Complexity

Volume 2018, Article ID 2342650, 22 pages

https://doi.org/10.1155/2018/2342650

## Analytical Reduction of Nonlinear Metabolic Networks Accounting for Dynamics in Enzymatic Reactions

Université Côte d’Azur, Inria, INRA, CNRS, UPMC Univ Paris 06, Biocore Team, Sophia Antipolis, France

Correspondence should be addressed to Claudia López Zazueta; rf.airni@ateuzaz-zepol.aidualc

Received 24 November 2017; Revised 29 March 2018; Accepted 26 April 2018; Published 12 August 2018

Academic Editor: Alain Vande Wouwer

Copyright © 2018 Claudia López Zazueta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Metabolic modeling has been particularly efficient to understand the conditions affecting the metabolism of an organism. But so far, metabolic models have mainly considered static situations, assuming balanced growth. Some organisms are always far from equilibrium, and metabolic modeling must account for their dynamics. This leads to high-dimensional models in which metabolic fluxes are no more constant but vary depending on the intracellular concentrations. Such metabolic models must be reduced and simplified so that they can be calibrated and analyzed. Reducing these models of large dimension down to a model of smaller dimension is very challenging, specially, when dealing with nonlinear metabolic rates. Here, we propose a rigorous approach to reduce metabolic models using quasi-steady-state reduction based on Tikhonov’s theorem, with a characterized and bounded reduction error. We assume that the metabolic network can be represented with Michaelis-Menten enzymatic reactions that evolve at different time scales. In this simplest approach, some metabolites can accumulate. We consider the case with a continuous varying input in the model, such as light for microalgae, so that the system is never at a steady state. Furthermore, our analysis proves that metabolites in the slow part of the metabolic system reach higher concentrations (by one order of magnitude) than metabolites in the fast part under some flux conditions. A simple example illustrates our approach and the resulting accuracy of the reduction method.

#### 1. Introduction

Metabolic models have considerably helped in understanding the metabolism of an organism and enhancing its production capability. These models are based on simplified metabolic networks and generally include several hundreds of reactions associated to many metabolic compounds. For example, metabolic models to better understand the production of triacylglycerols and carbohydrates from microalgae (both compounds can then be turned into biofuel) [1] use between 56 and 2190 reactions and between 46 and 1862 metabolites, depending on authors and studies. In order to manage the large dimension of these models, some simplifying assumptions are generally necessary.

The most classical hypothesis is balanced growth, that is, global steady-state assumption (SSA). This means that the derivatives, with respect to time, of all variables are put to zero. For instance, flux balance analysis (FBA) [2] or macroscopic bioreaction models (MBM) [3] are based on linear algebra to solve the equation , where is the stoichiometric matrix and is the vector of intracellular reaction rates.

Yet, metabolisms of microalgae and cyanobacteria are directly related to solar light providing the energy for incorporating CO_{2} through Calvin cycle. Periodic fluctuation of light induces unstationarity and permanent accumulation and reuse of metabolites (specially lipids and carbohydrates). Therefore, such metabolisms are never at a steady state, and the classical approaches based on balanced growth hypothesis cannot be used to describe their metabolisms.

Here, we propose a rigorous mathematical approach to reduce the dimension of a dynamical metabolic system, in order to analyze its behavior and calibrate it. The reduction that we propose allows to characterize the approximation error, and it is appropriate to model-based control strategies. The idea is to keep some dynamical components of the model that are necessary specially when dealing with microalgae and cyanobacteria.

A first attempt in this direction was carried out with the Dynamic Reduction of Unbalanced Metabolism (DRUM) method [4]. DRUM considers subnetworks in quasi-steady state (QSS), which are interconnected by metabolites that can accumulate. Then, elementary flux modes (EFM) are computed in each subnetwork to reduce them using quasi-steady-state assumption (QSSA). As a result, the dynamics of accumulative metabolites form a reduced system of ordinary differential equations (ODE). It provided sound results, specially to describe accumulation of lipids and carbohydrates in microalgae. However, as almost all the methods, it also relies on a series of assumptions whose mathematical bases have not been rigorously established [5]. Beyond QSSA which is not rigorously defined from a mathematical viewpoint, these approaches also neglect intracellular dilution due to growth.

Models of metabolic networks are nonlinear and high-dimensional systems, which make their dynamical behavior difficult to determine and calibrate. The main objective of our work is to provide mathematical foundations for the reduction of metabolic networks down to low-dimensional dynamical models.

Here, we study a class of metabolic models of dimension , where the enzymatic reaction rates are represented by Michaelis-Menten reactions. This class of models is the simplest nonlinear one to get accumulation of some intermediate compounds. The objective is to reduce this model accounting for a permanently fluctuating input and rigorously including dilution of the metabolic compounds due to the growth rate. The system is not closed and never reaches a steady state. At the end, we can express a slow dynamical system of small dimension and a fast system as a function of the variables of the slow system. The error in this reduction is characterized and bounded.

In Section 2, we introduce the class of models we consider, which is composed of two (general) subnetworks of fast reactions connected by metabolites with slow dynamics. In Section 3, we develop a mathematical model for these metabolic systems.

In Section 4, with proper mathematical hypotheses, after a change of variables for the metabolites with fast dynamics, the system becomes a slow-fast system. The conditions for applying Tikhonov’s theorem for singularly perturbed systems are verified and we end up with a reduced dynamical model and a bound of the approximation error.

In Section 5, we prove that metabolites in QSS have a concentration one order of magnitude lower than slow metabolites. Additionally, in Section 6, we propose an identification algorithm to estimate the parameters of the reduced system from available data.

Finally, we apply our method to a toy metabolic model in Section 7. This simple model is forced by a periodic input and includes standard bricks in metabolic networks: combination of reversible and nonreversible reactions, with chains and cycles.

#### 2. Network of Enzymatic Reactions

In this section, we present the class of metabolic networks studied all over the paper, which are illustrated in Figure 1. These networks are composed of two subnetworks of fast reactions, which are interconnected by several metabolites with slow rates of consumption. The subnetworks have an arbitrary finite number of metabolites and reactions between them.