¹³C metabolic flux analysis: Classification and characterization from the perspective of mathematical modeling and application in physiological research of neural cell

Abstract

¹³C metabolic flux analysis (¹³C-MFA) has emerged as a forceful tool for quantifying in vivo metabolic pathway activity of different biological systems. This technology plays an important role in understanding intracellular metabolism and revealing patho-physiology mechanism. Recently, it has evolved into a method family with great diversity in experiments, analytics, and mathematics. In this review, we classify and characterize the various branch of ¹³C-MFA from a unified perspective of mathematical modeling. By linking different parts in the model to each step of its workflow, the specific technologies of ¹³C-MFA are put into discussion, including the isotope labeling model (ILM), isotope pattern measuring technique, optimization algorithm and statistical method. Its application in physiological research in neural cell has also been reviewed.

Introduction

Metabolic flux refers to the in vivo conversion rate of metabolites, including the rate of the enzymatic reaction and the transport rate between different compartments (Hui et al., 2020). Flux information deepens our understanding of cell growth and maintenance in response to environmental changes (Becker and Wittmann, 2018; O’sullivan et al., 2019; Van Gastel and Carmeliet, 2021). It is also crucial for revealing the sites and mechanisms of metabolic regulation (Zamboni et al., 2015; Mei et al., 2021).

Accurately estimating flux within complex metabolic networks requires ¹³C metabolic fluxomics (Niedenführ et al., 2015; Varanasi et al., 2019; Lawson et al., 2021). ¹³C metabolic fluxomics has been applied to a number of important studies in recent years, strongly pushing forward the frontiers of metabolic research (Varanasi et al., 2019; Van Gastel and Carmeliet, 2021). This technique can identify changes in metabolic pathway activity and discover novel metabolic pathways (Zhang et al., 2018; Wang et al., 2020; Cobbold et al., 2021). Therefore, it is widely used to reveal metabolic changes in various pathogenic processes, such as colorectal adenocarcinomas (Wang et al., 2018), diabetes (Neinast et al., 2019), retinal degenerative disease (Yam et al., 2019) and immune cells (Varanasi et al., 2019). The technique can characterize the metabolic features of multiple plant organs, such as maize embryos (Cocuron et al., 2019), Arabidopsis leaves (Ma et al., 2014) and developing camelina seeds (Carey et al., 2020). It has also been applied in metabolic engineering to guide the optimization of the synthesis of target products, such as acetaldehyde (Cheah et al., 2020), isopropanol (Okahashi et al., 2017) and vitamin B2 (Schwechheimer et al., 2018).

In this review, metabolic fluxomics methods were first classified from the perspective of data modeling. Then, we introduced the components of fluxomics, including the modeling framework, experimental measuring technique, and optimization techniques. Finally, the application of ¹³C metabolic flux analysis (¹³C-MFA) in neural cell was reviewed.

Classification of ¹³C metabolic fluxomics

Recently, ¹³C-based metabolic fluxomics have evolved into a large family of diverse methods as shown in Figure 1 and Table 1 (Niedenführ et al., 2015). The major categories are as follows:

FIGURE 1

Qualitative fluxomics (isotope tracing)

In qualitative fluxomics, an isotope-labeled tracer is incorporated into the metabolic system, leading to variation in the isotopic pattern of the metabolites (Faubert et al., 2017; Jang et al., 2018). Qualitative pathway activity changes can be deduced by comparing isotopic data (Ma et al., 2017; Liang et al., 2021). For instance, feeding labeled glucose results in M+3 triose phosphates. M+3 fructose bisphosphate reflects the reversibility of aldolase, while M+3 glucose-6-phosphate reflects fructose bisphosphatase activity (Hackett et al., 2016).

