On a quest for metabolic fluxes: sampling and inference tools using thermodynamics,
metabolome and labelling data
Taborda Saldida Alves, Joana
DOI:
10.33612/diss.157440136
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Publication date: 2021
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Taborda Saldida Alves, J. (2021). On a quest for metabolic fluxes: sampling and inference tools using thermodynamics, metabolome and labelling data. University of Groningen.
https://doi.org/10.33612/diss.157440136
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CHAPTER 1
INFERENCE OF STEADY-STATE METABOLIC FLUXES
Metabolic fluxes as key quantifiers of metabolism
Living cells are open systems in non-equilibrium that create and maintain order through streams of chemical reactions that constitute their metabolism. A continuous flow of energy and material to the inside and outside of the cell provide the dynamics in me-tabolism. Organic molecules are taken apart, energy is generated and building blocks are created to allow for cell replication and maintenance. All the chemical reactions occurring in the cell compose a complex network where metabolites interact among each other governed by physical laws to allow the cell to grow and adapt to changing environmental conditions (Alberts et al., 2010).
Metabolism can be divided in sub-sets of chemical reactions that form diverse metabolic pathways with specific purposes and complex interactions. Metabolic pathways are mainly studied with the goal of describing the diverse chemical changes and the internal controls that govern the functioning of cells. While the structure of metabolic pathways is well known for a large number of organisms (Duarte et al., 2007; Lu et al., 2019; Oh et al., 2007; Orth et al., 2011), the interaction between the different players, genes, transcription factors, enzymes and metabolites, is largely unknown and is therefore under intensive study for many years.
A key goal in the study of metabolism is to unravel the interplay between genotype and phenotype behaviour, considering intermediate stages of transcriptome, pro-teome and metabolome. The final metabolic output of this complex layered process is the metabolic rate, or flux, that is tightly linked to cell physiology. In other words, metabolic fluxes characterise the metabolic state of a cell and adjust according to the internal and environmental conditions of the cell. This fact makes metabolic fluxes the essential quantity to obtain when studying metabolism (Blank, 2016).
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The study of metabolism through quantification of metabolic fluxes has applica-tions in the fields of biochemistry, biotechnology and medicine. Particularly, the ability to predict intracellular fluxes and, consequently, infer cellular behaviour is helpful for the study of diseases and development of medical treatments (Antoniewicz, 2018; Cheah et al., 2017). The knowledge of how fluxes adapt to new environmental conditions or upon genetic manipulation also guides metabolic engineering in strain improvement strategies (Ghosh et al., 2016; Kim et al., 2008) and constitutes a tool to study diversity between species (Nidelet et al., 2016). Essentially, metabolic fluxes are estimated with the purpose of developing solutions for health, environmental and industrial problems, as well as answering fundamental research questions.
Although techniques to quantify abundance of molecules, such as used with transcriptomics, proteomics and metabolomics, are widely available and a target of constant improvement in the last decade (Biswapriya et al., 2019), similar methods are not applicable to quantify metabolic fluxes as these are rates of metabolite trans-formation over time. While extracellular fluxes can be inferred from quantification of extracellular metabolites (substrates or end products) over time (e.g. in batch experi-ments), changes in the levels of metabolic intermediates do not quantify intracellular fluxes (Fernie et al., 2005). Therefore, metabolic fluxes cannot be directly measured with experimental techniques or inferred from other omics data. To overcome this, Systems Biology crosses information of quantitative measurements with advanced computational and statistical approaches to estimate metabolic fluxes (Sweetlove et al., 2014).
Constraint-based modelling to infer metabolic fluxes
The set of reactions that compose cellular metabolism can be represented by a metabolic network with metabolites as nodes and reactions as edges. The analysis of metabolic networks is essential to study microorganisms and has been increasingly used with the boost in collection of data in genome sequencing and molecular interactions (Brent, 2008). Metabolic network reconstruction is a procedure in which the annotation of genes, proteins and reactions is used to identify and categorise relationships between genome, proteome and fluxome (complete set of metabolic fluxes of a cell) in order to build a metabolic network. Genome-scale network reconstructions enumerate all of the known metabolic reactions in an organism and the genes that encode each enzyme. These reconstructions can be used to study metabolic functions and effect of gene deletions (O’Brien et al., 2015).
