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Metabolism is one of the best studied fields of biochemistry, but its regulation involves processes on many different levels, some of which are still not understood well enough to allow for quantitative modeling and prediction. Glycolysis in yeast is a good example: although high-quality quantitative data are available, well-established mathematical models typically only cover direct regulation of the involved enzymes by metabolite binding. The effect of various metabolites on the enzyme kinetics is summarized in carefully developed mathematical formulae. However, this approach implicitly assumes that the enzyme concentrations themselves are constant, thus neglecting other regulatory levels--e.g. transcriptional and translational regulation--involved in the regulation of enzyme activities. It is believed, however, that different experimental conditions result in different enzyme activities regulated by the above mechanisms. Detailed modeling of all regulatory levels is still out of reach since some of the necessary data--e.g. quantitative large scale enzyme concentration data sets--are lacking or rare. Nevertheless, a viable approach is to include the regulation of enzyme concentrations into an established model and to investigate whether this improves the predictive capabilities. Proteome data are usually hard to obtain, but levels of mRNA transcripts may be used instead as clues for changes in enzyme concentrations. Here we investigate whether including mRNA data into an established model of yeast glycolysis allows to predict the steady state metabolic concentrations for different experimental conditions. To this end, we modified an established ODE model for the glycolytic pathway of yeast to include changes of enzyme concentrations. Presumable changes were inferred from mRNA transcript level measurement data. We investigate how this approach can be used to predict metabolite concentrations for steady-state yeast cultures at five different oxygen levels ranging from anaerobic to fully aerobic conditions. We were partly able to reproduce the experimental data and present a number of changes that were necessary to improve the modeling result.

We demonstrate an approach to automatically generating kinetic models of metabolic networks. In a first step, the metabolic network is characterised by its stoichiometric structure. Then to each reaction a kinetic equation is associated describing the metabolic flux. For the kinetics we use a formula that is universally applicable to reactions with arbitrary numbers of substrates and products. Last, the kinetics of the reactions are assigned parameters. The resulting model in SBML format can be fed into standard simulation tools. The approach is applied to the sulphur-glutathione-pathway in Saccharomyces cerevisiae.

We describe a workflow to translate a given metabolic network into a kinetic model; the model summarises kinetic information collected from different data sources. All reactions are modelled by convenience kinetics; where detailed kinetic laws are known, they can also be incorporated. Confidence intervals and correlations of the resulting model parameters are obtained from Bayesian parameter estimation; they can be used to sample parameter sets for Monte-Carlo simulations. The integration method ensures that the resulting parameter distributions are thermodynamically feasible. Here we summarise different previous works on this topic: we give an overview over the convenience kinetics, thermodynamic criteria for parameter sets, Bayesian parameter estimation, the collection of kinetic data, and different machine learning techniques that can be used to obtain prior distributions for kinetic parameters. All methods have been assembled into a workflow that facilitates the integration of biochemical data and the modelling of metabolic networks from scratch.

Background: Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism; they can become quite involved and may contain a large number of parameters. Rate laws and enzyme parameters are still unknown for most enzymes., Results: We introduce a simple and general rate law called "convenience kinetics". It can be derived from a simple random-order enzyme mechanism. Thermodynamic laws can impose dependencies on the kinetic parameters. Hence, to facilitate model fitting and parameter optimisation for large networks, we introduce thermodynamically independent system parameters: their values can be varied independently, without violating thermodynamical constraints. We achieve this by expressing the equilibrium constants either by Gibbs free energies of formation or by a set of independent equilibrium constants. The remaining system parameters are mean turnover rates, generalised Michaelis-Menten constants, and constants for inhibition and activation. All parameters correspond to molecular energies, for instance, binding energies between reactants and enzyme., Conclusion: Convenience kinetics can be used to translate a biochemical network--manually or automatically--into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimisation. Parameter estimates can be easily computed from a least-squares fit to Michaelis-Menten values, turnover rates, equilibrium constants, and other quantities that are routinely measured in enzyme assays and stored in kinetic databases.