¹³C flux ratios

Based on the differences between the isotopic compositions of the metabolic precursor and the product, the relative fraction of metabolic fluxes converging to a node can be directly calculated (Sauer et al., 1999; Nanchen et al., 2007). A dozen such ratios can be identified from the isotope labeling patterns of amino acids or organic acids (Shen et al., 2013). This ratio estimation method can be performed when isotope labeling is dynamic (Hörl et al., 2013). Currently, the ratios can be estimated from ¹³C measurements by dedicated machine learning predictors (Kogadeeva and Zamboni, 2016). The metabolic flux ratio (FR) method has a unique advantage when the overall network topology is unclear and metabolite outflow rate measurements are difficult to detect and determine (Rantanen et al., 2008).

¹³C kinetic flux profiling

The kinetic flux profiling (KFP) method assumes that the labeled fraction of the metabolite pool changes exponentially during the labeling process. As long as the pool size is accurately measured, this method can estimate the absolute flux through sequential linear reactions according to the kinetic elution equation (Yuan et al., 2008). KFP is extended for quantifying fluxes within subnetworks encompassing convergent nodes and bounded by a unidirectional linear reaction (Szecowka et al., 2013; Heise et al., 2014). It was used to detect kinetic parameters such as the incorporation rate of [6-¹³C] glucose into phospholipids and the turnover rate of acylglycerol to determine the effect of deleting the cg6718 gene in Drosophila melanogaster (Schlame et al., 2020).

¹³C metabolic flux analysis

As a major component of metabolic fluxomics, ¹³C-MFA can accurately determine the absolute value of the flux of the global metabolic network (Antoniewicz, 2015), making it a unique tool for metabolic research. In the carbon labeling experiment, the isotopic distribution of these metabolites depends on the isotopic distribution of the substrate and metabolic flux values (Cheah and Young, 2018). Flux values can be estimated after the isotopic labeling values of measured metabolites are optimally fitted. The flux estimation process can be formalized as the following optimization problem:

v represents the vector of the metabolic flux, s represents the stoichiometric matrix of the metabolic network, and M⋅v ≥ b provides additional constraints from physiological parameters or excretion metabolite measurement. y_iⁱⁿ represents vectors of the isotope labeled substrate. X_n is a matrix, and its rows are the vectors of the isotope labeling model (ILM) of the corresponding elementary metabolite unit (EMU) fragment with n carbon atoms. Y_n is a matrix similar to X_n, in which its rows are the vectors of the ILM of the corresponding input substrate and/or the calculated EMU fragment with 1 ∼ (n-1) carbon atoms. In the objective function, x is the vector of the isotope-labeled molecules in X₁,…, X_n, and x^M is the experimental counterpart to x. A_n and B_n represent the system matrix determined by the corresponding metabolic reaction topology and atomic transfer relationship. Σ_e represents the covariance matrix of the measured values.

We briefly described the workflow and principles of ¹³C-MFA as depicted in Figure 2 and then discussed the technical details used in the process. The ¹³C-MFA method first involves a carbon labeling experiment (Hollinshead et al., 2019). Specific ¹³C-labeled substances are chosen as carbon sources for cell culture experiments, depending on the cell type. For example, early ¹³C-MFA approaches often used various mixtures of [1-¹³C] glucose, [U-¹³C] glucose and unlabeled glucose as substrates (Leighty and Antoniewicz, 2013). In carbon labeling experiments, the isotope label material is gradually distributed to various metabolites in the metabolic pathway. Since the amount and location of ¹³C in metabolites are closely related to metabolic flux, different metabolic flux distributions produce different isotope labeling levels. Then, the labeling status of the substrate is determined based on a specific mathematical relationship between the metabolic flux distribution and the isotopic labeling status of the metabolites in vivo (Wang et al., 2020), which can be described by formula (1). Based on this relationship, we can obtain the distribution of the metabolic flux by accurately measuring the isotope labeling levels of the metabolites. The accurate method used for determining the status of the ¹³C isotope label includes mass spectrometry (GC−MS and LC−MS) and nuclear magnetic resonance (NMR) spectroscopy (Rahim et al., 2022). Then, we can estimate the level of metabolic flux. Specifically, we can first determine the random distribution of the metabolic flux, and then, based on the value from formula (1), we can calculate the corresponding theoretical isotope labeling status of each metabolite. Then, the calculated labeling status is compared with the measured labeling status. According to the difference between the two, the given metabolic flux distribution can be repeatedly adjusted until the difference between the two is less than a specific threshold. The resulting metabolic flux distribution is the true distribution of the estimates (Weitzel et al., 2013).

