Dynamical and Structural Modularity of Discrete Regulatory Networks

A biological regulatory network can be modeled as a discrete function that contains all available information on network component interactions. From this function we can derive a graph representation of the network structure as well as of the dynamics of the system. In this paper we introduce a method to identify modules of the network that allow us to construct the behavior of the given function from the dynamics of the modules. Here, it proves useful to distinguish between dynamical and structural modules, and to define network modules combining aspects of both. As a key concept we establish the notion of symbolic steady state, which basically represents a set of states where the behavior of the given function is in some sense predictable, and which gives rise to suitable network modules. We apply the method to a regulatory network involved in T helper cell differentiation

Symbolic Steady States and Dynamically Essential Subnetworks of Discrete Regulatory Networks

Analyzing complex networks is a difficult task, regardless of the chosen modeling framework. For a discrete regulatory network, even if the number of components is in some sense manageable, we have to deal with the problem of analyzing the dynamics in an exponentially large state space. A well known idea to approach this difficulty is to identify smaller building blocks of the system the study of which in isolation still renders information on the dynamics of the whole network. In this talk, we introduce the notion of symbolic steady state which allows us to identify such building blocks. We state explicit rules how to derive attractors of the network from subnetwork attractors valid for synchronous as well as asynchronous dynamics. Illustrating those rules, we derive general conditions for circuits embedded in the network to transfer their behavioral characteristics pertaining number and size of attractors observed in isolation to the complex network

Relating attractors and singular steady states in the logical analysis of bioregulatory networks.

Abstract. In 1973 R. Thomas introduced a logical approach to modeling and analysis of bioregulatory networks. Given a set of Boolean functions describing the regulatory interactions, a state transition graph is constructed that captures the dynamics of the system. In the late eighties, Snoussi and Thomas extended the original framework by including singular values corresponding to interaction thresholds. They showed that these are needed for a refined understanding of the network dynamics. In this paper, we study systematically singular steady states, which are characteristic of feedback circuits in the interaction graph, and relate them to the type, number and cardinality of attractors in the state transition graph. In particular, we derive sufficient conditions for regulatory networks to exhibit multistationarity or oscillatory behavior, thus giving a partial converse to the well-known Thomas conjectures

Accelerated search for biomolecular network models to interpret high-throughput experimental data

<p>Abstract</p> <p>Background</p> <p>The functions of human cells are carried out by biomolecular networks, which include proteins, genes, and regulatory sites within DNA that encode and control protein expression. Models of biomolecular network structure and dynamics can be inferred from high-throughput measurements of gene and protein expression. We build on our previously developed fuzzy logic method for bridging quantitative and qualitative biological data to address the challenges of noisy, low resolution high-throughput measurements, i.e., from gene expression microarrays. We employ an evolutionary search algorithm to accelerate the search for hypothetical fuzzy biomolecular network models consistent with a biological data set. We also develop a method to estimate the probability of a potential network model fitting a set of data by chance. The resulting metric provides an estimate of both model quality and dataset quality, identifying data that are too noisy to identify meaningful correlations between the measured variables.</p> <p>Results</p> <p>Optimal parameters for the evolutionary search were identified based on artificial data, and the algorithm showed scalable and consistent performance for as many as 150 variables. The method was tested on previously published human cell cycle gene expression microarray data sets. The evolutionary search method was found to converge to the results of exhaustive search. The randomized evolutionary search was able to converge on a set of similar best-fitting network models on different training data sets after 30 generations running 30 models per generation. Consistent results were found regardless of which of the published data sets were used to train or verify the quantitative predictions of the best-fitting models for cell cycle gene dynamics.</p> <p>Conclusion</p> <p>Our results demonstrate the capability of scalable evolutionary search for fuzzy network models to address the problem of inferring models based on complex, noisy biomolecular data sets. This approach yields multiple alternative models that are consistent with the data, yielding a constrained set of hypotheses that can be used to optimally design subsequent experiments.</p

Combining Boolean Networks and Ordinary Differential Equations for Analysis and Comparison of Gene Regulatory Networks

