Studying the effect of cell division on expression patterns of the segment polarity genes

Global dynamics for switching systems and their extensions by linear differential equations

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

Combinatorial representation of parameter space for switching networks

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network ( DSGRN) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations is a regulatory network, i.e. a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piece-wise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.

A Minimal Regulatory Network of Extrinsic and Intrinsic Factors Recovers Observed Patterns of CD4+ T Cell Differentiation and Plasticity

CD4+ T cells orchestrate the adaptive immune response in vertebrates. While both experimental and modeling work has been conducted to understand the molecular genetic mechanisms involved in CD4+ T cell responses and fate attainment, the dynamic role of intrinsic (produced by CD4+ T lymphocytes) versus extrinsic (produced by other cells) components remains unclear, and the mechanistic and dynamic understanding of the plastic responses of these cells remains incomplete. In this work, we studied a regulatory network for the core transcription factors involved in CD4+ T cell-fate attainment. We first show that this core is not sufficient to recover common CD4+ T phenotypes. We thus postulate a minimal Boolean regulatory network model derived from a larger and more comprehensive network that is based on experimental data. The minimal network integrates transcriptional regulation, signaling pathways and the micro-environment. This network model recovers reported configurations of most of the characterized cell types (Th0, Th1, Th2, Th17, Tfh, Th9, iTreg, and Foxp3-independent T regulatory cells). This transcriptional-signaling regulatory network is robust and recovers mutant configurations that have been reported experimentally. Additionally, this model recovers many of the plasticity patterns documented for different T CD4+ cell types, as summarized in a cell-fate map. We tested the effects of various micro-environments and transient perturbations on such transitions among CD4+ T cell types. Interestingly, most cell-fate transitions were induced by transient activations, with the opposite behavior associated with transient inhibitions. Finally, we used a novel methodology was used to establish that T-bet, TGF-β and suppressors of cytokine signaling proteins are keys to recovering observed CD4+ T cell plastic responses. In conclusion, the observed CD4+ T cell-types and transition patterns emerge from the feedback between the intrinsic or intracellular regulatory core and the micro-environment. We discuss the broader use of this approach for other plastic systems and possible therapeutic interventions.

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

Boolean networks with multiexpressions and parameters

To model biological systems using networks, it is desirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have been proposed in the literature. However, analytical study of these more general Boolean networks models is lagging. This paper aims to develop a concise theory for these different Boolean logic-based modeling methods. Boolean models for networks where each node can have more than two levels of expression and Boolean models with parameters are defined algebraically with examples provided. Certain classes of random asynchronous Boolean networks and deterministic moduli asynchronous Boolean networks are investigated in detail using the setting introduced in this paper. The derived theorems provide a clear picture for the attractor structures of these asynchronous Boolean networks.

Boolean modeling of biological regulatory networks: a methodology tutorial

Given the complexity and interactive nature of biological systems, constructing informative and coherent network models of these systems and subsequently developing efficient approaches to analyze the assembled networks is of immense importance. The integration of network analysis and dynamic modeling enables one to investigate the behavior of the underlying system as a whole and to make experimentally testable predictions about less-understood aspects of the processes involved. In this paper, we present a tutorial on the fundamental steps of Boolean modeling of biological regulatory networks. We demonstrate how to infer a Boolean network model from the available experimental data, analyze the network using graph-theoretical measures, and convert it into a predictive dynamic model. For each step, the pitfalls one may encounter and possible ways to circumvent them are also discussed. We illustrate these steps on a toy network as well as in the context of the Drosophila melanogaster segment polarity gene network.

Modelling the Drosophila embryo

I provide a historical overview on the use of mathematical models to gain insight into pattern formation during early development of the fruit fly Drosophila melanogaster. It is my intention to illustrate how the aims and methodology of modelling have changed from the early beginnings of a theoretical developmental biology in the 1960s to modern-day systems biology. I show that even early modelling attempts addressed interesting and relevant questions, which were not tractable by experimental approaches. Unfortunately, their validation was severely hampered by a lack of specificity and appropriate experimental evidence. There is a simple lesson to be learned from this: we cannot deduce general rules for pattern formation from first principles or spurious reproduction of developmental phenomena. Instead, we must infer such rules (if any) from detailed and accurate studies of specific developmental systems. To achieve this, mathematical modelling must be closely integrated with experimental approaches. I report on progress that has been made in this direction in the past few years and illustrate the kind of novel insights that can be gained from such combined approaches. These insights demonstrate the great potential (and some pitfalls) of an integrative, systems-level investigation of pattern formation.

Logical modelling of cell cycle control in eukaryotes: a comparative study

Dynamical modelling is at the core of the systems biology paradigm. However, the development of comprehensive quantitative models is complicated by the daunting complexity of regulatory networks controlling crucial biological processes such as cell division, the paucity of currently available quantitative data, as well as the limited reproducibility of large-scale experiments. In this context, qualitative modelling approaches offer a useful alternative or complementary framework to build and analyse simplified, but still rigorous dynamical models. This point is illustrated here by analysing recent logical models of the molecular network controlling mitosis in different organisms, from yeasts to mammals. After a short introduction covering cell cycle and logical modelling, we compare the assumptions and properties underlying these different models. Next, leaning on their transposition into a common logical framework, we compare their functional structure in terms of regulatory circuits. Finally, we discuss assets and prospects of qualitative approaches for the modelling of the cell cycle.

Biological switches and clocks

To introduce this special issue on biological switches and clocks, we review the historical development of mathematical models of bistability and oscillations in chemical reaction networks. In the 1960s and 1970s, these models were limited to well-studied biochemical examples, such as glycolytic oscillations and cyclic AMP signalling. After the molecular genetics revolution of the 1980s, the field of molecular cell biology was thrown wide open to mathematical modellers. We review recent advances in modelling the gene-protein interaction networks that control circadian rhythms, cell cycle progression, signal processing and the design of synthetic gene networks.