms, to count individuals and compare traits across taxa, and to distinguish growth from reproduction. Most of the proposed criteria for individuality were designed for 'unitary' or 'paradigm' organisms: contiguous, functionally and physiologically integrated, obligately sexually reproducing multicellular organisms with a germ line sequestered early in development. However, the vast majority of the diversity of life on Earth does not conform to all of these criteria. We consider the issue of individuality in the 'minor' multicellular taxa, which collectively span a large portion of the eukaryotic tree of life, reviewing their general features and focusing on a model species for each group. When the criteria designed for unitary organisms are applied to other groups, they often give conflicting answers or no answer at all to the question of whether or not a given unit is an individual. Complex life cycles, intimate bacterial symbioses, aggregative development, and strange genetic features complicate the picture. The great age of some of the groups considered shows that 'intermediate' forms, those with some but not all of the traits traditionally associated with individuality, cannot reasonably be considered ephemeral or assumed transitional. We discuss a handful of recent attempts to reconcile the many proposed criteria for individuality and to provide criteria that can be applied across all the domains of life. Finally, we argue that individuality should be defined without reference to any particular taxon and that understanding the emergence of new kinds of individuals requires recognizing individuality as a matter of degree.

A next-generation sequencing approach to study the transcriptomic changes during the differentiation of physarum at the single-cell level

Physarum polycephalum is a unicellular eukaryote belonging to the amoebozoa group of organisms. The complex life cycle involves various cell types that differ in morphology, function, and biochemical composition. Sporulation, one step in the life cycle, is a stimulus-controlled differentiation response of macroscopic plasmodial cells that develop into fruiting bodies. Well-established Mendelian genetics and the occurrence of macroscopic cells with a naturally synchronous population of nuclei as source of homogeneous cell material for biochemical analyses make Physarum an attractive model organism for studying the regulatory control of cell differentiation. Here, we develop an approach using RNA-sequencing (RNA-seq), without needing to rely on a genome sequence as a reference, for studying the transcriptomic changes during stimulus-triggered commitment to sporulation in individual plasmodial cells. The approach is validated through the obtained expression patterns and annotations, and particularly the results from up- and downregulated genes, which correlate well with previous studies.

Futile attempts to differentiate provide molecular evidence for individual differences within a population of cells during cellular reprogramming

The heterogeneity of cell populations and the influence of stochastic noise might be important issues for the molecular analysis of cellular reprogramming at the system level. Here, we show that in Physarum polycephalum, the expression patterns of marker genes correlate with the fate decision of individual multinucleate plasmodial cells that had been exposed to a differentiation-inducing photostimulus. For several hours after stimulation, the expression kinetics of PI-3-kinase, piwi, and pumilio orthologs and other marker genes were qualitatively similar in all stimulated cells but quantitatively different in those cells that subsequently maintained their proliferative potential and failed to differentiate accordingly. The results suggest that the population of nuclei in an individual plasmodium behaves synchronously in terms of gene regulation to an extent that the plasmodium provides a source for macroscopic amounts of homogeneous single-cell material for analysing the dynamic processes of cellular reprogramming. Based on the experimental findings, we predict that circuits with switch-like behaviour that control the cell fate decision of a multinucleate plasmodium operate through continuous changes in the concentration of cellular regulators because the nuclear population suspended in a large cytoplasmic volume damps stochastic noise.

