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Thermodynamic principles governing metabolic operation : inference, analysis, and predictionNiebel, Bastian

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Thermodynamic principles governing metabolic operation

Inference, analysis, and prediction

Bastian Niebel

Cover design: Sarah Stratmann

Inspired by M. C. Escher’s Metamorphosis series

The work published in this thesis was carried out in the research group Molecular Systems Biology of the Groningen Biomolecular Sciences and Biotechnology Institute (GBB) of the University of Groningen, The Netherlands. The research was financially supported by the Netherlands Organization for Scientific Research (NWO) within the frame of the Systems Biology Centre for Energy Metabolism and Aging.

Copyright © 2014 Bastian Niebel

All rights reserved. No part of this publication may be produced, stored in a retrieval system of any nature, or transmitted in any form or by any means, electronic, mechanical, including photocopying and recording, without prior written permission of the author.

Printed by: Ipskamp Drukkers

ISBN: 978-94-625-9521-7

Thermodynamic principles governing metabolic operation

Inference, analysis, and prediction

PhD thesis

to obtain the degree of PhD at theUniversity of Groningenon the authority of the

Rector Magnificus Prof. E. Sterkenand in accordance with

the decision by the College of Deans.

This thesis will be defended in public on

Monday 19 January 2015 at 14:30 hours

by

Bastian Niebel

born on 23 February 1985in Schwäbisch Hall, Germany

Supervisors

Prof. dr. M. Heinemann

Prof. dr. E.C. Wit

Assessment committee

Prof. dr. B.M. Bakker

Prof. dr. A.J.M. Driessen

Prof. dr. U. Sauer

ContentsChapter 1: Thermodynamic constraints on metabolic operations 9

General overview 10

Background 11

Thermodynamic principles 11

Thermodynamic data 12

Previous work 14

Thermodynamic analysis of metabolic networks 14

Thermodynamic analysis of growth processes 15

Analysis of metabolic networks with the loop-law 15

Connections between kinetics and thermodynamics 15

This work 16

Research question 16

Outline of this thesis 16

Chapter 2: Entropy transfer constrains cellular metabolism 19

Introduction 20

Results and discussion 20

Methods 24

Methods 1: Stoichiometric metabolic network model for Saccharomyces cerevisiae 24

Methods 2: Cellular Gibbs energy balance and the cellular entropy production rate σcell

26

Methods 3: Gibbs energies of metabolic and exchange processes 27

Methods 4: Second law of thermodynamics for intracellular processes and biomass for-mation 28

Methods 5: Formulation of the thermodynamic constraint-based model 30

Methods 6: Estimation of σcell and standard Gibbs energies of reactions using nonlinear regression analysis 31

Methods 7: Condition-independent bounds for the concentrations and reaction’s Gibbs energies 34

Methods 8: Flux balance analyses with the thermodynamic constraint-based model 35

Supplementary figures 37

Supplementary tables 41

Chapter 3: Procedure for developing a thermodynamic metabolic model 63

Introduction 64

Workflow 64

Step I. Adding biochemical information to the stoichiometric network 66

Step II. Determining Gibbs energies changes of metabolic processes 69

Step III. Reducing the stoichiometric network 70

Step IV. Training the model 72

Discussion 72

Methods 73

Thermodynamic metabolic model 73

Determining chemical species and Gibbs free energies of formation using the component contribution method 73

Flux variability analysis and flux coupling analysis 74

Chapter 4: Accurate estimation of metabolic fluxes based on a thermodynamic metabolic network model and isotopomer balancing 75

Introduction 76

Results and discussion 77

Estimating metabolic fluxes using the thermodynamic metabolic model and isotopomer balancing 7713C labeling patterns validate the flux estimates of the thermodynamic metabolic network model 79

The ratio of the forward over the backward enzymatic rates correlate with the Gibbs en-ergy of reaction 82

Conclusions 84

Materials and Methods 85

Yeast strains 85

Batch cultivation conditions in minimal medium 85

Determination of biomass 85

Determination of glucose and extracellular metabolites 85

Estimation of extracellular rates 86

Quantification of mass isotopomer patterns in 13C labeling experiments 86

Thermodynamic metabolic network model and regression analysis 86

Network for isotopomer balancing 86

Isotopomer balancing using 13Cflux2 87

Weighted estimations of statistical values 87

Supplementary figures 88

Supplementary tables 89

Conclusions and future perspectives 105

References 109

Summary 121

Samenvatting 123

Zusammenfassung 125

Acknowledgments 127

9

Chapter 1

Thermodynamic constraints on metabolic operations

Bastian Niebel

10

Chapter 1

General overviewMetabolism converts nutrients into energy and biomass precursors. These conversions (or metabolic processes) are the enzyme-catalyzed chemical transformations and the trans-ports of metabolites across cellular membranes. But metabolic operations are not solely a consequence of the individual enzyme mechanisms. Instead, concerted action of metabolic processes are important for the functioning of metabolism. To unravel the complex system behavior of metabolic processes operating in an orchestrated manner, systems biology ap-proaches have been successfully applied and generated new insights into the operation of metabolism (1-4). In systems biology, mathematical models of metabolism or other cellular processes are developed, and the formal analysis of these models typically leads to new in-sights into the operation of these processes (2).

Given the fact that metabolic networks contain about thousand different metabolic processes, it is difficult to develop models that describe the mechanism of every metabolic process in great detail, i.e. with kinetics of the enzymatically catalyzed reactions. Thus, constraint-based models have been used, circumventing the need for detailed kinetic information on every metabolic process (4-6). In constraint-based models, metabolism is defined by a set of con-straints, where the solution space of these constraints describes the different possible met-abolic operations. Generally, the basis for these constraints are steady-state mass balances for the metabolites, which state that for every metabolite in the metabolic network the con-sumption rate equals the production rate, and thereby ensuring the conservation of mass. Adding—besides steady-state mass balances—more constraints to the model reduces the solution space and thereby can influence the quality of the predictions made with the model (6). During the last decades, numerous additional constraints have been identified ranging from constraints based on physical principles, e.g. thermodynamic constraints applied to the metabolic operations, or heuristically motivated constraints such as enzyme solvent capacity or transcriptional regulation (5, 6).

Thermodynamic constraints have the advantage of a physical foundation. Also, because ther-modynamic principles need to apply irrespective of the cellular growth conditions, these constraints are independent of the conditions. Therefore, thermodynamic constraints do not require readjustments when the model is applied to different conditions, unlike for most heu-ristic constraints, where for every new condition the constraint has to be reevaluated. Because of the physical foundations and condition-independence of thermodynamic constraints, they are being thought of as major constraints on the evolution of metabolism (7-13).

Constraint-based models can be analyzed using a multitude of different analysis techniques (6). The tool of flux balance analysis (FBA) uses a cellular objective functions, e.g. growth rate, to predict metabolic fluxes (14). This approach has been extended to explore possible cellular objectives using multi-objective optimization (15). Further, the shape of the solution space has been explored using flux variability analysis (16) and sampling based approaches (6, 15, 17, 18), while the topology of the metabolic network has been studied using extreme pathways (19) and elementary mode analysis (20). All these analyses depend on the quality of the constraint-based model. Thus, this calls for carefully curated and assembled models sup-plemented with a maximum number of hard physical constraints, but with a minimal number of heuristic ad-hoc constraints. Here, thermodynamics could offer such physical constraints.

11

Background

In the following sections (cf. Fig. 1), we briefly review the background of thermodynamic principles used in biochemistry and methods to determine thermodynamic data. Then, we discuss how these principles and data were used in previous work to get new insights in met-abolic operations. Specifically, we review the topics ‘thermodynamic analysis of metabolic networks’, ‘thermodynamic analysis of growth processes’, ‘analysis of metabolic networks with the loop-law’, and ‘connections between kinetics and thermodynamics’. We conclude this in-troduction with deriving the research question underlying this thesis, and provide an outline of the following chapters.

BackgroundThermodynamic principles

Thermodynamics of metabolic processes can be described by changes in the Gibbs energies, ∆rG (Fig. 2). ∆rG describes the difference in the Gibbs energy stored in the substrates and products of a metabolic processes (the so-called Gibbs formation energy). For a metabolic process to obey the second law of thermodynamics, ∆rG has to be negative (21). Strictly spo-ken, for a metabolic processes to proceed according to the second law of thermodynamics, entropy has to be produced. However, at the conditions, at which the metabolic process take place (i.e. constant pressure and temperature), the production of entropy is proportional to ∆rG.

Metabolic processes consist of chemical transformations and metabolite transport. Knowing Gibbs formation energies for the substrates and products of a process, it is possible to deter-mine the metabolic processes’ changes in the Gibbs energies, ∆rG. For chemical transforma-tions (superscript c), e.g. A → B, ∆rGc is defined by ∆rGc

A→B = ∆fGB - ∆fGA, where ∆fG is the Gibbs energies of formation of the chemical compounds, i.e. substrates and products of the transformations. ∆fG of a compound X is calculated by, ∆fG = ∆fGo + RT ln aX, where ∆fGo is the compound’s standard Gibbs energy of formation, a the compound’s chemical activity, R the Gas constant, and T the temperature. ∆fGo of a compound describes its change in Gibbs energy with respect to a common reference state, which is indicated by the superscript o. Note: The standard Gibbs energies of formations for a chemical conversion can be combined in the standard Gibbs energy of reaction ∆rGo, e.g. ∆rGo

A→B = ∆fGoB - ∆fGo

A, therefore the ∆rGc

Thermodynamic principles

Thermodynamicdata

Thermodynamicanalysis ofmetabolicnetworks

Analysis of metabolic

networks withthe loop-law

Thermodynamicanalysis of

growth processes

Connections between

kinetics and thermodynamics

Research question and outline

Background

Previous work

This work

Figure 1. Overview of topics covered in this introduction.

12

Chapter 1

becomes, ∆rGcA→B = ∆rGo

A→B + RT (ln aB - ln aA).

Metabolic networks not only consist of chemical transformations, but they also involve trans-port of metabolites across membranes. The change of Gibbs energy ∆rGt of a metabolite trans-port (superscript t), e.g. the transport of a compound X from the outside (superscript out) to the inside (superscript in) of a cellular compartment, Xout → Xin, is defined by, ∆rGt

Xout → Xin = RT (ln aXout - ln aXin) + F zX φ, where z is the charge of X, φ is the electrical membrane poten-tial, and F is the Faraday constant. One feature of many metabolic processes is that different mechanisms are mechanistically and thermodynamically coupled (22), e.g. the coupling of a chemical transformation, A → B, to the transport of a compound, Xout → Xin. The change of Gibbs energy, ∆rG, of this coupled metabolic process is then defined by the sum of the change in Gibbs energy of the chemical transformation, ∆rGc, and the transport process, ∆rGt, ∆rG = ∆rGc

A→B + ∆rGtXout → Xin.

Thermodynamic data

In order to specify the Gibbs energy changes of metabolic processes, we need standard Gibbs energies of formation (or of reactions). These standard Gibbs energies can be inferred from the experimental equilibrium constant of chemical conversions (21, 23-27)—also referred as reactant-contribution method (RC) (27)—or predicted using a group contribution method (GC) (26-30). Both methods rely on collections of experimentally determined equilibrium constants for a set of about 400 of enzymatic reactions, which has been collected and curated from 1000 different articles into a database (31). These two approaches deliver different cov-erages: With the RC method, one can obtain standard energies Gibbs energies of reactions for 11 % (~600) of all relevant reactions (full chemical description and chemically balanced) in the KEGG database, and 88 % (~4800) with GC (27). The median root mean square error of the estimated standard Gibbs energies of reactions has been determined to 1 KJ mol-1 with RC and 5.5 KJ mol-1 with GC (27). Another approach has been developed, which infers Gibbs energy changes based on similarities in the reactions (32), but the high average root mean square error of 10 KJ mol-1 and the ability to only infer the standard Gibbs energies of 106 reactions, renders this approach not practical.

Recently, different approaches to combine both the RC and GC methods have been developed (26, 27, 33, 34). When combining both methods, one is faced with the challenge of different reference states that were used for the estimated (RC) or predicted (GC) Gibbs energies of formations. If Gibbs energies are used in the thermodynamic constraints within a metabolic network model, then different reference states would lead to a violation of the first law of ther-modynamics, and thus all used standard Gibbs energies of formations must have the same thermodynamic reference state (27). This problem has been solved by the component con-tribution method (CC) by ensuring that all standard Gibbs energies of formations are within

A

B

Gib

bs e

nerg

y

Progress of metabolic process

∆GA B∆G = 0

A B∆G < 0

A B∆G > 0

Figure 2. Second law of thermodynamics for metabolic pro-cesses. A metabolic process consisting of a chemical transfor-mation of the metabolite A to B can happen when the change in the Gibbs energy ∆G is negative. ∆G describes the differ-ence in the Gibbs energy between the initial (A) and the final state (B) of the process.

13

Background

the null space of the stoichiometric network, i.e. fulfill the loop-law (27). To this end, CC also allows the exact determination of the estimation errors of the standard Gibbs energies.

Because the chemical conditions—at which metabolic processes take place within the cellular compartments—can be approximated by a dilute mixture of compounds in an electrolyte solution with constant ionic strength, I, and constant pH, we can transform the standard Gibbs energies to take into account theses chemical conditions. The assumption of a dilute electrolyte solution allows to approximate the chemical activity, a, by the molar concentra-tion, C, using the extended Debye-Hückel theory (21), i.e. a = γ(I)C , where γ is the activity coefficient determined as a function of I. Often, the activity coefficients are included in the standard Gibbs energy of formations, ∆fGo(I) = ∆fGo + RT ln γ(I), and thereby ∆fGo becomes a function of the ionic strength I.

In aqueous solutions, in which biochemical processes take place, metabolites (i.e. the re-actants) are typically present as different chemical species (i.e. differently protonated). The distribution between the abundance of these species is often pH-dependent. As handling of individual species would be cumbersome, we only consider reactants in biochemical thermo-dynamics. In order to determine the pH-dependent Gibbs energy of a reactant, we first make the standard Gibbs energy of formation of the species ι, ∆fGo

ι(I), pH-dependent using the Legendre transformation (21), ∆fG’oι = ∆fGo

ι(I) – NH (∆fGoH+(I)- RT pH ln 10), where NH is the

number of hydrogen atoms of the species, and ∆fGoH+(I) the standard Gibbs energy of forma-

tion of hydrogen ions (protons, H+). This Legendre transformation transforms the standard Gibbs energy of the species to the biochemical reference state (indicated by the apostrophe ‘), and thereby making ∆fGo

ι(I) a function of the pH and the ionic strength I. The standard Gibbs energy of formation of a reactant is then determined from all its chemical species ι, using the relationship (21) ∆fG’o = -RT ln[∑ιexp(-∆fG’oι/RT)], where we assume an equilibrium between the differently protonated species. Note, that while this concept is very practical, it compli-cates things once it comes to the thermodynamic description of transport processes, where we still also need to consider, for instance, the charge of individual species. For a detailed treatment of thermodynamics of transport processes, however, the reader is referred to the work of Jol et al. (35).

Gibbs energy changes of metabolic processes are also dependent on the processes’ reactant concentrations. In microorganisms and mammalian cells, metabolite concentrations typi-cally ranges between 1 uM to 10 mM (36-39). When using thermodynamic constraints on metabolic network operation, one can constrain metabolite concentrations to such generic physiological bounds. Notably, in some studies wider concentration ranges were used (34, 40, 41). Likely, metabolite concentration bounds had to be relaxed, because of inconsisten-cies in the reference state of the Gibbs energies of formation or missing adjustments of the thermodynamic data to pH or ionic strength (42, 43). Such inconsistencies in the thermo-dynamic data are regretful in the first place, because they might lead to wrong conclusions. But, furthermore, these inconsistencies then typically required researchers to apply large me-tabolite concentration ranges to get the network feasible at all. Consequently the constraints imposed by the Gibbs energies are relaxed significantly and might not be active after all. Therefore, it is key to use standard Gibbs energies estimated by methods which ensure the same thermodynamic reference state, such as component contribution (27). Then, narrow

14

Chapter 1

concentration ranges between 1 uM and 10 mM can be used and consequently thermody-namic constraints are more active.

Previous workThermodynamic analysis of metabolic networks

The second law of thermodynamics together with detailed information of the Gibbs energies of reactions and concentration ranges have been used to determine the feasibility of metabolic pathways (9-13, 42-46) and metabolic networks (33, 34, 38, 40, 41, 47-54). Early studies fo-cused on the thermodynamic feasibilities of metabolic pathways, where especially reactions were identified that serve as thermodynamic bottlenecks in glycolysis for different concentra-tion ranges of the substrates and products of the glycolytic pathway (44, 45). Also the sensitiv-ity of the thermodynamic bottlenecks with respect to pH, ionic strength, and magnesium has been studied (42, 43). More recent thermodynamic feasibility studies have been carried out to unravel the biochemical logic behind the glycolytic pathway (11, 13) and other pathways in central metabolism (13). Thermodynamic feasibility statements have been also applied for industrial applications. Here, feasibility of thermodynamic pathways have been used to op-timize the penicillin production of Penicillium chrysogenum (46), and for the design of new carbon fixation pathways (9, 10, 12).

The second law of thermodynamics, applied to the reactions of a metabolic network model, has been used in thermodynamic analysis of metabolic networks (38) to predict ranges for the Gibbs energies of reactions and concentrations in Escherichia coli (33, 38, 49-51, 54), Geobacter sulfurreducens (52), Saccharomyces cerevisiae (34, 38, 40, 41) and mammalian cells (34, 48, 51, 53) and to further check the thermodynamic consistency of metabolome data for E. coli and S. cerevisiae (38). Based on the predicted ranges of the Gibbs energies of reactions, potential regulatory reactions in metabolic networks have been identified (38, 41, 50, 52). By varying concentrations of toxic compounds, thermodynamic analysis of metabolic networks has been used to predict the responses of intracellular metabolite concentrations and Gibbs energies of reactions to different dosages of this toxic compounds (48). Also, the second law has been directly integrated as a mixed integer constraint allowing the computationally trac-table integration of network thermodynamics and the second law of thermodynamics (49, 51). This mixed integer constraint has been used together with flux balance analysis to predict flux distributions based on measured intracellular and extracellular metabolic concentrations (51), and to study the effect of thermodynamic constraints on the feasible solution space of metabolic networks (54).

Thermodynamic analysis of metabolic networks has further been used to define the direc-tionality of reactions within the metabolic network. In constraint-based models reactions are either classified as irreversible or reversible, where typically the majority of the reac-tions are classified as irreversible. This irreversibility classifications have been incorrectly referred to as thermodynamic constraints (55, 56). Thermodynamic analysis of metabolic networks allowed to determine which reactions can be correctly classified as irreversible on the basis of thermodynamics. Therefore, we collected from different studies that classified the irreversibility of reactions using thermodynamic analysis, the fraction of reactions that were correctly classified as irreversible. With this literature summary, we found that for dif-

15

Previous work

ferent genome scale metabolic reconstructions of E. coli, 28% (47), 29% (57), 34% (33), and mammalian reconstructions 39% (53), 29% (34), were correctly classified as irreversible on the basis of thermodynamics. In summary, on average 30% of the reactions classified as irre-versible in this genome scale metabolic reconstructions are correctly classified on thermody-namic basis.

Thermodynamic analysis of growth processes

Thermodynamic principles not only apply to intracellular metabolic processes, but also to the overall growth process, i.e the conversion of substrates, S, e.g. glucose, phosphate, oxy-gen, into biomass, B, and by-products, P, e.g. carbon-dioxide, ethanol, acetate, lactate, by the growth process, S → B + P. Similar to every metabolic process, the growth process has to fulfill the second law of thermodynamics, and thus, the change in Gibbs energy associated with the growth process needs to be negative, ∆rGS→B+P. To analyze these Gibbs energy changes of the growth process, black box models have been developed that determine ∆rGS→B+P on the basis of measured extracellular rates and Gibbs energies of formation of the different substrates, products and the biomass as input (8, 58-67). The Gibbs energy of formation of the biomass has been determined using low temperature calorimetry, statistical mechanics, or empirical relationships (67-69). Using these black box models that describe ∆rGS→B+P, the thermody-namic efficiency of different growth condition has been analyzed (59, 60, 62, 65, 67), and a number of different empirical relationships have been developed to predict the biomass yield based on measured ∆rGS→B+P (63, 65-67).

Analysis of metabolic networks with the loop-law

For metabolic networks, the first law of thermodynamics, i.e. conservation of energy, ensures that no energy can be produced or destroyed. Therefore, the first law is the energetic analogue to the mass conversation. For the operation of a metabolic network, this means that the Gibbs energy changes of a cyclic series of metabolic processes, e.g. loops of chemical conversions, A → B → C → A, of the metabolites, A, B, and C, must be zero, e.g. ∆rGA→B + ∆rGB→C + ∆rGC→A = 0. Combining the first law with the second law of thermodynamics forbids a metabolic flux through a loop of metabolic processes (in the following, we refer to this as the loop-law (70)). These loops in the metabolic network are defined by the null space of the stoichiometric matrix, i.e. the mathematical representation of the metabolic network (70). Using the loop-law does not require any information about Gibbs energies of formation. It only requires the calculation of the null space of the stoichiometric matrix. The loop-law was used in a series of constraint-based models as constraints and was used to exclude loops from the flux dis-tributions (18, 19, 71-74), to predict Gibbs energies of formations (73), and was included in flux variability analysis (75). It has been mathematically proven that using the loop-law only constrains thermodynamic infeasible loops and does not remove thermodynamically possi-ble flux distributions (76).

Connections between kinetics and thermodynamics

For elemental reaction steps, the kinetics of these reactions are described by the law of mass action and therefore the Gibbs energy is proportional to the natural logarithm of the ratio of the forward, v+, and the backward rate, v-, of this elementary reaction step, i.e. ∆rG = RT ln(v+/v-) (77, 78). This relationship between ∆rG and ln(v+/v-) has been integrated in con-

16

Chapter 1

straint-based models (72, 79-81) and been used to identify kinetically limited reactions in metabolic pathways (13). But since this relationship assumes mass action kinetics, the gen-erality of this relationship is questionable, since enzymatic reactions are not described by a single elementary reactions but consist of a series of elementary reactions, which make up the enzymatic mechanism. To this end, several attempts have been made to extend this relation-ship and develop new kinetic relationships between the Gibbs energy and reaction and the enzymatic rates (82, 83).

This workResearch question

Thermodynamic principles apply to every aspect of metabolic operations, although from pre-vious work it remains unclear to what degree metabolic operations are constrained by ther-modynamics beyond pure feasibility statements. Metabolic operations that generally occur in most organisms, e.g. bacteria (84), fungi (85), mammalian cells (86, 87), and even plants (88), are respiration and aerobic fermentation (89). While with respiration, ATP is generated at high yields, with aerobic fermentation ATP is only generated at low yields through sub-strate-level phosphorylation (90). Different research fields postulated numerous explanations why cells under aerobic conditions choose an ATP-inefficient fermentative metabolism over an ATP-efficient respiratory metabolism, amongst which are economics of enzyme produc-tion (89), ‘make-accumulate-consume’ strategy (85), intracellular crowding (84), limited nu-trient transport capacity (91), and adjustments to growth-dependent requirements (86, 87). These explanations have all the short coming that they do not give a detailed mechanistic reason, therefore they could not be quantified and validated by experimental data.

We asked whether—in contrast to the previously proposed explanations—rather a common inevitable principle would underlie the specific choice of metabolic operation, with fermen-tation being seemingly connected with high, and respiration with low rates of glucose up-take and glycolysis (92). Specifically we investigated, using constraint-based modeling (4, 6), whether thermodynamic constraints could be the cause for the ubiquitously observed and obviously sugar uptake rate-dependent choice between respiration and aerobic fermentation. Here, we aimed to identify these constraints by integrating previous work of thermodynamic analysis of metabolic networks, thermodynamic analysis of cellular growth, and the use of the loop-law to formulate a new thermodynamic metabolic network model.

