Mathematical Modeling of the λ Switch

Gene regulation plays a central role in the development and functioning of living organisms. Developing a deeper qualitative and quantitative understanding of gene regulation is an important scientific challenge. The Lambda switch is commonly used as a paradigm of gene regulation. Verbal descriptions of the structure and functioning of the Lambda switch have appeared in biological textbooks. We apply fuzzy modeling to transform one such verbal description into a well-defined mathematical model. The resulting model is a piecewise-quadratic, second-order differential equation. It demonstrates functional fidelity with known results while being simple enough to allow a rather detailed analysis. Properties such as the number, location, and domain of attraction of equilibrium points can be studied analytically. Furthermore, the model provides a rigorous explanation for the so-called stability puzzle of the Lambda switch.

Modelling Gene Regulatory Networks Using Computational Intelligence Techniques

This chapter presents modelling gene regulatory networks (GRNs) using probabilistic causal model and the guided genetic algorithm. The problem of modelling is explained from both a biological and computational perspective. Further, a comprehensive methodology for developing a GRN model is presented where the application of computation intelligence (CI) techniques can be seen to be significantly important in each phase of modelling. An illustrative example of the causal model for GRN modelling is also included and applied to model the yeast cell cycle dataset. The results obtained are compared for providing biological relevance to the findings which thereby underpins the CI based modelling techniques.

What are Gene Regulatory Networks?

This book deals with algorithms for inferring and analyzing Gene Regulatory Networks using mainly gene expression data. What precisely are the Gene Regulatory Networks that are inferred by such algorithms from this type of data? There is still much confusion in the current literature and it is important to start a book about computational methods for Gene Regulatory Networks with a definition that is as unambiguous as possible. In this chapter, I provide a definition and try to clearly explain what Gene Regulatory Networks are in terms of the underlying biochemical processes. To do the latter in a formal way, I will use a linear approximation to the in general non-linear kinetics underlying interactions in biochemical systems and show how a biochemical system can be ‘condensed’ into the more compact description of Gene Regulatory Networks. Important differences between the defined Gene Regulatory Networks and other network models for gene regulation, such as Transcriptional Regulatory Networks and Co-Expression Networks, will be highlighted.

A Linear Programming Framework for Inferring Gene Regulatory Networks by Integrating Heterogeneous Data

There exist many heterogeneous data sources that are closely related to gene regulatory networks. These data sources provide rich information for depicting complex biological processes at different levels and from different aspects. Here, we introduce a linear programming framework to infer the gene regulatory networks. Within this framework, we extensively integrate the available information derived from multiple time-course expression datasets, ChIP-chip data, regulatory motif-binding patterns, protein-protein interaction data, protein-small molecule interaction data, and documented regulatory relationships in literature and databases. Results on synthetic and real experimental data both demonstrate that the linear programming framework allows us to recover gene regulations in a more robust and reliable manner.

Improved Model Checking Techniques for State Space Analysis of Gene Regulatory Networks

A better understanding of the behavior of a cell, as a system, depends on our ability to model and understand the complex regulatory mechanisms that control gene expression. High level, qualitative models of gene regulatory networks can be used to analyze and characterize the behavior of complex systems, and to provide important insights on the behavior of these systems. In this chapter, we describe a number of additional functionalities that, when supported by a symbolic model checker, make it possible to answer important questions about the nature of the state spaces of gene regulatory networks, such as the nature and size of attractors, and the characteristics of the basins of attraction. We illustrate the type of analysis that can be performed by applying an improved model checker to two well studied gene regulatory models, the network that controls the cell cycle in the yeast S. cerevisiae, and the network that regulates formation of the dorsal-ventral boundary in D. melanogaster. The results show that the insights provided by the analysis can be used to understand and improve the models, and to formulate hypotheses that are biologically relevant and that can be confirmed experimentally.

Complexity of the BN and the PBN Models of GRNs and Mappings for Complexity Reduction

Constructing computational models of genomic regulation faces several major challenges. While the advances in technology can help in obtaining more and better quality gene expression data, the complexity of the models that can be inferred from data is often high. This high complexity impedes the practical applications of such models, especially when one is interested in developing intervention strategies for disease control, for example, preventing tumor cells from entering a proliferative state. Thus, estimating the complexity of a model and designing strategies for complexity reduction become crucial in problems such as model selection, construction of tractable sub-network models, and control of the dynamical behavior of the model. In this chapter we discuss these issues in the setting of Boolean networks and probabilistic Boolean networks – two important classes of network models for genomic regulatory networks.