FIGURE 2

Classification of ¹³C metabolic flux analysis

According to formula (1), metabolic flux analysis can be divided into three categories, stationary state¹³C metabolic flux analysis (SS-MFA) (Weitzel et al., 2013), isotopically instationary ¹³C metabolic flux analysis (INST-MFA) (Wahl et al., 2008; Young et al., 2008) and metabolically instationary ¹³C metabolic flux analysis (MNST-MFA) in Figure 3 (Abate et al., 2012; Van Heerden et al., 2014).

FIGURE 3

SS-MFA refers to the case when all dX_i/dt of the constraint conditions in formula (1) are equal to 0 and v is constant. The method is suitable for systems in which metabolic flux and isotope labeling state do not change with time (Wiechert and Nöh, 2005). Such system is usually at end of a long-term labeling process and is also called stable labeling. The flux distribution can be deduced by measuring the isotopic labeling state of metabolites in the steady state.

INST-MFA refers to the case when not all dX_i/dt in the constraint conditions in formula (1) are equal to 0 and v is constant. It can be applied to systems in which the metabolic flux and metabolite concentration do not change with time while the isotopic labeling fraction changes with time (Gopalakrishnan et al., 2018). In such a system, the isotope labeling process is in an early dynamic stage and does not reach a stationary state. Therefore, the labeling state is a function of not only the substrate labeling fraction and metabolic flux but also the labeling time. It is necessary to carry out multiple dynamic measurements and to solve the isotopic differential equations to deduce the metabolic flux.

MNST-MFA refers to the case when not all dX_i/dt in the constraint conditions in formula (1) are equal to 0 and v is a time-varying function. This method fits well with systems in which the metabolite concentration, metabolic flux and isotope labeling fraction all change with time (Abate et al., 2012). Its output is not just a flux value but a flux profile over a period of time (Quek et al., 2020).

Technology of ¹³C metabolic flux analysis

The necessary technology of ¹³C-MFA includes isotope labeling modeling, isotope labeling state measurement, flux optimization and statistical analysis. The details are as follows:

Isotope labeling model

The quantitative interpretation of isotope-labeled data, that is, the estimation of metabolic flux, requires an algorithm that can describe the accurate relationship between metabolic flux and isotope-labeled fractions (Zamboni, 2011). We call this the ILM, that is, the content of X in formula (1). The same ¹³C labeling experiment can be simulated with different ILMs (Weitzel et al., 2007). Different ILMs characterize the same labeling system from different perspectives.

If there is any ¹³C incorporated into the molecule, the same compound will be distinguished due to the number and position of ¹³C. ¹³C isotopomer is any isomer of an organic compound differing only in the number and position of ¹³C (Table 2). Isotopomers corresponds to basic isotopically labeled molecules (Schmidt et al., 1997). The isotopomer I₁₀₀ in Figure 3 represent a labeled molecule where the first carbon is ¹³C and the second and third carbons are ¹²C. Cumomers is another form of Isotopomer, and its coded appearance is exactly the same as that of Isotopomer. But the 0 in Isotopomer code means ¹²C, and the 0 in Cummer means “¹²C or ¹³C,” which is a collection where both ¹²C and ¹³C are allowed. Cumomers contain the same information as isotopomers, and the number of model particles is the same as that of isotopomers (Wiechert and De Graaf, 1997). The complete set of cumomers and the complete set of isotopomers are equivalent. The information of mass isotopomers and isotopic positional enrichment is coarser than that of isotopomers, and correspondingly, the number of model particles is lower than that of isotopomers (Hellerstein and Neese, 1992; Zupke and Stephanopoulos, 1994). Bondmer is an ILM in the case of uniformly labeled substrates. Each individual bondmer is a set of specific isotopomer whose fraction can calculated by a binomial distribution as shown in Figure 3 (Van Winden et al., 2002). The EMU framework is a framework suitable for simplifying and accelerating the simulation of various ILMs (Antoniewicz et al., 2007). It can accelerate various ILMs and has been widely adopted. Figure 4 shows the concept and corresponding relationship of various ILMs by taking a compound containing 3 carbon atoms as an example.