This thesis is concerned with different groups of qualitative models of gene regulatory networks. Four types of models will be considered: interaction graphs, Boolean networks, models based on differential equations and discrete abstractions of differential equations. We will investigate the relations between these modeling frameworks and how they can be used in the analysis of individual models. The focus lies on the mathematical analysis of these models. This thesis makes several contributions in relating these different modeling frameworks. The first approach concerns individual Boolean models and parametrized families of ordinary differential equations (ODEs). To construct ODE models systematically from Boolean models several automatic conversion algorithms have been proposed. In Chapter 2 several such closely related algorithms will be considered. It will be proven that certain invariant sets are preserved during the conversion from a Boolean network to a model based on ODEs. In the second approach the idea of abstracting the dynamics of individual models to relate structure and dynamics will be introduced. This approach will be applied to Boolean models and models based on differential equations. This allows to compare groups of models in these modeling frameworks which have the same structure. We demonstrate that this constitutes an approach to link the interaction graph to the dynamics of certain sets of Boolean networks and models based on differential equations. The abstracted dynamics – or more precisely the restrictions on the abstracted behavior – of such sets of Boolean networks or models based on differential equations will be represented as Boolean state transitions graphs themselves. We will show that these state transition graphs can be considered as asynchronous Boolean networks. Despite the rather theoretical question this thesis tries to answer there are many potential applications of the results. The results in Chapter 2 can be applied to network reduction of ODE models based on Hill kinetics. The results of the second approach in Chapter 4 can be applied to network inference and analysis of Boolean model sets. Furthermore, in the last chapter of this thesis several ideas for applications with respect to experiment design will be considered. This leads to the question how different asynchronous Boolean networks or different behaviours of a single asynchronous Boolean network can be distinguishedDiese Arbeit beschäftigt sich mit unterschiedlichen Typen von qualitativen Modellen genregulatorischer Netzwerke. Vier Typen von Modellen werden betrachtet: Interaktionsgraphen, Boolesche Netzwerke, Modelle, die auf Differentialgleichungen basieren und diskrete Abstraktionen von Differentialgleichungen. Wir werden mehr über die Beziehungen zwischen diesen Modellgruppen lernen und wie diese Beziehungen genutzt werden können, um einzelne Modelle zu analysieren. Der Schwerpunkt liegt hierbei auf der mathematischen Analyse dieser Modellgruppen. In dieser Hinsicht leistet diese Arbeit mehrere Beiträge. Zunächst betrachten wir Boolesche Netzwerke und parametrisierte Familien von gewöhnlichen Differentialgleichungen (ODEs). Um solche ODE-Modelle systematisch aus Booleschen Modellen abzuleiten, wurden in der Vergangenheit verschiedene automatische Konvertierungsalgorithmen vorgeschlagen. In Kapitel 2 werden einige dieser Algorithmen näher untersucht. Wir werden beweisen, dass bestimmte invariante Mengen bei der Konvertierung eines Booleschen Modells in ein ODE-Modell erhalten bleiben. Der zweite Ansatz, der in dieser Arbeit verfolgt wird, beschäftigt sich mit diskreten Abstraktionen der Dynamik von Modellen. Mit Hilfe dieser Abstraktionen ist es möglich, die Struktur – den Interaktionsgraphen – und die Dynamik der zugehörigen Modelle in Bezug zu setzen. Diese Methode wird sowohl auf Boolesche Modelle als auch auf ODE-Modelle angewandt. Gleichzeitig erlaubt dieser Ansatz Mengen von Modellen in unterschiedlichen Modellgruppen zu vergleichen, die dieselbe Struktur haben. Die abstrahierten Dynamiken (genauer die Einschränkungen der abstrahierten Dynamiken) der Booleschen Modellmengen oder ODE-Modellmengen können als Boolesche Zustandsübergangsgraphen repräsentiert werden. Wir werden zeigen, dass diese Zustandsübergangsgraphen wiederum selber als (asynchrone) Boolesche Netzwerke aufgefasst werden können. Trotz der theoretischen Ausgangsfrage werden in dieser Arbeit zahlreiche Anwendungen aufgezeigt. Die Ergebnisse aus Kapitel 2 können zur Modellreduktion benutzt werden, indem die Dynamik der ODE-Modelle auf den zu den Booleschen Netzwerken gehörigen “trap spaces” betrachtet wird. Die Resultate aus Kapitel 4 können zur Netzwerkinferenz oder zur Analyse von Modellmengen genutzt werden. Weiterhin werden im letzten Kapitel dieser Arbeit einige Anwendungsideen im Bezug auf Experimentdesign eingeführt. Dies führt zu der Fragestellung, wie verschiedene asynchrone Boolesche Netzwerke oder unterschiedliche Dynamiken, die mit einem einzelnen Modell vereinbar sind, unterschieden werden können

Formalization of molecular interaction maps in systems biology; Application to simulations of the relationship between DNA damage response and circadian rhythms

Quantitative exploration of biological pathway networks must begin with a qualitative understanding of them. Often researchers aggregate and disseminate experimental data using regulatory diagrams with ad hoc notations leading to ambiguous interpretations of presented results. This thesis has two main aims. First, it develops software to allow researchers to aggregate pathway data diagrammatically using the Molecular Interaction Map (MIM) notation in order to gain a better qualitative understanding of biological systems. Secondly, it develops a quantitative biological model to study the effect of DNA damage on circadian rhythms. The second aim benefits from the first by making use of visual representations to identify potential system boundaries for the quantitative model. I focus first on software for the MIM notation - a notation to concisely visualize bioregulatory complexity and to reduce ambiguity for readers. The thesis provides a formalized MIM specification for software implementation along with a base layer of software components for the inclusion of the MIM notation in other software packages. It also provides an implementation of the specification as a user-friendly tool, PathVisio-MIM, for creating and editing MIM diagrams along with software to validate and overlay external data onto the diagrams. I focus secondly on the application of the MIM software to the quantitative exploration of the poorly understood role of SIRT1 and PARP1, two NAD+-dependent enzymes, in the regulation of circadian rhythms during DNA damage response. SIRT1 and PARP1 participate in the regulation of several key DNA damage-repair proteins and are the subjects of study as potential cancer therapeutic targets. In this part of the thesis, I present an ordinary differential equation (ODE) model that simulates the core circadian clock and the involvement of SIRT1 in both the positive and negative arms of circadian regulation. I then use this model is then used to predict a potential role for the competition for NAD+ supplies by SIRT1 and PARP1 leading to the observed behavior of primarily phase advancement of circadian oscillations during DNA damage response. The model further predicts a potential mechanism by which multiple forms of post-transcriptional modification may cooperate to produce a primarily phase advancement

Deterministic Intracellular Modeling

The United States Air Force is interested in the potential side effects at the cellular level from exposure to mission-essential chemicals. Presently, Air Force toxicology studies are conducted to help shed light in identifying potential hazards to workers. However, it takes a considerable amount of money, resources, and time to obtain and analyze experimental results from toxicology studies. The necessity for innovative methods that enable researchers to more effectively generate and analyze data is apparent