Outline of this thesis

In Chapter 2, we develop a constraint-based model for Saccharomyces cerevisiae, which com-bines mass and charge balances with comprehensive description of the biochemical thermo-dynamics governing metabolic operations. As the model does not use any heuristic irrevers-ibility assignments of the intracellular rates, therefore all conclusions drawn from the model are based on thermodynamic principles. We use this model together with experimental phys-iological and metabolome data of S. cerevisiae and identify a global thermodynamic con-straint, i.e. a limit in the cellular rate of entropy production. Using this constraint in a flux balance analysis together with a cellular objective of maximizing the growth rate, we correctly predict the intracellular and extracellular rates of S. cerevisiae for a wide range of different

17

This work

glucose uptake rates.

In Chapter 3, we present a computational workflow to develop thermodynamic metabol-ic network models similar to the one developed in Chapter 2. Here, we especially focus on the potential pitfalls, when gathering the necessary biochemical information from different sources. Further, we give advice on potential model reductions and how to gather the neces-sary data to train such a model.

In Chapter 4, we use the thermodynamic metabolic model of S. cerevisiae and develop a sta-tistical workflow to accurately quantify the metabolic operation based on experimental data, where we combine the thermodynamic metabolic model with isotopomer balancing. We use this workflow to infer intracellular metabolic fluxes and also backward fluxes from measured extracellular rates, metabolomics data, standard Gibbs energies of reactions, and measured isotopomer patterns.

Lastly, we conclude this work by a short discussion of the impact of the here identified prin-ciples on metabolic operations and suggest potential following studies.

19

Chapter 2

Entropy transfer constrains cellular metabolism

Bastian Niebel and Matthias Heinemann

(Manuscript submitted)

The principles governing metabolic flux are poorly understood. Because diverse organisms show similar metabolic flux patterns, we hypothesized that fundamental thermodynamic constraints might shape cellular metabolism. We developed a constraint-based model for Saccharomyces cerevisiae that included a comprehensive description of biochemical ther-modynamics and a Gibbs energy balance. Nonlinear regression analyses of quantitative metabolome and physiology data showed that there is an upper limit for the cellular en-tropy transfer rate. Applying this limit in flux balance analyses with growth maximiza-tion as the objective, our model correctly predicted the physiology, intracellular metabolic fluxes, and maximal growth rates for different glucose uptake rates and carbon sources. Thus, reaction stoichiometry, fundamental thermodynamic constraints, and the objective of growth maximization shape metabolic fluxes in yeast.

BN and MH designed the study. BN developed and implemented the computational model. BN carried out the com-putational simulations, analyzed the data, and made the figures. BN and MH wrote the manuscript.

20

Chapter 2

IntroductionRespiration and aerobic fermentation are common metabolic processes in most organisms, including bacteria (84), fungi (85), mammals (86, 87), and plants (88). Respiration generates high yields of ATP, but, due to excretion of pyruvate-derived compounds (e.g. ethanol, ace-tate, or lactate), aerobic fermentation generates low yields of ATP through only substrate-lev-el phosphorylation (90). The question arises: Under aerobic conditions, why do cells use ATP-inefficient fermentative metabolism rather than ATP-efficient respiratory metabolism (84-87, 89-91)? Possible explanations include the economics of enzyme production (89), a ‘make-accumulate-consume’ strategy (85), intracellular crowding (84), limited nutrient trans-port capacity (91), and adjustments to growth-dependent requirements (86,87). The preva-lence of this metabolic behavior prompted us to ask whether a fundamental thermodynamic principle determines the metabolic mode, with fermentation occurring at high glucose up-take rates (GURs) and respiration at low GURs (86, 92). Specifically, we hypothesized that the rate at which entropy is generated by cellular processes and subsequently transferred to the environment must not exceed a certain limit and could determine the metabolic mode.

Results and discussionTo test this hypothesis, we developed a stoichiometric metabolic network model of Saccharo-myces cerevisiae: The model included 156 metabolites (table S1) and 241 metabolic processes (i.e. chemical conversions and metabolite transport; table S2, fig. S1), their compartmental localization, and a comprehensive description of the biochemical thermodynamics of each processes. In addition to mass-balances (tables S3-S5, methods 1), pH-dependent proton-bal-ances (table S6), and charge-balances (table S7), we introduced a Gibbs energy balance (Fig. 1A). This states that the cellular rate of entropy generation, σcell, i.e. the sum of the entropy production rates of the individual metabolic processes, is directly proportional to the rates at which Gibbs energy is exchanged with the environment, gEXG (methods 2). The rate of entro-py production of a given metabolic process, σ, was made a function of its metabolic flux, v, and its Gibbs energy, ∆rG’, according to σ = -(∆rG’·v) ⁄ T, where T is the temperature. In turn, the Gibbs energy of a metabolic process was defined as a function of the metabolite concen-tration, c, the standard Gibbs energy of the reaction ∆rG’o, and/or the Gibbs energy of the metabolite’s transmembrane transport, ∆rG’t. We obtained the standard Gibbs energies of the reactions using the component contribution method (27) and transformed them according to the compartmental pH values (methods 3). Finally, we applied the second law of thermo-dynamics by stating that the entropy production rate σ must be positive for every metabolic process carrying flux (methods 4). To avoid heuristic bias, we did not use directionality con-straints on any metabolic process.

To determine the cellular entropy production rate, σ cell, for different growth conditions, we formulated a constraint-based model (6) (methods 5) that we used for nonlinear regression analysis (methods 6) of physiological and metabolome data determined from eight different glucose-limited chemostat cultures (83). In these cultures, metabolism ranged from respira-tion at low GURs to aerobic fermentation with ethanol production at high GURs. We used the regression analysis to also determine a thermodynamically consistent set of standard Gibbs energies (i.e. standard Gibbs energies with identical thermodynamic reference state, table S8)

21

Results and discussion

− =∑T g v ccellEXGσ

( , )

S v vEXG∑ =

Growth rate (h-1)

5

10

15

0.0 0.2 0.4

A BMass balances

Gibbs energy balance

Additionally proton & charge balances second law of thermodynamics transport thermodynamics

0

σ cellup =12.3

= ( )∑σ v G cr, ( )∆

Entro

py p

rodu

ctio

n ra

te(J

K-1 g

CDW

-1 h

-1)

σce

ll

Figure 1. (A) Key balances of the constraint-based model for determining the cellular entro-py production rate, σcell (methods 1-5). Exchange processes (subscript EXG) are defined such that they are positive for transfer of metabolites from the cell to the environment. (B) Entropy production rate, σcell (black dots), as determined by regression analysis using the model and physiological and metabolome data (83), reaches an upper limit, which coincides with the onset of aerobic fermentation (indicated by the grey shaded area). σcellup was determined from the σcell values, at which aerobic fermentation occurred. The solid red line represents the median and the dashed red lines the 97.5% confidence interval. The dotted black line shows the linear increase of σcell during respiration. Note that although the regression was largely underdetermined (107 degrees of freedom, table S10), σcell could be estimated with high confidence because σcell could also be estimated directly using perfect physiological rate measurements (cf. Eq. 4 in methods 2). Errors represent the 97.5% confidence intervals.

based on the experimental data and the standard Gibbs energies obtained from the compo-nent contribution method (27). To improve the accuracy of subsequent analyses, we further determined the condition-independent ranges of the metabolite concentrations, c (table S9), and the Gibbs energies, ∆rG’ (table S8) (methods 7).

The regression analyses confirmed that the model could be fitted to the experimental data obtained under the different conditions (fig. S2; table S10). Focusing on the cellular rates of entropy generation estimated for the different conditions, we found that σcell first increased linearly with increasing growth rate µ and then plateaued at a µ of approximately 0.3 h-1 (Fig. 1B). The plateau of σcell above a certain µ suggested that there is an upper limit, σcellup, at which entropy produced in the cell could be transferred to the environment (at 12.3 J K-1 gCDW-1 h-1, Fig. 1B). Because the growth rate at which this limit is reached coincided with the onset of ethanol excretion (Fig. 1B), we speculated that this limit might cause the switch to fermen-tation at high GURs.

To investigate this, we constrained the model using the identified upper limit of σcell, σcellup, and the determined condition-independent ranges for the metabolite concentrations c and for ∆rG’. We then performed flux balance analyses (FBA) (6) with different GURs as input and growth maximization as the objective function (methods 8). Here, we predicted growth rates that perfectly matched the values observed in the glucose-limited chemostat cultures and values from glucose batch cultures (Fig. 2A). The model further correctly predict-ed respiratory metabolism at low GURs (< 3 mmol gCDW-1 h-1, Fig. 2B-D). Despite using growth maximization as the objective function, it also correctly predicted aerobic fermen-tation at a GUR > 3 mmol gCDW-1 h-1 (Fig. 2C) accompanied by declining oxygen transfer rates (Fig. 2B) and production of glycerol at GURs > 18 mmol gCDW-1 h-1 (Fig. 2D) (93).

22

Chapter 2

Gro

wth

rate

(h-1)

Rate

(mm

ol g

CDW

-1 h

-1)

20

0 10 20

0

0

10

20 D Glycerolproduction

EtOHproduction

C

0.0

0.2

0.4A

Biomassproduction

GUR (mmol gCDW-1 h-1)

Rate

(mm

ol

gCDW

-1 h

-1)

0

5 O2uptake

B

300 10 20 30

40

10

Figure 2. Flux balance analysis (FBA) predictions of the model constrained by σcellup agree with ex-perimentally determined physiological data. (A–D) Predictions of physiological rates for S. cerevisiae growth on glucose (solid black line) with growth maximization as an objective and the identified up-per limit in the cellular entropy transfer rate, σcellup, of 12.3 J K-1 gCDW-1 h-1. Red circles represent exper-imentally determined values from glucose-limited chemostat, cultures (83,98) and red triangles values from glucose batch cultures (93,98). The black arrow points to the GUR at which the maximum growth rate was observed; solid grey lines represent predic-tions above this GUR. The dotted black lines rep-resent FBA simulations with growth maximization as an objective, but without a constraint on σcell. The dashed black lines represent predictions with the ‘minimal sum of absolute fluxes’ as an objective and the σcellup-constraint.

At a GUR of 22 mmol gCDW-1 h-1, the model predicted a maximum growth rate, µmax, of 0.52 h-1 (cf. black arrow in Fig. 2A), which is in good agreement with the value of 0.46 h-1 found in batch cultures (93). At GURs > 22 mmol gCDW-1 h-1, growth rates and ethanol production rates decreased while glycerol production rates further increased (cf. grey solid lines in Fig. 2A-D). Maximal growth rates predicted for growth and ethanol and acetate were also in good agreement (µmax=0.28 h-1 versus experimental value µ=0.29 h-1 (94)); µmax=0.32 h-1 versus ex-perimental value µ=0.23 h-1 (95), respectively). In contrast, FBA with growth maximization as an objective, but without a constraining σcellup

and an otherwise identical model predicted a respiratory metabolism for all GURs (cf. dotted lines in Fig. 2A–D) and without reaching a growth rate maximum. Also, FBA with the ‘minimal sum of absolute fluxes’ as an objective, and with a constraining σcelluppredicted for all GURs metabolic operations that are different to the experimentally observed ones (cf. dashed lines in Fig. 2A–D). We thus concluded that fermentative metabolism at high GURs and the maximum growth rate, µmax, must be caused by a limit on σcell and the objective of growth rate maximization.

Next, we asked whether we could also predict intracellular fluxes with the σcellup-constrained model. To first identify the boundaries of each flux, we performed flux variability analyses by fixing µ to the value obtained in the FBA optimizations (methods 8). With increasing GURs, we found a global decrease in the flux variability of the 144 linear independent processes in the model (identified by flux coupling analysis (96)), albeit with discontinuities in the variability at specific GURs (Fig. 3A), which coincided with changes in the directionalities of many metabolic processes (Fig. 3B). Because flux ranges do not represent the true multivariate distributions of metabolic fluxes, we next sampled the solution space established by the constraints of the mod-el to estimate these distributions (methods 8). We found that the predicted multivariate distri-bution of fluxes were in excellent agreement with results from 13C-metabolic flux analysis as shown for key metabolic reactions located at branch points in central metabolism (Fig. 3C–F). The predictions showed the expected reorganization patterns; for instance, redirection of flux from the pentose-phosphate pathway to glycolysis with increasing GUR (Fig. 3C, E). Thus, the model constrained by σcellup also correctly predicted intracellular metabolic fluxes.

23

Results and discussion

Rate

nor

mal

ized

to G

UR

(-)

0.5

0

1

0SUCOAS1m

C

GND

B

PGI

F

0

-1

0.5

0

D

Variability of all 144 linear independent processes0.4

0.2 E

A

0 PDHm2 -2

1 1

GUR (mmol gCDW-1 h-1)GUR (mmol gCDW-1 h-1)0 10 20

Flux

var

iabi

lity

norm

alize

d to

GU

R (-)

Frac

tion

ofun

idire

ctio

nal

proc

esse

s (-)

0 10 20 0 10 200.60.70.8

Figure 3. Flux balance analysis (FBA) predictions of the model constrained by σcellup agree with experimentally de-termined intracellular flux data. (A) The overall flux variability of the 144 linear independent processes (determined by flux coupling analysis (96)) decreases with increasing GURs (black bars indicate the 0% and 100% quartiles, blue bars the 25% and 75% quartiles and black dots the median flux variability). (B) The fraction of unidirectional pro-cesses of the linear independent processes for different GURs determined from the directionality patterns of these processes (fig. S4) suggest discrete changes in metabolic operations. (C–F) Predicted and measured intracellular fluxes at key branch points in central metabolism. The graphs show flux boundaries from flux variability analyses (light grey areas) and the multivariate distribution of intracellular fluxes obtained by sampling the solution space of the σcellup-constrained model for optimal growth rates, with the black lines representing medians and the dark blue areas the 97.5% confidence intervals. The symbols denote fluxes determined by 13C-based metabolic flux analysis; diamonds data, (99); squares, (100); triangles, (101); circles, (102). Note that these fluxes were determined with small metabolic networks and heuristic assumptions on the reversibility on metabolic reactions. Therefore, these estimates may contain errors and biases as discussed in (99). PGI: glucose-6-phosphate isomerase; GND: phosphogluconate dehydrogenase; PDHm: pyruvate dehydrogenase; SUCOAS1m: succinate-CoA ligase.

To investigate how σcellup specifically governs metabolism, we determined the entropy produc-tion rates, σ, from the FBA results for each metabolic process at different GURs. From these rates, we identified seven clusters of metabolic processes that showed similar trends in entro-py production with increasing GURs (fig. S3). At GURs < 3 mmol gCDW-1 h-1, the processes in the two clusters that contributed 45% to the still relatively low total cellular entropy pro-duction rate, are related with respiration (respiration and energy metabolism cluster in Fig. 4A). When GUR increases and σcellup is reached, cells apparently redirect metabolic fluxes from entropy-intense pathways to pathways with lower entropy production, i.e. to fermentative processes which produce about 40% of the σcell at high GURs (> 20 mmol gCDW-1 h-1, pyru-vate decarboxylase and pyruvate kinase cluster in Fig. 4A). Flux redirection happens not only between respiration and fermentation, but also between other processes as indicated by the changes in the directionality patterns (fig. S4). Such flux redirection apparently allows cells to exploit higher GURs to achieve higher growth rates - although at lower carbon efficiencies (Fig. 4B) - while not exceeding σcellup.

ConclusionsOur work answers the long-standing question of what shapes metabolic fluxes, limits growth rate, and causes the growth rate/yield trade-off in cellular metabolism (15, 84-87, 89-91). With an upper limit in cellular entropy production and growth maximization, we identified the mechanistic basis underlying these phenomena, also offering a mechanistic explanation of

24

Chapter 2

GUR (mmol gCDW-1 h-1)0 10 20

0

10

5

Entro

py p

rodu

ctio

n ra

te (J

K-1 g

CDW

-1 h

-1)

00.20.4

A

B

Biom

ass

yiel

d (g

CDW

g-1) Glycolysis (12)

Respiration (6)Pyruvate decarboxylase (1)Biomass synthesis (86)

Other processes (132)Energy metabolism (3)

Pyruvate kinase (1)

Figure 4. (A) The limit in the entropy transfer rate causes flux redistribution with increasing GURs, globally leading to a change from respiratory to aer-obic-fermentative pathways. Seven clusters of meta-bolic processes were revealed by cluster analysis using the Euclidean distance between the average entropy production rates of metabolic processes at different GURs (for details of the processes in the clusters re-fer to fig. S3). The average entropy production rates of the metabolic processes were obtained by sampling the solution space of the σcellup-constrained model for optimal growth. The numbers in brackets indicate the number of processes in each cluster. (B) The shift to less carbon-efficient pathways is indicated by reduced biomass yield with increasing GURs.

the recently described Pareto-optimality in metabolism (15) (fig. S5). Applying the identified principles in FBA now allows to make excellent predictions of metabolic fluxes. An upper limit in the cellular entropy production rate suggests that higher values cannot be sustained, because otherwise cells would accumulate entropy, i.e. heat and small molecules, which might have detrimental consequences for cell function. Our findings imply that cells need to redis-tribute their metabolic fluxes in response to, for instance, increased substrate uptake rates. Thus, cells need to be able to sense intracellular fluxes, which indeed has been shown recently (92, 97). An upper limit on σcell might be a universal thermodynamic constraint on metabo-lism. The here identified principles might even explain the Warburg effect in cancer cells (86, 87).

MethodsMethods 1: Stoichiometric metabolic network model for Saccharomyces cerevisiae

On the basis of the stoichiometric model presented in (41), we developed a stoichiometric metabolic network model for the yeast Saccharomyces cerevisiae. The stoichiometric network model describes the steady-state mass balances for the metabolites i (table S1),

S v v iij jj MET

i EXG∈

∈∑ = ∀ , (Eq. 1)

where S is the stoichiometric matrix, whose elements are the stoichiometric coefficients Sij (ta-ble S3) of the metabolic (j ∈ MET) and exchange processes (i ∈ EXG) (table S2); vj∈MET are the rates of the metabolic processes, i.e. the chemical conversions and/or metabolite (incl. proton) transport; and vi∈EXG are the rates of the exchange processes, which describe transfer of me-tabolites across the system boundary, where the system boundary is between the extracellular space and the environment. Note that we define for the exchange processes the transfer of a metabolite from the inside to the outside of a cell as the positive direction. The here developed stoichiometric network includes the metabolic processes of glycolysis, gluconeogenesis, tri-carboxylic acid cycle, amino acid-, nucleotide-, sterol-synthesis and considers the processes’ location in the cytosol, mitochondria and extracellular space. The stoichiometric coefficients of the biomass synthesis reaction (table S4) are based on an earlier determined biomass com-

25

Methods

position (103). We included the precursors of membrane biosynthesis and an ATP-demand for protein polymerization of 23.9 mmol gCDW-1 (104) directly into the biomass synthesis reaction. Overall, the model contains 156 metabolites, 241 metabolic processes, and 15 ex-change processes.

The translocation of charge and protons across cellular membranes—for instance in the respi-ratory chain or the ATP synthase—is an important contributor to cellular energetics. Thus, we carefully modelled charge- and proton-dependent metabolite transport, and included charge and pH-dependent proton balances. In biochemistry, we typically only work with reactants i (for instance, the reactant ATP). However, reactants actually consist of different chemical species ι (e.g. the reactant ATP consists of the chemical species ATP3+ and ATP4+). Because the thermodynamics and the number of protons/charge translocated in metabolite transport depend on chemical species, here, we used chemical species to model the transport of me-tabolites according to an earlier described approach (35). Further, because the exactly trans-ported species and types of transport mechanism are often not known, we also included for transported reactants a number of different mechanisms (e.g. proton symporters or antiport-ers) with additionally including also variants for transport of different species (table S5). In this way, given the existing uncertainty in the biochemistry of metabolite transport, we did not over-constrain the model by assuming one fixed transport option, but in fact, allowing the model to choose between options. Further, we included a detailed model of the respiratory chain, where we took into account the translocation of electrons and protons in the different complexes of the respiratory chain (table S5, and fig. S1).

For each intra-cellular compartment separately, we included steady-states pH-dependent proton balances, enforcing that the metabolic fluxes are such that the pH in the respective compartment is kept constant. To formulate these proton balances, we determined the com-partment-specific stoichiometric coefficients of proton (h+) appearance or disappearance connected with each metabolic process (table S6). These stoichiometric coefficients were determined based on changes in proton abundance due to the following sub-processes: (i) chemical conversions; (ii) transports of species between compartments with different pH val-ue and the concomitant release or binding of protons caused by the protonation or de-proton-ation of the transported species; (iii) translocations of protons by proton sym-/anti-porters or proton pumps. Combining all these changes (note: depending on the metabolic process, multiple sub-processes operate simultaneously, such as in the ATP synthase or the respira-tory chain complexes), the stoichiometric coefficients for the appearance or disappearance of protons h+ in the respective compartment (the cytosol cyt, the mitochondria mit, and the extracellular space ext) due to metabolic process j ∈ MET become,

S S N s Nh j ij i

H

i cyt mit ext

i

jH

cyt mit ext+ = − +

∈∑

[ / / ] [ / / ]

( )

ι ι −−( )∈( )∧ ≠( )+

∑ Ni ofH

cyt mit ext h

ii

cyt mit ext

ιι ι[ / / ]

( )

[ / / ]

+ ⋅ ∀ ∈+ +s N j MET

h j hH

iii

cyt mit ext[ / / ]

( )

, (Eq. 2)

where Sij is stoichiometric coefficient of the ith reactant for the chemical conversion of j ∈ MET; sιj is the chemical species’ ɩ stoichiometric coefficient for the metabolite transport of j ∈ MET;

26

Chapter 2

NH i is the number of hydrogen atoms H of reactant i; NHɩ is the number of hydrogen atoms of

the species ɩ. The number of hydrogen atoms of the reactants NH i were determined from the dissociation constants of the metabolites and the pH values in the compartments, where we used a pH of 5.0 in the extracellular space, 7.0 in the cytosol and 7.4 in the mitochondria (83). The dissociation constants were predicted using the ChemAxon software (Chemaxon, Buda-pest, Hungary). For the number of hydrogen atoms in the biomass NH biomass we used a value of 67 mmol gCDW-1 (83). We also included an exchange process for the transfer of protons across the system boundary to allow for a change in the pH of the extra-cellular environment.

Finally, we introduced steady-state charge balances for the two intra-cellular compartments. These balances ensure that the membrane potentials across the mitochondrial and the plasma membrane are kept constant. To this end, we defined the stoichiometric coefficients for the changes in the total charge Q[cyt,mit] in the intra-cellular compartments due to the transport of metabolites by processes j ∈ MET (table S7) as

S s z j METQ cyt mit j jcyt mit

[ / ][ / ]

= ∀ ∈∑ ι ιι in

, (Eq. 3)

where zɩ is the charge of the metabolic species ι (table S5). Note, to not constraint the model by an incomplete charge balance, we modeled the respiratory chain and the ion transporters by introducing an unspecific unit-charge species (cf. table S5). This unspecific unit-charge species allowed us to account for the changes in total charges associated with the transfer of electrons across the mitochondrial membrane by the respiratory chain (fig. S1). Further, this unspecific unit-charge species allowed us to not distinguish in the model between specific ions such as potassium, calcium or magnesium, but instead to introduced unspecific ion un-iporters and ATP-driven unspecific ion pumps in the mitochondrial and plasma membrane (table S5), which account for the changes in total charge associated with transport of these ions.

Note, the proton and the charge balances are included into the model by adding the stoichi-ometry coefficients for the changes in the protons (Eq. 2) and the charge (Eq. 3) occurring in each compartment to the stoichiometric matrix S.