Diverse methods to estimate metabolic fluxes have been developed in the last decades, and were applied to a wide range of biological systems. Metabolic net-works with different levels of detail, from a dozen reactions to full genome-scale, constitute the framework for constraint-based modelling that allows fast calcula-tion of steady-state metabolic fluxes. Constraint-based models are used to estimate metabolic fluxes within metabolic flux analysis (MFA) that applies mass balance principles to estimate metabolic fluxes based on the mathematical representation of metabolic networks through stoichiometry. The metabolic network is represented by a stoichiometric matrix in which rows represent metabolites and columns rep-resent reactions and that matrix contains information on the amount of metabolites used in each reaction (stoichiometry). Additional constraints can be added to the MFA problem based on physical laws and heuristic assumptions. The most common assumption is stationarity where it is assumed that metabolite concentrations are stable through time when a steady-state is reached. Assuming steady-state condi-tions, considering the flux vector as the variable (unknown) v and the stoichiometric matrix S as the frame of the linear system of equations to solve, the infinite set of flux possibilities defined by Sv = 0 is called the flux solution space. Flux bounds are commonly assigned to make the flux solution space a finite linear set, also called a polytope. Diverse types of constraints can be added to the basic MFA problem and different directions can be taken in terms of computational approaches, including optimisation and sampling.
One of the most common metabolic flux analysis approaches is to fit the stoichio-metric model to available experimental data, specifically extracellular fluxes (An-toniewicz, 2015b) that can be inferred through evaluation of metabolite dynamics in the cell environment. In the fitting, flux values are adjusted to produce the closest fit as possible to the measured fluxes while taking experimental error into account, which results in the optimal flux point in the polytope. This method only requires simple linear algebra calculations and relatively robust experimental measurements (Murphy & Young, 2013), making it easy to apply and solve. However, because the number of reactions is always significantly larger than the number of metabolites, the system to solve is highly underdetermined even with measurements on extracellular fluxes. The large number of degrees of freedom of the problem results in uncertainty in the flux estimations (Antoniewicz, 2015b). Another method to estimate metabolic fluxes, Flux Balance Analysis (FBA) (Orth et al., 2010), is based on the optimisation of a trait which also results in one single optimal point, usually at the edge of the
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tope. This method does not require experimental data and it assumes that cells operate with a particular objective and allocate their resources to achieve it. Commonly used objectives are optimal growth or energy production and several studies investigate the use of multiple biologically relevant objectives to account for the complexity of cell operation (Budinich et al., 2017; Costanza et al., 2012; Patané et al., 2019).
Since the basic MFA problem is underdetermined, often even with the addition of extra constraints beyond mass balance, it is common to calculate the flux limits through Flux Variability Analysis (FVA) (Mahadevan & Schilling, 2003). FVA determines the maximum and minimum values fluxes can achieve under a system of constraints through optimisation. FVA can be used to estimate alternative optimal solutions for a particular FBA optimality condition. Specifically, multiple intracellular flux solutions can achieve the same value of optimal trait (e.g. maximal biomass production) and the flux bounds of such solutions can be determined through FVA. Through applica-tion of principal component analysis to the alternative soluapplica-tions computed by FVA, Thiele et al. identified metabolic flux couplings that were used to infer gene expression through an expression matrix, and the results were ultimately used to validate gene essentiality data (Thiele et al., 2010). In a metabolic redundancy study, reactions were categorised in essential, inactive, variable, substitutable and unbounded according to their flux ranges, obtained from FVA, across different nutritional conditions (Hay & Schwender, 2011). These examples of FVA application demonstrate its importance in the study of metabolism.
Thermodynamics helps tackling uncertainty in metabolic
flux inference
The typical metabolic flux analysis problem has a large number of degrees of freedom. The quality of the flux predictions obtained from the polytope improves with the addition of constraints beyond stoichiometry and flux bounds. MFA was previously combined with thermodynamic constraints (Henry et al., 2007), enzymatic constraints (Sánchez et al., 2017), transcriptomics (Covert & Palsson, 2002) and metabolomics data (Töpfer et al., 2015). Focussing on the first case, constraints based on thermody-namic principles have been increasingly applied to MFA problems in the last decades as a way to decrease uncertainty and improve flux predictions by imposing constraints on flux directions. Thermodynamic constraints can be applied to MFA problems at network level through the loop law or on a single reaction basis through the second law of thermodynamics.
