Abstract
Motivation: One of the challenging questions in modelling biological systems is to characterize the functional forms of the processes that control and orchestrate molecular and cellular phenotypes. Recently proposed methods for the analysis of metabolic pathways, for example, dynamic flux estimation, can only provide estimates of the underlying fluxes at discrete time points but fail to capture the complete temporal behaviour. To describe the dynamic variation of the fluxes, we additionally require the assumption of specific functional forms that can capture the temporal behaviour. However, it also remains unclear how to address the noise which might be present in experimentally measured metabolite concentrations.Results: Here we propose a novel approach to modelling metabolic fluxes: derivative processes that are based on multiple-output Gaussian processes (MGPs), which are a flexible non-parametric Bayesian modelling technique. The main advantages that follow from MGPs approach include the natural non-parametric representation of the fluxes and ability to impute the missing data in between the measurements. Our derivative process approach allows us to model changes in metabolite derivative concentrations and to characterize the temporal behaviour of metabolic fluxes from time course data. Because the derivative of a Gaussian process is itself a Gaussian process, we can readily link metabolite concentrations to metabolic fluxes and vice versa. Here we discuss how this can be implemented in an MGP framework and illustrate its application to simple models, including nitrogen metabolism in Escherichia coli.Availability and implementation: R code is available from the authors upon request.Contact: j.norkunaite@imperial.ac.uk; m.stumpf@imperial.ac.ukSupplementary information: Supplementary data are available at Bioinformatics online.
Highlights
It is generally impossible to simultaneously measure the abundance of all the molecular entities making up biological systems
In Gaussian process regression (GPR), we place a Gaussian processes (GP) (Haykin and Moher, 2010; McKay, 1998) prior over the functions fðxÞ, f $ GP, meaning that at any finite number of input points x1, . . . , xn the values fðxiÞ have a multivariate Gaussian distribution with zero mean and covariance function, K, 1⁄2fðx1Þ, . . . , fðxnÞT$ N ð0, Kðx, x0ÞÞ: Different functional forms can be chosen for the covariance function (Rasmussen and Williams, 2006), either to simplify computations or to reflect constrains imposed by the data
The estimates for a set of fluxes are obtained in a point-wise manner at discrete time points
Summary
It is generally impossible to simultaneously measure the abundance of all the molecular entities making up biological systems. Estimates for intracellular fluxes can be obtained by tracking products from isotope-labeled (13C and 15N metabolic flux analysis) metabolites through the metabolic network (Blank and Ebert, 2012; Zamboni, 2011) Such an approach is restricted to a metabolically steady-state analysis and is not appropriate for capturing dynamical flux variations. We provide a new framework that allows us to model metabolic fluxes and their dynamics, and which deals with the missing data problem in metabolic analysis in a flexible and consistent manner. MGPs can be used to infill the sparsely sampled data (Boyle and Frean, 2004) This means that by using MGPs, it is possible to impute the missing data in between the metabolic measurements more efficiently. This in turn enables us to treat time course data on metabolites and monitor the changes that occur in fluxes, e.g. over the course of physiological responses, such as to changes in the environment (Bryant et al, 2013)
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