Methods 2: Cellular Gibbs energy balance and the cellular entropy production rate σcell

Next to the mass, charge and proton balances, we also introduced a Gibbs energy balance (105), which states that the cellular entropy production rate σcell (Eq. 5) is directly proportional to the Gibbs energy exchange rates gi∈EXG (Eq 7) according to,

− =∈∑T gcell

ii EXGσ , (Eq. 4)

where T is the temperature (here, considered to be 303.15 K). The rate of entropy production by all cellular processes σcell is the sum of the entropy production rates σj∈MET (Eq. 6) over all metabolic processes operating in the cell (105),

σ σcelljj MET

=∈∑ . (Eq. 5)

In turn, the rates of entropy production σj∈MET are defined by,

27

Methods

σ jr j jG v

Tj MET= − ∀ ∈

∆ ’, (Eq. 6)

where vj∈MET are the rates of the metabolic processes, and ∆rG’i are the Gibbs energies of re-action (Eq. 8).

The Gibbs energy exchange rates gi∈EXG depend on the metabolite exchange rates vi∈EXG and the Gibbs energies of formation ∆fG’i∈EXG of the metabolites transferred across the system boundary by the exchange processes,

g G v i EXGi f i i= ∀ ∈∆ ’ . (Eq. 7)

Note, because the rates of the exchange processes do not describe chemical conversions or a metabolite transport, Gibbs energies of formations are used to determine the Gibbs energy exchange rates of the exchange processes, i.e. the transfers of metabolites across the system boundary with their corresponding Gibbs energies of formation.

Methods 3: Gibbs energies of metabolic and exchange processes

The metabolic processes have Gibbs energies of reactions, ∆rG’j∈MET, which are due to chem-ical conversions and/or metabolite transport. Here, we defined the Gibbs energies ∆rG’j∈MET according to,

∆ ∆ ∆r j ij ii hr jo

r jtG RT S cG G j MET’ ln’ ’= ++ ∀ ∈

∉ +∑ , (Eq. 8)

where ∆rG’oj∈MET (Eq. 9) is the standard Gibbs energy of the chemical conversions, ∆rG’tj∈MET (Eq. 10) is the Gibbs energy of the metabolite transports, ln ci is the natural logarithm of the concentration ci of the reactants i (i.e. metabolites), Sij is the stoichiometric coefficient of j ∈ MET, T is the temperature, and R is the universal gas constant. For the Gibbs energy exchange rates gi∈EXG (Eq. 7), we used Gibbs energies of formations, ∆fG’i∈EXG, of the respective reactants i ∈ EXG that are transferred across the system boundary,

∆ ∆f i f io

iG G RT c i EXG’ ’ ln= + ∈ , (Eq. 9)

where ∆fG’oi ∈ EXG are the transformed standard Gibbs energies of formation of the metabolites i ∈ EXG. Note, because the relationships for ∆rG’ (Eq. 8) and ∆fG’ (Eq. 9) are linear in the natural logarithm of the concentrations c, we used ln c as variables in these relationships.

The standard Gibbs energies of reactions, ∆rG’oj∈MET, were calculated by,

∆ ∆r jo

iji h f ioG S G j MET’ ’= ∀ ∈

∉ +∑ , (Eq. 10)

where ∆fG’oi are the standard Gibbs energies of formation of reactants i.

The changes in Gibbs energies accompanying metabolite transport, ∆rG’oj∈MET are due to (i) the transport of species ɩ between compartments with different pH values and the concomitant release or binding of protons caused by the protonation or de-protonation of the transported species, (ii) the translocations of protons by proton sym-/anti-porters or proton pumps; (iii) the transport of charged metabolites across electrical membrane potentials. The Gibbs ener-

28

Chapter 2

gies associated with metabolite transport were calculated as was previously done (35),

∆r jt

jh

i

h in j h in h ouG RT s RT s c c’ ln ln

( )

[ ] [ ] [= + −

∉ + + + +∑ ι ιιγ

tt

ii

Q in j j

iii

FS j MET]

( )

[ ]

( )

( ) + ∀ ∈

∆ϕ, (Eq. 11)

where γι is the fraction of the species ι in the reactant i determined from the dissociation con-stants of the metabolites and the pH in the compartment; sιj is the stoichiometric coefficient for the change in the chemical species ι due to the metabolite transports of j∈MET (table S5); SQj are the stoichiometric coefficients for the changes of the total charges in the intracellular compartments due to transport associated with j ∈ MET (table S7); ∆φj is the membrane potential, where we used a membrane potential of 60 mV across the cytoplasmic membrane (106), and a membrane potential of 160 mV across the inner mitochondrial membrane (107); [in] indicates the compartment at the inner side, and [out] indicates the compartment at the outer side of the membrane, where the inner- and outer-side is defined to match the positive direction of the membrane potential ∆φj; F is Faraday’s constant.

All Gibbs energies used were values transformed (108) (indicated by the apostrophe) to the pH values in the respective, where we used a pH of 5.0 for the extracellular space, 7.0 for the cytosol and 7.4 for the mitochondria (83). Further, we used the extended Debye-Hückel equa-tion to take into account the effect of electrolyte solution on charged metabolites (108), where we used an ionic strength of 0.2 M for all compartments (109). The standard Gibbs energies of formation, ∆fG’o, were estimated from measured equilibrium constants of the enzymatic reac-tions (31) and from the group-contribution method (29) using the component-contribution method (27). With the component contribution method, we also determined standard errors for the estimated standard Gibbs energies of reaction, ∆rG’o. As outlined below, we used these standard errors to later determine a consistent set of the standard Gibbs energies of reaction (table S7). The Gibbs energy of formation of the biomass taken from (110) was transformed to pH 7.0 with an average number of hydrogen atoms in the biomass NH biomass of 67 mmol gCDW-1 (83), and normalized to gram cell dry weight, resulting in a transformed standard Gibbs energy of formation of the biomass of -265.5 J gCDW-1.

Methods 4: Second law of thermodynamics for intracellular processes and biomass formation

The directionalities of the fluxes through the metabolic processes j ∈ MET need to obey the second law of thermodynamics (38), according to,

σ j

r j j

j

G vT

j MET BMSYN H Ot H Otm v

= −

> ∀ ∈( ) ∧ ≠( )∆ ’

\ , ,

0 2 2 0, (Eq. 12)

where the entropy production rate σj of j ∈ MET\BMSYN,H2Ot,H2Otm has to be greater than zero, in case there is flux through this metabolic process. Note, we excluded in (Eq. 12) the biomass synthesis reaction (BMSYN), since we enforce the second law of thermodynam-ics on the biomass synthesis separately (Eq. 14) and further we assumed the water transports (H2Ot, H2Otm) to be fully reversible.

29

Methods

Because a formulation as in (Eq. 12) cannot be used for mathematical optimizations (because optimizations do not allow strict inequalities), we reformulated the second law of thermody-namics as,

= −

≥ ∧ ≥ ∀ ∈σ j

r j j

r j

G vT

G j MET BMSYN H Ot H Otm

∆

∆

’

’ . \ , ,

0 0 5 2 2, (Eq. 13)

where we constrained the absolute value of the Gibbs energies of a reaction |∆rG’j| by a lower bound of 0.5 kJ mol-1 for j ∈ MET\BMSYN,H2Ot,H2Otm. This constraint, |∆rG’j| ≥ 0.5, ensured a positive rate of entropy production, when there is a flux through the metabolic process, and a zero rate of entropy production, when there is no flux through the metabolic process. Note, we choose the technical lower bound of 0.5 kJ mol-1, therefore enforcing in-equality in (Eq. 13) did not introduce numerical instabilities, but was still small enough to not perturb the actual ∆rG’j.

Also the biomass synthesis reaction σBMSYN has to obey the second law of thermodynamics, i.e. it needs to have a positive entropy production rate. Here, since the exact amount of ATP needed for the synthesis of new biomass from precursors is not known (111), we coupled the entropy production rates of the biomass synthesis σBMSYN with that of an ATP hydrolysis reac-tion σATPH, and stated that the sum of the entropy production rates of these processes (i.e. the growth process) has to be positive,

σ σBMSYN ATPH+ ≥ 0 . (Eq. 14)

Note that this approach is similar to the concept of “ATP-maintenance” requirement com-monly used in traditional constraint-based models (111). The differences are that neither a constant ATP hydrolysis rate is used (corresponding to the non-growth associated mainte-nance term used in classical FBA) nor an ATP hydrolysis rate that is stoichiometrically cou-pled to the biomass synthesis (corresponding to the growth associated maintenance term used in classical FBA). By coupling the entropy production rates of the biomass synthesis and the ATP hydrolysis, the ATP demand for the growth process is only enforced by the second law of thermodynamics (Eq. 14).

30

Chapter 2

Methods 5: Formulation of the thermodynamic constraint-based model

We formulated a constraint-based metabolic network model M(v,ln c) ≤ 0, which is a set of equalities and inequalities of the variables v, i.e. the rates of the metabolic processes j ∈ MET and the exchange processes i ∈ EXG and ln c, i.e. the natural logarithm of the concentrations of the metabolites i:

M v c

S v v i

T g

ij jj MET i EXG

cellii EXG

celljj

( , ln ) ≤

= ∀

− =

=

∈ ∈

∈

∑∑

0

σ

σ σ∈∈∑

= − ∀ ∈

∀ ∈

+

=

=

MET

jG v

T

f i i

r jo

r j

i

r j

r j j j MET

G v i EXG

G G

g

G

σ∆

∆

∆ ∆∆

’

’

’ ’’ tt

f i f io

i

j r j

ij ii hRT S c j MET

G G RT c i EXG

G

+ ∀ ∈

= + ∈

> ≥

∉ +∑ ln

’ ’ ln

’

∆ ∆

∆σ 0 0.. \ , , 5 2 2

0

∀ ∈

+ ≥

j MET BMSYN H Ot H Otm

BMSYN ATPHσ σ

, (Eq. 15)

where we combined the relevant equations mentioned above: the mass balances including charge and proton balances (Eq. 1), the cellular Gibbs energy balance (Eq. 4), the equation to calculate the cellular entropy production rate σcell (Eq. 5), the equations to calculate the entropy production rates σj∈MET (Eq.6), the equation to calculate the Gibbs energy exchange rates gi∈EXG (Eq. 7), the equation to calculate the Gibbs energies of reactions ∆rG’j∈MET (Eq. 8), the equation to calculate the Gibbs energies of formation ∆fG’i∈EXG of the metabolites i that are transferred across the system boundary (Eq. 9), the second law of thermodynamics for j ∈ MET (Eq. 13) and the growth process (Eq. 14).

The constraint-based metabolic network model M(v,ln c) ≤ 0 (Eq. 15) together with a set of bounds, B(v,ln c) ≤ 0, on the variables v and ln c, define the solution space Ω. Ω contains the space of mass-, proton- and charge-balanced and thermodynamically-feasible steady-state solutions, in terms of rates v and metabolite concentrations ln c,

Ω = ≤ ∧ ≤ ( , ln ) | ( , ln ) ( , ln )v C M v C B v C0 0 . (Eq. 16)

The set of bounds B(v,ln c) ≤ 0 consist of (combinations are possible) constraints on the rates of the extracellular processes, e.g. the uptake rate of a carbon source, which specifies the growth condition, the physiological ranges of the intracellular metabolite concentrations, ln c, or of the Gibbs energies of reactions, ∆rG’, an upper limit on the cellular entropy production rate, σcell. Note that ∆rG’ and σcell are functions of v and/or ln c, therefore the solution space is defined only in the variables v and ln c.

31

Methods

Here, we analyzed the solution space of the metabolic network model Ω (Eq. 16) using math-ematical optimization, where we formulated different optimization problems, e.g. flux bal-ance-, variability-, and regression analyses. Generally, these optimization problems, which optimize an objective function f of the variables v and ln c in the solution space Ω, had the following form,

f v C f v C v C( *, ln *) min ( , ln ) : ( , ln )= ∈ Ω , (Eq. 17)

where the superscript * indicates the optimal solution for the variables with respect to the objective function f and the solution space Ω of the metabolic network model M(v,ln c) ≤ 0 (Eq. 15). Because Ω is non-convex and non-linear, the optimization problems (Eq. 17) can contain multiple local optima. In order to efficiently solve these problems, we first determined an approximate solution by solving a linear relaxation of the optimization problem with the mixed integer programming solver CPLEX 12 (IBM ILOG, Armonk, USA). This relaxation was based on the mixed integer reformulation of the second law of thermodynamics (Eq. 13) as done in (50), and linear convex hulls (112) of the entropy production rates (Eq. 6) and Gibbs energy exchange rates (Eq. 7). Then, we used this approximate solution as starting point for the solution of the optimization problem (Eq. 17) with the global optimization solver ANTIGONE (113).

Generally, we implemented all these optimization problems in the mathematical program-ming system GAMS (GAMS Development Corporation. General Algebraic Modeling System (GAMS) Release 24.2.2. Washington, DC, USA). The optimization problems were solved on computational clusters, where we used for the model development and testing a small test cluster, which consisted of 30 cores. For the large scale studies, where we solved > 100000 of optimization problems, we set up a cluster in Amazon’s Elastic Compute Cloud, which con-sisted of 1248 cores. Solving these optimization problems typically took between 30 minutes and 10 hours.

Methods 6: Estimation of σcell and standard Gibbs energies of reactions using nonlinear regression analysis

We estimated the cellular rates of entropy production, σcell, (Fig. 1A), and a thermodynamic consistent set of standard Gibbs energies of reactions, ∆rG’o (table S8), i.e. a set of ∆rG’o with the same thermodynamic reference state, from experimental data and the constraint-based model M(v,ln c) ≤ 0 (). The experimental training data consisted of (i) measured extracellu-lar physiological rates v and (ii) intracellular metabolite concentrations c, both determined for glucose-limited chemostat cultures of S. cerevisiae CEN.PK-7D at eight different dilu-tion rates, ranging from 0.02 to 0.39 h-1 (83), and (iii) standard Gibbs energies of reactions, ∆rG’o determined from the component contribution method (27). (Note that the component contribution method cannot determine standard Gibbs energies for all chemical conversions within the metabolic network. Therefore, we determined a thermodynamic consistent set of standard Gibbs energies of reactions through the regression analysis as outlined in the fol-lowing.)

32

Chapter 2

For the regression analysis with the thermodynamic constraint-based model M(v,ln c) ≤ 0, we formulated the solution space of the regression analysis, Ωreg,

Ω ∆ ∆reg k kr

o k k kr

o METv C G M v C G Null S= ∧( , ln , ’ ) | ( , ln , ’ ) (( ) ( ) ( ) ( ) ( ) )) ’

l,( ), ,( ) ,( ),

⋅ =( )∧ ≤ ≤ ∀ ∈( ) ∧

∆ro

iexg k lo

iexg k

iexg k up

G

v v v i GC

0

nn ln ln( ), ( ) ( ),C C C i EMik lo

ik

ik up≤ ≤ ∀ ∈( ) , (Eq. 18)

where we indexed the thermodynamic constraint-based model (including its variables v and ln c) over the different experimental conditions k (i.e. different chemostat cultures). Further, we considered the ∆rG’o as variables and stated that they have to be within the null space of the stoichiometric matrix SMET (which only includes the stoichiometry of the metabolic process-es). This null space constraint enforced the same thermodynamic reference state for all ∆rG’o (27). Additionally, the exchange fluxes were constrained by the growth condition (GC), where we allowed the uptake of glucose, oxygen, phosphate, ammonium, water, protons, sulfate (re-sembling of what was available in the growth medium). Also, we stated that the extracellular metabolite concentrations needed to be within the concentration ranges measured in the me-dium of chemostat cultures at the respective growth condition, where the upper and lower bounds of these concentrations were set to the lowest and highest concentrations measured in the respective replicate experiments (83).

On the basis of the solution space Ωreg and the training data, we formulated a nonlinear regres-sion analysis that was regularized by the Lasso method (114). This regularization—done to prevent over fitting the data—included a regularization parameter α, which was determined by model selection (see below). The regression consisted of two steps: (i) determining the minimal training error errα(y*) (* indicates a value at a determined optimality) as a function of α; (ii) determining the goodness of fit using the reduced chi square χ2

red,α as a function of α. The model selection was performed by repeating these two steps for different α and selecting the α with a reduced chi square χ2

red,α of 1 (here, we found that an α of 0.05 gave the right χ2red,α,

cf. table S10), which means that the model and the data fit each other. In the following, the two steps will be explained in detail:

(i) The training error err(y) is the average loss of the model over the training data using a squared loss (corresponding to the mean squared error) as a measure for the error between the model and the training data (114). Here we determined the squared loss of all standard-ized (by the standard error) measured quantities using,

err yn

y yy

nv v

v

n nmean

nSE

n

ik

ik mean

i

( )#

#

( ) ( ),

=−

=−

∑1

1

2

(( ),,

ln ( ),

( ),

[ ]( )

k SEk i PR

ci

k mean

ik SE

e cc

i ck

+−

∈∑

2

++ −

∈∑

k i MC

C Ci

k mee e ci ck

i mk

,

ln ln ( ),. .[ ]( )

[ ]( )

1

2

0 9 0 1

aan

ik SE

k i MC

r jo

r jo mean

r jo Sc

G G

G

( ),,

,

,

’ ’

’

+

−

∈∑

2

2∆ ∆

∆ EEj CC

∈

∑2

, (Eq. 19)

33

Methods

where yn are the model values, which correspond to the n (#n= 644) measured quantities yn (with means and standard errors SE), i.e. physiological rates v(k)

i∈PR, (PR means physiological rates), intracellular metabolite concentrations c(k)

i (i ∈ MC1 ∪ MC2, see below), and stan-dard Gibbs energies of reactions ∆rG’o j ∈ CC (CC means determined by component contribution method). In order to formulate err(y) (Eq. 19), we transformed the logarithmic concentra-tions ln c back to the linear scale c. Further, for those metabolites that can be present in the cytosol and the mitochondria (i ∈ MC1 metabolites present in one compartment, i ∈ MC2 metabolites present in two compartments), we specified (as it was previously done (38)) that the sum of the metabolite concentrations in the respective compartment weighted by the fractional compartmental volume had to be equal to the measured (cell-averaging) metabolite concentration. Here, we used a fractional compartmental volume of 0.1 for the mitochondria and 0.9 for the cytosol (115). Then, we determined the minimal training error errα(y*) as a function of the regularization parameter α. Here, we minimized the training error errα(y) with an additional Lasso regularization for the standard Gibbs energies of reactions, for which no values could be estimated by the component contribution method, ∆rG’o j∉CC,

err y err v c Gn

Gj PRk

ik

r j CCo

r jo

j Cα

α( ) min ( , ln , ’ )#

’* ( ) ( )= +∈ ∈∉

∆ ∆ CC

k kr

o regv C G

∑∈

:

( , ln , ’ )( ) ( ) ∆ Ω, (Eq. 20)

(ii) The goodness of fit of the regression analysis was determined as a function of the regular-ization parameter α using reduced chi square χ2

red,α,

χ αα

αred

n EPEDF,

#2 =⋅

, (Eq. 21)

where EPEα is the expected prediction error, and DFα is the degree of freedom of the minimal training error errα(y*) (Eq. 20). When the model fits the data, then the reduced chi square is 1. If it is below 1, then the model overfits the data, and if above 1, then the model underfits the data. To estimate the reduced chi square, we first generated using parametric bootstrap (114). To this end we generated (#b=2000) new training data sets y(b) using the optimal model quantities y* from Eq. 20,

yy y y sd

y ynb n

b meann n

SEbn bn

nb SE

nSE

( )( ),

( ),

* , ( , *)=

= + ∼

=

ε ε N 0

∀( , )b n , (Eq. 22)

where the Gaussian noise ε was drawn (using a random number generator) from a normal distribution N with the standard deviation sd*. The standard deviation sd* was determined from the normalized residuals of minimal training error errα(y*) (Eq. 20) with,

sd Ny y

y Ny y

y

meas

SE

meas

SE* * *= − ( )( )−− −∑∑1

11

2

. (Eq. 23)

34

Chapter 2

With the newly generated training data sets y(b) (Eq. 22), we then determined b new minimal training errors errα(y(b)*) by solving Eq. 20 with the training data. Then, based on the b new optimal model quantities y(b)*

n (from the new minimal training errors errα(y(b)*)) and the orig-inal training data y, we estimated the expected prediction error EPEα (114) with,

EPEb n

y yy

nb

nmean

nSE

nbα =

−

∑∑1 1

2

# #*( )

, (Eq. 24)

and the degree of freedom using the effective degrees of freedom (114) with,

DF nsd

EPE err yαα

α α= −( )#*

( *)2 . (Eq. 25)

Additionally, we used the b new optimal model quantities y(b)* to determine the confidence intervals and medians for these model variables. The 97.5% confidence intervals were deter-mined from the 1.25% and 98.75% quantiles of y(b)* and the medians were determined from the 50% quantile of y(b)*.

For several reasons, the optimization problem (Eq. 20) is huge: First, it includes all experi-mental conditions k at once, because the set of thermodynamic consistent standard Gibbs en-ergies of reaction has to be the same across all conditions. Second, the exponential function, which was introduced to transform the logarithmic concentrations to concentrations on the linear scale, introduces additional non-linearity. Therefore, we solved full problem (Eq. 20) in three steps. First, we determined an approximated estimate for the thermodynamic con-sistent set of standard Gibbs energies of reactions by minimizing the training error (cf. Eq. 19) excluding the measured metabolite concentrations (avoiding the exponential functions). Second, we used this approximate estimate for the standard Gibbs energies of reactions to decompose the full optimization problem (Eq. 20) into smaller sub-problems. The model was decomposed by fixing the standard Gibbs energies and then minimizing the training error (cf. Eq. 19) for each experimental conditions independently. Third, we used the approximate solution determined in the second step as a starting point, i.e. approximated model values for the standard Gibbs energies of reactions, metabolite concentrations and metabolic rates, and solved the full optimization problem (Eq. 20), using the local optimization solver CONOPT3 (116). Note, the optimization problems for the parametric bootstrap only required step (iii), since the solution of (Eq. 20) was used as a starting point for these optimizations.

Methods 7: Condition-independent bounds for the concentrations and reaction’s Gibbs energies

Next, we determined the condition-independent physiological bounds for the metabolite concentrations ci (table S9) and for the Gibbs energies ∆rG’j∈MET of the metabolic processes j ∈ MET (table S8). These physiological bounds (lower lo, and upper up), were defined by the infimum and supremum, i.e. the smallest and greatest values, of c and ∆rG’ across all experi-mental conditions k of the training data set,

c c k c c k iilo

ik

iup

ik= = ∀inf : , sup :( ),min ( ),max . (Eq. 26)

35

Methods

and

∆ ∆ ∆ ∆r jlo

r jk

r jup

r jkG G k G G k j MET= = ∀ ∈inf : , sup :( ),min ( ),max . (Eq. 27)

where the superscripts min/max indicate the extreme values of c and ∆rG’ at condition k.

To determine the extreme values for c and ∆rG’ at the different experimental conditions k, we formulated the optimal regression solution space Ωreg*,

Ω Ωreg k k k k regi

ki

kv C v C v v* , ln | , ln( ) ( ) ( ) ( ) ( ), . % ( )= ( ) ( )∈ ∧ ≤ ≤1 25 vv i PR

G G j MET C

ik

r jo

r jo

i ck

( ), . %

, %[ ]( ),’ ’ ln

98 75

50

∀ ∈( )∧ = ∀ ∈( ) ∧∆ ∆ 11 25 98 75 1

1 25 0

. %[ ]( )

[ ]( ), . %

( ), . %

ln ln

.

