In brief, constraint-based modelling incorporates information about reaction stoichiometry and thermodynamics [

1], while kinetic modelling requires adding kinetic parameters and regulatory mechanisms in addition to what is required in constraint-based modelling [

2,

3,

4]. As a consequence, kinetic modelling precisely describes the dynamic behavior of a system. However, it is limited to relatively small-size metabolic networks due to the lack of kinetic parameters and computational challenges of integration of big sets of ordinary differential equations. At the same time, constraint-based modelling enables the analysis of large-scale metabolic systems at steady state. However, it cannot model the dynamic response of metabolism to a perturbation. In the following, we will briefly describe how metabolic networks are formalized via either kinetic modelling or constraint-based modelling (

Figure 1).

#### 2.1. Constraint-Based Modelling

Constraint-based modelling is one of the most widely used family of methods to study metabolism [

1]. It allows researchers to understand the possible states of the system (i.e., cell metabolism) based on prior knowledge such as the occurring reactions, their substrates and products (i.e., metabolites), their stoichiometry and reversibility. Based on this information, a metabolic network is built and formalized as a stoichiometric matrix (S). In this matrix, each row represents a metabolite and each column corresponds to a reaction, while the entries represent the stoichiometric coefficient of a metabolite in a specific reaction. Metabolites consumed in a reaction have a negative stoichiometric coefficient, while metabolites produced by the reaction have a positive stoichiometric coefficient for that specific reaction.

Given a vector describing the flux of each reaction (

$v$) and the stoichiometric matrix, one can determine how the metabolite concentration varies along time (

$\frac{dx}{dt}$) by applying the following equation:

where the changes in metabolite concentrations over time (

$\frac{dx}{dt}$) are given by the reaction fluxes and the system’s stoichiometric matrix. Yet, if we assume that the system is at steady state, i.e., the metabolite concentrations do not change over time, then we can describe the system as follows:

This ensures that the sum of fluxes for the reactions that produce a given metabolite is equal to the sum of fluxes that consume it, such that the metabolite concentration remains constant. In other words, there is no accumulation or depletion of intracellular metabolites over time. By solving this system of equations, it is possible to determine the flux distribution. The overall nutrient uptake and release are represented by exchange reactions, which describe the in- and out-fluxes that cross the boundaries of the system. Here, the system is usually defined as the cell, but can also be the cell and the media, depending on the modelling goals. These exchange reactions can be, for instance, metabolite uptake or secretion rates. Transport reactions describe the transport of a metabolite from one cellular compartment to another or between cell and media when the media is the part of the modelled system [

5].

In addition to equality constraints (

$Sv=\frac{dx}{dt}$), other kinds of constraints can be also added in the form of inequalities [

6]. These are typically used to define lower and upper boundaries for reaction fluxes that can describe an enzyme’s maximum capability [

7] or reversibility (reactions with a non-negative lower boundary are considered irreversible) [

8]. These boundaries can be inferred from exometabolomics measurements and constrain the space of feasible flux solutions of the system. Constraint-based modelling formulations allow us to model large metabolic network systems without the need to specify kinetic parameters or enzyme/metabolite concentrations. However, because there are more variables (reaction fluxes) than equations (

$Sv=0$), the described system is underdetermined, which leads to many possible solutions (flux distributions). A variety of methods has been proposed to find a solution space, i.e., a reduced set of feasible solutions, depending on the problem, some of which are described below.

Constraint-based modelling can be applied to problems of different scales. The applicability of this method to the genome scale is particularly important, since it provides an overview of the whole metabolism. With the development and wide usage of genome sequences, methods to automatically annotate the genome and to derive genome-scale models were created [

9,

10,

11,

12]. The process of annotation is based on so-called gene–protein–reaction rules (GPRs), using information that links enzymes and protein identifiers, reaction names and reacting metabolites and more recently stoichiometric GPRs (S-GPRs) that define the number of transcripts required to generate a catalytically active unit (enzyme) [

13]. After automatic annotation and manual curation, a draft of the plausible metabolic capabilities of the cell is assembled. This significantly reduces the amount of time needed to create an organism-specific model.

