Complexity

Volume 2018, Article ID 6879013, 20 pages

https://doi.org/10.1155/2018/6879013

## Distributed Finite-Time State Estimation of Interconnected Complex Metabolic Networks

^{1}SEPI-UPIBI, Instituto Politécnico Nacional, 07340 Mexico City, Mexico^{2}Department of Bioprocesses, UPIBI, Instituto Politécnico Nacional, 07340 Mexico City, Mexico

Correspondence should be addressed to Isaac Chairez; moc.liamg@ozeriahci

Received 16 November 2017; Accepted 5 March 2018; Published 9 May 2018

Academic Editor: Danilo Comminiello

Copyright © 2018 Alfonso Sepulveda-Galvez 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

A set of distributed robust finite-time state observers was developed and tested to estimate the main biochemical substances in interconnected metabolic networks with complex structure. The finite-time estimator was designed by composing several supertwisting based step-by-step state observers. This segmented structure was proposed accordingly to the partition of metabolic network obtained as a result of applying the observability analysis of the model used to represent metabolic networks. The observer was developed under the assumption that a sufficient and small number of intracellular compounds can be obtained by some feasible analytic techniques. These techniques are enlisted to demonstrate the feasibility of designing the proposed observer. A set of numerical simulations was proposed to test the observer design over the hydrogen producing metabolic behavior of* Escherichia coli*. The numerical evaluations showed the superior performance of the observer (on recovering immeasurable state values) over classical approaches (high gain). The variations of internal metabolites inserted in the hydrogen productive metabolic networks were collected from databases. This information supplied to the observer served to validate its ability to recover the time evolution of nonmeasurable metabolites.

#### 1. Introduction

Metabolic engineering (ME) represents one of the most relevant disciplines in bioengineering [1]. The set of methods and techniques integrated in this novel discipline seeks to optimize the metabolic circuits and regulatory mechanisms in different cells that increase their production of relevant metabolites [2]. ME is closely connected to genetic engineering and molecular biology [3] that can use complex interconnected network models with diverse structures.

ME takes into consideration the tools of applied mathematics to obtain models of the metabolic networks under study [4, 5]. The correct application of these models could save lots of resources and reduce the time to introduce productive modified cells with their remarkable secondary metabolites to the market.

There are different options to generate the model including Boolean structures, algebraic relationships, or time dependent descriptions based on ordinary or partial differential equations (ODE/PDE) [6]. This last option seems to be the most complete because the transient behavior of the cell can be captured in the ODE model. However, two natural problems arise when ODEs are used: (1) the data density needed to characterize the model is bigger than all other cases and (2) the number of parameters included in the model could be so large that existing parametric identification methods cannot be powerful enough to get a complete validation of the model [7].

The identification of uncertain parameters from time-series measurements of interesting biochemical compounds is a major aspect in system biology and ME. The majority of existing methods aimed at obtaining the parameters require the information of all compounds continuously or at least with some periodicity. On the other hand, best-fit parameter estimates are ill-posed due to issues related to data informativeness, problem formulation, and parameter sensitiveness [8, 9]. Even when the so-called canonical power-law formalism (using relative simple structures in the right-hand side of the ODE representing the metabolic network) is considered to obtain the model, the parametric identification solution remains as a complicated problem. This characteristic is emphasized when regulatory interactions among metabolites are considered. Despite the benefits of power-law formalism, the number of parameters increases if the metabolic network is described more precisely (including more metabolites and regulation interactions).

One major additional issue that must be solved to get accurate parameter values in the metabolic network representation is the necessity of measuring all the metabolites in the network. This is one of the most challenging aspects when ODEs are used to generate the model. Several experimental options have been proposed as the so-called metagenomics or using real-time polymerase chain reaction (PCR). Nevertheless, the expensiveness of applying these techniques may limit their application on characterizing metabolic networks. The number of experiments needed to characterize metabolic networks accurately is usually large. Nevertheless, in actual metabolic networks, only a small fraction of intracellular metabolites can be directly measured, and therefore initial conditions should be also estimated. Undoubtedly, in metabolic networks, lack of experimental data is unavoidable. This condition compromises the accurateness of parameters estimated by any feasible method [10].

State estimation in metabolic engineering has become an option to recover the time variation of compounds in metabolic networks. Different options have been proposed to solve the reconstruction of the metabolites concentration over time. However, the complexity of these networks limits the application of global observers for the entire network. One popular strategy recently explored is the divide-and-conquer scheme where a set of low-order state observes are running in parallel. Each of these observers is applied on a certain section of the network where observability property holds [11–13].

State observers to recover immeasurable information from biological and biotechnological systems have been developed for many years [14]. The application of these observers can provide information that can be eventually used to regulate the metabolic network and, in consequence, optimize the production of some metabolites. In this sense, a state observer is known as a software sensor. That is, a suitable and well-designed state estimator can be used as an accurate artificial sensor of some specific variables. The artificial measurements provided by the observer correspond to variables that cannot be measured online or their measuring cost is relatively high [15].

This study describes the design of a so-called step-by-step decentralized observer to estimate the unknown states of the selected metabolic network. This state estimator is composed of a set of robust high-order sliding mode differentiators. In particular, the decentralized observer is applied on a system representing hydrogen () production by a strain of* Escherichia coli*.

Notice that, in this study, no general characteristics of metabolic networks are used for the construction of the observers or for proposing a method to estimate their state variables. We only considered a simplified metabolic network that cannot generalize all the constraints, restrictions, and characteristics of general metabolic networks. Any possible solution for such problem requires a deeper understanding of internal interactions between metabolites, enzymes, and genes. However, a possible solution of that problem is beyond the scope of this article.

#### 2. Notation

This section introduces the nomenclature used in this manuscript. Acronyms of all substances considered to construct the model of metabolic network are listed as follows. : Biomass Glc: Glucose G6p: Glucose 6-phosphate 2gp: 2-Phospho-D-glycerate Pep: Phosphoenolpyruvate Oxa: Oxalacetate Mal: Malate Suc: Succinate Cit: Citrate -kg: -Ketoglutarate Pyr: Pyruvate Acoa: Acetyl coenzyme A Aad: Acetaldehyde EtOH: Ethanol Acp: Acetylphosphate Acet: Acetate For: Formate Lac: Lactate H_{2}: Hydrogen CO_{2}: Carbon dioxide.

#### 3. Hydrogen Producing Metabolic Network of* E. coli*

The model used in this study was built using 18 ODEs that represent the dynamic behavior of all metabolites involved in the metabolic pathway presented in Figure 1. Notice that this system was selected just as an example that demonstrates the application of the state estimation technique proposed in this study. The first ODE of the model represents the production of biomass and the rate of the reaction is modeled as Monod kinetics containing two different parameters: and . The subsequent reactions correspond to intracellular and extracellular metabolites of the metabolic network. These equations consider the production and consumption of each metabolite due to the action of the enzymes that catalyze the reactions. Each enzyme has a specific reaction rate depending on the concentration of its substrate, product, or any other metabolite that participates in the regulation mechanisms and can take a positive sign to express production or a negative sign to express consumption. The cell takes up Glc from the culture media by the action of the PTS system. This system is activated by the presence of Pep and takes Glc as substrate forming G6p as a product. According to [17], an activated mechanism can be modeled as a bisubstrate reaction (a second-order one).