A survey on methods for modeling and analyzing integrated biological networks

Understanding how cellular systems build up integrated responses to their dynamically changing environment is one of the open questions in Systems Biology. Despite their intertwinement, signaling networks, gene regulation and metabolism have been frequently modeled independently in the context of well-defined subsystems. For this purpose, several mathematical formalisms have been developed according to the features of each particular network under study. Nonetheless, a deeper understanding of cellular behavior requires the integration of these various systems into a model capable of capturing how they operate as an ensemble. With the recent advances in the "omics" technologies, more data is becoming available and, thus, recent efforts have been driven toward this integrated modeling approach. We herein review and discuss methodological frameworks currently available for modeling and analyzing integrated biological networks, in particular metabolic, gene regulatory and signaling networks. These include network-based methods and Chemical Organization Theory, Flux-Balance Analysis and its extensions, logical discrete modeling, Petri Nets, traditional kinetic modeling, Hybrid Systems and stochastic models. Comparisons are also established regarding data requirements, scalability with network size and computational burden. The methods are illustrated with successful case studies in large-scale genome models and in particular subsystems of various organisms.

Petri Nets - A Mathematical Formalism to Analyze Chemical Reaction Networks

In this review we introduce and discuss Petri nets - a mathematical formalism to describe and analyze chemical reaction networks. Petri nets were developed to describe concurrency in general systems. We find most applications to technical and financial systems, but since about twenty years also in systems biology to model biochemical systems. This review aims to give a short informal introduction to the basic formalism illustrated by a chemical example, and to discuss possible applications to the analysis of chemical reaction networks, including cheminformatics. We give a short overview about qualitative as well as quantitative modeling Petri net techniques useful in systems biology, summarizing the state-of-the-art in that field and providing the main literature references. Finally, we discuss advantages and limitations of Petri nets and give an outlook to further development.

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.

Modularization of biochemical networks based on classification of Petri net t-invariants

Background

Structural analysis of biochemical networks is a growing field in bioinformatics and systems biology. The availability of an increasing amount of biological data from molecular biological networks promises a deeper understanding but confronts researchers with the problem of combinatorial explosion. The amount of qualitative network data is growing much faster than the amount of quantitative data, such as enzyme kinetics. In many cases it is even impossible to measure quantitative data because of limitations of experimental methods, or for ethical reasons. Thus, a huge amount of qualitative data, such as interaction data, is available, but it was not sufficiently used for modeling purposes, until now. New approaches have been developed, but the complexity of data often limits the application of many of the methods. Biochemical Petri nets make it possible to explore static and dynamic qualitative system properties. One Petri net approach is model validation based on the computation of the system's invariant properties, focusing on t-invariants. T-invariants correspond to subnetworks, which describe the basic system behavior.

With increasing system complexity, the basic behavior can only be expressed by a huge number of t-invariants. According to our validation criteria for biochemical Petri nets, the necessary verification of the biological meaning, by interpreting each subnetwork (t-invariant) manually, is not possible anymore. Thus, an automated, biologically meaningful classification would be helpful in analyzing t-invariants, and supporting the understanding of the basic behavior of the considered biological system.

Methods

Here, we introduce a new approach to automatically classify t-invariants to cope with network complexity. We apply clustering techniques such as UPGMA, Complete Linkage, Single Linkage, and Neighbor Joining in combination with different distance measures to get biologically meaningful clusters (t-clusters), which can be interpreted as modules. To find the optimal number of t-clusters to consider for interpretation, the cluster validity measure, Silhouette Width, is applied.

Results

We considered two different case studies as examples: a small signal transduction pathway (pheromone response pathway in Saccharomyces cerevisiae) and a medium-sized gene regulatory network (gene regulation of Duchenne muscular dystrophy). We automatically classified the t-invariants into functionally distinct t-clusters, which could be interpreted biologically as functional modules in the network. We found differences in the suitability of the various distance measures as well as the clustering methods. In terms of a biologically meaningful classification of t-invariants, the best results are obtained using the Tanimoto distance measure. Considering clustering methods, the obtained results suggest that UPGMA and Complete Linkage are suitable for clustering t-invariants with respect to the biological interpretability.

Conclusion

We propose a new approach for the biological classification of Petri net t-invariants based on cluster analysis. Due to the biologically meaningful data reduction and structuring of network processes, large sets of t-invariants can be evaluated, allowing for model validation of qualitative biochemical Petri nets. This approach can also be applied to elementary mode analysis.