Flux balance analysis (FBA) [

6,

14,

15] is one of the most widely used constraint-based methods developed for simulating metabolism at the genome scale. FBA is a computational method that is usually used to predict flux distributions while optimizing a given cellular function, or a combination of them (i.e., maximize growth). FBA yields a specific solution, i.e., a flux distribution, where the objective function is guaranteed to be optimal. The idea that cells follow a specific functional objective, and if so which, is being debated, but thought to have an evolutionary reasoning [

16,

17,

18,

19]. One of the most widely used objective functions for actively dividing cells is the growth rate. In rich media, this approach is in good agreement with experimental data [

20]. Other common optimization functions are the maximization of ATP produced, minimization of consumed glucose, or a complex combination of such functions. However, choosing an objective function describing a given cell phenotype is not always obvious, since it depends on many factors, for instance, on the type of cells modelled, the growth conditions or the availability of nutrients [

18].

To explore alternative and suboptimal solutions as well as study the network flexibility, a variant of FBA, flux variability analysis (FVA) [

16], can be used. FVA can determine a range of fluxes for each reaction such that the minimum growth rate should be at least 90% (or other specified percentage) of the maximum growth rate [

21,

22]. FVA is also applied to the metabolism of byproducts and secondary metabolites, where FBA solutions are unreliable [

23].

While FBA with the objective of maximizing growth results in reasonable solutions for wild-type cells, it does not so for (unevolved) gene knockout mutants. While in principle wild-type cells optimize their growth, unevolved cells with knockouts do not. Since homeostasis governs metabolic reprogramming [

24,

25], we cannot assume that the cell will follow a common objective, such as maximizing its growth. To acknowledge the requirement for metabolic homeostasis, the minimization of metabolic adjustment (MOMA) approach was proposed [

26]. The main idea behind this method is that, to maintain homeostasis, the difference in fluxes before and after the perturbation should be minimal. MOMA predicts the fluxes of a knockout strain by assuming that the cell will have a minimal redistribution of fluxes compared to its ancestor [

26]. Another similar approach is the regulatory on/off minimization (ROOM), which minimizes the number of fluxes that are significantly different from the wild type [

27]. Wild-type fluxes, determined by FBA or other methods, need to be known in order to use these approaches.

For problems which cannot be addressed by traditional stoichiometric methods, such as —changing concentrations over time and/or changing steady states, —dynamic methods were proposed. The idea behind them is to solve multiple constraint-based problems, each at a different time point, assuming steady state at each of them, which allows the modelling of metabolic changes over time. This approach, called dynamic FBA (dFBA), was originally proposed by Varma and Palsson to predict the metabolism of

Escherichia. coli throughout batch cultivations in different conditions [

28].

In this review, we described constrained-based methods that are necessary to understand state-of-the-art approaches of metabolomics data integration. To obtain more information about the variability of approaches of constraint-based methods, we refer the readers to the review by Lewis et al. [

29]. We also refer to other articles covering other powerful and widely used techniques such as the GECKO method [

7], a method that enhances constraint-based methods to account for enzyme maximum capacity by incorporating kinetic constants, metabolism and expression models (ME models) [

30] and protein allocation models [

31,

32,

33,

34].

A different way of approaching the analysis of metabolic networks is the elementary flux modes (EFM) analysis [

35,

36]. EFMs are minimal sets of reactions that lead to a steady state flux solution. Minimal refers here to the fact that they do not contain any loops in the metabolic network and that no reaction can be removed while still allowing for a steady state flux [

37]. All the EFMs found in a EFM analysis (EFMA) for a given network describe the possible paths that can be taken through it, e.g., all the ways a substrate could take to be turned into a product under steady state conditions.

Among the major limitations of constraint-based models in their classical implementation is that they rely only on fluxes and stoichiometry and not metabolite concentration. Thus, under the steady state assumption, fluxes occur irrespective of the metabolite pools—so it is neither possible to incorporate nor predict concentrations in the system. The choice of objective function and the applicability of the steady state assumption have an important impact on the final solution [

18]. Another limitation is that it does not have the ability to account for allosteric regulation, which depends on metabolite concentrations [

38]. These limitations are addressed by imposing additional constraints or using different modelling approaches such as kinetic modelling [

39].

