The hidden sensitivity of non-smooth dynamics

Switches in dynamical systems are known to exhibit wildly different behaviours depending on how they are modelled, for instance whether they occur as smooth transitions or abrupt jumps, and whether they involve delays or discrete perturbations. These differences arise be- cause because switches are sensitive to perturbation, but there is lim- ited knowledge about where this sensitivity comes from. Here we take a switch in a simple one-dimensional system, then discretize time and introduce a small delay. The resulting system is described by a piecewise-linear map with incredibly complex dynamics, and is sensi- tive to parameter changes, even if the time-steps and delays are made infinitesimally small. We show that this sensitivity reveals itself in the more versatile numerical tool of the transition count, which captures the likelihood of switching occurring at any instant in a simulation. We then us this transition count to show that sensitivity to param- eters persists in a system with two switches, and we show why this sensitivity then has large-scale dynamical effects. The use of transi- tion counts in this way may prove a versatile tool for studying more complex switching processes in general

Artificial Ontogenies: A Computational Model of the Control and Evolution of Development

Understanding the behaviour of biological systems is a challenging task. Gene regulation, development and evolution are each a product of nonlinear interactions between many individual agents: genes, cells or organisms. Moreover, these three processes are not isolated, but interact with one another in an important fashion. The development of an organism involves complex patterns of dynamic behaviour at the genetic level. The gene networks that produce this behaviour are subject to mutations that can alter the course of development, resulting in the production of novel morphologies. Evolution occurs when these novel morphologies are favoured by natural selection and survive to pass on their genes to future generations. Computational models can assist us to understand biological systems by providing a framework within which their behaviour can be explored. Many natural processes, including gene regulation and development, have a computational element to their control. Constructing formal models of these systems enables their behaviour to be simulated, observed and quantified on a scale not otherwise feasible. This thesis uses a computational simulation methodology to explore the relationship between development and evolution. An important question in evolutionary biology is how to explain the direction of evolution. Conventional explanations of evolutionary history have focused on the role of natural selection in orienting evolution. More recently, it has been argued that the nature of development, and the way it changes in response to mutation, may also be a significant factor. A network-lineage model of artificial ontogenies is described that incorporates a developmental mapping between the dynamics of a gene network and a cell lineage representation of a phenotype. Three series of simulation studies are reported, exploring: (a) the relationship between the structure of a gene network and its dynamic behaviour; (b) the characteristic distributions of ontogenies and phenotypes generated by the dynamics of gene networks; (c) the effect of these characteristic distributions on the evolution of ontogeny. The results of these studies indicate that the model networks are capable of generating a diverse range of stable behaviours, and possess a small yet significant sensitivity to perturbation. In the context of developmental control, the intrinsic dynamics of the model networks predispose the production of ontogenies with a modular, quasi-systematic structure. This predisposition is reflected in the structure of variation available for selection in an adaptive search process, resulting in the evolution of ontogenies biased towards simplicity. These results suggest a possible explanation for the levels of ontogenetic complexity observed in biological organisms: that they may be a product of the network architecture of developmental control. By quantifying complexity, variation and bias, the network-lineage model described in this thesis provides a computational method for investigating the effects of development on the direction of evolution. In doing so, it establishes a viable framework for simulating computational aspects of complex biological systems

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Extended physics as theoretical framework for system biology?