The Gibbs free energy of a reaction (ΔrG) is the sum of the Gibbs formation energies (ΔfG) of each metabolite m involved in that reaction, ∆r G = ∑m S ∆f G, where S is the stoichiometric matrix. The loop law states that the thermodynamic driving forces in a metabolic loop, i.e. the ΔrG of the reactions involved in such loop, sum up to zero (Palsson et al., 2002). According to the second law of thermodynamics, fluxes must flow from reactants with higher to lower chemical potential (i.e. Gibbs formation energy of metabolites, ΔfG). In mathematical notation, the second law of thermody-namics states that the ΔrG of a reaction must have opposite sign to its flux, since the flux sign of a reaction is defined relatively to a standard direction. Applying the loop law to metabolic networks implies that the sum of ΔrG in a closed loop must be zero, which imposes constraints on flux (and ΔrG) sign configurations (Figure 1). Due to contradictory ΔfG inequalities (in red in Figure 1), some flux sign configurations are thermodynamically infeasible. When applied to the polytope of flux solutions generated in metabolic flux analysis, the loop law creates infeasible sections inside the feasible space and thus breaks its convexity. Although extensive enumeration of infeasible loops is computationally intensive (Beard et al., 2004), a few methods were developed to identify and remove infeasible loops in metabolic flux analysis problems. In these
Figure 1: Example of thermodynamically infeasible loop with two reactions. A loop
of two reactions has four different flux sign configurations: A, B, C, D. 1 and 2 repre-sent two reactions, a and b two metabolites and v the flux through a reaction. ΔrG and
ΔfG represent reaction and formation Gibbs energies, respectively. The sign of a flux
(whether it is positive or negative) merely indicates the direction of the reaction. For a thermodynamically favourable reaction, ΔrG should have the opposite sign of v.
Ac-cording to the relationship between Gibbs energies of formation and reaction: ΔrG1 =
ΔfGb - ΔfGa and ΔrG2 = ΔfGa - ΔfGb. The infeasibility of the ΔfG inequalities makes flux
sign configurations A and C thermodynamically impossible. The red areas in the flux space of v1 and v2 represent infeasible areas of the flux solution space.
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methods, flux solutions where infeasible loops are identified are removed from the solution space (Price et al., 2006) or converted to thermodynamically feasible through optimisation while fulfilling mass balance constraints (De Martino et al., 2013).
Another way to employ thermodynamic principles to constrain metabolic fluxes is through the explicit implementation of the second law of thermodynamics (Henry et al., 2007) that directly relates the flux of each reaction with its Gibbs free energy of reaction (ΔrG), through ∆rG × v ≤ 0 where v represents the flux of a reaction. This law constrains the direction of a reaction based on thermodynamic information in the form of ΔrG. ΔrG is related to metabolite concentrations through ∆r G = ∆r G0+RT ∑
m lncm where ∆r G0 is the standard Gibbs energy of reaction, R is the gas constant, T the temperature of the system and lnc the logarithm of the concentration of metabolite m that participates in the reaction. Through these equations, metabolite concentration measurements can be used to infer flux directions. Even though the second law of thermodynamics applies to single reactions individually, when used in a metabolic network, where the same metabolites participate in diverse reactions, it has a network-wide effect with reaction directions depending on other reactions. It is important to note that the enforcement of the second law of thermodynamics, together with mass balances, automatically enforce the loop law constraint by preventing reactions of having infeasible flux sign configurations. Thermodynamic constraints have been widely applied in metabolic flux analysis to determine directions of reactions (Küm-mel et al., 2006; Yang et al., 2005), find potential regulatory sites (Garg et al., 2010; Kümmel et al., 2006), and estimate realistic metabolite concentration ranges (Henry et al., 2007; Martínez et al., 2014). Overall, thermodynamic constraints are a power-ful tool to decrease uncertainty in metabolic fluxes.
