System estimation from metabolic time-series data

Abstract

At the center of computational systems biology are mathematical models that capture the dynamics of biological systems and offer novel insights. The bottleneck in the construction of these models is presently the identification of model parameters that make the model consistent with observed data. Dynamic flux estimation (DFE) is a novel methodological framework for estimating parameters for models of metabolic systems from time-series data. DFE consists of two distinct phases, an entirely model-free and assumption-free data analysis and a model-based mathematical characterization of process representations. The model-free phase reveals inconsistencies within the data, and between data and the alleged system topology, while the model-based phase allows quantitative diagnostics of whether--or to what degree--the assumed mathematical formulations are appropriate or in need of improvement. Hallmarks of DFE are the facility to: diagnose data and model consistency; circumvent undue compensation of errors; determine functional representations of fluxes uncontaminated by errors in other fluxes and pinpoint sources of remaining errors. Our results suggest that the proposed approach is more effective and robust than presently available methods for deriving metabolic models from time-series data. Its avoidance of error compensation among process descriptions promises significantly improved extrapolability toward new data or experimental conditions.

Fig. 1.

DFE approach to metabolic system estimation from an in vivo time-series data. Starting with experimental time series, the data are simultaneously balanced and smoothed for constant total mass throughout the time series. Then the slopes are estimated using published methods. Combined with the knowledge of the system topology, the slope information yields a linear system of fluxes. The system is solved, using linear algebra techniques, yielding dynamic profiles of all extra- and intra-cellular fluxes in the system. Next, functional assumptions are formulated on how to best represent the processes mathematically. These functions result in symbolic flux representations that can be independently fitted with regression methods to the respective dynamic flux profiles. When combined with knowledge of the system topology, the numerical flux functions are integrated as a single unified system model to obtain time courses.

Fig. 2.

Results of Case study 1. (a) Fermentation pathway in L.et al. Dark arrows show flow of material. Dashed arrows indicate leakage of material into secondary pathways. Enzyme activation and inhibition are indicated by light gray arrows. G6P, glucose 6-phosphate; FBP, fructose 1,6-bisphosphate; 3-PGA, 3-phosphoglycerate; PEP, phosphoenolpyruvate; ATP, adenosine triphosphate; ADP, adenosine diphosphate; Pi, inorganic phosphate; NAD+, nicotinamide adenine dinucleotide (oxidized); NADH, nicotinamide adenine dinucleotide (reduced). (b) Dynamic metabolic profiles. Time-series data of major metabolites in the primary pathway (symbols). Solid lines indicate fits with a model derived using DFE. (c) Dynamic flux profiles. The symbols show the time series of flux profiles estimated solely from data and the system stoichiometry using DFE. The solid lines indicate fitting of a power-law model to the dynamic flux data.

Fig. 3.

Results of Case study 2. (a) Dynamic metabolic profiles. Metabolic time-series data with added artificial noise (symbols). The solid lines represent the smoothed and balanced time series. (b) Dynamic mass balance. The random noise leads to mass imbalance which is successfully restored after optimization and smoothing. (c) Dynamic flux profiles. The linear system of fluxes is solved to obtain unique flux profiles (symbols). Power-law models are fitted to each flux time series independently (solid lines). (d) Results from the numerical model. Using DFE, a fully parametric kinetic model is derived from noisy metabolic time-series data (symbols). The results of the model (solid lines) closely match the original dynamic metabolic data.

Fig. 4.

Results of Case Study 3. (a)Sigmoidal glucose uptake. This type of uptake dynamics has been observed in experiments (symbols) and is difficult to represent with a simple power-law function (solid line). (b) Dynamic metabolic profiles. Time-series data of the major metabolites that result from sigmoidal glucose uptake. (c) and (d) Flux substrate plots. The ‘experimental’ flux profile (gray), obtained using DFE, is plotted against the corresponding flux obtained by fitting a power-law model (black). (c) shows systematic error when flux v1 is fitted with a power-law model. On the other hand, a power-law model accurately reproduces other fluxes like v3 in the same system (d).

Fig. 5.

Results of Case Study 4. (a) Dynamic metabolic profiles. Measured dynamics of metabolite pools in L.lactis following a 20 mM [6-13C] glucose bolus (symbols). (b) Dynamic mass balance. Systematic mass imbalance in the experimental data was attributable to missing information about secondary metabolites. The balance was successfully restored by accounting for secondary fluxes. (c) Dynamic flux profiles. The linear system of fluxes is solved to obtain the unique flux profiles (symbols). Power-law models are independently fitted to each flux time series, using linear and non-linear regression (solid lines). (d) Results from the numerical model. Using DFE, a fully parametric kinetic model is derived from the actual metabolic time-series data (symbols). The results of the model (solid lines) closely match the data.