David S. Broomhead

Pulsatile Stimulation Determines Timing and Specificity of NF-κB-Dependent Transcription

The nuclear factor κB (NF-κB) transcription factor regulates cellular stress responses and the immune response to infection. NF-κB activation results in oscillations in nuclear NF-κB abundance. To define the function of these oscillations, we treated cells with repeated short pulses of tumor necrosis factor–α at various intervals to mimic pulsatile inflammatory signals. At all pulse intervals that were analyzed, we observed synchronous cycles of NF-κB nuclear translocation. Lower frequency stimulations gave repeated full-amplitude translocations, whereas higher frequency pulses gave reduced translocation, indicating a failure to reset. Deterministic and stochastic mathematical models predicted how negative feedback loops regulate both the resetting of the system and cellular heterogeneity. Altering the stimulation intervals gave different patterns of NF-κB–dependent gene expression, which supports the idea that oscillation frequency has a functional role.

Something from nothing − bridging the gap between constraint‐based and kinetic modelling

Two divergent modelling methodologies have been adopted to increase our understanding of metabolism and its regulation. Constraint‐based modelling highlights the optimal path through a stoichiometric network within certain physicochemical constraints. Such an approach requires minimal biological data to make quantitative inferences about network behaviour; however, constraint‐based modelling is unable to give an insight into cellular substrate concentrations. In contrast, kinetic modelling aims to characterize fully the mechanics of each enzymatic reaction. This approach suffers because parameterizing mechanistic models is both costly and time‐consuming. In this paper, we outline a method for developing a kinetic model for a metabolic network, based solely on the knowledge of reaction stoichiometries. Fluxes through the system, estimated by flux balance analysis, are allowed to vary dynamically according to linlog kinetics. Elasticities are estimated from stoichiometric considerations. When compared to a popular branched model of yeast glycolysis, we observe an excellent agreement between the real and approximate models, despite the absence of (and indeed the requirement for) experimental data for kinetic constants. Moreover, using this particular methodology affords us analytical forms for steady state determination, stability analyses and studies of dynamical behaviour.

Delay Embeddings for Forced Systems.II. Stochastic Forcing

Abstract Takens’ Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens’ Theorem assume that the underlying system is autonomous (and noise-free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens’ Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochastically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.

Information-theoretic sensitivity analysis: a general method for credit assignment in complex networks

Most systems can be represented as networks that couple a series of nodes to each other via one or more edges, with typically unknown equations governing their quantitative behaviour. A major question then pertains to the importance of each of the elements that act as system inputs in determining the output(s). We show that any such system can be treated as a ‘communication channel’ for which the associations between inputs and outputs can be quantified via a decomposition of their mutual information into different components characterizing the main effect of individual inputs and their interactions. Unlike variance-based approaches, our novel methodology can easily accommodate correlated inputs.

A model of yeast glycolysis based on a consistent kinetic characterisation of all its enzymes

We present an experimental and computational pipeline for the generation of kinetic models of metabolism, and demonstrate its application to glycolysis in Saccharomyces cerevisiae. Starting from an approximate mathematical model, we employ a “cycle of knowledge” strategy, identifying the steps with most control over flux. Kinetic parameters of the individual isoenzymes within these steps are measured experimentally under a standardised set of conditions. Experimental strategies are applied to establish a set of in vivo concentrations for isoenzymes and metabolites. The data are integrated into a mathematical model that is used to predict a new set of metabolite concentrations and reevaluate the control properties of the system. This bottom‐up modelling study reveals that control over the metabolic network most directly involved in yeast glycolysis is more widely distributed than previously thought.

Relating individual behaviour to population dynamics

How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? Wepresent a stochastic individual–based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual–based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.

A new approach to dimensionality reduction: theory and algorithms

This paper applies Whitneys embedding theorem to the data reduction problem and introduces a new approach motivated in part by the (constructive) proof of the theorem. The notion of a good projection is introduced which involves picking projections of the high-dimensional system that are optimized such that they are easy to invert. The basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases. A method for constructing the inverse projection is detailed and its properties, including a new measure of complexity, are discussed. Finally, well-known methods of data reduction are compared with our approach within the context of Whitneys theorem.

Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems.

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov-Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling. (c) 1999 American Institute of Physics.

Exact analysis of the sampling distribution for the canonical particle swarm optimiser and its convergence during stagnation

Several theoretical analyses of the dynamics of particle swarms have been offered in the literature over the last decade. Virtually all rely on substantial simplifications, including the assumption that the particles are deterministic. This has prevented the exact characterisation of the sampling distribution of the PSO. In this paper we introduce a novel method, which allows one to exactly determine all the characteristics of a PSOs sampling distribution and explain how they change over any number of generations, in the presence stochasticity. The only assumption we make is stagnation, i.e., we study the sampling distribution produced by particles in search for a better personal best. We apply the analysis to the PSO with inertia weight, but the analysis is also valid for the PSO with constriction.

The Whitney Reduction Network: A Method for Computing Autoassociative Graphs

This article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m-dimensionalmanifold initially residing in an n-dimensional Euclidean space where n m. Motivated by Whitneys embedding theorem, the network is capable of training the identity mapping employing the idea of the graph of a function. In theory, a reduction to a dimension d that retains the differential structure of the original data may be achieved for some d 2m + 1. To implement this network, we propose the idea of a good-projection, which enhances the generalization capabilities of the network, and an adaptive secant basis algorithm to achieve it. The effect of noise on this procedure is also considered. The approach is illustrated with several examples.