# Computational Strategies for a System-Level Understanding of Metabolism

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## Abstract

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## 1. Introduction

- Establish from the beginning the scientific question that motivates the development of the model. Consequently, the analysis of the model is expected to increase the current knowledge on the system, thanks to novel predictions on its functioning and to their experimental validation. In this phase, initial experimental data are necessary to define a plausible mathematical model, since they can aid to discriminate among different hypotheses on the structure of the system.
- Identify the proper level of abstraction necessary to formally describe the components of thesystem and their mutual interactions. In particular, the model should take into account all available knowledge on the biochemical, physical or regulatory properties of all system components and interactions. In so doing, any detectable emergent property of the system—either in the physiological state or in response to genetic, chemical or environmental perturbations—can be discovered with the appropriate computational methods. The choice of the level of abstraction will bring to the definition of either fine-grained (e.g., mechanism-based) or coarse-grained (e.g., interaction-based or constraint-based) models. Typically, the mechanism-based approach deals with toy or core models, while the interaction-based and constraint-based approaches are more suited for the analysis of genome-wide or core models. A schematic overview of the three main modeling approaches is given in Figure 1, including a list of their principal dichotomic features [23], such as quantitative vs. qualitative, static vs. dynamic, parameterized vs. non parameterized, single volume vs. compartmental, well-stirred vs. heterogeneous (diffusion), etc.
**Figure 1.**Schematic overview of the main modeling approaches for biological systems, together with their principal characteristics and differences. Moving from the coarse-grained (interaction-based, constraint-based) to the fine-grained (mechanism-based) approach, models vary with respect to: (i) the size of the system, defined in terms of the number of components and respective interactions included in the model, which decrease from genome-wide to core models (Section 2.1); (ii) the computational costs required for the analysis of the model, which increase from the analysis of the topological properties of the network typical of interaction-based models (Section 3.1), to the study of flux distributions typical of constraint-based models (Section 3.2), to the investigation of the system dynamics typical of mechanism-based models (Section 3.3); (iii) the nature of the computational results together with the predictive capability, which changes from qualitative to quantitative while moving from interaction-based models (characterized by a high level of abstraction) to mechanism-based models (fully parameterized and describing the system at the level of the functional chemical interactions). - Choose the most appropriate mathematical formalism. A one-to-one correspondence between each modeling approach and a specific modeling purpose would facilitate the choice of the most suitable strategy to be employed. Unfortunately, a sharp-cutting separation is not always possible. In general, mechanism-based (dynamical) models—which are usually defined as systems of differential equations—are considered the most likely candidates to achieve a detailed comprehension of cellular processes. Nonetheless, the usual lack of quantitative parameters represents a limit to a wide applicability of this approach for large metabolic networks. Various attempts have been proposed for the automatic estimation of missing parameters or the characterization of the parameters space [24,25,26].On the other side of the spectrum of modeling approaches, interaction-based models are characterized by a simplified representation of the biological process and allow to achieve qualitative knowledge only. These models can be analyzed by using, for instance, graph theory or topological analysis to investigate the “design principles” of metabolic networks, that can be considered transversal to different organisms [27]. Moreover, they allow to easily identify the so-called hubs (highly interconnected components, essential for the existence of several metabolic processes), as well as the metabolites and reactions connecting them, which can be of particular interest within the scope of, e.g., drug target discovery [28].Considering the limitations of these modeling approaches, the common practice for the computational investigation of metabolism usually relies on constraint-based models. These models are based on the definition and manipulation of stoichiometric matrices, whose native application pertains to the field of metabolic engineering. In this case, the methodologies that were initially developed for the optimization of microbial strains or for the maximization of some product yields in biotechnological applications, are now widely used with different goals in the study of metabolic networks.

**Figure 2.**General scheme of the computational investigation of metabolism, from network reconstruction to in silico analysis. Violet arrows indicate the relationships between experimental data and reverse engineering methods used to reconstruct the metabolic network and to identify the stoichiometry of the reactions (see Section 2.1 and Section 2.3) Red arrows indicate the relationships between experimental data and the methods to derive or estimate the unknown parameters (see Section 2.2 and Section 2.4). Green arrows indicate the computational analyses that can be performed on metabolic networks and models (see Section 3 and Section 4). Blue arrows indicate specific computational analyses that can be carried out on different types of models (see, in particular, Section 3.2 and Section 3.3).

