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.

Components of the neural signal underlying congenital nystagmus.

Congenital nystagmus is an involuntary bilateral horizontal oscillation of the eyes that develops soon after birth. In this study, the time constants of each of the components of the neural signal underlying congenital nystagmus were obtained by time series analysis and interpreted by comparison with those of the normal oculomotor system. In the neighbourhood of the fixation position, the system generating the neural signal is approximately linear with 3 degrees of freedom. The shortest time constant was in the range of 7-9 ms and corresponds to a normal saccadic burst signal. The other stable time constant was in the range of 22-70 ms and corresponds to the slide signal. The final time constant characterises the unidentified neural mechanism underlying the unstable drift component of the oscillation cycle and ranges between 31 and 32 ms across waveforms. The characterisation of this unstable time constant poses a challenge for the modelling of both the normal and abnormal oculomotor control system. We tentatively identify the unstable component with the eye position signal supplied to the superior colliculus in the normal eye movement system and explore some of the implications of this hypothesis.

A max-plus model of ribosome dynamics during mRNA translation.

We examine the dynamics of the translation stage of cellular protein production, in which ribosomes move uni-directionally along an mRNA strand, building amino acid chains as they go. We describe the system using a timed event graph-a class of Petri net useful for studying discrete events, which have to satisfy constraints. We use max-plus algebra to describe a deterministic version of the model, where the constraints represent steric effects which prevent more than one ribosome reading a given codon at a given time and delays associated with the availability of the different tRNAs. We calculate the protein production rate and density of ribosomes on the mRNA and find exact agreement between these analytical results and numerical simulations of the deterministic model, even in the case of heterogeneous mRNAs.

A systematic survey of the response of a model NF-κB signalling pathway to TNFα stimulation.

White's lab established that strong, continuous stimulation with tumour necrosis factor-α (TNFα) can induce sustained oscillations in the subcellular localisation of the transcription factor nuclear factor κB (NF-κB). But the intensity of the TNFα signal varies substantially, from picomolar in the blood plasma of healthy organisms to nanomolar in diseased states. We report on a systematic survey using computational bifurcation theory to explore the relationship between the intensity of TNFα stimulation and the existence of sustained NF-κB oscillations. Using a deterministic model developed by Ashall et al. in 2009, we find that the system's responses to TNFα are characterised by a supercritical Hopf bifurcation point: above a critical intensity of TNFα the system exhibits sustained oscillations in NF-kB localisation. For TNFα below this critical value, damped oscillations are observed. This picture depends, however, on the values of the model's other parameters. When the values of certain reaction rates are altered the response of the signalling pathway to TNFα stimulation changes: in addition to the sustained oscillations induced by high-dose stimulation, a second oscillatory regime appears at much lower doses. Finally, we define scores to quantify the sensitivity of the dynamics of the system to variation in its parameters and use these scores to establish that the qualitative dynamics are most sensitive to the details of NF-κB mediated gene transcription.

Interactions among oscillatory pathways in NF-kappa B signaling.

BACKGROUND: Sustained stimulation with tumour necrosis factor alpha (TNF-alpha) induces substantial oscillations--observed at both the single cell and population levels--in the nuclear factor kappa B (NF-kappa B) system. Although the mechanism has not yet been elucidated fully, a core system has been identified consisting of a negative feedback loop involving NF-kappa B (RelA:p50 hetero-dimer) and its inhibitor I-kappa B-alpha. Many authors have suggested that this core oscillator should couple to other oscillatory pathways. RESULTS: First we analyse single-cell data from experiments in which the NF-kappa B system is forced by short trains of strong pulses of TNF-alpha. Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency. Using a recent model of the NF-kappa B system due to Caroline Horton, we carried out extensive numerical simulations to analyze the response frequencies induced by trains of pulses of TNF-alpha stimulation having a wide range of frequencies and amplitudes. These studies suggest that for sufficiently weak stimulation, various nonlinear resonances should be observable. To explore further the possibility of probing alternative feedback mechanisms, we also coupled the model to sinusoidal signals with a wide range of strengths and frequencies. Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics. CONCLUSIONS: Our numerical results suggest that the entrainment phenomena observed in pulse-stimulated experiments is a consequence of the high intensity of the stimulation. Computational studies based on current models suggest that resonant interactions between periodic pulsatile forcing and the system's natural frequencies may become evident for sufficiently weak stimulation. Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

Systematic integration of experimental data and models in systems biology.

BACKGROUND: The behaviour of biological systems can be deduced from their mathematical models. However, multiple sources of data in diverse forms are required in the construction of a model in order to define its components and their biochemical reactions, and corresponding parameters. Automating the assembly and use of systems biology models is dependent upon data integration processes involving the interoperation of data and analytical resources. RESULTS: Taverna workflows have been developed for the automated assembly of quantitative parameterised metabolic networks in the Systems Biology Markup Language (SBML). A SBML model is built in a systematic fashion by the workflows which starts with the construction of a qualitative network using data from a MIRIAM-compliant genome-scale model of yeast metabolism. This is followed by parameterisation of the SBML model with experimental data from two repositories, the SABIO-RK enzyme kinetics database and a database of quantitative experimental results. The models are then calibrated and simulated in workflows that call out to COPASIWS, the web service interface to the COPASI software application for analysing biochemical networks. These systems biology workflows were evaluated for their ability to construct a parameterised model of yeast glycolysis. CONCLUSIONS: Distributed information about metabolic reactions that have been described to MIRIAM standards enables the automated assembly of quantitative systems biology models of metabolic networks based on user-defined criteria. Such data integration processes can be implemented as Taverna workflows to provide a rapid overview of the components and their relationships within a biochemical system.