≤ ≤ ∀ ∈( )∧ ≤

C C i MC

C

i ck

i ck

ik 99 0 1 98 75 2e e C i MCC C

iki c

ki m

kln ln ( ), . %[ ]( )

[ ]( )

.+ ≤ ∀ ∈( ). (Eq. 28)

where we further constrained the solution space of the regression analysis Ωreg (Eq. 18) by fixing the standard Gibbs energies of reactions to the thermodynamic consistent set ∆rG’o,50% (which we had identified by parametric bootstrap, table S8), and constrained to the physio-logical rates i PR and the metabolite concentrations i ∈ MC1 ∪ MC2 by the 97.5% confidence intervals (which we had identified by parametric bootstrap). Then, we determined the ex-treme values of intracellular concentrations c by solving,

C C v C iik

ik k k reg( ),min/max ( ) ( ) ( ) *min/ max ln : , ln= ( )∈ ∀Ω . (Eq. 29)

and the Gibbs energies of the reaction ∆rG’ by solving,

∆ ∆ Ωr jk

r jk k k regG G v C j MET( ),min/max ( ) ( ) ( ) *min/ max : , ln= ( )∈ ∀ ∈ . (Eq. 30)

Please note, since the optimal regression solution space Ωreg* (Eq. 28) had a fixed set of stan-dard Gibbs energies of reactions, it could be decomposed for the different conditions k to facilitate the solution time of these optimization problems (Eq. 29) and (Eq. 30).

Methods 8: Flux balance analyses with the thermodynamic constraint-based model

For different growth conditions, i.e. glucose uptake rates, we predicted metabolic fluxes using the thermodynamic constraint-based model M(v,ln c) ≤ 0 (Eq. 15). Here, we defined solution spaces of the flux balance analysis (FBA) ΩFBA

GUR for M(v,ln c) ≤ 0 (Eq. 15) with varying glu-cose uptake rates (GUR),

Ω

∆ ∆GURFBA

lo upr

lor

v C M v C

C C C G G

= ≤∧ ≤ ≤( ) ∧ ≤ ≤

( , ln ) | ( , ln )

ln ln ln ’ ’

0

∆∆rup

cell cellilo

i iup

glc D EX

G

v v v i GC vup

’

_

( )∧ ≤ ≤( ) ∧ ≤ ≤ ∀ ∈( ) ∧ =−0 σ σ −−( )GUR

. (Eq. 31)

where the metabolite concentrations ln c and the Gibbs energies of the metabolic process-es ∆rG’ were constrained by the identified physiological bounds (tables S8 and S9), the upper limit of the cellular entropy production σcell was constrained by its identified maxi-mum σcellup (Fig. 1B), the extracellular rates were constrained by the growth condition (GC),

36

Chapter 2

such that oxygen, phosphate, ammonium, water, protons, sulfate (resembling of what was available in the growth medium) could be taken up, and all other compounds could be ex-creted.

Then, we used flux balance analyses (FBA) (6), where we maximized the growth rate, µ, in the solution space ΩFBA

GUR (Eq.15),

µGUR BMSYN GURFBAv v C* max : ( , ln )= ∈ Ω . (Eq. 32)

where µGUR* is the optimal growth rate at a specific glucose uptake rate, and BMSYN is the bio-mass synthesis reaction (cf. table S4). We solved the flux balance analysis problems for GURs ranging from .25 to 30 mmol gCDW-1 h-1, where we used intervals of 0.25 mmol gCDW-1 h-1. Thus, we solved the optimization problem in total for 130 GURs. The solution of this optimi-zation problems typically took around 10 hours using a 4 CPUs.

We characterized the solution space ΩFBAµ*(GUR) for optimal growth rates at a given GUR,

Ω Ωµ

µ* ( )( , ln ) |( , ln ) *

GURFBA

GURFBA

BMSYN GURv C v C v= ∈ ∧ =( ) . (Eq. 33)

using flux variability analyses, and, as done earlier (6, 15, 18), using Markov Chain Monte Carlo (MCMC) sampling. Where we first used flux variability analysis to determine the lower and upper values (lo/up) in the solution space for optimal growth rates ΩFBA

µ*(GUR) (Eq. 33) for different model quantities x, i.e. v, ∆rG’, σ, σcell, g, and ln c,

x x v CGUR lo upGUR

FBAµµ

* ( ), /*( )min/ max : ( , ln )= ∈ Ω . (Eq. 34)

Because it is computationally not feasible to sample the non-linear solution space ΩFBAµ*(GUR),

we sampled from the linear convex hull of this solution space ΩFBA,LINµ*(GUR). ΩFBA,LIN

µ*(GUR) was formulated using the extreme values of the different model quantities x, the linear equations of the constraint-based metabolic network model, M(v,ln c) ≤ 0 (Eq. 15), and the linear con-vex hulls (112) of the entropy production rates (Eq. 6) and Gibbs energy exchange rates (Eq. 7). Then, we used artificial centering hit and run sampling (117) to sample the linear con-strained space ΩFBA,LIN

µ*(GUR). In order to rigorously sample this solution space, we generated 10000 sampled points in ΩFBA,LIN

µ*(GUR) representing quantities xLIN,(s), i.e. vLIN,(s), ∆rG’LIN,(s), σLIN,(s), σcell,LIN,(s), gLIN,(s), and ln cLIN,(s)) of the linearized model per GUR. Because the sampling algo-rithm always performed 1000 steps between each sampled point, we generated for each GUR in total 10’000’000 points in the linearized solution space. Then, we estimated for the model quantities x the 97.5% confidence intervals with the 1.25% and 98.75% quantiles of xLIN,(s), and the median with the 50% quantile of xLIN,(s). Similarly, the standard deviation and the average of x were also estimated from the sampled xLIN,(s).

37

Supplementary figures

inner mitochondrial-membrane

inter-membranespace

outer mitochondrial-membrane

cytosol

mitochondrialmatrix

pH 7

pH 7.4

160 mV

- - - - -

++++

pH 7

CoQ

CoQH2

2 e-

2 H+

2 CytC2+ 2 CytC3+ 2 CytC2+

4 H+

2 H+

2 H+

H2O1/2 O2 + 4 H+

2 e-

Complex IVComplex III

CoQ CoQH2

2 e-

FADFADH2

2 e-

FumarateSuccinate

Complex II

CoQ CoQH2

2 H+

2 e-

NAD+NADH + H+

Internal NADHdehydrogenase

2 H+2 e-

NAD+NADH + H+

External NADHdehydrogenase

Glycerol-3-phosphate dehydrogenase

DHAPGlyc-3-P

2 e-

FADFADH2

2 e-

3 ADP3 ATP

10 H+

10 H+

ATP synthase

0 10-10

10

0

-100 50 100

0

50

100

Measured value (mM)

Mod

el v

alue

(mM

)

Measured value(mmol gCDW-1 h-1)

Mod

el v

alue

(mm

ol g

CDW

-1 h

-1)

Extracellular ratesIntracellular

concentrations

a b

Supplementary figures

Figure S1. Model of the respiratory chain that was included into the metabolic network. The ubiquinone pool and the cytochrome c were considered to be localized in the cytosol (118), and we assumed that the internal and exter-nal NADH dehydrogenases do not pump protons according to (119). Further, the inter-membrane space had the same pH as the cytosol. The black dashed line indicates the cytosolic or mitochondrial localization of metabolites in the model.

Figure S2. Model values versus measured values (a: extracellular rates; b: intracellular metabolite concentrations) from the regression analysis with a regularization factor a of 0.05. The plot shows the values for all conditions.

38

Chapter 2

ALCD2xALCD2m, ETOHtm+0

CO2t-1+HCO2tm-1+H

MDHmICDHym

AKGMALtm-2-2MALPItm-2-2

FUMmCITMALtm-3-2

MDHPYRtm-1+H

PPCKACONT

MEmPItm-2+H

ADK1m, PPAm, ACSm3MOPtm-1+H

SUCCt-2+HgmpSYN1

ACONTmH2Otm+0

GLXtm-1+H

OAAtm-2+2H

DehydrogenaseMitochondrial transporterPeriplasmatic transporter

Hydro-lyaseOthers

Type of process:

0 5 10 15 20GUR (mmol gCDW-1 h-1)

unidirectionalbidirectionalProcess directionality:

Figure S3. Clusters of linear independent metabolic processes with similar trends in the entropy production rate. The average entropy production rates, σ, of the processes at a given GUR were determined from the sampled points of the solution space (methods 8) and then normalized to the average cellular entropy production rate σcell at the given GUR. We identified the clusters using consensus clustering (120) with partitioning around me-doids (121) as the underlying cluster algorithm, where we used the Euclidean distance of the σcell -normalized entropy production rates as a distance measure.

39

Supplementary figures

Figure S4. Changes in the σcellup-con-strained FBA predicted directional-ity (unidirectional/bidirectional) of the linear independent metabolic processes. The directionality of a metabolic process was determined from the σcellup-constrained FBA predicted flux variability at a given GUR. Note, the plot only shows processes which change their direc-tionality between GURs.PYK

PYRDC

CO2t-1ATPASE

ADPATPtm-4-3

NADH2-u6mComplexIII, ComplexIV, O2tm+0

G3PD2NADH2-u6i

CO2tm-1

FBA

G3PD1

O2t

ADPATPtm-4-2

ATPS3m

PGL, G6PDH2, GND

MDH

ALCD2m, ETOHtm+0

ACALDtm+0

PItm-2+H

PGI

TPI

CSm

ComplexII

NH3t+1

ICDHxm

ETOHt+0

PPCK

ALDD2ym

CO2tm-1+H

CO2t-1+H

AKGDm, SUCOAS1m

PDHm

PGPS

ALDD2xm

ATPHYD

G3PTGLYK

ACOAH

ORNtm+1

NH3tm+1

POSQtm, IONPUMPm

PYRtm-1

OAAtm-2+H

GLUtm-1

OAAtm-2

ACS

ALDDy

MALPItm-2-2

MEm

ICDHym

PYRtm-1+H

PItm-2+2H

GLUDy

ASPK, ASAD, HSDy

GLNS

PPA

MDHm

ASPTAm, ASPGLUtm-1-1

OAAtm-2+2H

FUMmFUM

PItm-2

SUCCPItm-2-2

ALATA_Lm

ADK1m, PPAm, ACSm

GLUtm-1+H

ASPTA

2OBUTtm-1

3MOBtm-1+H

Htm

3MOPtm-1+H

ALAtm+0

ALATA_L

SUCFUMtm-2-2

CITMALtm-3+H-2

3MOPtm-1

GLXtm-1+H

AKGMALtm-2-2

2OBUTtm-1+H3MOBtm-1

ACtm-1+H

GLXtm-1

ACtm-1

CITMALtm-3-2

SUCCt-2+H

ACONTm

CITICITtm-3-3

ICDHy

ICL

ACONT

AGTm, GLYtm+0

FDH

CS

SO4t-2

MALS

FBP

PIt-1+H

GLYCt+0

FTHFLMTHFC, MTHFD

GHMT2

GLUDym

NH3tm+1-H

GLUSxm, GLNtm+0

ORNtm+1-H

PC

TALA, TKT1

THRA

PIt-1

gmpSYN1

RPI

TKT2

gmpSYN2

RPE

ampSYN1ampSYN2

POSQt

SO4t-2+H

ADK1

Ht

PYRt-1+H

ACt-1

PYRt-1

NH3t+1-H

SUCCt-2

SO4t-2+2H

ACt-1+H

SUCCt-2+2H

HSK, THRS

IONPUMPACLD2x

HEX1, GLCt+0

ENO, PGM

GAPD, PGK

Reaction group 1*PFK

0 5 10 15 20GUR (mmol gCDW-1 h-1)

Glycolysis (12)

Respiration (6)

Pyruvate decarboxylase (1)Biomass synthesis (86)

Other processes (132)

Energy metabolism (3)

Pyruvate kinase (1)

0.0 0.1 0.2

σ / σcell (-)

* Reaction group 1:BMSYN, ASNS1, GLCS2, PGMT,TRESYN, GALU,ACGKm, ACOTAm,ARGSL, ARGSS, CBPS,GLU5K, G5P5, AGPRm, OCBT, ORNTACm,cmpSYN, dampSYN, dcmpSYN, dgmpSYN,dtmpSYN, ergstSYN,umpSYN, zymstSYN,BPNT, SO4SO3, SULR, MTHFR3, ANS, CYSTS,PRMICIi, ATPPRT,HISTD, HISTP, HSTPT,IGPDH, PRATPAMPC,PRPPS, METS, CYSTGL,HSERTA, AHSERL2,NDPK2, HCITSm, AATA,MCITDm, HACNHm,HICITDm, AASAD1,THRD_Lm, OXAGm,SACCD2, SACCD1,DHQT, DHQS, DDPA,PSCVTi, ANPRT,CHORM, CHORS,IGPS, PHETA1, PRAIi,PPNDH, PPND2, SHK3D,SHKK, TRPS1, TYRTA,ACHBSm, IPPMIb, IPPS,OMCDC, IPPMIa, IPMD, KARA1m, ACLSm,DHAD1m, DHAD2m, ILETA, KARA2m, LEUTA, VALTA, THRtm+0,2OXOADPAKGtm-2-2

40

Chapter 2

0.0 0.2 0.4 0.6 0.8Distance to Pareto surface (-)

Samples of thecomplete solution space

Samples of the σcellup-constrainedFBA predictions at a GUR of 15 mmol gCDW-1 h-1

0

0.1

0.2

0.3

Frac

tion

of s

ampl

es (-

)

Figure S5. Sampled points of the σcellup-constrained FBA predictions at a GUR of 15 mmol gCDW-1 h-1 are closer to the Pareto surface (comprised of three biological rel-evant objective functions, i.e. maximization of biomass yield, maximization of ATP yield and minimization of the sum of absolute fluxes) than random sampled points of the thermodynamic constraint-based model M(v,ln c) ≤ 0 (Eq. 15) without the σcellup-constraint. The Pareto surface of M(v,ln c) ≤ 0 (σcellup-constrained) was determined using the ε-constraint method, where we used 2500 grid points to describe the Pareto surface. The distance of a sampled point was defined by the Euclidean distance to the Pareto surface were we weighed each objective by its minimum and maximum value. For further details on the Pareto op-timality analysis refer to (15).

41

Supplementary tables

Supplementary tables

Abbreviation Metabolite Molecular formulaKEGG

ID

10fthf10-formyl-

tetrahydrofolateC20H23N7O7 C00234

13dpg3-phospho-D-

glyceroyl phosphateC3H8O10P2 C00236

23dhmb(R)-2,3-Dihydroxy-3-methylbutanoate

C5H10O4 C04272

23dhmp(R)-2,3-Dihydroxy-3-methylpentanoate

C6H12O4 C06007

2ahbut(S)-2-aceto-2-

hydroxybutanoateC6H10O4 C06006

2cpr5p

1-(2-carboxy-phenylamino)-

1-deoxy-D-ribulose 5-phosphate

C12H16NO9P C01302

2dda7p2-dehydro-3-deoxy-D-arabino-heptonate

7-phosphateC7H13O10P C04691

2ippm 2-isopropylmaleate C7H10O4 C02631

2obut 2-oxobutanoate C4H6O3 C00109

2oxoadp 2-oxoadipate C6H8O5 C00322

2pg2-phospho-D-

glycerateC3H7O7P C00631

34hpp3-

(4-hydroxyphenyl)-pyruvate

C9H8O4 C01179

3c2hmp(2R,3S)-3-

IsopropylmalateC7H12O5 C04411

3c3hmpAlpha-isopropyl

malateC7H12O5 C02504

3c4mop(2S)-2-isopropyl-3-

oxosuccinateC7H10O5 C04236

3dhq 3-dehydroquinate C7H10O6 C00944

3dhsk 3-dehydroshikimate C7H8O5 C02637

3ig3pIndole-glycerol

phosphateC11H14NO6P C03506

3mob3-methyl-2-

oxobutanoic acidC5H8O3 C00141

3mop(S)-3-methyl-2-

oxopentanoic acidC6H10O3 C00671

3pg3-phospho-D-

glycerateC3H7O7P C00197

3psme5-O-(1-

Carboxyvinyl)-3-phosphoshikimate

C10H13O10P C01269

4mop4-methyl-2-

oxopentanoateC6H10O3 C00233

4pasp4-phospho-L-

aspartateC4H8NO7P C03082

5mthf5-methyl-

tetrahydrofolateC20H25N7O6 C00440

Abbreviation Metabolite Molecular formulaKEGG

ID

6pgc6-phospho-D-

gluconateC6H13O10P C00345

6pglD-glucono-1,5-

lactone 6-phosphateC6H11O9P C01236

ac Acetate C2H4O2 C00033

acald Acetaldehyde C2H4O C00084

accoa Acetyl-CoA C23H38N7O17P3S C00024

acg5pN-acetyl-L-glutamate

5-phosphateC7H12NO8P C04133

acg5saN-acetyl-L-glutamate

5-semialdehydeC7H11NO4 C01250

acglu N-acetyl-L-glutamate C7H11NO5 C00624

achmsO-acetyl-L-homoserine

C6H11NO4 C01077

acorn N-acetylornithine C7H14N2O3 C00437

adp ADP C10H15N5O10P2 C00008

akg 2-oxoglutarate C5H6O5 C00026

alac-S (S)-2-acetolactate C5H8O4 C06010

ala-L L-alanine C3H7NO2 C00041

amp AMP C10H14N5O7P C00020

anth Anthranilate C7H7NO2 C00108

arg-L L-arginine C6H14N4O2 C00062

argsucN-(L-arginino)-

succinateC10H18N4O6 C03406

asn-L L-asparagine C4H8N2O3 C00152

asp-L L-aspartate C4H7NO4 C00049

aspsaL-aspartate

4-semialdehydeC4H7NO3 C00441

atp ATP C10H16N5O13P3 C00002

b124tc(Z)-but-1-ene-1,2,4-

tricarboxylateC7H8O6 C04002

biomass

cbpCarbamoyl phosphate

CH4NO5P C00169

charge unspecific charge

chor Chorismate C10H10O6 C00251

cit Citrate C6H8O7 C00158

citr-L L-citrulline C6H13N3O3 C00327

cmp CMP C9H14N3O8P C00055

co2totDissolved carbon

dioxideHCO3 C00288

coa CoA C21H36N7O16P3S C00010

coq UbiquinoneC14H18O4(C5H8)

nC00399

coqh2 UbiquinolC14H20O4(C5H8)

nC00390

cys-L L-cysteine C3H7NO2S C00097

cyst-L L-cystathionine C7H14N2O4S C02291

damp dAMP C10H14N5O6P C00360

Table S1. Metabolites of the model.

table continues next page

42

Chapter 2

Abbreviation Metabolite Molecular formulaKEGG

ID

dcmp dCMP C9H14N3O7P C00239

dgmp dGMP C10H14N5O7P C00362

dhap Glycerone phosphate C3H7O6P C00111

dtmp dTMP C10H15N2O8P C00364

e4pD-erythrose 4-phosphate

C4H9O7P C00279

eig3pD-erythro-1-

(Imidazol-4-yl)-glycerol 3-phosphate

C6H11N2O6P C04666

ergst Ergosterol C28H44O C01694

etoh Ethanol C2H6O C00469

f6pD-Fructose 6-phosphate

C6H13O9P C00085

fdpD-Fructose

1,6-bisphosphateC6H14O12P2 C00354

ficytc Ferricytochrome cC42H44

FeN8O8S2R4C00125

focytc Ferrocytochrome cC42H44

FeN8O8S2R4C00126

for Formate CH2O2 C00058

fumarate Fumarate C4H4O4 C00122

g1pD-glucose

1-phosphateC6H13O9P C00103

g3pD-glyceraldehyde

3-phosphateC3H7O6P C00118

g6pD-glucose

6-phosphateC6H13O9P C00092

glc-D D-glucose C6H12O6 C00031

gln-L L-glutamine C5H10N2O3 C00064

glu5pL-glutamyl

5-phosphateC5H10NO7P C03287

glu-L L-glutamate C5H9NO4 C00025

glx Glyoxylate C2H2O3 C00048

gly Glycine C2H5NO2 C00037

glyc Glycerol C3H8O3 C00116

glyc3pSn-glycerol 3-phosphate

C3H9O6P C00093

glycogen Glycogen C24H42O21 C00182

gmp GMP C10H14N5O8P C00144

h H+ H C00080

h2o H2O H2O C00001

h2s Hydrogen sulfide H2S C00283

hcit(R)-2-

hydroxybutane-1,2,4-tricarboxylate

C7H10O7 C01251

hcys-L L-homocysteine C4H9NO2S C00155

hicit Homoisocitrate C7H10O7 C05662

his-L L-histidine C6H9N3O2 C00135

hispL-histidinol phosphate

C6H12N3O4P C01100

hist L-histidinol C6H11N3O C00860

Abbreviation Metabolite Molecular formulaKEGG

ID

hom-L L-homoserine C4H9NO3 C00263

icit Isocitrate C6H8O7 C00311

ile-L L-isoleucine C6H13NO2 C00407

imacp3-(Imidazol-4-yl)-2-oxopropyl phosphate

C6H9N2O5P C01267

L2aadp L-2-aminoadipate C6H11NO4 C00956

L2aadp6saL-2-aminoadipate 6-semialdehyde

C6H11NO3 C04076

leu-L L-leucine C6H13NO2 C00123

lys-L L-lysine C6H14N2O2 C00047

mal-L (S)-malate C4H6O5 C00149

methf5,10-methenyl-tetrahydrofolate

C20H22N7O6 C00445

met-L L-methionine C5H11NO2S C00073

mlthf5,10-methylene-tetrahydrofolate

C20H23N7O6 C00143

nad NAD+ C21H28N7O14P2 C00003

nadh NADH C21H29N7O14P2 C00004

nadp NADP+ C21H29N7O17P3 C00006

nadph NADPH C21H30N7O17P3 C00005

nh3 NH3 NH3 C00014

o2 Oxygen O2 C00007

oaa Oxaloacetate C4H4O5 C00036

orn L-ornithine C5H12N2O2 C00077

oxag Oxaloglutarate C7H8O7 C05533

papAdenosine

3',5'-bisphosphateC10H15N5O10P2 C00054

pepPhosphoenol-

pyruvateC3H5O6P C00074

phe-L L-phenylalanine C9H11NO2 C00079

phomO-phospho-L-

homoserineC4H10NO6P C01102

phpyr Phenylpyruvate C9H8O3 C00166

pi Orthophosphate H3PO4 C00009

pphn Prephenate C10H10O6 C00254

ppi Diphosphate H4P2O7 C00013

pranN-(5-phospho-D-

ribosyl)-anthranilateC12H16NO9P C04302

prbatp1-(5-phospho-D-

ribosyl)-ATPC15H25N5O20P4 C02739

prfp

5-(5-phospho-D-ribosyl-amino-

formimino)-1-(5-phosphoribosyl)-

imidazole-4-carboxamide

C15H25N5O15P2 C04896

Table S1. Metabolites of the model (continued).

table continues next page

43

Supplementary tables

Abbreviation Metabolite Molecular formulaKEGG

ID

prlp

N-(5’-phospho-D-1’-ribulosyl-formimino)-5-amino-1-(5’’-

phospho-D-ribosyl)-4-imidazole-carboxamide

C15H25N5O15P2 C04916

pro-L L-proline C5H9NO2 C00148

prpp5-phospho-alpha-D-ribose 1-diphosphate

C5H13O14P3 C00119

pyr Pyruvate C3H4O3 C00022

r5pAlpha-D-Ribose

5-phosphateC5H11O8P C03736

ru5p-DD-ribulose

5-phosphateC5H11O8P C00199

s7pSedoheptulose 7-phosphate

C7H15O10P C05382

saccrp-LN6-(L-1,3-

dicarboxypropyl)-L-lysine

C11H20N2O6 C00449

ser-L L-serine C3H7NO3 C00065

skm Shikimate C7H10O5 C00493

Abbreviation Metabolite Molecular formulaKEGG

ID

skm5pShikimate

5-phosphateC7H11O8P C03175

so3 Sulfite H2SO3 C00094

so4 Sulfate H2SO4 C00059

succ Succinate C4H6O4 C00042

succoa Succinyl-CoA C25H40N7O19P3S C00091

thf Tetrahydrofolate C19H23N7O6 C00101

thr-L L-threonine C4H9NO3 C00188

treAlpha,alpha-

TrehaloseC12H22O11 C01083

trp-L L-tryptophan C11H12N2O2 C00078

tyr-L L-tyrosine C9H11NO3 C00082

udp UDP C9H14N2O12P2 C00015

udpg UDP-glucose C15H24N2O17P2 C00029

ump UMP C9H13N2O9P C00105

utp UTP C9H15N2O15P3 C00075

val-L L-valine C5H11NO2 C00183

xu5p-DD-xylulose

5-phosphateC5H11O8P C00231

zymst Zymosterol C27H44O C05437

Table S1. Metabolites of the model (continued).