FIGURE 4

The cascade equation solution can reduce the computational complexity and has been used until now (Wiechert et al., 1997). Isotopic differential equations were developed as a basic framework for isotopically instationary flux analysis (Wahl et al., 2008; Young et al., 2008), which significantly expanded the scope of flux analysis. The modeling of mass isotopomers can be rearranged like that of cumomers and gradually parallelized (Zhang et al., 2020). In addition, a framework for metabolically instationary flux analysis was proposed and applied to the temporal flux reconfiguration of the glucose pathway of adipocytes in response to insulin (Abate et al., 2012; Quek et al., 2020).

Isotope molecule measurement

The measurement of X in formula (1) determines the objective function. The appearance of ¹³C at different positions classifies a molecule as different isotopic molecules. Differences in the physical and chemical properties of ¹³C and ¹²C are the basis for discriminating these isotopic molecules. They have at least two differences. One is that ¹³C has a half integer nuclear spin, so it can be detected by NMR (Reardon et al., 2016). The other is that ¹³C has a greater mass number than ¹²C, so their difference can be detected by mass spectrometry.

1D NMR of ¹H and ¹³C can provide information on positional labeling enrichment as stated in ILM (Vinaixa et al., 2017; Deja et al., 2020; see Figure 5). Correlation Spectroscopy (COSY) is a 2D NMR technique that displays correlations between J-coupled nuclear by stepping up the delay between two 90°-proton pulses. Two-dimensional heteronuclear correlation spectroscopy (2D-COSY) (Szyperski, 1995), three-dimensional heteronuclear correlation spectroscopy (3D-COSY) (Boisseau et al., 2013) and proton-detected 2D heteronuclear single-quantum coherence (HSQC) (Lane et al., 2019) can detect the fraction of a specific subset of isotopomers. The set is a fragment of 3 continuous carbon atoms. Its middle carbon is ¹³C, while the carbons on both sides are ¹²C or ¹³C. 2D heteronuclear multiple quantum coherence-total correlation spectroscopy (HMQC-TOCSY) (Carvalho et al., 1998) and 3D total correlation-heteronuclear single-quantum coherence (TOCSY-HSQC) can detect another type subset of isotopomers (Reardon et al., 2016). This set is a fragment of continuous carbon atoms in the same spin system. One of its ends is ¹³C, and the other may be ¹²C or ¹³C. A non-uniform sampling technique can be used to improve the sensitivity and resolution of the fine structure in NMR spectrometry (Lee et al., 2017).

FIGURE 5

Gas chromatography-electron ionization (GC-EI)-quadrupole can quantify the mass isotopomers of amino acids (Dauner and Sauer, 2000), organic acids and phosphate sugars (Jung and Oh, 2015; Evers et al., 2021). With the development of tandem mass spectrometry, gas chromatography-electron ionization-triple quadrupole (GC-EI-tripleQ) (Okahashi et al., 2016), gas chromatography-chemical ionization- quadrupole time of flight (GC-CI-QToF) (Mairinger et al., 2015), liquid chromatography-electron spray ionization-triple quadrupole (LC-ESI-tripleQ) (Jeffrey et al., 2002; Rühl et al., 2012), and liquid chromatography-electron spray ionization-quadrupole time of flight (LC-ESI-QToF) (Kappelmann et al., 2017) have been used to measure position enrichment and mass isotopomers. Hundreds of metabolites can be detected, and more accurate results can be obtained. To simplify the sample processing, capillary electrophoresis QToF has also been added to the lineup (Toya et al., 2007). Gas chromatography-combination isotope ratio mass spectrometry (GC-C-IRMS) is suitable for the accurate determination of very low isotopic enrichment abundance (Yuan et al., 2010; Croyal et al., 2016). The data of mass spectrometry should be corrected for naturally occurring heavy isotopes before used by flux analysis (Jeong et al., 2021).