## 2. From Experimental Data to Models

#### 2.1. Metabolic Network Reconstruction

**Genome-wide models.**One of the main purposes in the reconstruction of genome-wide (GW) models is to summarize all the current knowledge concerning metabolic processes at the level of single gene annotation, trying to take into account every single reaction that is known to occur within an organism. These reconstructions can be exploited in a dual way: on the one hand, they represent the scaffolds for different computational analyses, such as those described in the next sections; on the other hand, they work as repositories of all the collected knowledge about metabolic pathways [33].

**Toy and core models.**An opposite strategy to the reconstruction of GW models is the definition of models with a simple structure and including a limited number of components, which are often referred to as toy models. The goal of these models is not to describe in molecular terms a specific pathway, but rather to highlight some major regulatory properties. In a toy model, many pathways and thousands of reactions are thus summarized in a few steps, in order to more easily identify the most relevant components of the system. Good examples can be found in [60,61], where toy models were defined to probe the relationship between the type of energetic metabolism (respirative vs. fermentative) and cellular growth.

**Figure 3.**Example of core model representing the main metabolic pathways of yeast (modified from [65]). The pathways included in this example are: glycolysis (green arrows), ethanol fermentative pathway (blue arrows), pentose phosphate pathway (light blue arrows), fatty acids biosynthesis (violet arrows), tricarboxylic acid (TCA) cycle (red arrows) and oxidative phosphorylation pathway (orange arrows).

#### 2.2. Parameter Estimation

^{+}/Ca

^{2+}cycle and the K

^{+}cycle, were estimated exploiting a Monte Carlo optimization algorithm based on simulated annealing (SiAn) [81], and refined using a gradient-based method.

#### 2.3. Reverse Engineering

#### 2.4. Ensemble Modeling

## 3. From Models to in Silico Data

#### 3.1. Topological Analysis

#### 3.2. Flux Balance Analysis

_{i}v

_{i}, where c

_{i}is a weight indicating how much the flux v

_{i}of reaction i contributes to the OF.

#### 3.3. Simulation of the Dynamics

**Ordinary differential equations.**When the molecular species involved in a system are present in large amounts, as it is usually assumed in metabolic networks, the effect of noise can be neglected. In these case, it is therefore convenient to apply a deterministic modeling approach, where each molecular species is described by means of a continuous variable (representing its cellular concentration) and its variation in time is formally described through ODEs. ODEs represent the standard mathematical framework for the simulation of the dynamics of biological systems: given the initial state of the system (i.e., the concentration values of all species), together with the kinetic parameters, the dynamics of the system is usually obtained by numerically solving the set of differential equations. Being one of the most applied approaches in almost any area of the science, there exist many efficient numerical integrators to rely on [150,151].

_{1}complex (also called complex III), leading to the production of reactive oxygen species (ROS) involved in the pathophysiology of several diseases. This small-scale mathematical model was able to reproduce the experimental data of the activity of the mitochondrial electron transfer chain, and allowed to investigate the response of complex III in the presence of the inhibitor antimycin A, as well as the effect of different values of the membrane potential on ROS generation. The behavior of the electron transfer chain was also analyzed with a novel composable kinetic rate equation (called the chemiosmotic rate law) in [154], solving some limits of previous ODEs models of the same mitochondrial process [155,156], therefore possibly facilitating the comparative analysis of the function of these structures in different pathological conditions.

**Reaction-Diffusion models.**The classic modeling approaches for the description of a biochemical system do not usually take into account the diffusion of molecules. However, if the effect of chemical species localization plays a relevant role on the system dynamics (see, e.g., [160]), then the system can be modeled by means of Partial Differential Equations (PDEs), that define a Reaction-Diffusion (RD) model [161,162,163]. PDEs provide a continuous time and space domain description of the system, where the mass transport, the chemical kinetics and the conservation laws, together with the boundary conditions, are embedded within the same set of equations that can be solved analytically or numerically.