Sampling as a tool to characterise the flux solution space
and flux uncertainty
Metabolic flux analysis methods typically use optimisation to produce one single solution (Figure 2). The uncertainty of such solution varies according to the type and amount of constraints and experimental data employed. Besides the fact that, often, more than one flux solution can achieve the same objective (Mahadevan & Schilling, 2003; Schellenberger & Palsson, 2009), in the case of FBA (Figure 2) there is also no consensus on the most relevant objective(s) to optimise (Martínez et al., 2014). Therefore, it is useful to consider the whole solution space defined by constraints and experimental data, which represents all the possible solutions under the stated problem.
Characterisation of the flux solution space through sampling (Figure 2) is a useful method to quantify the inherent uncertainty of metabolic flux analysis problems. Sam-pling the flux solution space generates a set of flux solutions that fulfil all constraints and these solutions provide information on flux ranges and probabilities (Wiback et al., 2004). Sampling approaches applied to the study of metabolic networks predominantly use a Markov Chain Monte Carlo method, called Hit-and-Run (Chen & Schmeiser, 1996), to sample the flux solution space constrained by linear equations (a polytope). As this method was extensively applied, diverse techniques were recently developed to improve its efficiency (De Martino et al., 2015; Haraldsdóttir. et al., 2017; Megchelenbrink et al., 2014). Other sampling approaches use optimisation procedures in their workflow to specifically sample the solution space when some fluxes are set to optimal values from an FBA solution (Binns et al., 2015; Chaudhary et al., 2016). As an alternative to the explicit sampling techniques, Bayesian Analysis models the metabolic network in probabilistic terms producing a multivariate posterior distribution to represent fluxes (Heinonen et al., 2018). A machine learning method (Braunstein et al., 2017) uses Expectation Propagation (Minka, 2001) to compute an analytical approximation of the flux solution space in a fast and flexible method. Analysis of metabolic networks through sampling was effectively applied to, among others, determination of the shape of solution spaces (Wiback et al., 2004), design of experiments (Price et al., 2004), identification of key regulators (Bordel et al., 2010), and estimation of flux coupling under different scenarios (Gomes de Oliveira Dal’Molin et al., 2015).
A significant contribution to the field was the inclusion of thermodynamic constraints while sampling. Thermodynamic constraints introduce non-convexities in the
solu-Figure 2: Different ways to approach the flux solution space. In metabolic flux
analy-sis (MFA), the model is fitted to data and the optimal solution corresponds to the best fit which is one point in the polytope that represents the flux solution space. In flux balance analysis (FBA), an objective is defined and maximised or minimised (e.g. maximisation of biomass production) and the optimal solutions is an extreme point of the polytope. In sampling, several solutions cover the polytope defining the shape of the solution space.
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tion space which make sampling an extra challenging computational task. The loop law was taken into account in sampling studies by removal of thermodynamically infeasible loops (Price et al., 2006) and replacement of infeasible solutions by the nearest feasible solution (Schellenberger et al., 2011). Furthermore, a box (convex) approximation of the non-convex flux solution space was employed to sample the flux solution space with subsequent rejection of infeasible flux solutions (Saa & Nielsen, 2016). However, while explicitly applying the loop law and the second law of thermodynamics helps tackling the uncertainty in flux predictions, no sampling technique is yet available to efficiently deal with the non-convexity and dimensional-ity of such flux solution space.
Labelling experiments as an improvement of metabolic
flux analysis
With improvement of experimental techniques, that resulted in high-throughput data, and of computational routines, MFA has been applied to more complex biological systems (Nöh et al., 2008). Specifically, data from labelling experiments was incorporated with mathematical models in a method called 13C metabolic flux analysis (13C-MFA) (Wiechert, 2001). In labelling experiments, a partly labelled 13C-substrate is fed to the cells and the labelled carbons are propagated through the network of reactions creating a pattern in the metabolic intermediates and products (Figure 3a). The labelled outputs, often amino acids, are measured using Mass Spectrometry and Gas/Liquid Chromatography or Nuclear Magnetic Resonance (Wiechert, 2001). During the measuring procedure, amino acids are broken down into fragments of different mass, where each fragment can contain a different labelled state. An isotope-isomer, called an isotopomer, has the same number of each isotope of each atom but with a different labelled position. The ultimate measurement output of labelling experiments is the fraction of abundance of isotopomers for each amino acid fragment (Figure 3b). To relate the isotopomer abundances, often named labelling, to the metabolic fluxes, a model with mass and isotopomer balances is used (Figure 3c). Mass balances make use of the stoichiometric matrix while isotopomer balances are enforced through an atom transition network that accounts for the path of individual atoms through the metabolic network. Because the labelling can travel in either direction of a reaction – forward or backward – isotopomer models have to include both fluxes as variables. Often, forward and backward fluxes are modelled as net (vnet = vforward - vbackward) and ‘labelling’ exchange fluxes (vexchange =
data (Figure 3d) with the objective of minimising the variance-weighted sum of squared residuals between measured and predicted isotopomer abundances (Wiechert, 2001).
