Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits

In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An $n$-dimensional arabesque system has $n$ 2-element circuits, but in addition, it displays by construction, two $n$-element circuits which are both positive vs one positive and one negative, depending on the parity (even or odd) of the dimension $n$. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they can not possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever the dimension is, in agreement with the order of the nonlinearity. All three are unstable, as there can not be any attractor in their state-space. The 3D variant (that we call for short "A1\_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting $n$ cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2\_$n$D". A2\_3D behaves essentially as A1\_3D, in agreement with the fact that the signs of the circuits remain identical. A2\_4D, as well as other arabesque systems with even dimension, has two positive $n$-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao

Diversity and Plasticity of Th Cell Types Predicted from Regulatory Network Modelling

Alternative cell differentiation pathways are believed to arise from the concerted action of signalling pathways and transcriptional regulatory networks. However, the prediction of mammalian cell differentiation from the knowledge of the presence of specific signals and transcriptional factors is still a daunting challenge. In this respect, the vertebrate hematopoietic system, with its many branching differentiation pathways and cell types, is a compelling case study. In this paper, we propose an integrated, comprehensive model of the regulatory network and signalling pathways controlling Th cell differentiation. As most available data are qualitative, we rely on a logical formalism to perform extensive dynamical analyses. To cope with the size and complexity of the resulting network, we use an original model reduction approach together with a stable state identification algorithm. To assess the effects of heterogeneous environments on Th cell differentiation, we have performed a systematic series of simulations considering various prototypic environments. Consequently, we have identified stable states corresponding to canonical Th1, Th2, Th17 and Treg subtypes, but these were found to coexist with other transient hybrid cell types that co-express combinations of Th1, Th2, Treg and Th17 markers in an environment-dependent fashion. In the process, our logical analysis highlights the nature of these cell types and their relationships with canonical Th subtypes. Finally, our logical model can be used to explore novel differentiation pathways in silico

Graph properties of biological interaction networks

This thesis considers two modelling frameworks for interaction networks in biology. The first models the interacting species qualitatively as discrete variables, with the regulatory graphs expressing their mutual influence. Circuits in the regulatory structure are known to be indicative of some asymptotic behaviours. We investigate the relationship between local negative circuits and sustained oscillations, presenting new examples of Boolean networks without local negative circuits and admitting a cyclic attractor. We then show how regulatory properties of Boolean networks can be investigated via satisfiability problems, and use the technique to examine the role of local negative circuits in networks of small dimension. To enable the application of Boolean techniques to the study of multivalued networks, a mapping of discrete networks to Boolean can be considered. The Boolean version, however, is defined only on a subset of the Boolean states. We propose a method for extending the Boolean version that preserves both the attractors and the regulatory structure of the network. Chemical reaction network theory models the dynamics of species concentrations via systems of ordinary differential equations, establishing connections between the network structure and the dynamics. Some results assume mass action kinetics, whereas biochemical models often adopt other rate forms. We propose algorithms for elimination of intermediate species, that can be used to find whether a mass action network simplifies to a given chemical system. We then consider the problem of identification of generalised mass action networks that give rise to a given mass action dynamics, while displaying useful structural properties, such as weak reversibility. In particular, we investigate systems obtained by preserving the reaction vectors of the mass action network, and outline a new algorithmic approach

Graph properties of biological interaction networks

This thesis considers two modelling frameworks for interaction networks in biology. The first models the interacting species qualitatively as discrete variables, with the regulatory graphs expressing their mutual influence. Circuits in the regulatory structure are known to be indicative of some asymptotic behaviours. We investigate the relationship between local negative circuits and sustained oscillations, presenting new examples of Boolean networks without local negative circuits and admitting a cyclic attractor. We then show how regulatory properties of Boolean networks can be investigated via satisfiability problems, and use the technique to examine the role of local negative circuits in networks of small dimension. To enable the application of Boolean techniques to the study of multivalued networks, a mapping of discrete networks to Boolean can be considered. The Boolean version, however, is defined only on a subset of the Boolean states. We propose a method for extending the Boolean version that preserves both the attractors and the regulatory structure of the network. Chemical reaction network theory models the dynamics of species concentrations via systems of ordinary differential equations, establishing connections between the network structure and the dynamics. Some results assume mass action kinetics, whereas biochemical models often adopt other rate forms. We propose algorithms for elimination of intermediate species, that can be used to find whether a mass action network simplifies to a given chemical system. We then consider the problem of identification of generalised mass action networks that give rise to a given mass action dynamics, while displaying useful structural properties, such as weak reversibility. In particular, we investigate systems obtained by preserving the reaction vectors of the mass action network, and outline a new algorithmic approach

Computational physics of the mind

In the XIX century and earlier such physicists as Newton, Mayer, Hooke, Helmholtz and Mach were actively engaged in the research on psychophysics, trying to relate psychological sensations to intensities of physical stimuli. Computational physics allows to simulate complex neural processes giving a chance to answer not only the original psychophysical questions but also to create models of mind. In this paper several approaches relevant to modeling of mind are outlined. Since direct modeling of the brain functions is rather limited due to the complexity of such models a number of approximations is introduced. The path from the brain, or computational neurosciences, to the mind, or cognitive sciences, is sketched, with emphasis on higher cognitive functions such as memory and consciousness. No fundamental problems in understanding of the mind seem to arise. From computational point of view realistic models require massively parallel architectures

Computational Models of Intracellular and Intercellular Processes in Developmental Biology

Systems biology takes a holistic approach to biological questions as it applies mathematical modeling to link and understand the interaction of components in complex biological systems. Multiscale modeling is the only method that can fully accomplish this aim. Mutliscale models consider processes at different levels that are coupled within the modeling framework. A first requirement in creating such models is a clear understanding of processes that operate at each level. This research focuses on modeling aspects of biological development as a complex process that occurs at many scales. Two of these scales were considered in this work: cellular differentiation, the process of in which less specialized cells acquired specialized properties of mature cell types, and morphogenesis, the process in which an organism develops its shape and tissue architecture. In development, cellular differentiation typically is required for morphogenesis. Therefore, cellular differentiation is at a lower scale than morphogenesis in the overall process of development. In this work, cellular differentiation and morphogenesis were modeled in a variety of biological contexts, with the ultimate goal of linking these different scales of developmental events into a unified model of development. Three aspects of cellular differentiation were investigated, all united by the theme of how the dynamics of gene regulatory networks (GRNs) control differentiation. Two of the projects of this dissertation studied the effect of noise and robustness in switching between cell types during differentiation, and a third deals with the evaluation of hypothetical GRNs that allow the differentiation of specific cell types. All these projects view cell types as high-dimensional attractors in the GRNs and use random Boolean networks as the modeling framework for studying network dynamics. Morphogenesis was studied using the emergence of three-dimensional structures in biofilms as a relatively simple model. Many strains of bacteria form complex structures during growth as colonies on a solid medium. The morphogenesis of these structures was modeled using an agent-based framework and the outcomes were validated using structures of biofilm colonies reported in the literature