## 4. From In Silico Data to Experimental Hypothesis

#### 4.1. Model Validation

**Figure 4.**Schematic representation of the validation process of mathematical models. A model draft undergoes an iterative process in which the in silico outcomes are compared to the experimental data to validate the model, and to formulate new hypotheses about the functioning of the underlying biological process. A validated model can then be used for deeper computational investigations.

#### 4.2. Sensitivity Analysis

_{max}of 10-formyltetrahydrofolate synthase varies between 100 and 486,000 µM/h.

#### 4.3. Control Theory

## 5. Computational Strategies at Work: Gaining Novel Insights on Metabolism

#### 5.1. Increase, Integrate and Validate Biological Knowledge

#### 5.2. Generate Experimentally Testable Hypotheses: Identify Regulatory Nodes and Drug Targets

#### 5.3. Design Microbial Strains for Metabolic Engineering and Industrial Applications

## 6. Conclusions and Perspectives

**Table 1.**Overview of some recent literature papers on the modeling and computational analysis of metabolism.

Pathway/Aim ofthe Model | Cell Type/Organ | Organism | Modeling Approach &Methodology | ExperimentalData | Reference |
---|---|---|---|---|---|

Glycolysis | - | T. brucei | CM, ODE | L | Achcar et al. [159] |

GW metabolic network and succinic acid production | - | S. cerevisiae | GW, FBA | M | Agren et al. [58] |

GW metabolic network | - | A. niger | GW, FBA | L | Andersen et al. [173] |

Mitochondrial energy metabolism, Na^{+}/Ca^{2+} cycle, K^{+} cycle | Heart, liver | B. taurus, S. scrofa, R. norvegicus | CM, DAE, PE, SA | L, M | Bazil et al. [80] |

OXPHOS | Cardiomyocytes | R. norvegicus | CM, ODE | L | Beard [156] |

Electron transport chain | Heart homogenates | R. norvegicus | CM, ODE, CRL | L, M | Chang et al. [154] |

Glycolysis, OXPHOS | Not specified | Eukaryotic, H. sapiens | CM, Control theory | L | Cloutier et al. [200] |

Bow-tie architecture of metabolism | Not specified | H. sapiens | GW, Topological analysis | L | Csete et al. [118] |

Central metabolism | - | Yeast | CM, FBA | L | Damiani et al. [65] |

Energy metabolism | Skeletal muscle cell | Mammal | CM, PDE | L | Dasika et al. [165] |

Glycolysis and pentose phosphate pathway | - | E. coli | CM, ODE, SA | L | Degenring et al. [179] |

Glycolysis and pentose phosphate pathway | - | E. coli | CM, ODE, SA | L | Degenring et al. [179] |

Biosynthesis of valine and leucine | - | C. glutaminicum | CM, ODE, SDE | M | Dräger et al. [76] |

Anabolic, catabolic, chemiosmosis pathways | - | E. coli | GW, Control theory | M | Federowicw et al. [202] |

Small world behavior of metabolism | Not specified | H. sapiens | GW, Topological analysis | L | Fell et al. [116] |

GW metabolic network | Not specified | H. sapiens | GW, FBA | L | Duarte et al. [39] |

GW metabolic network | - | E. coli MG1655 | GW, FBA | M | Edwards and Palsson [175] |

GW metabolic network | - | H. influenzae | GW, FBA | L | Edwards et al. [36] |

Cancer metabolic networks | Various (NCI-60 collection) | H. sapiens | Network reconstruction, FBA, gene (pair) analysis | L | Folger et al. [208] |

GW metabolic network HepatoNet1 | Hepatocytes | H. sapiens | GW. Network reconstruction | L | Gille et al. [220] |

Cytochrome bc1 complex, ROS production | Muscle, heart, liver, kidney, brain | R. norvegicus | CM, ODE | L | Guillaud et al. [153] |

GW metabolic network EHMN | Not specified | H. sapiens | GW, Network reconstruction | L | Hao et al. [221] |

GW metabolic network | - | S. cerevisiae S288c | GW, Network reconstruction, FBA | L | Heavner et al. [3] |