The incorporation of labelling patterns into metabolic flux analysis introduced the possibility of estimating fluxes in parallel pathways, bidirectional reaction steps,
Figure 3: 13C metabolic flux analysis workflow. (a) A labelled carbon source (e.g. glucose) is fed to a culture that will metabolise it to different products, including amino acids. Empty and filled black circles represent unlabelled and labelled carbon atoms, respectively. (b) The isotopomer abundance fraction is measured through Mass Spec-trometry and Gas/Liquid Chromatography or Nuclear Magnetic Resonance. M0 rep-resents unlabelled metabolite, M1 metabolite labelled in one position, etc. Isotopom-ers are isomIsotopom-ers with isotopic atoms. Metabolite pool example refIsotopom-ers to a three carbon atom molecule. The pool of that molecule with one labelled position (M1) contains all three types of labelling, where the first, second or third atom is labelled and the rest is unlabelled. The same logic applies to the other pools with 0 (M0), 2 (M2) and 3 (M3) labelled atoms. The isotopomer abundance of the metabolite pools is referred to in the text as labelling or labelling data. (c) A network state is simulated with a mathematical model including mass and isotopomer balances. To build isotopomer balances, an atom transition network maps carbon atom distributions over the metabolic reactions. The isotopomer model estimates a flux solution, represented by the numbers in the network, and isotopomer abundances. (d) The measured and predicted abundances are compared. According to how well the predicted abundance compares to the measured abundance,
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and metabolic cycles, which was not possible before (Calheiros, 2012). Fitting data from multiple parallel labelling experiments has been proven to increase the resolu-tion of fluxes (Antoniewicz, 2015a). Particularly, many 13C-MFA studies in yeast use, as substrate, glucose with different labelling positions (Table 1). 13C-MFA has been refined over the years with the improvement of measurement techniques, introduc-tion of better experimental methods and applicaintroduc-tion of more efficient computaintroduc-tional routines (Antoniewicz, 2013, 2015a; Antoniewicz et al., 2007; van Winden et al., 2002). An extensive set of software’s is currently available to perform steady-state 13C-MFA (He et al., 2016; Quek et al., 2009; Weitzel et al., 2013; Zamboni et al., 2005), see Table 1 for examples, and there is continued work on improving solvers and statistical analysis tools. Such software’s facilitate the implementation of 13 C-MFA workflows that are applied in diverse fields, mostly in engineering (Lange et al., 2017; McAtee et al., 2015), disease (Schwartz et al., 2015; Yang et al., 2008) and fundamental research on metabolism (Christen & Sauer, 2011; Kleijn, 2006). The most common approach of 13C-MFA, where metabolic and isotopic steady-states are assumed, is currently considered to be the most reliable method to estimate metabolic fluxes in steady-state with accuracy (Cheah et al., 2017).
15960801 16269086 NA 19684065 21205161 25311863 27761435 29188182 30969019 PMID 2005 2005 2007 2009 2011 2015 2016 2017 2019 Year Blank L Frick O Kleijn R Heyland J Christen S Wasylenko T Ghosh A Lehnen M Jessop-Fabre MM 1st author Sauer/Blank Wittmann Winden Blank Sauer Stephanopoulos García Martín Blank Blank/Keasling/ Borodina PI 34/21 32/25 24/23 34/21 34/21 94/76 1075/687 139/101 101/83 # reactions / # defined net flux
directions Yes Yes Yes Yes Yes Yes Yes Yes Yes Compartments Reversibility as 2 distinct reactions Net and labelling
exchange fluxes Net and labelling
exchange fluxes Reversibility as 2 distinct reactions Reversibility as 2 distinct reactions Net and labelling
exchange fluxes Forward,
back-ward, net and exchange fluxes Reversibility as 2 distinct reactions Net and labelling
exchange fluxes
Labelling exchange flux
modelling 13C-MFA model specifications
Table 1: Overview of steady-state 13C metabolic flux analysis studies with Saccharomyces cerevi-siae in the last 15 years. NA in the PMID column refers to the article ‘Metabolic flux analysis of a glycerol-overproducing Saccharomyces cerevisiae strain based on GC-MS, LC-MS and NMR-derived C-13-labelling data’ which does not have a PubMed ID. PI refers to Principal Investigator which is commonly the last author of the paper. Number (#) of defined net flux directions denotes the number of reactions for which the net flux is assumed to be in a particular direction and therefore its lower bound is zero. The rest of the reactions are allowed to be positive or negative in the 13C-MFA optimisation.