Background: Translating a known metabolic network into a dynamic model requires reasonable guesses of all enzyme parameters. In Bayesian parameter estimation, model parameters are described by a posterior probability distribution, which scores the potential parameter sets, showing how well each of them agrees with the data and with the prior assumptions made., Results: We compute posterior distributions of kinetic parameters within a Bayesian framework, based on integration of kinetic, thermodynamic, metabolic, and proteomic data. The structure of the metabolic system (i.e., stoichiometries and enzyme regulation) needs to be known, and the reactions are modelled by convenience kinetics with thermodynamically independent parameters. The parameter posterior is computed in two separate steps: a first posterior summarises the available data on enzyme kinetic parameters; an improved second posterior is obtained by integrating metabolic fluxes, concentrations, and enzyme concentrations for one or more steady states. The data can be heterogeneous, incomplete, and uncertain, and the posterior is approximated by a multivariate log-normal distribution. We apply the method to a model of the threonine synthesis pathway: the integration of metabolic data has little effect on the marginal posterior distributions of individual model parameters. Nevertheless, it leads to strong correlations between the parameters in the joint posterior distribution, which greatly improve the model predictions by the following Monte-Carlo simulations., Conclusion: We present a standardised method to translate metabolic networks into dynamic models. To determine the model parameters, evidence from various experimental data is combined and weighted using Bayesian parameter estimation. The resulting posterior parameter distribution describes a statistical ensemble of parameter sets; the parameter variances and correlations can account for missing knowledge, measurement uncertainties, or biological variability. The posterior distribution can be used to sample model instances and to obtain probabilistic statements about the model's dynamic behaviour.

Dynamic modeling and simulation of signal transduction pathways is an important topic in systems biology and is obtaining growing attention from researchers with experimental or theoretical background. Here we review attempts to analyze and model specific signaling systems. We review the structure of recurrent building blocks of signaling pathways and their integration into more comprehensive models, which enables the understanding of complex cellular processes. The variety of mechanisms found and modeling techniques used are illustrated with models of different signaling pathways. Focusing on the close interplay between experimental investigation of pathways and the mathematical representations of cellular dynamics, we discuss challenges and perspectives that emerge in studies of signaling systems.

E. coli is widely used for systems biology research; there exists a need, however, for tools that can be used to accurately and comprehensively measure expression dynamics in individual living cells. To address this we present a library of transcriptional fusions of gfp to each of about 2,000 different promoters in E. coli K12, covering the great majority of the promoters in the organism. Each promoter fusion is expressed from a low-copy plasmid. We demonstrate that this library can be used to obtain highly accurate dynamic measurements of promoter activity on a genomic scale, in a glucose-lactose diauxic shift experiment. The library allowed detection of about 80 previously uncharacterized transcription units in E. coli, including putative internal promoters within previously known operons, such as the lac operon. This library can serve as a tool for accurate, high-resolution analysis of transcription networks in living E. coli cells.

Values of enzyme kinetic parameters are a key requisite for the kinetic modelling of biochemical systems. For most kinetic parameters, however, not even an order of magnitude is known, so the estimation of model parameters from experimental data remains a major task in systems biology. We propose a statistical approach to infer values for kinetic parameters across species and enzymes making use of parameter values that have been measured under various conditions and that are nowadays stored in databases. We fit the data by a statistical regression model in which the substrate, the combination enzyme-substrate and the combination organism-substrate have a linear effect on the logarithmic parameter value. As a result, we obtain predictions and error ranges for unknown enzyme parameters. We apply our method to decadic logarithmic Michaelis-Menten constants from the BRENDA database and confirm the results with leave-one-out crossvalidation, in which we mask one value at a time and predict it from the remaining data. For a set of 8 metabolites we obtain a standard prediction error of 1.01 for the deviation of the predicted values from the true values, while the standard deviation of the experimental values is 1.16. The method is applicable to other types of kinetic parameters for which many experimental data are available.