GW metabolic network | - | S. cerevisiae | Network reconstruction | L | Herrgård et al. [43] |

Topological properties of metabolism | - | 43 different organisms | GW, Topological analysis | L | Jeong et al. [27] |

Glycolysis, OXPHOS | - | Not specified | CM, ODE, Game theory | - | Kareva [189] |

Whole-cell life cycle model | - | M. genitalium | GW, FBA, ODE | L, M | Karr et al. [204] |

Glycolysis, pentose phosphate pathway | - | T. brucei | CM, ODE | L | Kerkhoven et al. [64] |

Energy metabolism | Colorectal cells | H. sapiens | CM, FBA, EM | M | Khazaei et al. [214] |

GW metabolic network | - | Synechocystis sp. PCC 6803 | GW, FBA | L | Knoop et al. [37] |

Glycolysis, gluconeogenesys, glycogen metabolism | Hepatocytes | H. sapiens | CM, ODE | L | König et al. [157] |

Adenine nucleotide translocase | Heart mitochondria | B. taurus | CM, ODE, PE, SA | L | Metelkin et al. [152] |

GW metabolic network | - | Z. mays L. subsp. mays | GW, Network reconstruction | L | Monaco et al. [40] |

Xylose metabolism | - | L. lactis IO-1 | CM, ODE, SA | M | Oshiro et al. [183] |

GW metabolic network | - | S. cerevisiae | GW, Network reconstruction, FBA | L | Österlund et al. [222] |

GW metabolic network and succinic acid production | - | S. cerevisiae | GW, FBA | M | Otero et al. [57] |

Topological properties of metabolism | - | 43 different organisms, E. coli | GW, Topological analysis | L | Ravasz et al. [122] |

One-carbon metabolism, trans-sulfuration pathway, synthesis of glutathione | Hepatocyte | H. sapiens | CM, ODE | L | Reed et al. [158] |

Glycolysis, TCA cycle, pentose phosphate pathway, glutaminolysis, OXPHOS | HeLa cell | H. sapiens | CM, FBA | M | Resendis-Antonio et al. [67] |

Modularity of metabolism | Not specified | H. sapiens | GW, Topological analysis | L | Resendis-Antonio et al. [120] |

GW metabolic network | Not specified | H. sapiens | GW, Network reconstruction | L | Sahoo et al. [223] |

Acetone, butanol and ethanol production | - | C. acetobutylicum | CM, ODE, SA | M | Shinto et al. [184] |

Cancer metabolic networks | Various (NCI-60 collection) | H. sapiens | FBA | L | Shlomi et al. [206] |

GW metabolic network | - | S. cerevisiae | GW, FBA | L | Simeonidis et al. [130] |

Glycolysis | - | S. cerevisiae | CM, ODE | M | Teusink et al. [62] |

GW metabolic network | Not specified | H. sapiens | GW, FBA | L | Thiele et al. [4] |

Primary metabolism | - | E. coli | CM, ODE, EM | - | Tran et al. [101] |

Fueling reaction network | - | E. coli W3110 | CM, FBA | M | Varma et al. [174] |

Reduced model of cell metabolism | - | - | CM, FBA | L | Vazquez et al. [61] |

Small-world property of metabolism | - | E. coli | GW. Topological analysis | L | Wagner et al. [117] |

GW metabolic network | - | C. glabrata | GW, FBA | L | Xu et al. [38] |

Erythrocyte metabolism | Red blood cell | H. sapiens | Hybrid: ODE + MFA | - | Yugi et al. [166] |

Mitochondrial energy metabolism | Various tissues | Mammal | CM, ODE | - | Yugi [224] |

Modularity of metabolism | Not specified | H. sapiens | GW, Topological analysis | L | Zhao et al. [119] |

ROS-induced ROS release in mitochondria network | Cardiomyocytes | C. porcellus | CM, ODE, PDE, RD, Finite Difference Method | M | Zhou et al. [164] |

Tool name | Purpose | Interaction-based | Constraint-Based | Mechanism-Based | Reference |
---|---|---|---|---|---|