Table S2. Names of the metabolic processes of the model.

Abbreviation Metabolic process Protein

ASNS1 Asparagine synthase(Asn2) or (Asn1) or

(Asn3)

ASPK Aspartate kinase Hom3

ASPTAAspartate transaminase,

cytosolicAat2

ASPTAmAspartate transaminase,

mitochondrialAat1-m

ASADAspartate-semi-aldehyde

dehydrogenaseHom2

ALATA_LL-alanine transaminase,

cytosolicAlt1

ALATA_LmL-alanine transaminase,

mitochondrialAlt2-m

GLCS2 Glycogen synthase(Gsy1) or

(Gsy2)

PGMT Phosphoglucomutase(Pgm1) or

(Pgm2)

TRESYN trehalose synthase

GALUUTP-glucose-1-phosphate

uridylyltransferase(Ugp1) or

(Ugp2)

FBP Fructose-bisphosphatase Fbp1

ICL Isocitrate lyase Icl1

MALS Malate synthase

MEmMalic enzyme, mitochondrial

Mae1-m

PPCKPhosphoenolpyruvate

carboxykinasePck1

Abbreviation Metabolic process Protein

PC Pyruvate carboxylase (Pyc1) or (Pyc2)

ACGKmAcetylglutamate kinase,

mitochondrialArg5-m

ACOTAmActeylornithine transaminase, mitochondrial

Arg8-m

ARGSL Argininosuccinate lyase Arg4

ARGSSArgininosuccinate

synthaseArg1

CBPSCarbamoyl-phosphate

synthase Cpa

GLU5K glutamate 5-kinase Pro1

G5P5

glutamate-5-semialdehyde dehydrogenase/L-

glutamate 5-semialdehyde dehydratase/Pyrroline-5-

carboxylate reductase

(Pro2 and Pro3)

AGPRmN-acetyl-g-glutamyl-phosphate reductase,

mitochondrialArg5-m

OCBTOrnithine

carbamoyltransferaseArg3

ORNTACmOrnithine transacetylase,

mitochondrialEcm40-m

ampSYN1 AMP synthesis

ampSYN2 AMP synthesis

cmpSYN CMP synthesis

dampSYN dAMP synthesis

table continues next page

44

Chapter 2

Abbreviation Metabolic process Protein

dcmpSYN dCMP synthesis

dgmpSYN dGMP synthesis

dtmpSYN dTMP synthesis

ergstSYN Ergosterol synthesis

gmpSYN1 GMP synthesis

gmpSYN2 GMP synthesis

umpSYN UMP synthesis

zymstSYN Zymosterol synthesis

ACONTAconitate hydratase,

cytosolicAco1

ACONTmAconitate hydratase,

mitochondrial(Aco1-m) or

(Aco2-m)

AKGDmOxoglutarate

dehydrogenase, mitochondrial

(Kgd1-m) and (Kgd2-m) and

(PdE3-m)

CS Citrate synthase, cytosolic Cit2

CSmCitrate synthase, mitochondrial

(Cit1-m) or (Cit3-m)

ICDHxmIsocitrate dehydrogenase,

mitochondrialIdh-m

ICDHyIisocitrate dehydrogenase,

cytosolicIdp2

ICDHymIsocitrate dehydrogenase,

mitochondrialIdp1-m

SUCOAS1mSuccinate--CoA ligase,

mitochondrialLsc-m

BPNT3',5'-bisphosphate

nucleotidaseMet22

SO4SO3

Sulfate adenylyltransferase/adenylyl-Sulfate kinase/phosphoadenylyl-sulfate

reductase

((Met3 and Ipp1) and Met16

and Met14)

SULR Sulfite reductase (Ecm17) or

(Met10)

MTHFR35,10-Methylene-

tetrahydrofolatereductase(Met12) or

(Met13)

FTHFLFormate-tetrahydrofolate

ligaseAde3

MTHFCMethenyltetrahydrofolate

cyclohydrolaseAde3

MTHFDMethylenetetrahydrofolate

dehydrogenase Ade3

ANS Anthranilate synthase (Trp2) or (Trp3)

GLUDymGutamate dehydrogenase,

mitochondrialGdh1

GLUDyGlutamate dehydrogenase,

cytosolic(Gdh2) or

(Gdh3)

GLUSxmGlutamate synthase,

mitochondrial

GLNS Glutamine synthetase Gln1

ALCD2mAlcohol dehydrogenase,

mitochondrialAdh3-m

GLYK Glycerol kinase Gut1

Abbreviation Metabolic process Protein

G3PT Glycerol-3-phosphatase(Hor2) or

(Rhr2)

G3PD1glycerol-3-phosphate

dehydrogenaseGpd1

G3PD2glycerol-3-phosphate

dehydrogenaseGpd1

AGTmAlanine-glyoxylate

transaminase, mitochondrial

Agt1

CYSTSCystathionine beta-

synthaseCys4

GHMT2Glycine hydroxyl-methyl-

transferaseShm2

HSDyHomoserine

dehydrogenaseHom6

HSK Homoserine kinase Thr1

PGPS

Phosphoglycerate dehydrogenase/ phosphoserine transaminase/

phosphoserine phosphatase

(Ser1 and Ser2 (Ser3) or

(Ser33))

ENO Enolase(Err2) or (Eno1)

or (Eno2) or (Err1) or (Err3)

FBAFructose-bisphosphate

aldolaseFba1

PGIGlucose-6-phosphate

isomerasePgi1

GAPDGlyceraldehyde-3-

phosphate dehydrogenase

(Tdh1) or (Tdh2) or

(Tdh3)

PDHm Pyruvate dehydrogenase(PdE1-m and PdE2-m and

PdE3-m)

HEX1 Hexokinase(Hxk1) or (Hxk2) or

(Glk1)

PFK Phosphofructokinase Pfk

PGK Phosphoglycerate kinase Pgk1

PGM Phosphoglycerate mutase(Gpm1) or (Gpm2) or

(Gpm3)

PYK Pyruvate kinase(Cdc19) or

(Pyk2)

TPITriose-phosphate

isomeraseTpi1

PRMICIi

1-(5-phosphoribosyl)-5-[(5-phosphoribosylamino)

methylideneamino)imidazole-4-carboxamide

isomerase

His6

ATPPRTATP

phosphoribosyltransferaseHis1

HISTD Histidinol dehydrogenase His4

Table S2. Names of the metabolic processes of the model (continued).

table continues next page

45

Supplementary tables

Abbreviation Metabolic process Protein

HISTP Histidinol-phosphatase His2

HSTPTHistidinol-phosphate

transaminaseHis5

IGPDHImidazoleglycerol-

phosphate dehydrataseHis3

PRATPAMPC

Phosphoribosyl-ATP pyrophosphohydrolase /

phosphoribosyl

-AMP cyclohydrolase

His4

PRPPSPhosphoribosyl-

pyrophosphate synthetase

(Prs5) or (Prs4) or (Prs2) or

(Prs3) or (Prs1)

FDH Formate dehydrogenase(Fdh1) or (Fdh2) or

(Fdh3)

METS Methionine synthase Met6

CYSTGL Cystathionine g-lyase Cys3

HSERTAHomoserine O-trans-

acetylaseMet2

AHSERL2O-acetylhomoserine

(thiol)-lyaseMet17

ADK1 Adenylate kinase Adk1

ADK1mAdenylate kinase,

mitochondrialAdk2-m

ATPHYD ATP hydrolysis

NDPK2Nucleoside-diphosphate

kinase Ynk1

NADH2-u6iNADH dehydrogenase,

external(Nde1-m) or

(Nde2-m)

FUM Fumarase Fum1

FUMm Fumarase, mitochondrial Fum1-m

PPAInorganic diphosphatase,

cytosolicIpp1

PPAmInorganic diphosphatase,

mitochondrialPpa2-m

MDHMalate dehydrogenase,

cytosolicMdh2

MDHmMalate dehydrogenase,

mitochondrialMdh1-m

PGL6-phospho-

gluconolactonase

(Sol1) or (Sol3) or (Sol4) or

(Sol2)

G6PDH2Glucose 6-phosphate

dehydrogenaseZwf1

GNDPhosphogluconate

dehydrogenase(Gnd2) or

(Gnd1)

RPIRibose-5-phosphate

isomeraseRki1

RPERibulose 5-phosphate

3-epimeraseRpe1

TALA Transaldolase (Tal1) or (Tal2)

TKT2 Transketolase (Tkl1) or (Tkl2)

TKT1 Transketolase (Tkl2) or (Tkl1)

ACOAH Acetyl-CoA hydrolase Ach1

Abbreviation Metabolic process Protein

ACSAcetyl-CoA synthetase,

cytosolicAcs2

ACSmAcetyl-CoA synthetase,

mitochondrialAcs1-m

ALCD2x Alcohol dehydrogenase

(Adh4) or (Adh5) or (Sfa1)

or (Adh1) or (Adh2)

HCITSm Homocitrate synthase

PYRDC Pyruvate decarboxylase(Pdc6) or (Pdc5)

or (Pdc1)

AATA2-aminoadipate

transaminase

MCITDm2-methylcitrate

dehydratase, mitochondrial

HACNHmHomoacontinate

hydratase, mitochondrialLys4-m

HICITDmHomoisocitrate dehydrogenase, mitochondrial

Lys12-m

AASAD1L-aminoadipate-

semialdehyde dehydrogenase

Lys25

THRD_LmL-threonine deaminase,

mitochondrialIlv1-m

OXAGmNon-enzymatic reaction,

mitochondrial

SACCD2Saccharopine

dehydrogenaseLys1

SACCD1Saccharopine

dehydrogenaseLys9

THRA Threonine aldolase Gly1

THRS Threonine synthase Thr4

DHQT3-dehydroquinate

dehydrataseAro1

DHQS 3-dehydroquinate synthase Aro1

DDPA3-deoxy-D-arabino-

heptulosonate 7-phosphate synthase

(Aro4) or (Aro3)

PSCVTi3-phosphoshikimate

1-carboxyvinyltransferaseAro1

ALDD2yAldehyde dehydrogenase,

cytosolicAld6

ALDD2xmAldehyde dehydrogenase,

mitochondrialAld4-m

ALDD2ymAldehyde dehydrogenase,

mitochondrialAld4-m

ANPRTAnthranilate

phosphoribosyltransferaseTrp4

CHORM Chorismate mutase Aro7

CHORS Chorismate synthase Aro2

IGPSIndole-3-glycerol-

phosphate synthaseTrp3

Table S2. Names of the metabolic processes of the model (continued).

table continues next page

46

Chapter 2

Abbreviation Metabolic process Protein

PHETA1Phenylalanine transaminase

Aro9

PRAIiPhosphoribosyl-

anthranilate isomeraseTrp1

PPNDH Prephenate dehydratase Pha2

PPND2 Prephenate dehydrogenase Tyr1

SHK3D Shikimate dehydrogenase Aro1

SHKK Shikimate kinase Aro1

TRPS1 Ttryptophan synthase Trp5

TYRTA Tyrosine transaminase Aat2

ACHBSm2-aceto-2-

hydroxybutanoate synthase, mitochondrial

Ilv26-m

IPPMIb2-isopropylmalate

hydrataseLeu1

IPPS 2-isopropylmalate synthase (Leu4) or (Leu5)

OMCDC2-oxo-4-methyl-3-carboxypentanoate

decarboxylationBat2

IPPMIa3-isopropylmalate

dehydrataseLeu1

IPMD3-isopropylmalate

dehydrogenaseLeu2

KARA1mAcetohydroxy acid isomeroreductase,

mitochondrialIlv5-m

ACLSmAcetolactate synthase,

mitochondrialIlv26-m

DHAD1m

Dihydroxy-acid dehydratase

(2,3-dihydroxy-3-methylbutanoate),

mitochondrial

Ilv3-m

DHAD2m

Dihydroxy-acid dehydratase

(2,3-dihydroxy-3-methylpentanoate),

mitochondrial

Ilv3-m

ILETA isoleucine transaminase Bat2

KARA2m

Ketol-acid reductoisomerase

(2-Aceto-2-hydroxybutanoate),

mitochondrial

Ilv5-m

LEUTA Leucine transaminase Bat2

VALTA Valine transaminase Bat2

NADH2-u6mNADH dehydrogenase

mitochondrial(Ndi1-m) or

(Acp1-m)

ComplexIISuccinate dehydrogenase,

mitochondrial

(Sdh-m) or (Sdh2-m) or (Sdh3-m) or

(Sdh4-m)

ComplexIIIUbiquinol-6 cytochrome c

reductaseCbc1-m

Abbreviation Metabolic process Protein

ComplexIV Cytochrome c oxidase(Cco1-m) or

(Cco2-m)

ATPS3mATP synthase, mitochondrial

(Atps2-m) or (Atps1-m)

ATPASE ATPase, periplasmic(Pma1) or

(Pma2)

POSQtCharge transport,

periplasmic

POSQtmCharge transport,

mitochondrial

Ht Proton leak, periplasmic

Htm Proton leak, mitochondrial

IONPUMP ATPase, periplasmic(Pma1) or

(Pma2)

IONPUMPm ATPase, mitochondrial(Pma1) or

(Pma2)

ACt-1Acetate transport,

periplasmicADY2

ACt-1+HAcetate transport,

periplasmicADY2

CO2t-1CO2 transport,

periplasmic

CO2t-1+HCO2 transport,

periplasmic

ETOHt+0Ethanol transport,

periplasmic

GLCt+0Gglucose transport,

periplasmic

(Hxt4) or (Gal2) or (Hxt11) or

(Stl1) or (Hxt1) or (Hxt13) or (Hxt15) or (Hxt16)

or (Hxt10) or (Hxt17) or

(Hxt2) or (Hxt3) or (Hxt5) or

(Hxt6) or (Hxt7) or (Hxt8) or

(Hxt9)

GLYCt+0Glycerol transport,

periplasmicFps1

H2Ot+0H2O transport,

periplasmic

NH3t+1NH3 transport,

periplasmic

(Mep1) or (Mep2) or

(Mep3)

NH3t+1-HNH3 transport,

periplasmic

(Mep1) or (Mep2) or

(Mep3)

O2t O2 transport, periplasmic

PIt-1Phosphate transport,

periplasmic

(Pho84) or (Pho87) or (Pho89) or (Pho90) or (Pho91) or

(Git1)

Table S2. Names of the metabolic processes of the model (continued).

table continues next page

47

Supplementary tables

Abbreviation Metabolic process Protein

PIt-1+HPhosphate transport,

periplasmic

(Pho84) or (Pho87) or (Pho89) or (Pho90) or (Pho91) or

(Git1)

PYRt-1Pyruvate transport,

periplasmicJen1

PYRt-1+HPyruvate transport,

periplasmicJen1

SO4t-2+2HSulfate transport,

periplasmic

SO4t-2+HSulfate transport,

periplasmic(Sul1) or (Sul2)

or (Sul3)

SO4t-2Sulfate transport,

periplasmic(Sul1) or (Sul2)

or (Sul3)

SUCCt-2+2HSuccinate transport,

periplasmic

SUCCt-2+HSuccinate transport,

periplasmic

SUCCt-2Succinate transport,

periplasmic

2OBUTtm-12-oxobutanoate

transporter, mitochondrial

2OBUTtm-1+H2-oxobutanoate

transporter, mitochondrial

2OXOADPAKGtm-2-22-oxodicarboxylate

transporter, mitochondrialODC1

3MOBtm-13-methyl-2-oxobutanoate transport, mitochondrial

3MOBtm-1+H3-methyl-2-oxobutanoate transport, mitochondrial

3MOPtm-13-Methyl-2-oxopentanoate transport, mitochondrial

3MOPtm-1+H3-Methyl-2-oxopentanoate transport, mitochondrial

ACtm-1Acetate transport,

mitochondrial

ACtm-1+HAcetate transport,

mitochondrial

ACALDtm+0Indole-3-acetaldehyde

transport, mitochondrial

ALAtm+0Alanine transporter,

mitochondria

ADPATPtm-4-2ADP/ATP transporter,

mitochondrial

(Aac3-m) or (Pet9-m) or (Aac1-m)

ADPATPtm-4-3ADP/ATP transporter,

mitochondrial

(Aac3-m) or (Pet9-m) or (Aac1-m)

AKGMALtm-2-22-oxoglutarate/Malate

transporter, mitochondrial

ASPGLUtm-1-1Aspartate/Glutamate

transporter, mitochondrialAGC1

CITICITtm-3-3Citrate transport,

mitochondrialCtp1-m

Abbreviation Metabolic process Protein

CITMALtm-3-2Citrate transport,

mitochondrialCtp1-m

CITMALtm-3+H-2Citrate transport,

mitochondrialCtp1-m

CO2tm-1CO2 transport, mitochondrial

CO2tm-1+HCO2 transport, mitochondrial

ETOHtm+0Ethanol transport,

mitochondrial

GLNtm+0Glutamine transport,

mitochondrial

GLUtm-1Glutamate transport,

mitochondrialAGC1

GLUtm-1+HGlutamate transport,

mitochondrialAGC1

GLYtm+0Glycine transport,

mitochondrial

GLXtm-1 Glyoxylate, mitochondrial

GLXtm-1+H Glyoxylate, mitochondrial

H2Otm+0H2O transport, mitochondrial

MALPItm-2-2Malate transport,

mitochondrialDic1-m

NH3tm+1NH3 transport, mitochondrial

NH3tm+1-HNH3 transport, mitochondrial

O2tm+0O2 transport, mitochondrial

OAAtm-2+2HOxaloacetate transport,

mitochondrialOac1-m

OAAtm-2+HOxaloacetate transport,

mitochondrialOac1-m

OAAtm-2Oxaloacetate transport,

mitochondrialOac1-m

ORNtm+1Ornithine transport,

mitochondrialOrt1-m

ORNtm+1-HOrnithine transport,

mitochondrialOrt1-m

PItm-2+2HPhosphate transporter,

mitochondrialMir1-m

PItm-2+HPhosphate transporter,

mitochondrialMir1-m

PItm-2Phosphate transporter,

mitochondrialMir1-m

PYRtm-1Pyruvate transport,

mitochondrial

PYRtm-1+HPyruvate transport,

mitochondrial

SUCCPItm-2-2Succinate transport,

mitochondrialDic1-m

SUCFUMtm-2-2Succinate/fumarate

transport, mitochondrialSfc1-m

Table S2. Names of the metabolic processes of the model (continued).

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48

Chapter 2

Abbreviation Stoichiometry of the metabolic process

ASNS1asp-L[cyt] + atp[cyt] + gln-L[cyt] + h2o[cyt]

= amp[cyt] + asn-L[cyt] + glu-L[cyt] + ppi[cyt]

ASPK asp-L[cyt] + atp[cyt] = 4pasp[cyt] + adp[cyt]

ASPTA akg[cyt] + asp-L[cyt] = glu-L[cyt] + oaa[cyt]

ASPTAmakg[mit] + asp-L[mit] = glu-L[mit] +

oaa[mit]

ASAD4pasp[cyt] + nadph[cyt] = aspsa[cyt] +

nadp[cyt] + pi[cyt]

ALATA_L glu-L[cyt] + pyr[cyt] = akg[cyt] + ala-L[cyt]

ALATA_Lm glu-L[mit] + pyr[mit] = akg[mit] + ala-L[mit]

GLCS2h2o[cyt] + 4 udpg[cyt] = glycogen[cyt] + 4

udp[cyt]

PGMT g1p[cyt] = g6p[cyt]

TRESYNudpg[cyt] + g6p[cyt] + h2o[cyt] = tre[cyt] +

udp[cyt] + pi[cyt]

GALU g1p[cyt] + utp[cyt] = udpg[cyt] + ppi[cyt]

FBP fdp[cyt] + h2o[cyt] = f6p[cyt] + pi[cyt]

ICL icit[cyt] = glx[cyt] + succ[cyt]

MALSaccoa[cyt] + glx[cyt] + h2o[cyt] = coa[cyt]

+ mal-L[cyt]

MEmh2o[mit] + mal-L[mit] + nadp[mit] = co2tot[mit] + nadph[mit] + pyr[mit]

PPCKatp[cyt] + h2o[cyt] + oaa[cyt] = adp[cyt] +

co2tot[cyt] + pep[cyt]

PCatp[cyt] + co2tot[cyt] + pyr[cyt] = adp[cyt] +

oaa[cyt] + pi[cyt]

ACGKmacglu[mit] + atp[mit] = acg5p[mit] +

adp[mit]

ACOTAmacg5sa[mit] + glu-L[mit] = acorn[mit] +

akg[mit]

ARGSL argsuc[cyt] = arg-L[cyt] + fumarate[cyt]

ARGSSasp-L[cyt] + atp[cyt] + citr-L[cyt] = amp[cyt]

+ argsuc[cyt] + ppi[cyt]

CBPS2 atp[cyt] + co2tot[cyt] + gln-L[cyt] +

h2o[cyt] = 2 adp[cyt] + cbp[cyt] + glu-L[cyt] + pi[cyt]

GLU5K atp[cyt] + glu-L[cyt] = adp[cyt] + glu5p[cyt]

Abbreviation Stoichiometry of the metabolic process

G5P5glu5p[cyt] + 2 nadph[cyt] = pro-L[cyt] + 2

nadp[cyt] + pi[cyt] + h2o[cyt]

AGPRmacg5p[mit] + nadph[mit] = acg5sa[mit] +

nadp[mit] + pi[mit]

OCBT cbp[cyt] + orn[cyt] = citr-L[cyt] + pi[cyt]

ORNTACmacorn[mit] + glu-L[mit] = acglu[mit] +

orn[mit]

ampSYN1

2 10fthf[cyt] + 2 asp-L[cyt] + 6 atp[cyt] + co2tot[cyt] + 2 gln-L[cyt] + gly[cyt] + r5p[cyt] = 5 adp[cyt] + 2 amp[cyt] + 2

fumarate[cyt] + 2 glu-L[cyt] + 5 pi[cyt] + ppi[cyt] + 2 thf[cyt]

ampSYN2

10fthf[cyt] + asp-L[cyt] + atp[cyt] + gln-L[cyt] + prlp[cyt] = adp[cyt] + amp[cyt] + eig3p[cyt] + fumarate[cyt] + glu-L[cyt] +

h2o[cyt] + pi[cyt] + thf[cyt]

cmpSYNatp[cyt] + nh3[cyt] + ump[cyt] = adp[cyt] +

cmp[cyt] + pi[cyt]

dampSYNamp[cyt] + nadph[cyt] = damp[cyt] +

h2o[cyt] + nadp[cyt]

dcmpSYNcmp[cyt] + nadph[cyt] = dcmp[cyt] +

h2o[cyt] + nadp[cyt]

dgmpSYNgmp[cyt] + nadph[cyt] = dgmp[cyt] +

h2o[cyt] + nadp[cyt]

dtmpSYNdcmp[cyt] + h2o[cyt] + mlthf[cyt] +

nadph[cyt] = dtmp[cyt] + nadp[cyt] + nh3[cyt] + thf[cyt]

ergstSYN

2 atp[cyt] + met-L[cyt] + 3 nadph[cyt] + 2 o2[cyt] + zymst[cyt] = adp[cyt] + amp[cyt] + ergst[cyt] + 2 h2o[cyt] + hcys-L[cyt] + 3

nadp[cyt] + pi[cyt] + ppi[cyt]

gmpSYN1

2 10fthf[cyt] + asp-L[cyt] + 6 atp[cyt] + co2tot[cyt] + 3 gln-L[cyt] + gly[cyt] + 2

h2o[cyt] + nad[cyt] + r5p[cyt] = 4 adp[cyt] + 2 amp[cyt] + fumarate[cyt] + 3 glu-L[cyt] +

gmp[cyt] + nadh[cyt] + 4 pi[cyt] + 2 ppi[cyt] + 2 thf[cyt]

gmpSYN2

10fthf[cyt] + atp[cyt] + 2 gln-L[cyt] + h2o[cyt] + nad[cyt] + prlp[cyt] = amp[cyt]

+ eig3p[cyt] + 2 glu-L[cyt] + gmp[cyt] + nadh[cyt] + ppi[cyt] + thf[cyt]

Abbreviation Metabolic process Protein

THRtm+0Threonine transport,

mitochondrial

BMSYN Biomass synthesis

ac_EX Acetate exchange

biomass_EX Biomass exchange

co2tot_EX HCO3- exchange

etoh_EX Ethanol exchange

glc-D_EX D-Glucose exchange

glyc_EX Glycerol exchange

Abbreviation Metabolic process Protein

h_EX H+ exchange

h2o_EX H2O exchange

nh3_EX NH3 exchange

o2_EX Oxygen exchange

pi_EX Orthophosphate exchange

pyr_EX Pyruvate exchange

so4_EX Sulfate exchange

succ_EX Succinate exchange

Table S2. Names of the metabolic processes of the model (continued).