Optimization technique

Flux estimation solves the entire optimization problem of formula (1). This process depends on local or global optimization methods. As a heuristic method, an evolutionary algorithm was employed early in the optimization of S-MFA (Wiechert, 2001). Some of its derivatives, such as the convex evolutionary algorithm and adaptive evolutionary algorithm, were introduced later (Chen et al., 2007; Yang et al., 2007). Simulated annealing has also been successfully applied to S-MFA (Fu et al., 2015).

A hybrid algorithm combining the trust-region method and sequential quadratic programming (SQP) facilitates numerically stable and accurate flux estimation of S-MFA (Yang et al., 2008). The Levenberg Marquardt algorithm, one type of trust-region method, is widely used and can guarantee fast convergence for INST-MFA (Young, 2014).

The sum-of-squares residual error (SSE) between the measured and simulated data is the main form of the objective function in formula (1)(Wiechert, 2001). Additionally, the Akaike information criterion (AIC) can be used as an objective function in seeking the simplest candidate models sufficient to generate the observed data (Alger et al., 2021).

Flux uncertainty analysis

Since metabolic flux analysis is a parameter estimation problem, there are uncertainties in the obtained parameters. Flux uncertainty analysis should become part of routine analyses. Local linear error propagation based on the chain rule is an earlier method for determining the confidence interval of flux values (Wiechert et al., 1997). In contrast, a profile likelihood method would directly execute non-linear mapping over error propagation, which is preferred due to the corresponding high computational efficiency and accuracy (Antoniewicz et al., 2006). In addition to the error propagation method, another idea is to obtain the probability distribution of metabolic flux. The Monte Carlo method can be used to generate such a distribution based upon a Gaussian distribution or chi-square distribution assumption for the measurement error (Wittmann and Heinzle, 2002). In some experiments, the shape of this distribution can be posteriorly calculated using Markov chain Monte Carlo (MCMC) from isotopic data (Kadirkamanathan et al., 2006; Theorell et al., 2017).

Derivative method

Different constraints can be introduced in formula (1) to generate a new derivative of MFA, which can expand the scope of metabolic flux analysis or make it more accurate in some cases. Compartment-specific metabolic flux in mitochondria and cytosol can be quantified by a spatial-flux analysis with rapid subcellular fractionation and quenching of metabolism (Lee et al., 2019). A flux analysis method for local subnetworks was recently proposed and needs only the information of the subnetwork of interest, requiring no additional knowledge of the surrounding networks (Millard et al., 2020). ¹³C MFA has also been extended to solve metabolic networks at the genome scale by incorporating a large set of secondary metabolic reactions (Blank et al., 2005; Martín et al., 2015; Gopalakrishnan et al., 2018). In addition, flux analysis goes beyond the scope of cells (Hui et al., 2020; Liu et al., 2020). The flux from source cells to recipient cells through vesicles can be accessed by exosome-mediated metabolic flux analysis (Exo-MFA) (Achreja et al., 2017). Gebreselassie and Antoniewicz (2015) proposed a framework determining the metabolic flux of multiple bacteria coexisting in a mixed system. ¹³CO₂ and ¹⁵NH₄ labeling strategies can be used to unravel the short-term flux of plant-assimilated C and fungal-obtained N through an in situ ectomycorrhiza system (Gorka et al., 2019).