The Systems Biology Markup Language (SBML) is an XML-based format for representing mathematical models of biochemical reaction networks, and it is likely to become a main standard in the systems biology community. As published mathematical models in cell biology are growing in number and size, modular modelling approaches will gain additional importance. The main issue to be addressed in computer-assisted model combination is the specification and handling of model semantics. The software SBMLmerge assists the user in combining models of biological subsystems to larger biochemical networks. First, the program helps the user in annotating all model elements with unique identifiers pointing to databases such as KEGG or Gene Ontology. Second, during merging, SBMLmerge detects and resolves various syntactic and semantic problems. Typical problems are conflicting variable names, elements which appear in more than one input model, and mathematical problems arising from the combination of equations. If the input models make contradicting statements about a biochemical quantity, the user is asked to choose between them. In the end the merging process results in a new, valid SBML model.

Expression of metabolic enzymes increases the metabolic capabilities of the cell, but it also consumes resources, and the gene regulatory systems of cells have to handle this tradeoff. To study whether gene expression patterns reflect the varying metabolic needs of the cell, we translated gene expression profiles into sets of active biochemical reactions, which constitute the expressed metabolic subnetworks. The metabolic capacity of a carbon source denotes the number of metabolites that can be produced from this carbon source and some inorganic nutrients. We studied the metabolic subnetworks that are expressed during diauxic shift in yeast and found that the capacities of different carbon sources tend to decrease during the diauxic shift. The subnetwork expressed in the initial glucose environment shows a high capacity of glucose, much higher than networks of the same size arising in later stages of the diauxic shift or from a random selection of reactions. The same holds for the essential capacity, the number of important, constitutive compounds that can be produced from glucose. These findings indicate that gene regulation increases the range of essential compounds that can be obtained from the available nutrients, while minimising the number of expressed enzymes and therefore the burden of protein synthesis.

Physiological concentrations of metabolites can partly be explained by their molecular structure. We hypothesize that substances containing certain chemical groups show increased or decreased concentration in cells. We consider here, as chemical groups, local atomic configurations, describing an atom, its bonds, and its direct neighbor atoms. To test our hypothesis, we fitted a linear statistical model that relates experimentally determined logarithmic concentrations to feature vectors containing count numbers of the chemical groups. In order to determine chemical groups that have a clear effect on the concentration, we use a regularized (lasso) regression. In a dataset on 41 substances of central metabolism in different organisms, we found that the physical concentrations are increased by the occurrence of amino and hydroxyl groups, while aldehydes, ketones, and phosphates show decreased concentrations. The model explains about 22% of the variance of the logarithmic mean concentrations.

The modelling of biochemical networks becomes delicate if kinetic parameters are varying, uncertain or unknown. Facing this situation, we quantify uncertain knowledge or beliefs about parameters by probability distributions. We show how parameter distributions can be used to infer probabilistic statements about dynamic network properties, such as steady-state fluxes and concentrations, signal characteristics or control coefficients. The parameter distributions can also serve as priors in Bayesian statistical analysis. We propose a graphical scheme, the 'dependence graph', to bring out known dependencies between parameters, for instance, due to the equilibrium constants. If a parameter distribution is narrow, the resulting distribution of the variables can be computed by expanding them around a set of mean parameter values. We compute the distributions of concentrations, fluxes and probabilities for qualitative variables such as flux directions. The probabilistic framework allows the study of metabolic correlations, and it provides simple measures of variability and stochastic sensitivity. It also shows clearly how the variability of biological systems is related to the metabolic response coefficients.

Modelling of biochemical systems usually focuses on certain pathways, while the concentrations of so-called external metabolites are considered fixed. This approximation ignores feedback loops mediated by the environment, that is, via external metabolites and reactions. To achieve a more realistic, dynamic description that is still numerically efficient, we propose a new methodology: the basic idea is to describe the environment by a linear effective model of adjustable dimensionality. In particular, we (a) split the entire model into a subsystem and its environment, (b) linearize the environment model around a steady state, and (c) reduce its dimensionality by balanced truncation, an established method for large-scale model reduction. The reduced variables describe the dynamic modes in the environment that dominate its interaction with the subsystem. We compute metabolic response coefficients that account for complexity-reduced dynamics of the environment. Our simulations show that a dynamic environment model can improve the simulation results considerably, even if the environment model has been drastically reduced and if its kinetic parameters are only approximately known. The speed-up in computation gained by model reduction may become vital for parameter estimation in large cell models.

