Piecewise affine approximations of fluxes and enzyme kinetics from in vivo¹³C labeling experiments

Abstract

Owing to the ever increasing amount of available information on metabolic networks and, in particular, to the increase in information content from in vivo 13 C dynamic labeling experiments, this work investigates the problem of reconstructing dynamic fluxes and enzyme kinetics. The model structure is based on the use of piecewise affine approximations. The optimization procedure at the basis of the model identification is improved by separating the parameter estimation procedure into two different phases. As a first step, a dynamic flux profile in time is reconstructed using functions that are piecewise affine (in time). To achieve scalability for this step, several approaches have been developed and compared. Afterwards, the time-dependent profiles are embedded in the concentration space and the enzyme kinetic functions for single reactions are identified independently. This is an advantage compared with standard complete kinetic network approaches, which are typically characterized by hundreds of parameters, because now only a few need to be optimized simultaneously. Additionally, different kinetic formats can be rapidly compared. The overall approach is demonstrated using an informative in silico experiment. availability is not sufficient for a statistically reliable parameter estimation [14], and thus further motivated the development of models from data.To enhance the modeling of metabolic networks, four main challenges have to be tackled:(i) In vivo kinetic mechanisms. Many enzymes have been studied in much detail in vitro; however, it has been shown that their in vivo conditions can differ drastically [15]. In fact, physiological parameters and enzyme kinetic mechanisms are peculiar to the living cell. A variety of models with large parameter space have to be employed and tested [14,16] in order to select proper ones fitting in vivo data. (ii) Data availability. Data sets, even coming from rapid sampling [11] techniques, usually contain only few observations, typically compounded with noise. The gathering of data is impaired by technological limitations or sheer experimental costs. (iii) Limited excitation. In vivo experiments have typically limited dynamical ranges and, besides the substrate, only a few metabolites in the network can be fully excited. Additionally, the limited information content associated to time-series of metabolite concentrations leads to multiple possible outcomes for model identification problems. (iv) Computational overhead. Optimization procedures applied to large-scale kinetic modelsespecially when isotopomer states are included -are demanding because of size, existence of nonlinearities in the flux versus concentration dependence, and again presence of noise.With regards to the challenges in (ii) and (iii) (scarceness of data with limited information content), this study leverages dynamic datasets that are enriched by 13 C labeling measurements. Previous studies [17][18][19] have shown that adding 13 C labeled tracers increases the information content on pool turnover and with t...

How to measure metabolic fluxes: a taxonomic guide for 13 C fluxomics

“…In the case of moderately fast metabolic dynamics, flux profiles can be approximated through piecewise affine (PWA) functions [25]. This fluxomics method infers flux profile interpolations from labeling and concentration time series, thereby dispensing with the need for explicit enzyme kinetics.…”

On stability analysis of cascaded linear time varying systems in dynamic isotope experiments

“…This section introduces the formulation of cascaded linear dynamic models in dynamic isotope experiments. Method to approximately quantify intracellular fluxes and metabolite pools with the piecewise affine approximation of the intracellular fluxes is introduced in the section. Under the metabolic non‐steady state condition, the metabolic network balance equations can be established as follows, where N is the stoichiometric matrix represents the structure of the metabolic network, v is the vector denoting time varying external fluxes and intracellular fluxes in mmol/10 10 cell/hr, c denotes the time varying concentrations of the intracellular metabolic pools in mmol/10 10 cell.…”

“…Since the number of samples is finite, the piecewise affine approximation of the intracellular fluxes is adopted to estimate the fluxes and concentrations of the intracellular pools. Assume that the cell culture is at the metabolic steady‐state state at t 0 , such that C ( t 0 ) is measured and v ( t 0 ) is determined from the steady‐state isotope experiment.…”

“…Based on Equation , one can obtain the approximate intracellular fluxes and concentrations with piecewise affine approximation of intracellular fluxes Remark Following the quasi‐steady state assumption, such that, and intracellular fluxes are time varying, one can approximately obtain Equation as well.…”

“…This section introduces the formulation of cascaded linear dynamic models in dynamic isotope experiments. Method to approximately quantify intracellular fluxes and metabolite pools with the piecewise affine approximation of the intracellular fluxes is introduced 9,14 in the section. Under the metabolic non-steady state condition, the metabolic network balance equations can be established as follows,…”