BioMet Toolbox | Genome-wide metabolic model validation, FBA, probabilistic FBA, gene set analysis | √ | [225] | ||

Cobra Toolbox | FBA, FVA, dFBA, gap filling, network visualization | √ | [226] | ||

COPASI | Determinstic, stochastic and hybrid simulation, PE, SA, MCA | √ | [227,228] | ||

cupSODA | Deterministic simulations on GPUs | √ | [171] | ||

Cytoscape | Complex networks visualization and topological analysis | √ | [229,230] | ||

FAME | Web based FBA and FVA | √ | [231] | ||

FASIMU | FBA, FVA, gene deletion analysis, gap filling | √ | [232] | ||

OptFlux | FBA, FVA, EFM, gene deletion analysis | √ | [233] | ||

Pathway Tools | GW reconstruction, FBA, gap filling | √ | [234] | ||

Raven Toolbox | GW reconstructions, FBA, network analysis and visualization | √ | √ | [235] | |

SurreyFBA | FBA, FVA, EFM | √ | [236] |

**Table 3.**Principal databases collecting biological data or metabolic models, fundamental resources for the investigation of metabolism.

Database | Contents | Reference |
---|---|---|

BiGG | Genome-scale metabolic networks | [237] |

BioCyc | Collection of more than 3000 pathways / genome databases | [238] |

BioModels | SBML models of biological processes | [239] |

Brenda | Molecular and biochemical information on enzymes | [240] |

CellML | XML-based models of biological processes | [241] |

Ensembl | Genome browser for genomic information | [242] |

ExPASy | Portal to existing databases and tools categorized by life science areas | [243] |

GeneCards | Omics data on human genes | [244] |

HumanCyc | Human metabolism pathways | [245] |

Human Metabolic Atlas | Human metabolism models | [52] |

Human Protein Atlas | Human protein expression profiles with spatial localization in tissues and cells | [53] |

JWS | Curated models of biochemical pathways and simulation tools | [246] |

KEGG | Manually curated pathway maps integrating molecular-level information | [32] |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## List of Acronyms

BDO | 1,4-butanediol |

CM | Core Model |

DE | Differential Evolution |

dFBA | Dynamic Flux Balance Analysis |

dNTP | Deoxyribonucleotide Triphosphate |

EFM | Elementary Flux Modes |

EM | Ensemble Modeling |

FBA | Flux Balance Analysis |

FDG-PET | 18F-fluorodeoxyglucose–positron emission tomography |

FVA | Flux Variability Analysis |

GA | Genetic Algorithm |

GC | Gas Chromatography |

GC-MS | Gas Chromatography Mass Spectrometry |

GP | Genetic Programming |

GPU | Graphical Processing Unit |

GW | Genome-Wide |

MCA | Metabolic Control Analysis |

MFA | Metabolic Flux Analysis |

MID | Mass Isoptomers Distribution |

MS | Mass Spectrometry |

OAT | One-factor-at-A-Time |

ODE | Ordinary Differential Equation |

OF | Objective Function |

PE | Parameter Estimation |

PSO | Particle Swarm Optimization |

RD | Reaction-Diffusion |

RE | Reverse Engineering |

ROS | Reactive Oxygen Species |

SA | Sensitivity Analysis |

SiAn | Simulated Annealing |

SDE | Stochastic Differential Equation |

TCA | Tricarboxylic Acid |

## Appendix I: Experimental Methodologies for Metabolic Data Generation

^{13}C,

^{2}H,

^{15}N,

^{18}O and

^{34}S) are non-radioactive and present no risk to humans, in contrast to radionuclides commonly used in cancer diagnosis, as

^{18}F-fluorodeoxyglucose–positron emission tomography (FDG-PET). FDG-PET is clinically used to visualize the glucose uptake of cancer cells, but it suffers of some limitations: it allows to investigate only the first step of glycolysis metabolism, it is able to observe only cancer types with glycolytic phenotype, and it is not able to identify information related to downstream glucose metabolic pathways [253,254]. On the contrary, using appropriate stable isotope tracers we have an effective tool able to characterize the metabolic phenotype of organisms [255,256].