Parsimonious MFA utilized a dual optimization minimizing both the SSE and the sum of the flux values. This minimization can be weighted by gene expression to integrate gene expression data with ¹³C data (Foguet et al., 2019). In COMPLETE-MFA, multiple substrates are used in parallel to generate complementary labeling information to improve the accuracy of the flux estimation. Generally, the accuracy can be significantly improved by utilizing 2–3 parallel data (Leighty and Antoniewicz, 2013). A truncated multi-model MCMC method was adopted to infer the in vivo probability of bidirectional reactions and to determine whether they are unidirectional or bidirectional (Theorell and Nöh, 2020). This can expand the range of ¹³C MFA from parameter inference to structure inference.

The reaction thermodynamic information from metabonomic and physiological data can restrict the solution space of flux and avoid a priori hypotheses about the flux direction (Saldida et al., 2020). With this tool, higher-precision inference of the network structure and flux values can be achieved, and some new flux patterns can be identified. Conversely, the Gibbs free energy of the reaction can be deduced from the metabolic flux values (Park et al., 2016).

Application in neural cell research

Neurons and glial cells are the major cells of the nervous system. Neurons are a kind of specially differentiated cells with the ability to sense stimulation and conduct excitation, and is important to the functioning of the nervous system. Glial cells are not able to conduct impulses but nourish, insulate and protect neurons. Neurons provide electrical signaling to glial cells to fuel oxidative metabolism in the brain, while glial cells provide metabolic substrates to neurons. Metabolic activity in the brain requires a lot of energy. Cognitive phenotypes are related to neural metabolism, including neuronal mitochondrial mutation and neuron-glia metabolic crosstalk (Watts et al., 2018). Metabolic flux through both cell types is a factor modulating higher order phenotypes. Hence, it is rather interesting to discover how metabolic flux and neural cell function are related.

As outlined in Table 3, different types of ¹³C metabolic fluxomics have been successfully applied to study the metabolic reprogramming and its regulatory mechanism of neural cell (Lanz et al., 2013). INST-MFA was utilized to assess the metabolism shift upon differentiation of Neural Stem Cells (NSCs) into astrocytes, discovering an extensive decrease of central carbon metabolism and conversion of flux through TCA cycle to lac pathway during astrocytic differentiation (Sá et al., 2017). Isotope-tracing has been used to quantify the fraction of labeled monosaccharides in the glycans and glycosphingolipids of both pluripotent and neural NTERA-2 cells. It revealed that exogenous monosaccharide utilization would vary noticeably according to the cell differentiation state and different glycan structures (Wong et al., 2020). FR analysis and steady-state flux analysis was combined to find that more of glucose flux was channeled by glycolysis than that by pentose cycle of adherent cerebellar granule neurons. Meanwhile, it determined that 16% of glucose used by mitochondria comparing to 46% by lactate dehydrogenase (Gebril et al., 2016). By revising bi-directional reaction of the non-oxidative PPP pathway and TCA cycle, this method became more broadly applicable to different cell types (Jekabsons et al., 2017). Patel et al. (2021) investigated TCA cycle and neurotransmitter cycle fluxes ratio by FR method from a steady-state [2-¹³C] acetate experiment and the ¹³C turnover rates of neurotransmitter by a KFP-like fitting on labeling kinetics of amino acids in ¹³C glucose infusion experiment. The mitochondrial TCA flux of glutamatergic neurons and glutamate-glutamine cycle flux was declined in the cerebral cortex of aged mice. Isotope-tracing allows investigation of astrocyte-specific metabolic networks affected by Apolipoprotein (APOE) and observes an increase in flux through the pentose phosphate pathway, with subsequent increases in gluconeogenesis and lipid biosynthesis pathway in Apolipoprotein E4 astrocytes (Williams et al., 2020).

Conclusion

Stable isotope metabolic flux analysis has been successfully used in applications from homogeneous cell systems to heterogeneous cell systems, even at the level of animal and plant organs. It has now become the gold standard for measuring metabolic flux values. Several technologies that it includes are also developing rapidly, so the metabolic fluxomics method is evolving into a family with different members that have higher accuracy, wider coverage, more application scenarios, and shorter time consumption.

At present, the development of stable isotope metabolic flux analysis should focus on improving the temporal and spatial resolution. This requires the continuous introduction of updated detection technology.

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