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.

We investigate a model of optimal regulation, intended to describe large-scale differential gene expression. Relations between the optimal expression patterns and the function of genes are deduced from an optimality principle: the regulators have to maximise a fitness function which they influence directly via a cost term, and indirectly via their control on important cell variables, such as metabolic fluxes. According to the model, the optimal linear response to small perturbations reflects the regulators' functions, namely their linear influences on the cell variables. The optimal behaviour can be realised by a linear feedback mechanism. Known or assumed properties of response coefficients lead to predictions about regulation patterns. A symmetry relation predicted for deletion experiments is verified with gene expression data. Where the optimality assumption is valid, our results justify the use of expression data for functional annotation and for pathway reconstruction and suggest the use of linear factor models for the analysis of gene expression data.

Functional properties of biochemical networks depend on both the network structure and the kinetic parameters. Extensive data on metabolic network topologies have been collected in databases, but much less information is available about the kinetic constants or metabolite concentrations. Depending on the values of these parameters, metabolic fluxes and control coefficients may vary within a wide range. Nevertheless, some of the parameters may have little influence on the observables of interest. We address the question whether, despite uncertainty about kinetic parameters, probabilistic statements can be made about dynamic network features. To this end, we perform a variability analysis of the parameters: assuming that the parameters follow statistical distributions, we compute the resulting distributions of the network properties like metabolic fluxes, concentrations, or control coefficients by Monte Carlo simulation. In this manner, we study systematically the possible distributions arising from typical topologies of biochemical networks such as linear chains, branched networks, and signaling and gene expression cascades. This analysis reveals how much information about dynamic behavior can be drawn from structural knowledge.

Motivation: The expression of genes is controlled by specific combinations of cellular variables. We applied Independent Component Analysis (ICA) to gene expression data, deriving a linear model based on hidden variables, which we term 'expression modes'. The expression of each gene is a linear function of the expression modes, where, according to the ICA model, the linear influences of different modes show a minimal statistical dependence, and their distributions deviate sharply from the normal distribution., Results: Studying cell cycle-related gene expression in yeast, we found that the dominant expression modes could be related to distinct biological functions, such as phases of the cell cycle or the mating response. Analysis of human lymphocytes revealed modes that were related to characteristic differences between cell types. With both data sets, the linear influences of the dominant modes showed distributions with large tails, indicating the existence of specifically up- and downregulated target genes. The expression modes and their influences can be used to visualize the samples and genes in low-dimensional spaces. A projection to expression modes helps to highlight particular biological functions, to reduce noise, and to compress the data in a biologically sensible way.

We have developed a non-steady-state mathematical model describing post-translational protein translocation across the endoplasmic reticulum membrane. Movement of the polypeptide chain through the channel in the endoplasmic reticulum membrane is considered to be a stochastic process which is biased at the lumenal side of the channel by the binding of BiP (Kar2p), a member of the Hsp70 family of ATPases (ratcheting model). Assuming that movement of the chain through the channel is caused by passive diffusion (Brownian ratchet), the model describes all available experimental data. The optimum set of model parameters indicates that the ratcheting mechanism functions at near-maximum rate, being relatively insensitive to variations of the association or dissociation rate constants of BiP or its concentration. The estimated rate constant for diffusion of a polypeptide inside the channel indicates that the chain makes contact with the walls of the channel. Since fitting of the model to the data required that the backward rate constant be larger than the forward constant during early diffusion steps, translocation must occur against a force. The latter may arise, for example, from the unfolding of the polypeptide chain in the cytosol. Our results indicate that the ratchet can transport polypeptides against a free energy of about 25 kJ/mol without significant retardation of translocation. The modeling also suggests that the BiP ratchet is optimized, allowing fast translocation to be coupled with minimum consumption of ATP and rapid dissociation of BiP in the lumen of the ER. Finally, we have estimated the maximum hydrophobicity of a polypeptide segment up to which lateral partitioning from the channel into the lipid phase does not result in significant retardation of translocation., (Copyright 2001 Academic Press.)