Table S3. Stoichiometries of the metabolic processes of the model. The square brackets indicate the compartmental location of the metabolites, i.e. extracellular space (ext), the mitochondria (mit), and the cytosol (cyt).

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49

Supplementary tables

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Table S3. Stoichiometries of the metabolic processes of the model (continued).

Abbreviation Stoichiometry of the metabolic process

umpSYNasp-L[cyt] + 3 atp[cyt] + gln-L[cyt] + 0.5

o2[cyt] + r5p[cyt] = 2 adp[cyt] + amp[cyt] + glu-L[cyt] + 2 pi[cyt] + ppi[cyt] + ump[cyt]

zymstSYN

18 accoa[cyt] + 18 atp[cyt] + h2o[cyt] + 26 nadph[cyt] + 2 nad[cyt] + 10 o2[cyt] = 18 adp[cyt] + 8 co2tot[cyt] + 18 coa[cyt] + for[cyt] + 2 nadh[cyt] + 26 nadp[cyt] + 6

pi[cyt] + 6 ppi[cyt] + zymst[cyt]

ACONT cit[cyt] = icit[cyt]

ACONTm cit[mit] = icit[mit]

AKGDmakg[mit] + coa[mit] + h2o[mit] + nad[mit] =

co2tot[mit] + nadh[mit] + succoa[mit]

CSaccoa[cyt] + h2o[cyt] + oaa[cyt] = cit[cyt]

+ coa[cyt]

CSmaccoa[mit] + h2o[mit] + oaa[mit] = cit[mit]

+ coa[mit]

ICDHxmh2o[mit] + icit[mit] + nad[mit] = akg[mit] +

co2tot[mit] + nadh[mit]

ICDHyh2o[cyt] + icit[cyt] + nadp[cyt] = akg[cyt] +

co2tot[cyt] + nadph[cyt]

ICDHymh2o[mit] + icit[mit] + nadp[mit] = akg[mit]

+ co2tot[mit] + nadph[mit]

SUCOAS1matp[mit] + coa[mit] + succ[mit] = adp[mit] +

pi[mit] + succoa[mit]

BPNT h2o[cyt] + pap[cyt] = amp[cyt] + pi[cyt]

SO4SO3so4[cyt] + 2 atp[cyt] + nadph[cyt] = so3[cyt] + pap[cyt] + nadp[cyt] + adp[cyt] + ppi[cyt]

SULR3 h2o[cyt] + h2s[cyt] + 3 nadp[cyt] = 3

nadph[cyt] + so3[cyt]

MTHFR3mlthf[cyt] + nadph[cyt] = 5mthf[cyt] +

nadp[cyt]

FTHFLatp[cyt] + for[cyt] + thf[cyt] = 10fthf[cyt] +

adp[cyt] + pi[cyt]

MTHFC h2o[cyt] + methf[cyt] = 10fthf[cyt]

MTHFDmlthf[cyt] + nadp[cyt] = methf[cyt] +

nadph[cyt]

ANSchor[cyt] + gln-L[cyt] = anth[cyt] + glu-

L[cyt] + pyr[cyt]

GLUDymglu-L[mit] + h2o[mit] + nadp[mit] =

akg[mit] + nadph[mit] + nh3[mit]

GLUDyglu-L[cyt] + h2o[cyt] + nadp[cyt] = akg[cyt]

+ nadph[cyt] + nh3[cyt]

GLUSxmakg[mit] + gln-L[mit] + nadh[mit] = 2 glu-

L[mit] + nad[mit]

GLNSatp[cyt] + glu-L[cyt] + nh3[cyt] = adp[cyt] +

gln-L[cyt] + pi[cyt]

ALCD2metoh[mit] + nad[mit] = acald[mit] +

nadh[mit]

GLYK atp[cyt] + glyc[cyt] = adp[cyt] + glyc3p[cyt]

G3PT glyc3p[cyt] + h2o[cyt] = glyc[cyt] + pi[cyt]

G3PD1dhap[cyt] + nadh[cyt] = glyc3p[cyt] +

nad[cyt]

G3PD2glyc3p[cyt] + coq[cyt] = dhap[cyt] +

coqh2[cyt]

AGTm ala-L[mit] + glx[mit] = gly[mit] + pyr[mit]

Abbreviation Stoichiometry of the metabolic process

CYSTShcys-L[cyt] + ser-L[cyt] = cyst-L[cyt] +

h2o[cyt]

GHMT2ser-L[cyt] + thf[cyt] = gly[cyt] + h2o[cyt] +

mlthf[cyt]

HSDyaspsa[cyt] + nadph[cyt] = hom-L[cyt] +

nadp[cyt]

HSKatp[cyt] + hom-L[cyt] = adp[cyt] +

phom[cyt]

PGPS3pg[cyt] + glu-L[cyt] + nad[cyt] + h2o[cyt] =

ser-L[cyt] + akg[cyt] + nadh[cyt] + pi[cyt]

ENO 2pg[cyt] = h2o[cyt] + pep[cyt]

FBA fdp[cyt] = dhap[cyt] + g3p[cyt]

PGI g6p[cyt] = f6p[cyt]

GAPDg3p[cyt] + nad[cyt] + pi[cyt] = 13dpg[cyt]

+ nadh[cyt]

PDHmcoa[mit] + h2o[mit] + nad[mit] + pyr[mit] =

accoa[mit] + co2tot[mit] + nadh[mit]

HEX1 atp[cyt] + glc-D[cyt] = adp[cyt] + g6p[cyt]

PFK atp[cyt] + f6p[cyt] = adp[cyt] + fdp[cyt]

PGK 3pg[cyt] + atp[cyt] = 13dpg[cyt] + adp[cyt]

PGM 2pg[cyt] = 3pg[cyt]

PYK adp[cyt] + pep[cyt] = atp[cyt] + pyr[cyt]

TPI dhap[cyt] = g3p[cyt]

PRMICIi prfp[cyt] = prlp[cyt]

ATPPRT atp[cyt] + prpp[cyt] = ppi[cyt] + prbatp[cyt]

HISTDh2o[cyt] + hist[cyt] + 2 nad[cyt] = his-L[cyt]

+ 2 nadh[cyt]

HISTP h2o[cyt] + hisp[cyt] = hist[cyt] + pi[cyt]

HSTPT glu-L[cyt] + imacp[cyt] = akg[cyt] + hisp[cyt]

IGPDH eig3p[cyt] = h2o[cyt] + imacp[cyt]

PRATPAMPC2 h2o[cyt] + prbatp[cyt] = ppi[cyt] +

prfp[cyt]

PRPPS atp[cyt] + r5p[cyt] = amp[cyt] + prpp[cyt]

FDHfor[cyt] + h2o[cyt] + nad[cyt] = co2tot[cyt]

+ nadh[cyt]

METS5mthf[cyt] + hcys-L[cyt] = met-L[cyt] +

thf[cyt]

CYSTGLcyst-L[cyt] + h2o[cyt] = 2obut[cyt] + cys-

L[cyt] + nh3[cyt]

HSERTAaccoa[cyt] + hom-L[cyt] = achms[cyt] +

coa[cyt]

AHSERL2 achms[cyt] + h2s[cyt] = ac[cyt] + hcys-L[cyt]

ADK1 amp[cyt] + atp[cyt] = 2 adp[cyt]

ADK1m amp[mit] + atp[mit] = 2 adp[mit]

ATPHYD atp[cyt] + h2o[cyt] = adp[cyt] + pi[cyt]

NDPK2 udp[cyt] + atp[cyt] = utp[cyt] + adp[cyt]

NADH2-u6i nadh[cyt] + coq[cyt] = nad[cyt] + coqh2[cyt]

FUM fumarate[cyt] + h2o[cyt] = mal-L[cyt]

FUMm fumarate[mit] + h2o[mit] = mal-L[mit]

PPA h2o[cyt] + ppi[cyt] = 2 pi[cyt]

PPAm h2o[mit] + ppi[mit] = 2 pi[mit]

MDH mal-L[cyt] + nad[cyt] = nadh[cyt] + oaa[cyt]

50

Chapter 2

Abbreviation Stoichiometry of the metabolic process

MDHmmal-L[mit] + nad[mit] = nadh[mit] +

oaa[mit]

PGL 6pgl[cyt] + h2o[cyt] = 6pgc[cyt]

G6PDH2g6p[cyt] + nadp[cyt] = 6pgl[cyt] +

nadph[cyt]

GND6pgc[cyt] + h2o[cyt] + nadp[cyt] =

co2tot[cyt] + nadph[cyt] + ru5p-D[cyt]

RPI r5p[cyt] = ru5p-D[cyt]

RPE ru5p-D[cyt] = xu5p-D[cyt]

TALA g3p[cyt] + s7p[cyt] = e4p[cyt] + f6p[cyt]

TKT2 e4p[cyt] + xu5p-D[cyt] = f6p[cyt] + g3p[cyt]

TKT1 r5p[cyt] + xu5p-D[cyt] = g3p[cyt] + s7p[cyt]

ACOAH ac[cyt] + coa[cyt] = accoa[cyt] + h2o[cyt]

ACSac[cyt] + atp[cyt] + coa[cyt] = accoa[cyt] +

amp[cyt] + ppi[cyt]

ACSmac[mit] + atp[mit] + coa[mit] = accoa[mit] +

amp[mit] + ppi[mit]

ALCD2x etoh[cyt] + nad[cyt] = acald[cyt] + nadh[cyt]

HCITSmaccoa[mit] + akg[mit] + h2o[mit] = coa[mit]

+ hcit[mit]

PYRDC h2o[cyt] + pyr[cyt] = acald[cyt] + co2tot[cyt]

AATA2oxoadp[cyt] + glu-L[cyt] = akg[cyt] +

L2aadp[cyt]

MCITDm hcit[mit] = b124tc[mit] + h2o[mit]

HACNHm b124tc[mit] + h2o[mit] = hicit[mit]

HICITDmhicit[mit] + nad[mit] = nadh[mit] +

oxag[mit]

AASAD1atp[cyt] + L2aadp[cyt] + nadph[cyt] =

amp[cyt] + L2aadp6sa[cyt] + nadp[cyt] + ppi[cyt]

THRD_Lm thr-L[mit] = 2obut[mit] + nh3[mit]

OXAGmh2o[mit] + oxag[mit] = 2oxoadp[mit] +

co2tot[mit]

SACCD2h2o[cyt] + nad[cyt] + saccrp-L[cyt] =

akg[cyt] + lys-L[cyt] + nadh[cyt]

SACCD1glu-L[cyt] + L2aadp6sa[cyt] + nadph[cyt] =

h2o[cyt] + nadp[cyt] + saccrp-L[cyt]

THRA acald[cyt] + gly[cyt] = thr-L[cyt]

THRS h2o[cyt] + phom[cyt] = pi[cyt] + thr-L[cyt]

DHQT 3dhq[cyt] = 3dhsk[cyt] + h2o[cyt]

DHQS 2dda7p[cyt] = 3dhq[cyt] + pi[cyt]

DDPAe4p[cyt] + h2o[cyt] + pep[cyt] = 2dda7p[cyt]

+ pi[cyt]

PSCVTi pep[cyt] + skm5p[cyt] = 3psme[cyt] + pi[cyt]

ALDD2yacald[cyt] + h2o[cyt] + nadp[cyt] = ac[cyt]

+ nadph[cyt]

ALDD2xmacald[mit] + h2o[mit] + nad[mit] = ac[mit]

+ nadh[mit]

ALDD2ymacald[mit] + h2o[mit] + nadp[mit] = ac[mit]

+ nadph[mit]

ANPRT anth[cyt] + prpp[cyt] = ppi[cyt] + pran[cyt]

CHORM chor[cyt] = pphn[cyt]

CHORS 3psme[cyt] = chor[cyt] + pi[cyt]

Abbreviation Stoichiometry of the metabolic process

IGPS 2cpr5p[cyt] = 3ig3p[cyt] + co2tot[cyt]

PHETA1akg[cyt] + phe-L[cyt] = glu-L[cyt] +

phpyr[cyt]

PRAIi pran[cyt] = 2cpr5p[cyt]

PPNDH pphn[cyt] = co2tot[cyt] + phpyr[cyt]

PPND2h2o[cyt] + nadp[cyt] + pphn[cyt] =

34hpp[cyt] + co2tot[cyt] + nadph[cyt]

SHK3D3dhsk[cyt] + nadph[cyt] = nadp[cyt] +

skm[cyt]

SHKK atp[cyt] + skm[cyt] = adp[cyt] + skm5p[cyt]

TRPS13ig3p[cyt] + ser-L[cyt] = g3p[cyt] + h2o[cyt]

+ trp-L[cyt]

TYRTAakg[cyt] + tyr-L[cyt] = 34hpp[cyt] + glu-

L[cyt]

ACHBSm2obut[mit] + h2o[mit] + pyr[mit] =

2ahbut[mit] + co2tot[mit]

IPPMIb 2ippm[cyt] + h2o[cyt] = 3c3hmp[cyt]

IPPS3mob[cyt] + accoa[cyt] + h2o[cyt] =

3c3hmp[cyt] + coa[cyt]

OMCDC3c4mop[cyt] + h2o[cyt] = 4mop[cyt] +

co2tot[cyt]

IPPMIa 3c2hmp[cyt] = 2ippm[cyt] + h2o[cyt]

IPMD3c2hmp[cyt] + nad[cyt] = 3c4mop[cyt] +

nadh[cyt]

KARA1malac-S[mit] + nadph[mit] = 23dhmb[mit] +

nadp[mit]

ACLSmh2o[mit] + 2 pyr[mit] = alac-S[mit] +

co2tot[mit]

DHAD1m 23dhmb[mit] = 3mob[mit] + h2o[mit]

DHAD2m 23dhmp[mit] = 3mop[mit] + h2o[mit]

ILETA akg[cyt] + ile-L[cyt] = 3mop[cyt] + glu-L[cyt]

KARA2m2ahbut[mit] + nadph[mit] = 23dhmp[mit]

+ nadp[mit]

LEUTAakg[cyt] + leu-L[cyt] = 4mop[cyt] + glu-

L[cyt]

VALTAakg[cyt] + val-L[cyt] = 3mob[cyt] + glu-

L[cyt]

NADH2-u6mnadh[mit] + coq[cyt] = nad[mit] +

coqh2[cyt]

ComplexIIcoq[cyt] + succ[mit] = fumarate[mit] +

coqh2[cyt]

ComplexIII2 ficytc[cyt] + coqh2[cyt] = 2 focytc[cyt]

+ coq[cyt]

ComplexIV2 focytc[cyt] + 0.5 o2[mit] = 2 ficytc[cyt] +

h2o[mit]

ATPS3m3 adp[mit] + 3 pi[mit] = 3 atp[mit] + 3

h2o[mit]

ATPASE atp[cyt] + h2o[cyt] = adp[cyt] + pi[cyt]

IONPUMP atp[cyt] + h2o[cyt] = adp[cyt] + pi[cyt]

IONPUMPm atp[mit] + h2o[mit] = adp[mit] + pi[mit]

ACt-1 ac[ext] = ac[cyt]

ACt-1+H ac[ext] = ac[cyt]

CO2t-1 co2tot[ext] = co2tot[cyt]

Table S3. Stoichiometries of the metabolic processes of the model (continued).

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51

Supplementary tables

Table S3. Stoichiometries of the metabolic processes of the model (continued).

Abbreviation Stoichiometry of the metabolic process

CO2t-1+H co2tot[ext] = co2tot[cyt]

ETOHt+0 etoh[ext] = etoh[cyt]

GLCt+0 glc-D[ext] = glc-D[cyt]

GLYCt+0 glyc[cyt] = glyc[ext]

H2Ot+0 h2o[ext] = h2o[cyt]

NH3t+1 nh3[ext] = nh3[cyt]

NH3t+1-H nh3[ext] = nh3[cyt]

O2t o2[ext] = o2[cyt]

PIt-1 pi[ext] = pi[cyt]

PIt-1+H pi[ext] = pi[cyt]

PYRt-1 pyr[ext] = pyr[cyt]

PYRt-1+H pyr[ext] = pyr[cyt]

SO4t-2+2H so4[ext] = so4[cyt]

SO4t-2+H so4[ext] = so4[cyt]

SO4t-2 so4[ext] = so4[cyt]

SUCCt-2+2H succ[ext] = succ[cyt]

SUCCt-2+H succ[ext] = succ[cyt]

SUCCt-2 succ[ext] = succ[cyt]

2OBUTtm-1 2obut[cyt] = 2obut[mit]

2OBUTtm-1+H 2obut[cyt] = 2obut[mit]

2OXOADPAKGtm-2-22oxoadp[mit] + akg[cyt] = 2oxoadp[cyt] +

akg[mit]

3MOBtm-1 3mob[cyt] = 3mob[mit]

3MOBtm-1+H 3mob[cyt] = 3mob[mit]

3MOPtm-1 3mop[cyt] = 3mop[mit]

3MOPtm-1+H 3mop[cyt] = 3mop[mit]

ACtm-1 ac[cyt] = ac[mit]

ACtm-1+H ac[cyt] = ac[mit]

ACALDtm+0 acald[cyt] = acald[mit]

ALAtm+0 ala-L[cyt] = ala-L[mit]

ADPATPtm-4-2 adp[cyt] + atp[mit] = adp[mit] + atp[cyt]

ADPATPtm-4-3 adp[cyt] + atp[mit] = adp[mit] + atp[cyt]

AKGMALtm-2-2akg[cyt] + mal-L[mit] = akg[mit] + mal-

L[cyt]

ASPGLUtm-1-1asp-L[cyt] + glu-L[mit] = asp-L[mit] +

glu-L[cyt]

CITICITtm-3-3 cit[cyt] + icit[mit] = cit[mit] + icit[cyt]

CITMALtm-3-2 cit[cyt] + mal-L[mit] = cit[mit] + mal-L[cyt]

CITMALtm-3+H-2 cit[cyt] + mal-L[mit] = cit[mit] + mal-L[cyt]

CO2tm-1 co2tot[cyt] = co2tot[mit]

CO2tm-1+H co2tot[cyt] = co2tot[mit]

Abbreviation Stoichiometry of the metabolic process

ETOHtm+0 etoh[cyt] = etoh[mit]

GLNtm+0 gln-L[cyt] = gln-L[mit]

GLUtm-1 glu-L[cyt] = glu-L[mit]

GLUtm-1+H glu-L[cyt] = glu-L[mit]

GLYtm+0 gly[cyt] = gly[mit]

GLXtm-1 glx[cyt] = glx[mit]

GLXtm-1+H glx[cyt] = glx[mit]

H2Otm+0 h2o[cyt] = h2o[mit]

MALPItm-2-2 mal-L[cyt] + pi[mit] = mal-L[mit] + pi[cyt]

NH3tm+1 nh3[cyt] = nh3[mit]

NH3tm+1-H nh3[cyt] = nh3[mit]

O2tm+0 o2[cyt] = o2[mit]

OAAtm-2+2H oaa[cyt] = oaa[mit]

OAAtm-2+H oaa[cyt] = oaa[mit]

OAAtm-2 oaa[cyt] = oaa[mit]

ORNtm+1 orn[cyt] = orn[mit]

ORNtm+1-H orn[cyt] = orn[mit]

PItm-2+2H pi[cyt] = pi[mit]

PItm-2+H pi[cyt] = pi[mit]

PItm-2 pi[cyt] = pi[mit]

PYRtm-1 pyr[cyt] = pyr[mit]

PYRtm-1+H pyr[cyt] = pyr[mit]

SUCCPItm-2-2 pi[mit] + succ[cyt] = pi[cyt] + succ[mit]

SUCFUMtm-2-2fumarate[mit] + succ[cyt] = fumarate[cyt]

+ succ[mit]

THRtm+0 thr-L[cyt] = thr-L[mit]

ac_EX ac[ext] =

biomass_EX biomass =

co2tot_EX co2tot[ext] =

etoh_EX etoh[ext] =

glc-D_EX glc-D[ext] =

glyc_EX glyc[ext] =

h_EX h[ext] =

h2o_EX h2o[ext] =

nh3_EX nh3[ext] =

o2_EX o2[ext] =

pi_EX pi[ext] =

pyr_EX pyr[ext] =

so4_EX so4[ext] =

succ_EX succ[ext] =

52

Chapter 2

Abbreviation Stoichiometric coefficient

accoa[cyt] -0.4536

adp[cyt] 26.297

ala-L[cyt] -0.4588

amp[cyt] -0.0409

arg-L[cyt] -0.1607

asn-L[cyt] -0.1017

asp-L[cyt] -0.2975

atp[cyt] -26.3021

biomass 1

coa[cyt] 0.4536

cmp[cyt] -0.0447

co2tot[cyt] 0.0062

cys-L[cyt] -0.0066

damp[cyt] -0.0036

dcmp[cyt] -0.0024

dgmp[cyt] -0.0024

dtmp[cyt] -0.0036

Abbreviation Stoichiometric coefficient

ergst[cyt] -0.0007

f6p[cyt] -0.8079

g6p[cyt] -1.1401

gln-L[cyt] -0.1054

glu-L[cyt] -0.3018

gly[cyt] -0.2904

glyc3p[cyt] -0.0247

glycogen[cyt] -0.129625

gmp[cyt] -0.046

h2o[cyt] -23.87882

hcys-L[cyt] 0.0051

his-L[cyt] -0.0663

ile-L[cyt] -0.1927

leu-L[cyt] -0.2964

lys-L[cyt] -0.2862

met-L[cyt] -0.0558

nadp[cyt] 0.82824

Abbreviation Stoichiometric coefficient

nadph[cyt] -0.82824

o2[cyt] -0.03304

phe-L[cyt] -0.1339

pi[cyt] 24.3602

ppi[cyt] 1.9653

pro-L[cyt] -0.1647

ser-L[cyt] -0.1976

so4[cyt] -0.02

thr-L[cyt] -0.1914

tre[cyt] -0.0234

trp-L[cyt] -0.0284

tyr-L[cyt] -0.102

ump[cyt] -0.0599

val-L[cyt] -0.2646

zymst[cyt] -0.0015

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

NADH2-u6mnadh[mit] +

coq[cyt] = nad[mit] + coqh2[cyt]

2 charge-1[mit] + 2 h1[mit] = 2

charge-1[cyt] + 2 h1[cyt]

ComplexIIcoq[cyt] + succ[mit]

= fumarate[mit] + coqh2[cyt]

2 charge-1[mit] + 2 h1[mit] = 2

charge-1[cyt] + 2 h1[cyt]

ComplexIII

2 ficytc[cyt] + coqh2[cyt] = 2

focytc[cyt] + coq[cyt]

2 h1[mit] = 2 h1[cyt]

ComplexIV

2 focytc[cyt] + 0.5 o2[mit] = 2 ficytc[cyt] +

h2o[mit]

2 h1[mit] + 2 charge-1[cyt] =

2 h1[cyt] + 2 charge-1[mit]

ATPS3m3 adp[mit] + 3

pi[mit] = 3 atp[mit] + 3 h2o[mit]

10 h1[cyt] = 10 h1[mit]

ATPASEatp[cyt] + h2o[cyt] = adp[cyt] + pi[cyt]

h1[cyt] = h1[ext]

POSQt charge1[ext] = charge1[cyt]

POSQtm charge1[cyt] = charge1[mit]

Ht h1[ext] = h1[cyt]

Htm h1[cyt] = h1[mit]

IONPUMPatp[cyt] + h2o[cyt] = adp[cyt] + pi[cyt]

charge1[cyt] = charge1[ext]

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

IONPUMPmatp[mit] + h2o[mit] = adp[mit] + pi[mit]

charge1[mit] = charge1[cyt]

ACt-1 ac-1[ext] = ac-1

[cyt]

ACt-1+H ac-1[ext] + h1

[ext] = ac-1[cyt] + h1[cyt]

CO2t-1 co2tot-1[ext] = co2tot-1[cyt]

CO2t-1+H co2tot-1[ext] + h1

[ext] = co2tot-1[cyt] + h1[cyt]

ETOHt+0 etoh0[ext] = etoh0[cyt]

GLCt+0 glc-D0[ext] = glc-D0[cyt]

GLYCt+0 glyc0[cyt] = glyc0

[ext]

H2Ot+0 h2o0[ext] = h2o0

[cyt]

NH3t+1 nh31[ext] = nh31

[cyt]

NH3t+1-H nh31[ext] + h1

[cyt] = nh31[cyt] + h1[ext]

O2t o20[ext] = o20

[cyt]

PIt-1 pi-1[ext] = pi-1

[cyt]

Table S4. Stoichiometry of the biomass synthesis reaction (BMSYN) of the model. The biomass synthesis reaction was based on (103). The square brackets indicate the compartmental location of the metabolites, i.e. cytosol (cyt).