^{13}C) in the culture medium (i.e., [U

^{13}–C

_{6}]glucose or [U

^{13}–C

_{5}]glutamine), uptaken and metabolized into the cells, will lead to the production of molecules that contain

^{13}C atoms at various positions. The compounds produced by the

^{13}C tracer will be different only in the number of isotopic atoms incorporated, referred to as mass isotopomers. The MS generating ionized fragments, selectively detected by their mass-to-charge (m/z) ratio, allows to measure the Mass Isoptomer Distribution (MID) characterized as the fractional abundance of mass isotopomers, defined by M0, M1 to Mn, n ∈

**N**(see Figure A1).

^{13}C

_{5}] glutamine tracers (Figure A1). In mammalian cells two major metabolic fates have been identified for glutamine: glutaminolysis and reductive carboxylation. In both pathways glutamine (Gln) ([U–

^{13}C

_{5}]glutamine) is deaminated to glutamate (Glu) generating M5 glutamate, which is then converted to α-ketoglutarate (M5 Akg) by glutamate dehydrogenase. At this point Akg can be oxidized into TCA cycle (forward TCA cycle), generating M4 labeled succinate (Succ), fumarate (Fum), malate (Mal), oxaloacetate (OAA) and citrate (Cit). On the other side, if Akg is metabolized through reductive carboxylation, we can observe labeling of M5 Cit which, exported from the mitochondria into the cytoplasm, will generate the labeling of acetyl-CoA (M2 AcCoA) and later lipids. Therefore, by exploiting appropriate tracers as

^{13}C-glucose and

^{15}N-glutamine, which are routinely used to assess specific metabolic activities in mammalian cells, we may know the overall contribution of nutrients in the biological system under investigation [257]. The use of stable isotope tracers not only improves the resolution of metabolomic techniques, but it also allows to perform MFA, which includes enzyme function quantification and metabolic flux reconstruction [248,255,257,258,259,260,261,262,263].

^{13}C tracers provide a great amount of information regarding the dynamics of active metabolic pathways, as well as their participation in a specific metabolic pathway. As a result, the choice of selective tracer is extremely important, because it largely determines the precision with which one can estimate metabolic fluxes, especially in complex mammalian systems that require multiple substrates [256]. A very recent work has shown quantification and visualization in vivo of glucose metabolic flux in mouse brain after intraperitoneal injection of stable isotope-labeled glucose tracer ([

^{13}C

_{6}]glucose) [264], increasing the areas of application of metabolomics tools. Taken together, these evidences show the important role of metabolomics in the metabolic characterization of cell physiology and pathology.

**Figure A1.**The map illustrates the fate of [U–

^{13}C

_{5}]glutamine (Gln) in the TCA cycle. Red (white, respectively) circles denote labeled (non labeled, respectively) carbon atoms in each metabolite. See text for details.

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## Share and Cite

**MDPI and ACS Style**

Cazzaniga, P.; Damiani, C.; Besozzi, D.; Colombo, R.; Nobile, M.S.; Gaglio, D.; Pescini, D.; Molinari, S.; Mauri, G.; Alberghina, L.; Vanoni, M. Computational Strategies for a System-Level Understanding of Metabolism. *Metabolites* **2014**, *4*, 1034-1087.
https://doi.org/10.3390/metabo4041034

**AMA Style**

Cazzaniga P, Damiani C, Besozzi D, Colombo R, Nobile MS, Gaglio D, Pescini D, Molinari S, Mauri G, Alberghina L, Vanoni M. Computational Strategies for a System-Level Understanding of Metabolism. *Metabolites*. 2014; 4(4):1034-1087.
https://doi.org/10.3390/metabo4041034

**Chicago/Turabian Style**

Cazzaniga, Paolo, Chiara Damiani, Daniela Besozzi, Riccardo Colombo, Marco S. Nobile, Daniela Gaglio, Dario Pescini, Sara Molinari, Giancarlo Mauri, Lilia Alberghina, and Marco Vanoni. 2014. "Computational Strategies for a System-Level Understanding of Metabolism" *Metabolites* 4, no. 4: 1034-1087.
https://doi.org/10.3390/metabo4041034