How Arsenic Acts

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 heterogenous, 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. Published: 15 December 2006 Theoretical Biology and Medical Modelling 2006, 3:42 doi:10.1186/1742-4682-3-42 Received: 11 September 2006 Accepted: 15 December 2006 This article is available from: http://www.tbiomed.com/content/3/1/42 © 2006 Liebermeister and Klipp; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 1 of 11 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2006, 3:42 http://www.tbiomed.com/content/3/1/42 Background Dynamic simulation of metabolic systems Local perturbations of biochemical systems, e.g. by differential gene expression or drug treatment, can lead to global effects that are by no means self-evident. An intention of systems biology is to predict them by computer simulations, which requires mathematical models of the biochemical networks. The structure of metabolic networks has been characterised for many organisms [1-3], and metabolic fluxes in large networks [4-6] are successfully described by pathwayor constraint-based methods [710]. However, such methods do not explain how the fluxes are actually evoked by the activities of enzymes and how they respond to moderate perturbations. These questions can be answered by kinetic models, which employ differential equations to describe the temporal behaviour of the system. Kinetic models allow for bifurcation and control analysis [11-13]; parameter distributions [14-17] can be used to explore their variability and potential behaviour. Unfortunately, there is a disproportion between the high number of parameters contained in kinetic models and the relatively incomplete data available: kinetic laws are not known for most enzymes, and kinetic and metabolic data are sparse, uncertain, and dispersed over databases [18-20], models [21,22], and the literature [23,24]. Therefore, parameter estimation is an integral part of kinetic modelling, and model fitting is currently receiving increasing attention [25-29]. Interestingly, some dynamic properties are determined by the network structure alone, for instance, the sums of metabolic control coefficients described in summation theorems; other properties may be rather insensitive to the choice of parameters. Parameter ensembles [15,30] can be used to assess and distinguish the respective impact of structure and kinetics. Given a metabolic network, it would be desirable at least to know plausible ranges and correlations for all model parameters, in agreement with the known data. Here we suggest a way to achieve this by collecting and integrating heterogenous data in an automatic manner. Outline of the paper We aim at translating a metabolic network into a kinetic model, using the convenience kinetics described in the companion article [31]. For parameter estimation, we use Data integration pipeline Figure 1 Data integration pipeline. A metabolic network (A) is translated into a kinetic model. The model parameters are described by statistical distributions. Experimental values of enzyme parameters (B) are used to obtain a first, kinetics-based distribution of enzyme parameters (D). A fit to metabolic data (C) such as metabolite and enzyme concentrations and metabolic fluxes leads to a second, metabolics-based, distribution of system parameters (thermodynamic and kinetic parameters) and state parameters (metabolite and enzyme concentrations) (E). The system parameters describe the enzymatic reactions in general and remain constant for a given cell; fluxes and concentrations can fluctuate and depend on specific states of the cell; however, integrating metabolic data from several experiments can also improve the fit of kinetic parameters. Metabolite concentrations A B C Stoichiometric matrix Gene expression data Protein concentrations/ Reaction fluxes Enzyme data Metabolic data Structural model Turnover rates Equilibrium constants Reaction Gibbs energies Gibbs energies of formation Michaelis−Menten constants Activation and inhibition constants Regulatory interactions Reversible reactions (activation/inhibition)