#### 2.2. Metabolic Flux Analysis

Metabolic flux analysis (MFA) [

40] is a widely used tool for metabolic engineering, which has also become more common in other fields, such as biomedical applications [

41] and basic science [

42,

43]. MFA follows the same principles as FBA but does not assume any functional objective, e.g., the growth rate or ATP production. Instead, MFA is an optimization problem that minimizes the difference between simulated and experimental flux data for constraint-based models (

Figure 2). In other words, unlike FBA that uses metabolomics to define the boundaries of the system and constrain the space of feasible flux solution, MFA incorporates the metabolomic data in the objective function in order to set a metabolic flux profile able to minimize the difference between the predicted and calculated secretion and uptake rates. There are two main types of MFA, differing on the use of isotopic labelling.

Stoichiometric MFA (stMFA) [

44,

45] does not require labelling experiments. It is used to balance the fluxes of extracellular and intracellular metabolites according to the biochemical network when given experimental measurements such as glucose uptake rate, CO

_{2} secretion rate, or the growth rate. stMFA is based on minimizing the difference between the experimental and simulated secretion and uptake rates. Given stoichiometric constraints and measured fluxes, least square regression is performed. However, secretion and uptake rates are usually insufficient to resolve all the reaction fluxes in the model. Further, small-scale models are typically used, and therefore the reduction in genome-scale models is needed. Similar to dFBA [

46], the dynamic extension of FBA, dynamic MFA, is also used to address changing concentrations in the environment.

Another technique is

^{13}C-MFA [

47,

48], where isotope labelling patterns are used as additional constraints that help resolve the intracellular fluxes.

^{13}C-MFA is based on minimizing the difference between the experimental and simulated secretion, uptake rates and isotopic-labelling pattern. This more detailed metabolic information allows

^{13}C-MFA to infer metabolic reaction fluxes, which otherwise could not be resolved, such as cycles, bifurcations or the source of NADPH production. Thus, when such a part of the metabolism is of interest, researchers should consider adding isotope labelling to their experiments. However, this implies more complexity in terms of experimental design and data analysis [

40,

49].

There are different ways to approach labeling experiments, depending on the molecules/reactions of interest and the required resolution. Methods range from measuring the labeling patterns of distinct sets of metabolites, e.g., central carbon metabolism intermediates [

50], proteinogenic amino acids [

51] or further bound entities such as glycogen-bound glucose or RNA-bound ribose [

52], to increasing the analysis to the genome scale [

53]. For more specific parts of metabolism that are of interest, subsets can be investigated, as described by Liu et al. for NAD metabolism [

54].

It might not be possible to resolve the more complex metabolic architectures by a single labeling experiment and may require parallel setups with differentially labeled tracers, for example, a mixture of different tracers of a single substrate, such as (1-

^{13}C)glucose and (

^{13}C

_{6})glucose, or different labelled substrates, such as (

^{13}C

_{6})glucose and (

^{13}C

_{6})glutamine [

55,

56].

While measuring the labeling at metabolic and isotopic steady state is the commonly used approach, isotopically non-stationary metabolic flux analysis (INST-MFA) harbors distinct advantages in certain aspects. For instance, by taking time-dependent labeling of metabolites into account, metabolite pool sizes and reversible exchange fluxes can be estimated. This allows for comparison with (or inclusion of) metabolomics data [

8,

57].

The above-described methods use the same formalism as constraint-based models: linear combinations of reaction fluxes and the steady state assumption. Notably, this approach (minimization of the difference between observed and modelled fluxes and isotopic pattern) can be applied in the kinetic modelling framework and such an application was demonstrated [

58,

59,

60,

61,

62]. In this approach, kinetic parameters are refined and the flux distribution is determined. This approach was not widely explored, despite clear benefits: acquiring not only flux distribution that describes the metabolic phenotype, but also a kinetic model that has predictive and explanatory capacities.

For detailed information about MFA and a family of different MFA methods, we refer to this review [

40] and protocol [

52].