Reversibility accounts for reactions having a forward and backward flux. In the substrate column,
U-13C refers to uniformly labelled, 1-13C to labelled in the first carbon, and 1,2-13C
2 to labelled in the first
two carbons. It is important to note that, even though all studies involve compartments, many of them consider only a few metabolites as being in both cytosol and mitochondria (e.g. pyruvate).
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WT & Mutants WT Mutant (glycerol-over-producing) WT & Mutants WT & mutants Mutant (xylose-consuming) Mutants WT Mutants StrainsProteinogenic amino acids Proteinogenic amino acids
Proteinogenic amino acids
Proteinogenic amino acids Proteinogenic amino acids Organic acids and amino acids
Amino acids
Proteinogenic amino acids Proteinogenic amino acids
Measured in
Mix 20% [U-13C] & 80% unlabelled glucose
[1-13C] glucose Mix 10% [U-13C] & 90%
[1-13C] glucose Mix 20% [U-13C] & 80%
unlabelled glucose Mix 20% [U-13C] & 80%
unlabelled glucose [1,2-13C
2] glucose; [1,2-13C
2] xylose Mix 20% [U-13C] & 80%
[1-13C] glucose Mix 80% [1-13C] & 20%
[U-13C] glucose Mix 20% [U-13C] & 80%
unlabelled glucose Substrate METAFoR In-house software (MATLAB) MNA v3.0 (SpadIT) METAFoR FIATFLUX In-house EMU-based software 2S-13C MFA OpenFLUX INCA (v1.6) MATLAB toolbox Software Labelling data
Limitations of
13C metabolic flux analysis
13C labelling experiments gather a large number of redundant measurements that contribute to the accuracy of the flux estimations (Antoniewicz, 2015b). However, the size of the equation system and the limited labelling data makes 13C-MFA difficult to apply to complex systems. Additionally, this method is extremely resource inten-sive, both experimentally and computationally. Hence, 13C-MFA is mainly applied to simple central carbon metabolism networks (Crown & Antoniewicz, 2013) (Table 1) as opposed to genome-scale networks that take into account the relationship between different pathways. Compartmentation of eukaryotic organisms imposes another limitation on the application of 13C-MFA to determine metabolic fluxes (Antonie-wicz, 2015b). The consideration of compartments increases the complexity of the flux estimation problem through introduction of alternative pathways and inclusion of inter-compartment transports that can create cyclic routes. This increases the de-grees of freedom of the problem and, consequently, the uncertainty of flux predictions (Wahrheit et al., 2011). However, several studies in yeast included the differentiation between cytosolic and mitochondrial reactions and metabolite pools (Table 1) mostly applied to small networks.
In essence, compartmentation and network size highly affect the prediction ca-pabilities of 13C-MFA methods due to the lack of information contained in labelling measurements compared to the complexity of the mathematical model. Nevertheless, several studies attempted to ally labelling measurements to genome-scale models in order to provide a systems view of metabolism. Most of these studies attempted to validate predictions from Flux Balance Analysis models at genome-scale with labelling measurements (Chen et al., 2011; Kuepfer et al., 2005) instead of directly calculat-ing fluxes from labellcalculat-ing data. Others used labellcalculat-ing measurements as constraints to the model (Gopalakrishnan & Maranas, 2015a). Moreover, with the use of parallel labelling experiments with different labelled substrates and even different labelled molecules (Antoniewicz, 2015a), labelling measurements were directly incorporated into 13C-MFA genome-scale models to estimate fluxes in E. coli (Basler et al., 2018). In (Choi et al., 2007), relative fluxes of converging pathways derived from labelling data were used to constrain flux distributions estimated for a genome-scale model. A direct integration of labelling data in genome-scale models with compartmentation (García Martín et al., 2015; Ghosh et al., 2016) relied on the assumption that carbon flows from core to peripheral metabolism and not in the opposite direction. This method was also applied to the compartmentalised metabolism of the yeast Saccharomyces
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cerevisiae (Table 1). Overall, the high complexity and uncertainty associated with 13C-MFA applied to large scale networks requires the adoption of extra constraints based on heuristic assumptions.