Table S5. Stoichiometry for the metabolic processes in the model which contain transport processes.

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53

Supplementary tables

Table S5. Stoichiometry for the metabolic processes in the model which contain transport processes (continued).

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

PIt-1+H pi-1[ext] + h1

[ext] = pi-1[cyt] + h1[cyt]

PYRt-1 pyr-1[ext] = pyr-

1[cyt]

PYRt-1+H pyr-1[ext] + h1

[ext] = pyr-1[cyt] + h1[cyt]

SO4t-2+2H so4-2[ext] + 2 h1[ext] = so4-2[cyt] +

2 h1[cyt]

SO4t-2+H so4-2[ext] + h1

[ext] = so4-2[cyt] + h1[cyt]

SO4t-2 so4-2[ext] = so4-

2[cyt]

SUCCt-2+2H succ-2[ext] + 2 h1[ext] = succ-2[cyt]

+ 2 h1[cyt]

SUCCt-2+H succ-2[ext] + h1[ext] = succ-2[cyt]

+ h1[cyt]

SUCCt-2 succ-2[ext] = succ-

2[cyt]

2OBUTtm-1 2obut-1[cyt] = 2obut-1[mit]

2OBUTtm-1+H 2obut-1[cyt] + h1

[cyt] = 2obut-1[mit] + h1[mit]

2OXOADPAKGtm-2-2

2oxoadp-2[mit] + akg-2[cyt] =

2oxoadp-2[cyt] + akg-2[mit]

3MOBtm-1 3mob-1[cyt] = 3mob-1[mit]

3MOBtm-1+H 3mob-1[cyt] + h1

[cyt] = 3mob-1[mit] + h1[mit]

3MOPtm-1 3mop-1[cyt] = 3mop-1[mit]

3MOPtm-1+H 3mop-1[cyt] + h1

[cyt] = 3mop-1[mit] + h1[mit]

ACtm-1 ac-1[cyt] = ac-1

[mit]

ACtm-1+H ac-1[cyt] + h1

[cyt] = ac-1[mit] + h1[mit]

ACALDtm+0 acald0[cyt] = acald0[mit]

ALAtm+0 ala-L0[cyt] = ala-L0[mit]

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

ADPATPtm-4-2 adp-2[cyt] + atp-4[mit] = adp-2[mit]

+ atp-4[cyt]

ADPATPtm-4-3 adp-3[cyt] + atp-4[mit] = adp-3[mit]

+ atp-4[cyt]

AKGMALtm-2-2

akg-2[cyt] + mal-L-2[mit]

= akg-2[mit] + mal-L-2[cyt]

ASPGLUtm-1-1

asp-L-1[cyt] + glu-L-1[mit] = asp-L-1[mit] +

glu-L-1[cyt]

CITICITtm-3-3 cit-3[cyt] + icit-3[mit] = cit-3[mit] +

icit-3[cyt]

CITMALtm-3-2

cit-3[cyt] + mal-L-2[mit] = cit-3[mit] + mal-L-2

[cyt]

CITMALtm-3+H-2

cit-3[cyt] + h1[cyt] + mal-L-2

[mit] = cit-3[mit] + h1[mit] + mal-L-2

[cyt]

CO2tm-1 co2tot-1[cyt] = co2tot-1[mit]

CO2tm-1+H co2tot-1[cyt] + h1

[cyt] = co2tot-1[mit] + h1[mit]

ETOHtm+0 etoh0[cyt] = etoh0[mit]

GLNtm+0 gln-L0[cyt] = gln-L0[mit]

GLUtm-1 glu-L-1[cyt] = glu-L-1[mit]

GLUtm-1+H glu-L-1[cyt] + h1[cyt] = glu-L-1[mit]

+ h1[mit]

GLYtm+0 gly0[cyt] = gly0

[mit]

GLXtm-1 glx-1[cyt] = glx-1

[mit]

GLXtm-1+H glx-1[cyt] + h1

[cyt] = glx-1[mit] + h1[mit]

H2Otm+0 h2o0[cyt] = h2o0

[mit]

MALPItm-2-2 mal-L-2[cyt] + pi-2[mit] = mal-L-2[mit] + pi-2[cyt]

NH3tm+1 nh31[cyt] = nh31

[mit]

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54

Chapter 2

AbbreviationCompartment

ext cyt mit

ASNS1 1.8

ASPK 0.1

ASAD -1.4

GLCS2 2.8

TRESYN 0.5

GALU -0.2

FBP -0.2

MALS 1

MEm 1

PPCK 0.9

PC 0.7

ACGKm -0.1

ARGSS 1.8

CBPS 1.6

GLU5K -0.1

G5P5 -2.2

AGPRm -1

OCBT 0.8

ampSYN1 8.3

ampSYN2 1.7

cmpSYN 1.7

AbbreviationCompartment

ext cyt mit

dampSYN -1

dcmpSYN -1

dgmpSYN -1

dtmpSYN -2

ergstSYN -0.5

gmpSYN1 10.4

gmpSYN2 3.8

umpSYN 3.2

zymstSYN -2.4

AKGDm 1

CS 1

CSm 1

ICDHxm 1

ICDHy 1

ICDHym 1

SUCOAS1m -0.1

BPNT -0.2

SO4SO3 1.4

SULR 3.7

MTHFR3 -1

FTHFL -0.3

AbbreviationCompartment

ext cyt mit

MTHFC 1

ANS 1

GLUDym 1

GLUDy 1

GLUSxm -1

GLNS 0.7

ALCD2m 1

GLYK 0.9

G3PT -0.2

G3PD1 -1

HSDy -1

HSK 0.7

PGPS 0.8

GAPD 1.2

PDHm 1

HEX1 0.9

PFK 0.9

PGK -0.1

PYK -0.9

PRMICIi 1

ATPPRT 0.8

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

NH3tm+1-H nh31[cyt] + h1

[mit] = nh31[mit] + h1[cyt]

O2tm+0 o20[cyt] = o20

[mit]

OAAtm-2+2H oaa-2[cyt] + 2 h1

[cyt] = oaa-2[mit] + 2 h1[mit]

OAAtm-2+H oaa-2[cyt] + h1

[cyt] = oaa-2[mit] + h1[mit]

OAAtm-2 oaa-2[cyt] = oaa-

2[mit]

ORNtm+1 orn1[cyt] = orn1

[mit]

ORNtm+1-H orn1[cyt] + h1

[mit] = orn1[mit] + h1[cyt]

PItm-2+2H pi-2[cyt] + 2 h1

[cyt] = pi-2[mit] + 2 h1[mit]

AbbreviationStoichiometry of the:

Chemical conversion

Metabolite transport

PItm-2+H pi-2[cyt] + h1

[cyt] = pi-2[mit] + h1[mit]

PItm-2 pi-2[cyt] = pi-2

[mit]

PYRtm-1 pyr-1[cyt] = pyr-

1[mit]

PYRtm-1+H pyr-1[cyt] + h1

[cyt] = pyr-1[mit] + h1[mit]

SUCCPItm-2-2 pi-2[mit] + succ-2[cyt] = pi-2[cyt] +

succ-2[mit]

SUCFUMtm-2-2

fumarate-2[mit] + succ-2[cyt] =

fumarate-2[cyt] + succ-2[mit]

THRtm+0 thr-L0[cyt] = thr-L0[mit]

Table S5. Stoichiometry for the metabolic processes in the model which contain transport processes (continued).

Table S6. Stoichiometric coefficients of the proton changes in the metabolic process of the model. The stoichiometric coefficients were determined for the extracellular space (ext; pH 5.0), the mitochondria (mit; pH 7.4), and the cytosol (cyt; pH 7.0). Note that metabolic processes, in which the stoichiometric coefficients for the protons were all zero, are not shown this table. Further note that non-integer coefficients are caused by the reactants (pH-dependent) used in the chemical conversions.

table continues next page

55

Supplementary tables

AbbreviationCompartment

ext cyt mit

HISTD 3.1

HISTP -0.2

HSTPT -0.2

PRATPAMPC 1.6

PRPPS 1.2

FDH 1

ADK1 -0.4

ADK1m -0.2

ATPHYD 0.7

NADH2-u6i -1

PPAm 0.2

MDH 1

MDHm 1

PGL 1

G6PDH2 1

GND 1

ACOAH -1

ACS 0.8

ACSm 0.8

ALCD2x 1

HCITSm 1

HICITDm 1

AASAD1 -0.2

SACCD2 1

SACCD1 -1

DHQS -0.2

DDPA -0.2

PSCVTi -0.2

ALDD2y 2

ALDD2xm 2

ALDD2ym 2

AbbreviationCompartment

ext cyt mit

ANPRT 0.6

CHORS -0.2

PPND2 1

SHK3D -1

SHKK 0.9

IPPS 1

IPMD 1

KARA1m -1

KARA2m -1

NADH2-u6m -1

ComplexIII 4 -2

ComplexIV 2 -4

ATPS3m -10 7.3

ATPASE 1 -0.3

Ht -1 1

Htm -1 1

IONPUMP 0.7

IONPUMPm 0.9

ACt-1 0.1

ACt-1+H -0.9 1

CO2t-1 0.8

CO2t-1+H -0.2 1

NH3t+1-H 1 -1

PIt-1 0.8

PIt-1+H -1 1.8

PYRt-1+H -1 1

SO4t-2+2H -2 2

SO4t-2+H -1 1

SUCCt-2+2H -1.4 2

SUCCt-2+H -0.4 1

SUCCt-2 0.6

AbbreviationCompartment

ext cyt mit

2OBUTtm-1+H -1 1

3MOBtm-1+H -1 1

3MOPtm-1+H -1 1

ACtm-1+H -1 1

ADPATPtm-4-2 -0.9 0.9

ADPATPtm-4-3 0.1 -0.1

CITMALtm-3+H-2 -1 1

CO2tm-1+H -1 1

GLUtm-1+H -1 1

GLXtm-1+H -1 1

MALPItm-2-2 -0.2

NH3tm+1-H 1 -1

OAAtm-2+2H -2 2

OAAtm-2+H -1 1

ORNtm+1-H 1 -1

PItm-2+2H -1.8 2

PItm-2+H -0.8 1

PItm-2 0.2

PYRtm-1+H -1 1

SUCCPItm-2-2 -0.2

h_EX -1

BMSYN 15.4

Table S6. Stoichiometric coefficients of the proton changes in the metabolic process of the model (continued).

AbbreviationCompartment

ext cyt mit

ComplexIII 2 -2

ComplexIV 4 -4

ATPS3m -10 10

ATPASE 1 -1

POSQt -1 1

POSQtm -1 1

Ht -1 1

Htm -1 1

IONPUMP 1 -1

IONPUMPm 1 -1

ACt-1 1 -1

CO2t-1 1 -1

NH3t+1 -1 1

AbbreviationCompartment

ext cyt mit

PIt-1 1 -1

PYRt-1 1 -1

SO4t-2+H 1 -1

SO4t-2 2 -2

SUCCt-2+H 1 -1

SUCCt-2 2 -2

2OBUTtm-1 1 -1

3MOBtm-1 1 -1

3MOPtm-1 1 -1

ACtm-1 1 -1

ADPATPtm-4-2 -2 2

ADPATPtm-4-3 -1 1

CITMALtm-3-2 1 -1

AbbreviationCompartment

ext cyt mit

CO2tm-1 1 -1

GLUtm-1 1 -1

GLXtm-1 1 -1

NH3tm+1 -1 1

OAAtm-2+H 1 -1

OAAtm-2 2 -2

ORNtm+1 -1 1

PItm-2+H 1 -1

PItm-2 2 -2

PYRtm-1 1 -1

BMSYN 1.3

Table S7. Stoichiometric coefficients of the charge changes in the metabolic process of the model. The stoichiometric coefficients were determined for the extracellular space (ext), the mitochondria (mit), and the cytosol (cyt). Note that metabolic processes, in which the stoichiometric coefficients for the charges were all zero, are not shown this table.

56

Chapter 2

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

ASNS1 -31.4 0.0 -78.4 0.0

ASPK 19.0 0.0 -6.3 -0.5

ASPTA 3.7 0.0 -0.7 17.1

ASPTAm 3.7 0.0 -22.1 26.9

ASAD 16.6 0.0 -6.3 -0.5

ALATA_L -0.1 0.0 -25.3 2.5

ALATA_Lm -0.1 0.0 -32.3 19.9

GLCS2 -72.3 0.0 -124.9 -87.2

PGMT -5.6 0.0 0.5 3.9

TRESYN -23.7 0.0 -42.6 -24.2

GALU 1.5 0.0 -8.7 -0.5

FBP -10.2 0.0 -23.8 -14.4

ICL 9.0 0.0 -41.7 16.2

MALS -35.7 0.0 -71.5 -8.2

MEm 5.6 0.0 -29.1 19.3

PPCK 5.1 0.0 -9.9 11.7

PC -5.0 0.0 -26.4 -1.7

ACGKm 27.7 0.0 -19.4 -0.5

ACOTAm 14.2 0.0 -19.4 -0.5

ARGSL 11.9 0.0 -46.1 -0.5

ARGSS 0.2 0.0 -52.9 -0.5

CBPS -16.4 0.0 -57.6 -18.3

GLU5K 30.6 0.0 -12.4 -0.5

G5P5 -44.2 0.0 -44.5 -20.8

AGPRm -2.5 0.0 -19.4 -0.5

OCBT -26.5 0.0 -45.8 -0.5

ORNTACm 0.2 0.0 -19.4 -0.5

ampSYN1 -68.1 0.0 -217.1 -61.4

ampSYN2 -44.2 0.0 -138.9 -29.7

cmpSYN -10.0 0.0 -22.5 -0.5

dampSYN -57.6 0.0 -78.1 -43.2

dcmpSYN -52.8 0.0 -80.3 -29.4

dgmpSYN -53.9 0.0 -82.9 -30.7

dtmpSYN -86.0 0.0 -139.3 -56.5

ergstSYN -948.3 0.0 -1013.3 -887.6

gmpSYN1 29.1 0.0 -143.6 14.2

gmpSYN2 53.0 0.0 -65.0 56.5

umpSYN -258.0 0.0 -309.6 -272.2

zymstSYN 0.0 0.0 -679.4 -0.5

ACONT 7.6 0.0 -11.9 8.0

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

ACONTm 7.6 0.0 -15.6 29.5

AKGDm -34.3 0.0 -38.6 -0.5

CS -36.3 0.0 -40.8 -5.7

CSm -38.6 0.0 -68.4 -5.5

ICDHxm -2.4 0.0 -22.8 -0.5

ICDHy 0.3 0.0 -29.7 -0.5

ICDHym -2.0 0.0 -19.8 11.5

SUCOAS1m -2.3 0.0 -19.7 40.7

BPNT -9.4 0.0 -32.1 -2.5

SO4SO3 -12.6 0.0 -97.2 -30.0

SULR 121.8 0.0 98.9 162.4

MTHFR3 -38.6 0.0 -67.6 -23.2

FTHFL -5.3 0.0 -29.2 29.5

MTHFC -7.8 0.0 -23.1 -0.5

MTHFD 7.6 0.0 -15.7 -0.5

ANS -145.7 0.0 -192.9 -134.2

GLUDym 33.9 0.0 0.5 27.9

GLUDy 36.2 0.0 0.5 22.2

GLUSxm -45.2 0.0 -47.4 3.0

GLNS -14.7 0.0 -20.7 -11.4

ALCD2m 17.8 0.0 -15.9 40.8

GLYK -15.8 0.0 -32.5 -13.1

G3PT -10.6 0.0 -22.7 -3.6

G3PD1 -24.8 0.0 -21.9 -14.6

G3PD2 -56.5 0.0 -56.2 -44.4

AGTm -8.1 0.0 -51.5 38.3

CYSTS -27.1 0.0 -38.9 -0.5

GHMT2 -6.9 0.0 -26.2 -0.5

HSDy -21.5 0.0 -33.2 -0.5

HSK -25.4 0.0 -52.9 -2.9

PGPS 11.6 0.0 -37.3 -12.9

ENO -4.1 0.0 -3.9 -0.5

FBA 21.9 0.0 -9.6

PGI 2.8 0.0 -2.6 -0.5

GAPD 5.1 0.0 -3.4

PDHm -38.8 0.0 -54.7 -0.5

HEX1 -17.0 0.0 -20.8

PFK -16.2 0.0 -21.7

PGK 18.1 0.0 3.4

PGM -4.7 0.0 0.8 2.2

Table S8. Gibbs energies of the metabolic and exchange processes. ∆rG’o are the standard Gibbs energies of the chemical conversions and ∆rG’t are the Gibbs energies of metabolite transports. ∆rG are physiological bounds (lower lo, and upper up) on the Gibbs energies of the metabolic processes as determined in the regression analysis. ∆fG’o are the standard Gibbs energies of formation for the metabolites transferred across the system boundary by the exchange processes. The units of the Gibbs energies are in KJ mol-1 except for the biomass exchange (biomass_EXG) and the biomass synthesis reaction (BMSYN), which are in J gCDW-1. Note, the different units for the Gibbs energies are explained by the different units of the rates v, whose units are generally in mmol gCDW-1 h-1, except for the biomass exchange and the biomass synthesis reactions, which have the unit gCDW gCDW-1 h-1. The values correspond to the following pH values: extracellular space (pH 5.0), the mitochondria (pH 7.4), and the cytosol (pH 7.0).

table continues next page

57

Supplementary tables

Table S8. Gibbs energies of the metabolic and exchange processes (continued).