International audienceIn this essay we examine whether a theoretical and conceptual framework for systems biology could be built from the Bailly-Longo (2008-9) proposal. These authors aim to understand life as a coherent critical structure, and propose to develop an extended physical approach of evolution, as a diffusion of biomass in a space of complexity. Their attempt leads to a simple mathematical reconstruction of Gould's assumption (1989) concerning the bacterial world as a "left wall of least complexity" that we will examine. Extended physical systems are characterized by their constructive properties. Time is acting and new properties emerge by their history that can open the list of their initial properties. This conceptual and theoretical framework is nothing more than a philosophical assumption, but as such it provides a new and exciting approach concerning the evolution of life, and the transition between physics and biology

Visual analytics for relationships in scientific data

Domain scientists hope to address grand scientific challenges by exploring the abundance of data generated and made available through modern high-throughput techniques. Typical scientific investigations can make use of novel visualization tools that enable dynamic formulation and fine-tuning of hypotheses to aid the process of evaluating sensitivity of key parameters. These general tools should be applicable to many disciplines: allowing biologists to develop an intuitive understanding of the structure of coexpression networks and discover genes that reside in critical positions of biological pathways, intelligence analysts to decompose social networks, and climate scientists to model extrapolate future climate conditions. By using a graph as a universal data representation of correlation, our novel visualization tool employs several techniques that when used in an integrated manner provide innovative analytical capabilities. Our tool integrates techniques such as graph layout, qualitative subgraph extraction through a novel 2D user interface, quantitative subgraph extraction using graph-theoretic algorithms or by querying an optimized B-tree, dynamic level-of-detail graph abstraction, and template-based fuzzy classification using neural networks. We demonstrate our system using real-world workflows from several large-scale studies. Parallel coordinates has proven to be a scalable visualization and navigation framework for multivariate data. However, when data with thousands of variables are at hand, we do not have a comprehensive solution to select the right set of variables and order them to uncover important or potentially insightful patterns. We present algorithms to rank axes based upon the importance of bivariate relationships among the variables and showcase the efficacy of the proposed system by demonstrating autonomous detection of patterns in a modern large-scale dataset of time-varying climate simulation

Tipping points in the dynamics of speciation.

Speciation can be gradual or sudden and involve few or many genetic changes. Inferring the processes generating such patterns is difficult, and may require consideration of emergent and non-linear properties of speciation, such as when small changes at tipping points have large effects on differentiation. Tipping points involve positive feedback and indirect selection stemming from associations between genomic regions, bi-stability due to effects of initial conditions and evolutionary history, and dependence on modularity of system components. These features are associated with sudden 'regime shifts' in other cellular, ecological, and societal systems. Thus, tools used to understand other complex systems could be fruitfully applied in speciation research

An integrative computational model for intestinal tissue renewal

Objectives\ud \ud The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation.\ud \ud Methods\ud \ud At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole.\ud \ud Results\ud \ud We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data. \ud \ud Conclusions\ud \ud We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis

Parameter estimate of signal transduction pathways

BACKGROUND: The "inverse" problem is related to the determination of unknown causes on the bases of the observation of their effects. This is the opposite of the corresponding "direct" problem, which relates to the prediction of the effects generated by a complete description of some agencies. The solution of an inverse problem entails the construction of a mathematical model and takes the moves from a number of experimental data. In this respect, inverse problems are often ill-conditioned as the amount of experimental conditions available are often insufficient to unambiguously solve the mathematical model. Several approaches to solving inverse problems are possible, both computational and experimental, some of which are mentioned in this article. In this work, we will describe in details the attempt to solve an inverse problem which arose in the study of an intracellular signaling pathway. RESULTS: Using the Genetic Algorithm to find the sub-optimal solution to the optimization problem, we have estimated a set of unknown parameters describing a kinetic model of a signaling pathway in the neuronal cell. The model is composed of mass action ordinary differential equations, where the kinetic parameters describe protein-protein interactions, protein synthesis and degradation. The algorithm has been implemented on a parallel platform. Several potential solutions of the problem have been computed, each solution being a set of model parameters. A sub-set of parameters has been selected on the basis on their small coefficient of variation across the ensemble of solutions. CONCLUSION: Despite the lack of sufficiently reliable and homogeneous experimental data, the genetic algorithm approach has allowed to estimate the approximate value of a number of model parameters in a kinetic model of a signaling pathway: these parameters have been assessed to be relevant for the reproduction of the available experimental data