Direction and reversibility assumptions employed in
13C-MFA
A common assumption employed in 13C-MFA to decrease the degrees of freedom of the mathematical problem is related to the direction of reactions (Figure 4a). The assignment of flux directions is usually done during the step of metabolic network curation (Duarte et al., 2007; Heavner et al., 2012) and it is based on a priori ther-modynamic analysis of the network (Haraldsdóttir et al., 2012; Henry et al., 2006; Wiechert et al., 2001; Yang et al., 2005) or retrieved from data bases and textbooks (Karp et al., 2005). Generally, and particularly in the case of S. cerevisiae, it is notable that reactions with determined direction always comprise more than half, and often more than 2/3, of the total number of reactions in the network (Table 1). However, many assigned directions were found to be erroneous when a thermodynamic analysis was applied to the metabolic network using physiological metabolite concentration ranges (Martínez et al., 2014). Overall, one should be careful when fixing directions of net fluxes in a metabolic model as they can lead to wrong conclusions (Faria et al., 2010; Zamboni, 2011).
Another aspect of 13C-MFA that is worth mentioning concerns labelling exchange fluxes. In a reversible reaction, there is a forward and a backward flux, which are often modelled as net and labelling exchange fluxes in theisotopomer model. Inversely, an irreversible reaction carries zero labelling exchange (or backward) flux (Wiechert & de Graaf, 1997). In 13C-MFA studies, labelling exchange or backward fluxes are rarely mentioned, or only briefly (Table 1). This is likely because the information provided by labelling data through the isotopomer model is not sufficient to find accurate values of labelling exchange fluxes, often noted as undetermined from statistical analysis (Yang et al., 2005). Reversibility assumptions, i.e. assuming that certain reactions are irreversible (Figure 4b), are also commonly employed, in ad-dition to direction assumptions, to decrease the degrees of freedom of the problem (Gopalakrishnan & Maranas, 2015b; Zamboni et al., 2009). Such assumptions come from previous literature, are inferred from thermodynamics (Wiechert, 2007), or are iteratively added and assumed consistent if they improve the fit to the labelling data (Ando & García Martín, 2019; García Martín et al., 2015). However, varied choices in reversibility assignment will have an impact on the flux solutions obtained, even
if labelling exchange fluxes are deemed statistically undetermined (i.e. not possible to predict).
Figure 4: Direction and reversibility assumptions explained. (a) Direction
assump-tion is when the direcassump-tion of a net flux is defined a priori to go from metabolite a to me-tabolite b (or the opposite). The arrow direction represents the assumed net flux direc-tion. The double-sided arrow represents a net flux of which direction is not yet known, thus values of this net flux can assume positive or negative values in the MFA problem. Circles represent directions that were eliminated from the net fluxes. (b) Reversibility assumption refers to when the labelling exchange or backward flux of a reaction is assumed to carry zero flux and thus making such reaction irreversible. Longer arrows represent forward flux and shorter arrows represent backward flux. Crosses represent backward fluxes assumed to be zero.
On avoiding heuristic and arbitrary assumptions on reaction direction and revers-ibility, the integration of metabolic flux analysis methods with diverse types of data to increase the accuracy in flux estimation should come as the ‘obvious’ approach. Particularly, integration of thermodynamic constraints and metabolomics data with 13C-MFA would help avoid heuristic assumptions on reaction direction and revers-ibility. However, even though the connection between thermodynamics and flux pre-dictions from labelling experiments was explored (Jacobson et al., 2019; Wiechert, 2007), a more formal direct integration of the two was not yet formulated, due to the computational complexity of both thermodynamic constraints and isotopomer models. The non-linearities and large number of equations of the isotopomer model make 13C-MFA a very difficult mathematical problem to solve, while the addition of thermodynamic constraints that make the problem non-convex would immensely increase the computational burden of a joint approach. A new routine that includes both thermodynamic constraints and labelling data is required in the field of meta-bolic flux analysis to estimate metameta-bolic fluxes with less heuristic assumptions on individual reactions.