Pulsatile stimulation determines timing and specificity of NF-kappaB-dependent transcription.

The nuclear factor kappaB (NF-kappaB) transcription factor regulates cellular stress responses and the immune response to infection. NF-kappaB activation results in oscillations in nuclear NF-kappaB abundance. To define the function of these oscillations, we treated cells with repeated short pulses of tumor necrosis factor-alpha at various intervals to mimic pulsatile inflammatory signals. At all pulse intervals that were analyzed, we observed synchronous cycles of NF-kappaB 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-kappaB-dependent gene expression, which supports the idea that oscillation frequency has a functional role.

A consensus yeast metabolic network reconstruction obtained from a community approach to systems biology.

Genomic data allow the large-scale manual or semi-automated assembly of metabolic network reconstructions, which provide highly curated organism-specific knowledge bases. Although several genome-scale network reconstructions describe Saccharomyces cerevisiae metabolism, they differ in scope and content, and use different terminologies to describe the same chemical entities. This makes comparisons between them difficult and underscores the desirability of a consolidated metabolic network that collects and formalizes the 'community knowledge' of yeast metabolism. We describe how we have produced a consensus metabolic network reconstruction for S. cerevisiae. In drafting it, we placed special emphasis on referencing molecules to persistent databases or using database-independent forms, such as SMILES or InChI strings, as this permits their chemical structure to be represented unambiguously and in a manner that permits automated reasoning. The reconstruction is readily available via a publicly accessible database and in the Systems Biology Markup Language (http://www.comp-sys-bio.org/yeastnet). It can be maintained as a resource that serves as a common denominator for studying the systems biology of yeast. Similar strategies should benefit communities studying genome-scale metabolic networks of other organisms.

Dynamics of saccadic oscillations.

The brainstem circuitry underlying saccades is symmetrical with respect to the midline. The oculomotor behaviour generated by the circuitry depends on a combination of signals passed along fibre tracts and less easily identifiable connections, such as those across the midline. The midline crossing connections are often affected by developmental disorders which give rise to unstable eye movements (see J. Jen, this volume). The connections at the levels of the colliculus, pause cells, and neural integrator generate different dynamical mechanisms for the development of instabilities, which can be identified in eye movement recordings using phase space analysis techniques.

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.

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.

Bridging the gap between in silico and cell-based analysis of the nuclear factor-kappaB signaling pathway by in vitro studies of IKK2.

Previously, we have shown by sensitivity analysis, that the oscillatory behavior of nuclear factor (NF-kappaB) is coupled to free IkappaB kinase-2 (IKK2) and IkappaBalpha(IkappaBalpha), and that the phosphorylation of IkappaBalpha by IKK influences the amplitude of NF-kappaB oscillations. We have performed further analyses of the behavior of NF-kappaB and its signal transduction network to understand the dynamics of this system. A time lapse study of NF-kappaB translocation in 10,000 cells showed discernible oscillations in levels of nuclear NF-kappaB amongst cells when stimulated with interleukin (IL-1alpha), which suggests a small degree of synchronization amongst the cell population. When the kinetics for the phosphorylation of IkappaBalpha by IKK were measured, we found that the values for the affinity and catalytic efficiency of IKK2 for IkappaBalpha were dependent on assay conditions. The application of these kinetic parameters in our computational model of the NF-kappaB pathway resulted in significant differences in the oscillatory patterns of NF-kappaB depending on the rate constant value used. Hence, interpretation of in silico models should be made in the context of this uncertainty.

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway.

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.

Characterisation of congenital nystagmus waveforms in terms of periodic orbits.

Because the oscillatory eye movements of congenital nystagmus vary from cycle to cycle, there is no clear relationship between the waveform produced and the underlying abnormality of the ocular motor system. We consider the durations of successive cycles of nystagmus which could be (1) completely determined by the lengths of the previous cycles, (2) completely independent of the lengths of the previous cycles or (3) a mixture of the two. The behaviour of a deterministic system can be characterised in terms of a collection of (unstable) oscillations, referred to as periodic orbits, which make up the system. By using a recently developed technique for identifying periodic orbits in noisy data, we find evidence for periodic orbits in nystagmus waveforms, eliminating the possibility that each cycle is independent of the previous cycles. The technique also enables us to identify the waveforms which correspond to the deterministic behaviour of the ocular motor system. These waveforms pose a challenge to our understanding of the ocular motor system because none of the current extensions to models of the normal behaviour of the ocular motor system can explain the range of identified waveforms.

A new framework for investigating both normal and abnormal eye movements.

Any comprehensive framework for understanding eye movements has to include both normal and abnormal eye movement behaviour. One approach which is applicable to the entire range of oculomotor behaviour is provided by the techniques of nonlinear dynamics. The stability of models of the oculomotor system can be analysed in terms of the characteristics of their fixed points and periodic orbits, and the method of delays can be used to recover such parameters from measurements of eye position. Within this framework, quantitative comparisons can be made between the predictions of different models, and both normal and clinical eye movement recordings.