A method for estimation of elasticities in metabolic networks using steady state and dynamic metabolomics data and linlog kinetics

Abstract

Background: Dynamic modeling of metabolic reaction networks under in vivo conditions is a crucial step in order to obtain a better understanding of the (dis)functioning of living cells. So far dynamic metabolic models generally have been based on mechanistic rate equations which often contain so many parameters that their identifiability from experimental data forms a serious problem. Recently, approximative rate equations, based on the linear logarithmic (linlog) format have been proposed as a suitable alternative with fewer parameters.

Results: In this paper we present a method for estimation of the kinetic model parameters, which are equal to the elasticities defined in Metabolic Control Analysis, from metabolite data obtained from dynamic as well as steady state perturbations, using the linlog kinetic format. Additionally, we address the question of parameter identifiability from dynamic perturbation data in the presence of noise. The method is illustrated using metabolite data generated with a dynamic model of the glycolytic pathway of Saccharomyces cerevisiae based on mechanistic rate equations. Elasticities are estimated from the generated data, which define the complete linlog kinetic model of the glycolysis. The effect of data noise on the accuracy of the estimated elasticities is presented. Finally, identifiable subset of parameters is determined using information on the standard deviations of the estimated elasticities through Monte Carlo (MC) simulations.

Conclusion: The parameter estimation within the linlog kinetic framework as presented here allows the determination of the elasticities directly from experimental data from typical dynamic and/or steady state experiments. These elasticities allow the reconstruction of the full kinetic model of Saccharomyces cerevisiae, and the determination of the control coefficients. MC simulations revealed that certain elasticities are potentially unidentifiable from dynamic data only. Addition of steady state perturbation of enzyme activities solved this problem.

Figure 1

Response of the network to an increase in external glucose concentration. a) First five minutes, a "rapid sampling" experiment (black) and the simulation of the same perturbation with linlog kinetics (blue) using the elasticities estimated from the experimental data from dynamic perturbation. b) Long term response of the cells to the glucose pulse. The time is presented in minutes; intracellular metabolites are given in μmol gDW^-1; extracellular metabolites are given in mM.

Figure 2

Results of the estimation of the elasticities using dynamic data only, comparison of the theoretical elasticities (black), with the linlog elasticities (white).

Figure 3

The results of the new experimental design (The inlet and outlet of the reactor is blocked just after the glucose is increased. The biomass concentration is 15 gDW L^-1, see text for further details in the experimental design) (black) and simulation of the same perturbation with linlog kinetics (blue).

Figure 4

Linlog model simulation of the dynamic perturbation of Figure 1a using elasticities obtained from steady state data only. a) Experimental data (black) and the linlog simulation (blue). Units are the same as Figure 1. b) Comparison of the theoretical elasticities (black), with the linlog elasticities (white) estimated from steady state perturbation data.

Figure 5

Results of the final estimation. a) Experimental data (black) and the linlog simulation using final elasticities (blue). Units are the same as Figure 1. b) Comparison of the theoretical elasticities (black), initial estimates from steady state perturbation data (grey) and the final estimates (white).

Figure 6

Results of the estimation of the elasticities, after model reduction. a) Experimental data (black) and the linlog simulation with the reduced elasticity matrix (blue). Units are the same as Figure 1. b) Comparison of the theoretical elasticities (black), with the linlog elasticities (white).

Figure 7

Results of the cross validation study. An independent steady state perturbation is introduced, in which the activity of V_PFK is decreased by half (see text). A comparison of the normalized in silico experimental results (black) with normalized model prediction (white) are given. The concentrations are normalized with respect to the reference conditions, given in Table 3.

Figure 8

Comparison of the systemic properties, flux control coefficients (C^J0) of the ethanol flux. First column (black): theoretical C^J0s calculated using theoretical elasticities. Second column (grey): C^J0s calculated using estimated elasticities in Figure 5b. Third column (white): C^J0s calculated using the estimated elasticities of the reduced model (Figure 6b).

Figure 9

Comparison of the systemic properties, concentration control coefficients (C^x0) of the two branch point metabolites: G6P (upper panel) and FdP (lower panel). First column (black): theoretical C^x0s calculated using theoretical elasticities. Second column (grey): C^x0s calculated using estimated elasticities in Figure 5b. Third column (white): C^x0s calculated using the estimated elasticities of the reduced model (Figure 6b).

Figure 10

Glycolytic Pathway of Saccharomyces cerevisiae (Adapted from Galazzo and Bailey, 1990).

Figure 11

Nonzero entries of the elasticity matrix. Adapted from Galazzo and Bailey, 1990, with modifications in the V_IN reaction. For the modifications in the uptake reaction, see text.