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Statistical analysis applied to quantification of flux uncertainty from 13C-MFA estimates have received significantly less attention than computational routines to solve the main optimisation problem of minimising the distance between predicted and measured labelling. Most softwares, see Table 1 for examples, rely on linear statistics to calculate confidence intervals for the estimated fluxes. However, because the isotopomer model behaviour is non-linear, different methods used to quantify the uncertainty of 13C-MFA results were shown to produce different estimates (Theorell et al., 2017). This fact calls for new methods to quantify uncertainty in flux estimates.
Research question and outline of this thesis
The aim of this thesis was to develop new tools to estimate intracellular fluxes and their uncertainty under different metabolic conditions without use of heuristic as-sumptions on reaction direction and reversibility.
In chapter 2, we used a previously developed thermodynamic and stoichiometric model (Niebel et al., 2019), TSM, to infer metabolic flux distributions for two dis-tinct metabolic states of S. cerevisiae through a pipeline of computational tools that used information contained in different types of experimental data. With the TSM model and data on exchange fluxes, metabolite concentrations and Gibbs standard energies, we determined the limits of the feasible solution space of fluxes. To char-acterise the flux solution space by generating flux distributions, we developed a new sampling technique based on a reformulation of the mathematical problem to avoid its non-convexity. Specifically, we divided the solution space of fluxes, metabolite concentrations and Gibbs energies into flux polytope and a concentration polytope for each flux solution. Subsequently, we divided the flux polytope in sectors to avoid thermodynamically infeasible loops and used a Parallel Tempering scheme to aid the efficiency of the Hit-and-Run sampler. Finally, we ranked the different flux solutions, generated through sampling, by assigning each of them a score based on the fit to labelling data through 13C-MFA optimisation. With the novel sampling method, we were able to quantify the uncertainty of metabolic fluxes that are in agreement with thermodynamics and the experimental data. This was achievable through efficient sampling of the flux solution space constrained by thermodynamics and data, something that was not possible before. Furthermore, by allying together the constraining power of thermodynamic constraints with labelling data, we found previously unknown flux patterns through the yeast network. Additionally, we found that, statistically, labelling data had a small constraining effect on the flux solution space of two yeast strains
already constrained with thermodynamics, exchange fluxes and metabolomics data. In conclusion, we developed a method to estimate metabolic fluxes and their uncer-tainty, through sampling, without making use of heuristic assumptions on reaction direction and reversibility.
In chapter 3, we evaluated three sampling methods when applied to two metabolic networks of different complexity, using the sampling approach developed in chapter 2. Using the approach of splitting the non-convex solution space constrained by the TSM model into a flux polytope and a concentration polytope for each flux solution, we formulated strategies of Rejection Sampling and Conditional Sampling with and without Parallel Tempering to obtain flux distributions, with the aim of assessing the efficiency of each sampling method. The efficiency of each sampler was tested in two cases: when excluding and including the energy balance constraint. When excluding the energy balance constraint, we found that the Rejection Sampler was the most ef-ficient choice for the smaller metabolic network, while the Conditional Sampler was the best sampler for the larger network. When including the energy balance, only the Conditional Sampler with Parallel Tempering was applicable for both metabolic networks, as was the case in chapter 2. Overall, we demonstrated the applicability of our new approach to sample thermodynamically constrained fluxes with different mathematical complexity levels.
As an outlook of this work, we envision that our new flux inference method, that does not make assumptions on reaction direction and reversibility, will be a useful tool to validate previously stated fundamental hypotheses and to allow flux prediction for more complex systems. The extension of the physiology, metabolomics and labelling data sets used would potentially further decrease the uncertainty in the flux estima-tions. Beyond, the developed sampling method will be an asset for quantification of uncertainty and mathematical formulations of non-convex spaces.
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