Biochemical networks with uncertain parameters

Abstract

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. IntroductionCell simulations aspired to in systems biology [1] require knowledge of enzyme kinetic parameters. However, owing to a lack of measurements, measurement errors and biological variability, most of these parameters are still unknown or uncertain, which turns out to be a major obstacle in large-scale cell modelling. In this situation, a probabilistic description of the parameters can be helpful: to assess the effects of measurement errors, to find out which model results persist for a wide range of parameters, and to derive probabilities for different model outcomes. Besides this, parameter distributions can also be employed to study the natural variability and robustness of biological systems. The effects of temporal random fluctuations in gene expression [2,3] and metabolism [4,5] have been studied. Here, we focus on models with static yet uncertain parameters: the parameters are described by a probability distribution, and the standard deviation of the resulting variable distributions (their variability) reflects how strongly the variables respond to parameter variations. (In this paper, we use the term 'variable' quite generally for quantitative model results, such as concentrations or fluxes in steady state or as functions of time, or functions of them, such as signal amplitudes or durations [6].) Of course, this influence depends on the system, on the variable of interest and on the parameter: at bifurcation points, a small parameter change can even change the qualitative dynamic behaviour. In other cases, parameters can have a weak influence on the system behaviour. In fact, various biological systems seem to have evolved robustness, that is, low sensitivity and thus low variability, against a typical amount of parameter variation (see [7] and references therein) [8 -10].If the parameter v...

Bayesian inference for a discretely observed stochastic kinetic model

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Global characterization of in vivo enzyme catalytic rates and their correspondence to in vitro k cat measurements

“…Several studies grapple with missing k cat values by sampling from the distribution of k cat values measured in vitro or by using measurements of the same enzyme from related species (13)(14)(15)(16). These approximations systematically ignore any errors resulting from the differences between in vitro and in vivo environments.…”

Bringing metabolic networks to life: convenience rate law and thermodynamic constraints

“…To circumvent this problem, we introduce new, thermodynamically independent system parameters [18]. For each substance i , we define the dimensionless energy constant…”

“…In addition, the second law of thermodynamics implies constraints between the kinetic parameters: in a metabolic system, the Gibbs free energies of formation of the metabolites determine the equilibrium constants of the reactions [15]. This leads to constraints between kinetic parameters within reactions [16] and across the entire network [17,18] – a big disadvantage for all methods that scan the parameter space, such as parameter fitting, sampling, and optimisation. Also, if parameter values are guessed from experiments and then directly inserted into a model, this model is likely to be thermodynamically wrong.…”

“…To ensure thermodynamic correctness, we write the convenience kinetics in terms of thermodynamically independent parameters [18]. A short introduction to kinetic modelling is given in the methods section; a list of mathematical symbols and an illustrative example is also provided [See Additional file 1].…”