Inferring Gene Regulatory Networks from Genetical Genomics Data

In this chapter, we review the current state of Gene Regulatory Network inference based on ‘Genetical Genomics’ experiments (Brem & Kruglyak, 2005; Brem, Yvert, Clinton & Kruglyak, 2002; Jansen, 2003; Jansen & Nap, 2001; Schadt et al., 2003) as a special case of causal network inference in ‘Systems Genetics’ (Threadgill, 2006). In a Genetical Genomics experiment, a segregating or genetically randomized population is DNA marker genotyped and gene-expression profiled on a genomewide scale. The genotypes are regarded as natural, multifactorial perturbations resulting in different gene-expression ‘phenotypes’, and causal relationships can therefore be established between the measured genotypes and the gene-expression phenotypes. In this chapter, we review different computational approaches to Gene Regulatory Network inference based on the joint analysis of DNA marker and expression data and additionally of DNA sequence information if available. This includes different methods for expression QTL mapping, selection of regulator-target pairs, construction of an encompassing network, which strongly constrains the network search space, and pairwise and multivariate methods for Gene Regulatory Network inference, such as Bayesian Networks and Structural Equation Modeling.

Petri Nets and GRN Models

In this chapter, modeling of GRNs using Petri net theory is considered. It aims at providing a conceptual understanding of Petri nets to enable the reader to explore GRNs applying Petri net modeling and analysis techniques. Starting with an overview on modeling biochemical networks using Petri nets, the state-of-the-art with focus on GRNs is described. Other modeling techniques, for example, hybrid Petri nets are discussed. Basic concepts of Petri net theory are introduced involving special analysis techniques for modeling biochemical systems, for example, MCT-sets, T-clusters, and Mauritius maps. To illustrate these Petri net concepts, a more complex case study–the gene regulation in Duchenne Muscular Dystrophy–is explained in detail, considering the biological background and the interpretation of analysis results. Considering both, advantages and disadvantages, the chapter demonstrates the usefulness of Petri net modeling, in particular for GRNs.

Dynamic Links and Evolutionary History in Simulated Gene Regulatory Networks

In this chapter, we describe the use of evolutionary methods for the in silico generation of artificial gene regulatory networks (GRNs). These usually serve as models for biological networks and can be used for enhancing analysis methods in biology. We clarify our motivation in adopting this strategy by showing the importance of detailed knowledge of all processes, especially the regulatory dynamics of interactions undertaken during gene expression. To illustrate how such a methodology works, two different approaches to the evolution of small-scale GRNs with specified functions, are briefly reviewed and discussed. Thereafter, we present an approach to evolve medium sized GRNs with the ability to produce stable multi-cellular growth. The computational method employed allows for a detailed analysis of the dynamics of the GRNs as well as their evolution. We have observed the emergence of negative feedback during the evolutionary process, and we suggest its implication to the mutational robustness of the regulatory network which is further supported by evidence observed in additional experiments.

Generalized Boolean Networks

Random Boolean Networks (RBN) have been introduced by Kauffman more than thirty years ago as a highly simplified model of genetic regulatory networks. This extremely simple and abstract model has been studied in detail and has been shown capable of extremely interesting dynamical behavior. First of all, as some parameters are varied such as the network’s connectivity, or the probability of expressing a gene, the RBN can go through a phase transition, going from an ordered regime to a chaotic one. Kauffman’s suggestion is that cell types correspond to attractors in the RBN phase space, and only those attractors that are short and stable under perturbations will be of biological interest. Thus, according to Kauffman, RBN lying at the edge between the ordered phase and the chaotic phase can be seen as abstract models of genetic regulatory networks. The original view of Kauffman, namely that these models may be useful for understanding real-life cell regulatory networks, is still valid, provided that the model is updated to take into account present knowledge about the topology of real gene regulatory networks, and the timing of events, without loosing its attractive simplicity. According to present data, many biological networks, including genetic regulatory networks, seem, in fact, to be of the scale-free type. From the point of view of the timing of events, standard RBN update their state synchronously. This assumption is open to discussion when dealing with biologically plausible networks. In particular, for genetic regulatory networks, this is certainly not the case: genes seem to be expressed in different parts of the network at different times, according to a strict sequence, which depends on the particular network under study. The expression of a gene depends on several transcription factors, the synthesis of which appear to be neither fully synchronous nor instantaneous. Therefore, we have recently proposed a new, more biologically plausible model. It assumes a scale-free topology of the networks and we define a suitable semi-synchronous dynamics that better captures the presence of an activation sequence of genes linked to the topological properties of the network. By simulating statistical ensembles of networks, we discuss the attractors of the dynamics, showing that they are compatible with theoretical biological network models. Moreover, the model demonstrates interesting scaling abilities as the size of the networks is increased.