Boolean Models of Bistable Biological Systems

This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the life sciences. This paper is focused on dynamical systems that exhibit bistability and are desc ribedby delay equations. It is shown that if a certain motif including a feedback loop is present in the wiring diagram of the system, the Boolean model captures the bistability of molecular switches. The method is appl ied to two examples from biology, the lac operon and the phage lambda lysis/lysogeny switch

Network Topology as a Driver of Bistability in the lac Operon

The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits bistability. Further we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.Comment: 15 pages, 13 figures, supplemental information include

Modeling and Analysis of Germ Layer Formations Using Finite Dynamical Systems

The development of an embryo from a fertilised egg to a multicellular organism proceeds through numerous steps, with the formation of the three germ layers (endoderm, mesoderm, ectoderm) being one of the first. In this paper we study the mesendoderm (the tissue that collectively gives rise to both mesoderm and endoderm) gene regulatory network for two species, \textit{Xenopus laevis} and the axolotl (\textit{Ambystoma mexicanum}) using Boolean networks. We find that previously-established bistability found in these networks can be reproduced using this Boolean framework, provided that some assumptions used in previously-published differential equations models are relaxed. We conclude by discussing our findings in relation to previous work modeling gene regulatory networks with Boolean network models

A Boolean Network Model of the L-Arabinose Operon

The regulation of gene expression is essential for the maintenance of homeostasis within an organism. Thus, the ability to predict which genes are expressed and which are silenced based on the cellular environment is highly desired by molecular biologists. Mathematical models of gene regulatory networks have frequently been given in terms of systems of differential equations, which although useful for understanding the mechanisms of regulation, are not always as interpretable as discrete models when one wishes to analyze the global-level dynamics of the system. In particular, Boolean network models have been previously shown to be simple yet effective tools for modeling operons such as the lactose operon in \emph{Escherichia coli}. In this thesis, we propose a Boolean model of a similar nature for the arabinose operon. While this operon is also used by \emph{E. coli} to regulate sugar metabolism, it contains several unique biological features such as a positive inducible control mechanism that distinguish it from previously modeled gene networks. By treating the network model as a polynomial dynamical system, analysis of the system dynamics shows that our model accurately captures the biological behavior of the operon and also provides insight into interactions within the network

Boolean network-based model of the Bcl-2 family mediated MOMP regulation

Mitochondrial outer membrane permeabilization (MOMP) is one of the most important points, in majority of apoptotic signaling cascades. Decision mechanism controlling whether the MOMP occurs or not, is formed by an interplay between members of the Bcl-2 family. To understand the role of individual members of this family within the MOMP regulation, we constructed a boolean network-based mathematical model of interactions between the Bcl-2 proteins. Results of computational simulations reveal the existence of the potentially malign configurations of activities of the Bcl-2 proteins, blocking the occurrence of MOMP, independently of the incoming stimuli. Our results suggest role of the antiapoptotic protein Mcl-1 in relation to these configurations. We demonstrate here, the importance of the Bid and Bim according to activation of effectors Bax and Bak, and the irreversibility of this activation. The model further shows the distinct requirements for effectors activation, where the antiapoptic protein Bcl-w is seemingly a key factor preventing the Bax activation. We believe that this work may help to describe the functioning of the Bcl-2 regulation of MOMP better, and hopefully provide some contribution regarding the anti-cancer drug development research

Structurally robust biological networks

Background: The molecular circuitry of living organisms performs remarkably robust regulatory tasks, despite the often intrinsic variability of its components. A large body of research has in fact highlighted that robustness is often a structural property of biological systems. However, there are few systematic methods to mathematically model and describe structural robustness. With a few exceptions, numerical studies are often the preferred approach to this type of investigation. Results: In this paper, we propose a framework to analyze robust stability of equilibria in biological networks. We employ Lyapunov and invariant sets theory, focusing on the structure of ordinary differential equation models. Without resorting to extensive numerical simulations, often necessary to explore the behavior of a model in its parameter space, we provide rigorous proofs of robust stability of known bio-molecular networks. Our results are in line with existing literature. Conclusions: The impact of our results is twofold: on the one hand, we highlight that classical and simple control theory methods are extremely useful to characterize the behavior of biological networks analytically. On the other hand, we are able to demonstrate that some biological networks are robust thanks to their structure and some qualitative properties of the interactions, regardless of the specific values of their parameters