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

PYK -26.4 0.0 -34.5 -12.8

TPI 5.3 0.0 -3.7 -0.5

PRMICIi -11.5 0.0 -34.7 -0.5

ATPPRT 0.0 0.0 -39.2 -0.5

HISTD -12.9 0.0 -39.3 -8.2

HISTP -14.8 0.0 -46.5 -0.5

HSTPT 7.6 0.0 -43.9 -0.5

IGPDH -40.8 0.0 -64.0 -17.5

PRATPAMPC 0.0 0.0 -54.9 -0.5

PRPPS -0.1 0.0 -25.1 -0.5

FDH -19.7 0.0 -34.1 -4.3

METS -6.7 0.0 -40.2 -0.5

CYSTGL -0.6 0.0 -61.8 -10.0

HSERTA -16.5 0.0 -52.4 -0.5

AHSERL2 -68.8 0.0 -111.0 -22.3

ADK1 -1.7 0.0 -4.9 -0.5

ADK1m -1.3 0.0 -41.6 14.7

ATPHYD -26.4 0.0 -38.7 -35.0

NDPK2 -2.8 0.0 -3.4 -0.5

NADH2-u6i -81.3 0.0 -73.9 -64.0

FUM -3.5 0.0 -17.2 15.6

FUMm -3.5 0.0 -24.6 19.7

PPA -15.3 0.0 -8.8 -0.5

PPAm -15.0 0.0 -60.9 8.2

MDH 28.9 0.0 -1.6 27.7

MDHm 26.6 0.0 -2.4 37.8

PGL -21.8 0.0 -29.2 -5.2

G6PDH2 -3.9 0.0 -25.1 -0.5

GND 5.3 0.0 -12.1 -2.0

RPI 3.7 0.0 0.5 3.6

RPE -3.5 0.0 -1.2 -0.5

TALA -0.7 0.0 -8.4 -4.2

TKT2 -10.0 0.0 -6.9 -1.7

TKT1 -1.8 0.0 -7.7 -2.3

ACOAH 30.6 0.0 34.4 71.4

ACS -5.1 0.0 -41.9 -0.5

ACSm -7.2 0.0 -66.2 52.2

ALCD2x 20.1 0.0 -7.4 26.4

HCITSm -33.8 0.0 -57.8 -19.8

PYRDC -16.5 0.0 -47.5 -14.2

AATA 0.9 0.0 -31.2 -0.5

MCITDm 3.0 0.0 -6.0 -0.5

HACNHm 2.6 0.0 -6.0 -0.5

HICITDm 16.3 0.0 -6.0 -0.5

AASAD1 9.7 0.0 -59.1 -0.5

THRD_Lm -24.5 0.0 -82.4 -12.9

OXAGm -14.3 0.0 -28.7 -0.5

SACCD2 31.4 0.0 -17.0 -0.5

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

SACCD1 -28.8 0.0 -44.5 -0.5

THRA -10.4 0.0 0.5 31.5

THRS -30.4 0.0 -42.8 -14.6

DHQT -5.2 0.0 -28.4 -0.5

DHQS -118.1 0.0 -149.8 -102.7

DDPA -69.6 0.0 -65.3 -33.2

PSCVTi -13.4 0.0 -28.5 -0.5

ALDD2y -50.3 0.0 -69.7 -21.3

ALDD2xm -55.3 0.0 -84.3 -32.4

ALDD2ym -54.9 0.0 -83.9 -25.9

ANPRT -69.3 0.0 -111.1 -34.4

CHORM -33.8 0.0 -57.0 -10.6

CHORS -25.2 0.0 -56.8 -9.8

IGPS -79.0 0.0 -116.8 -66.9

PHETA1 -0.1 0.0 0.5 34.4

PRAIi -5.2 0.0 -28.4 -0.5

PPNDH -132.8 0.0 -170.5 -120.6

PPND2 -75.6 0.0 -112.9 -57.6

SHK3D -44.0 0.0 -73.0 -20.6

SHKK -4.4 0.0 -30.8 -0.5

TRPS1 -30.6 0.0 -57.9 -29.6

TYRTA 0.3 0.0 0.5 34.2

ACHBSm -26.3 0.0 -49.8 -0.5

IPPMIb -2.9 0.0 0.5 9.7

IPPS -26.7 0.0 -48.4 -0.5

OMCDC -21.0 0.0 -44.7 -8.8

IPPMIa -2.5 0.0 0.5 9.7

IPMD 18.7 0.0 -9.7 -0.5

KARA1m -16.5 0.0 -45.5 -0.5

ACLSm -35.7 0.0 -59.2 -0.5

DHAD1m -37.7 0.0 -60.9 -14.5

DHAD2m -37.7 0.0 -60.9 -14.5

ILETA -6.0 0.0 0.5 27.6

KARA2m -16.5 0.0 -45.6 -0.5

LEUTA -0.9 0.0 0.5 33.9

VALTA 2.3 0.0 0.5 32.3

NADH2-u6m -83.6 4.6 -79.7 -67.4

ComplexII -31.3 4.6 -50.9 -1.6

ComplexIII -25.4 35.5 -7.3 -0.5

ComplexIV -106.7 66.4 -25.0 -11.3

ATPS3m 84.6 -177.6 -57.1 -0.5

ATPASE -26.4 17.4 -21.3 -17.6

POSQt 0.0 -5.8 -5.8 -5.8

POSQtm 0.0 -15.4 -15.4 -15.4

Ht 0.0 -17.4 -17.4 -17.4

Htm 0.0 -17.8 -17.8 -17.8

IONPUMP -26.4 5.8 -32.9 -29.2

IONPUMPm -28.2 15.4 -43.6 -24.7

table continues next page

58

Chapter 2

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

ACt-1 0.0 6.2 -7.2 16.9

ACt-1+H 0.0 -11.2 -24.6 -0.5

CO2t-1 0.0 10.4 12.4 21.9

CO2t-1+H 0.0 -7.0 -5.0 4.5

ETOHt+0 0.0 0.0 0.5 22.1

GLCt+0 0.0 0.0 -20.5 0.0

GLYCt+0 0.0 0.0 -17.4 -0.5

NH3t+1 0.0 -5.8 -8.7 -5.8

NH3t+1-H 0.0 11.6 8.7 11.6

O2t 0.0 0.0 -14.3 -0.5

PIt-1 0.0 1.5 1.8 2.9

PIt-1+H 0.0 -15.9 -15.6 0.0

PYRt-1 0.0 5.8 0.5 25.9

PYRt-1+H 0.0 -11.6 -16.9 8.5

SO4t-2+2H 0.0 -23.2 -24.7 -20.6

SO4t-2+H 0.0 -5.8 -7.3 -3.2

SO4t-2 0.0 11.6 10.1 14.2

SUCCt-2+2H 0.0 -21.2 -32.8 -1.3

SUCCt-2+H 0.0 -3.8 -15.4 16.1

SUCCt-2 0.0 13.6 2.0 33.4

2OBUTtm-1 0.0 15.4 -7.8 17.3

2OBUTtm-1+H 0.0 -2.3 -25.5 -0.5

2OXOADPAKGtm-2-2 0.0 0.0 -20.7 -0.5

3MOBtm-1 0.0 15.4 0.5 38.7

3MOBtm-1+H 0.0 -2.3 -17.3 20.9

3MOPtm-1 0.0 15.4 0.5 38.7

3MOPtm-1+H 0.0 -2.3 -17.3 20.9

ACtm-1 0.0 15.4 -7.8 34.4

ACtm-1+H 0.0 -2.3 -25.5 16.7

ACALDtm+0 0.0 0.0 -23.2 23.2

ALAtm+0 0.0 0.0 -24.1 5.3

ADPATPtm-4-2 0.0 -33.2 -56.4 -33.7

ADPATPtm-4-3 0.0 -15.4 -38.6 -15.9

AKGMALtm-2-2 0.0 0.0 -10.6 41.2

ASPGLUtm-1-1 0.0 0.0 -30.9 10.9

CITICITtm-3-3 0.0 0.0 -37.2 20.2

CITMALtm-3-2 0.0 15.4 -14.8 40.0

CITMALtm-3+H-2 0.0 -2.3 -32.6 22.2

CO2tm-1 0.0 15.4 14.6 21.6

CO2tm-1+H 0.0 -2.3 -3.1 3.9

Abbreviation ∆rG’o ∆rG’t∆rG’

∆fG’o

lo up

ETOHtm+0 0.0 0.0 -27.9 21.6

GLNtm+0 0.0 0.0 -25.0 2.5

GLUtm-1 0.0 15.4 8.3 21.6

GLUtm-1+H 0.0 -2.3 -9.4 3.8

GLYtm+0 0.0 0.0 -21.0 8.7

GLXtm-1 0.0 15.4 -7.8 38.7

GLXtm-1+H 0.0 -2.3 -25.5 20.9

MALPItm-2-2 0.0 -0.5 -21.9 40.0

NH3tm+1 0.0 -15.4 -41.4 -15.4

NH3tm+1-H 0.0 2.3 -23.7 2.3

O2tm+0 0.0 0.0 -14.3 -0.5

OAAtm-2+2H 0.0 -4.6 -11.9 8.3

OAAtm-2+H 0.0 13.1 5.8 26.1

OAAtm-2 0.0 30.9 23.6 43.8

ORNtm+1 0.0 -15.4 -17.3 -11.7

ORNtm+1-H 0.0 2.3 0.5 6.0

PItm-2+2H 0.0 -4.2 -22.2 -1.5

PItm-2+H 0.0 13.6 -4.4 16.3

PItm-2 0.0 31.4 13.3 34.0

PYRtm-1 0.0 15.4 1.1 28.8

PYRtm-1+H 0.0 -2.3 -16.6 11.0

SUCCPItm-2-2 0.0 -0.5 -17.2 40.7

SUCFUMtm-2-2 0.0 0.0 -38.1 36.7

THRtm+0 0.0 0.0 -23.6 -0.5

BMSYN 0.0 0.0 -293.4 -145.4

ac_EX -279.8

co2tot_EX -564.0

biomass_EX -266.5

etoh_EX -0.7

glc-D_EX -558.9

glyc_EX -255.6

h_EX 29.0

h2o_EX -179.5

nh3_EX 35.9

o2_EX 16.4

pi_EX -1074.9

pyr_EX -378.2

so4_EX -748.4

succ_EX -564.5

Table S8. Gibbs energies of the metabolic and exchange processes (continued).

59

Supplementary tables

Table S9. Bounds on the metabolite concentrations. Default and physiological bounds (lower lo, and upper up) on the metabolite concentrations (methods 7). Default bounds were used for the regression analysis (text S7), whereas physiological bounds were used for the flux balance analysis (text S10). The units are mM, except for water and biomass, which had a fixed concentration of 1M corresponding to a chemical activity of 1. The default bounds for the metabolite concentrations were specified to be between 0.001 mM and 10 mM. For several metabolites, these default bounds were adjusted to minimum and maximum reported values found in the literature (cf. Reference column). Furthermore, we used for certain redox cofactors (nad/nadh, nadp/nadph, ficytc/ficytc, coq/coqh2) the ratio between the reduced and oxidized form instead of concentrations (practically by fixing the concentrations of the one partner of the redox couple to 1). The concentration of the dissolved carbon dioxide was determined on the basis of the Henry constants from (122) and a partial pressure for CO2 between 1 mbar and 35 mbar (83). Note, if a metabolite has an empty entry in a column (corresponding to a compartment), this metabolite was assumed to be not present in this compartment.

Abbre-viation

Default concentration bounds Physiological concentration bounds

extracellular cytosol mitochondria Refer-ence

extracellular cytosol mitochondria

lo up lo up lo up lo up lo up lo up

10fthf 0.001 10.000 0.001 10.000

13dpg 0.001 10.000 0.001 0.017

23dhmb 0.001 10.000 0.001 10.000

23dhmp 0.001 10.000 0.001 10.000

2ahbut 0.001 10.000 0.001 10.000

2cpr5p 0.001 10.000 0.001 10.000

2dda7p 0.001 10.000 0.001 10.000

2ippm 0.001 10.000 0.004 10.000

2obut 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

2oxoadp 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

2pg 0.001 10.000 0.069 0.430

34hpp 0.001 10.000 0.001 10.000

3c2hmp 0.001 10.000 0.007 10.000

3c3hmp 0.001 10.000 0.003 10.000

3c4mop 0.001 10.000 0.001 1.386

3dhq 0.001 10.000 0.001 10.000

3dhsk 0.001 10.000 0.001 10.000

3ig3p 0.001 10.000 0.001 10.000

3mob 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

3mop 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

3pg 0.001 10.000 1.000 3.866

3psme 0.001 10.000 0.001 10.000

4mop 0.001 10.000 0.001 10.000

4pasp 0.001 10.000 0.001 6.016

5mthf 0.001 10.000 0.001 10.000

6pgc 0.001 10.000 0.119 0.750

6pgl 0.001 10.000 0.001 10.000

ac 0.001 3.000 0.001 10.000 0.001 10.000 (41) 0.051 3.000 0.005 10.000 0.001 10.000

acald 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

accoa 0.001 0.150 0.010 1.500 (41) 0.002 0.150 0.010 1.500

acg5p 0.001 10.000 0.001 10.000

acg5sa 0.001 10.000 0.001 10.000

acglu 0.001 10.000 0.001 10.000

achms 0.001 10.000 0.001 10.000

acorn 0.001 10.000 0.001 10.000

adp 0.001 10.000 0.001 10.000 0.024 1.294 0.001 10.000

table continues next page

60

Chapter 2

Abbre-viation

Default concentration bounds Physiological concentration bounds

extracellular cytosol mitochondria Refer-ence

extracellular cytosol mitochondria

lo up lo up lo up lo up lo up lo up

akg 0.001 10.000 0.001 10.000 0.001 1.289 0.001 10.000

alac-S 0.001 10.000 0.001 10.000

ala-L 0.001 30.000 0.001 30.000 (38) 3.136 14.099 0.001 30.000

amp 0.001 10.000 0.001 10.000 0.001 0.342 0.001 3.720

anth 0.001 10.000 0.001 10.000

arg-L 0.001 10.000 0.001 10.000

argsuc 0.001 10.000 0.001 10.000

asn-L 0.001 10.000 1.570 2.851

asp-L 0.001 30.000 0.001 30.000 (38) 2.641 18.513 0.001 30.000

aspsa 0.001 10.000 0.001 9.636

atp 1.000 10.000 0.001 30.000 (123) 1.000 4.626 0.001 30.000

b124tc 0.001 10.000 0.001 10.000

biomass 1000 1000 1000 1000

cbp 0.001 10.000 0.001 10.000

charge

chor 0.001 10.000 0.001 10.000

cit 0.001 10.000 0.001 10.000 0.396 8.335 0.001 10.000

citr-L 0.001 10.000 0.001 10.000

cmp 0.001 10.000 0.001 10.000

co2tot 0.043 1.502 3.125 12.500 9.091 36.364 (122) 0.087 1.502 3.125 12.500 9.091 36.364

coa 0.001 0.050 0.001 0.500 (41) 0.001 0.050 0.001 0.500

coq 1.000 1.000 1.000 1.000

coqh2 0.100 10.000 (124) 0.100 10.000

cys-L 0.001 10.000 0.007 0.112

cyst-L 0.001 10.000 0.001 10.000

damp 0.001 10.000 0.001 10.000

dcmp 0.001 10.000 0.001 10.000

dgmp 0.001 10.000 0.001 10.000

dhap 0.001 10.000 0.095 0.988

dtmp 0.001 10.000 0.001 10.000

e4p 0.000 10.000 0.0005 0.0080

eig3p 0.001 10.000 0.001 10.000

ergst 0.001 10.000 0.001 10.000

etoh 0.001 65.000 0.001 65.000 0.001 65.000 (41) 0.001 53.304 0.001 65.000 0.001 65.000

f6p 0.001 10.000 0.301 1.437

fdp 0.001 10.000 0.059 5.364

ficytc 0.100 10.000 (124) 0.100 10.000

focytc 1.000 1.000 1.000 1.000

for 0.001 10.000 0.001 10.000

fumarate 0.001 10.000 0.001 10.000 0.001 0.459 0.001 5.003

g1p 0.001 10.000 0.072 0.283

g3p 0.010 10.000 0.010 0.042

g6p 0.001 10.000 1.099 5.883

glc-D 0.001 55.000 0.001 10.000 (41) 0.008 3.498 0.001 2.869

gln-L 0.001 30.000 0.001 30.000 (38) 8.358 20.407 0.001 30.000

glu5p 0.000 10.000 0.000 0.063

glu-L 0.001 200.000 0.001 200.000 (38) 16.200 90.046 0.001 200.000

Table S9. Bounds on the metabolite concentrations (continued).

table continues next page

61

Supplementary tables

table continues next page

Table S9. Bounds on the metabolite concentrations (continued).

Abbre-viation

Default concentration bounds Physiological concentration bounds

extracellular cytosol mitochondria Refer-ence

extracellular cytosol mitochondria

lo up lo up lo up lo up lo up lo up

glx 0.001 10.000 0.001 10.000 0.001 10.000 0.001 10.000

gly 0.001 10.000 0.001 10.000 0.051 4.178 0.001 10.000

glyc 0.001 1.000 0.001 10.000 (41) 0.001 1.000 0.001 10.000

glyc3p 0.001 10.000 0.025 0.143

glycogen 0.001 200 0.001 200.000

gmp 0.001 10.000 0.001 10.000

h 0.014 0.014 0.000 0.000 0.000 0.000 0.014 0.014 0.000 1000.000 0.000 0.000

h2o 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

h2s 0.001 10.000 0.001 10.000

hcit 0.001 10.000 0.001 10.000

hcys-L 0.001 10.000 0.001 10.000

hicit 0.001 10.000 0.004 10.000

his-L 0.001 10.000 2.273 5.420

hisp 0.001 10.000 0.001 10.000

hist 0.001 10.000 0.001 10.000

hom-L 0.001 10.000 0.001 10.000

icit 0.001 10.000 0.001 10.000 0.001 0.534 0.001 6.646

ile-L 0.001 10.000 1.156 1.837

imacp 0.001 10.000 0.001 10.000

L2aadp 0.001 10.000 0.001 10.000

L2aadp6sa 0.001 10.000 0.001 10.000

leu-L 0.001 10.000 0.669 1.246

lys-L 0.001 30.000 3.763 15.176

mal-L 0.001 10.000 0.001 10.000 0.001 2.389 0.001 10.000

methf 0.001 10.000 0.001 10.000

met-L 0.001 10.000 0.123 0.331

mlthf 0.001 10.000 0.001 10.000

nad 10.000 1000 1.000 10.000 (38) 19.375 106.973 1.000 10.000

nadh 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

nadp 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

nadph 0.100 10.000 0.100 10.000 (38) 0.545 10.000 0.100 10.000

nh3 30.000 32.000 10.000 30.000 0.001 10.000 (125) 30.000 32.000 10.000 30.000 0.001 10.000

o2 0.001 0.500 0.001 0.500 0.001 0.500 (38) 0.320 0.360 0.001 0.295 0.001 0.242

oaa 0.001 0.018 0.001 0.170 (41) 0.001 0.018 0.001 0.170

orn 0.001 10.000 0.001 10.000 2.132 9.821 1.494 10.000

oxag 0.001 10.000 0.001 2.387

pap 0.001 10.000 0.001 10.000

pep 0.001 10.000 0.070 1.668

phe-L 0.001 10.000 0.512 0.926

phom 0.001 10.000 0.001 10.000

phpyr 0.001 10.000 0.001 10.000

pi 25.000 30.000 35.000 75.000 0.001 100.000 (125) 25.000 30.000 35.000 44.895 0.007 100.000

pphn 0.001 10.000 0.001 10.000

ppi 0.001 100.000 0.001 100.000 0.003 0.252 0.001 100.000

pran 0.001 10.000 0.001 10.000

prbatp 0.001 10.000 0.001 10.000

prfp 0.001 10.000 0.001 10.000

62

Chapter 2

Abbre-viation

Default concentration bounds Physiological concentration bounds

extracellular cytosol mitochondria Refer-ence

extracellular cytosol mitochondria

lo up lo up lo up lo up lo up lo up

prlp 0.001 10.000 0.001 10.000

pro-L 0.001 10.000 1.461 3.722

prpp 0.001 10.000 0.001 10.000

pyr 0.001 0.500 0.050 10.000 0.010 10.000 (41, 123) 0.001 0.500 0.050 2.937 0.010 10.000

r5p 0.100 10.000 0.100 0.729

ru5p-D 0.001 10.000 0.019 0.271

s7p 0.001 10.000 0.671 3.539

saccrp-L 0.001 10.000 0.003 10.000

ser-L 0.001 30.000 (38) 1.538 10.980

skm 0.001 10.000 0.001 10.000

skm5p 0.001 10.000 0.001 10.000

so3 0.001 10.000 0.001 10.000

so4 3.500 4.500 2.500 10.000 (125) 3.500 4.500 2.500 10.000

succ 0.001 0.100 0.001 10.000 0.001 10.000 (41) 0.001 0.100 0.001 2.638 0.001 10.000

succoa 1.000 70.000 (41) 1.000 70.000

thf 0.001 10.000 0.001 10.000

thr-L 0.001 30.000 0.001 30.000 (38) 1.709 11.663 0.001 9.564

tre 0.001 200.000 (38) 0.001 144.159

trp-L 0.001 10.000 0.084 0.230

tyr-L 0.001 10.000 0.593 1.244

udp 0.001 10.000 0.124 0.588

udpg 0.001 10.000 0.521 1.832

ump 0.001 10.000 0.016 0.072

utp 0.001 10.000 0.573 1.495

val-L 0.001 10.000 5.017 9.933

xu5p-D 0.001 10.000 0.079 0.742

zymst 0.001 10.000 0.001 10.000

α errα(y) EPEα DFα χ2red,α

0 0.047 0.093 69 0.86

0.01 0.047 0.114 100 0.73

0.05 0.075 0.166 107 1.00

0.1 0.093 0.185 98 1.21

Table S9. Bounds on the metabolite concentrations (continued).

Table S10. Goodness of fit obtained in the regression analysis. The goodness of fit was determined by the reduced chi square statistics (methods 6), which we evaluated for different choices of the regularization parameter α. We found that the model fits the data the best for a α of 0.05 (χ2

red,α = 1.00).

63

Chapter 3

Procedure for developing a thermodynamic metabolic model

Bastian Niebel, Simeon Leupold, Matthias Heinemann

(Manuscript in preparation)

Recently, we developed a new flux balance analysis-based approach, which enabled us to predict extracellular and intracellular fluxes of Saccharomyces cerevisiae with unprecedent-ed accuracy. These predictions were performed with a thermodynamic metabolic network model including a comprehensive description of the biochemical thermodynamics: the second law of thermodynamic, a cellular Gibbs energy balance, and an upper limit on the cellular entropy production rate. Here, we present a workflow that outlines the steps to build such a thermodynamic metabolic network model from an existing metabolic recon-struction, and using thermodynamic data for the metabolic processes and a training data set. This workflow consists of the four steps, (I) adding biochemical information to the stoichiometric network, (II) determining Gibbs energies for the metabolic processes, (III) reducing the stoichiometric network, and (IV) training the model on experimental data. Here, in addition to the model for S. cerevisiae, which is described in Chapter 2, we apply steps (I-II) of this workflow on the most recent genome scale reconstruction for Escherichia coli.

BN and SL contributed equally to this work. BN developed the procedure. SL carried out the computational simu-lations and analyzed the data. BN and SL made the figures. BN wrote the manuscript. SL contributed to the writing. MH supervised the research and contributed to the writing

64

Chapter 3

IntroductionIn the past decades, metabolic networks of different organisms, ranging from bacteria, fungi, mammalian cells, to plants, have been reconstructed (126-129). These networks consist of the steady-state mass balances for the metabolites and describe the rates through metabolic pro-cesses, i.e. chemical conversions and metabolite transport. Metabolic network models have become a valuable tool in research and for applications (4, 6).

An often used tool to analyze these models is flux balance analysis (FBA) (14). FBA uses mathematical optimization to address the problem that these network models represent an underdetermined system of equations. Typically, an objective function, such as maximizing the cellular growth rate subject to the constraints defined by the mass balances of the meta-bolic network, is optimized. While FBA has been found to generate reasonable predictions in some conditions (4, 6, 14), so far no condition-independent set of constraints and objective functions could be identified that were able to correctly predict fluxes across different condi-tions (130).

Recently, we identified a thermodynamic constraint (Chapter 2), which, when implemented into a thermodynamic metabolic network model for Saccharomyces cerevisiae and used with the objective of biomass maximization, gave correct FBA-predictions across a wide range of different conditions. This identified constraint enforces that metabolic operation does not exceed a maximum rate of cellular entropy production. The maximal rate has to be deter-mined for the given organism from experimental data, i.e. physiological rates and metabolite levels. Then it can be used with a stoichiometric metabolic network model including a correct description of the biochemical thermodynamics to predict metabolic fluxes accurately under wide range of conditions.

A metabolic network model with a comprehensive description of the biochemical thermo-dynamics, i.e. a thermodynamic metabolic network model, contains (i) mass-, charge- and pH-dependent-proton balances, (ii) the second law of thermodynamics, and (iii) a cellular Gibbs energy balance (Chapter 2, methods 2), which ties the rates of Gibbs energy exchange with the environments to the cellular rate of entropy production, σcell, where later rate is the sum of the metabolic processes’ entropy production rates.

Here, we present a workflow consisting of four steps, with which a genome-scale model can be translated into a thermodynamic metabolic network model. This workflow was developed based on the model building process of Chapter 2, which resulted in a model for S. cerevisiae. We illustrate the first three steps on this workflow on the most recent genome scale recon-struction for Escherichia coli.

WorkflowIn order to use FBA with an upper limit in the cellular entropy production rate, σcellup-con-strained FBA, a metabolic network model needs to be augmented for a proper thermo-dynamic description and constraints. Such a thermodynamic metabolic network mod-el M(v,ln c) ≤ 0 (Chapter 2, methods 5) consists of a system of linear and non-linear equations with the variables denoting the rates v through the metabolic processes and

65

Workflow

I. Adding biochemical information

(iii) Calculate charge & proton balances

(ii) Specify transport stoichiometry

Stoichiometric metabolic network with a comprehensive description of the transport stoichiometry, charge balances and pH-dependent proton balances

II. Determine Gibbs energies of metabolic processes

Equilibrium constants of enzymatic reactions

III. Model extraction and reduction(i) Removing inactive metabolic processes

Medium composition

Reduced stoichiometric metabolic

(ii) Merging metabolic processes of linear pathways

IV. Model training and validation Training and validation data- extracellular rates- metabolite concentrations

(i) Regression analysis

(iii) Flux coupling analysis

(ii) Flux balance analysis

Validated thermodynamic constraint-based model

(iii) Statistical analysis

Steps Data

(i) Identify chemical species based on pKA and pH

Stoichiometric metabolic network

Structure of metabolitesCompartmental pH’s

Transport mechanisms

Compartmental pH’sand ionic strengths

Electical membrane potentials

Concentration ranges

Gibbs energy changes forthe metabolic processes

(iii) Specify concentration ranges

(ii) Calculate Gibbs energy of metabolite transports

(i) Estimate and transform standard Gibbs energies

Figure 1. Workflow diagram for the development of a thermodynamic metabolic network model in four steps.

66

Chapter 3

the metabolite concentrations ln c (on a logarithmic scale). The solution space of the ther-modynamic metabolic network model defines the possible metabolic operations. In order to define M(v,ln c) ≤ 0, the following data are necessary: the stoichiometric coefficients for the reactions/transporters that define the metabolic network, Gibbs energies changes of the meta-bolic processes, and rough condition-independent bounds on the metabolite concentrations, which can be obtained from a collection of quantitative metabolome measurements.

Here, we present a workflow (Fig. 1) to build such a thermodynamic metabolic network mod-el starting from a genome-scale metabolic reconstruction (131-134). This workflow consists of four steps: (I) adding biochemical information to the stoichiometric network, (II) deter-mining the Gibbs energy changes of the metabolic processes, (III) reducing the stoichiomet-ric network, and (IV) training the thermodynamic metabolic network model in order to iden-tify the unknown model parameters. Besides the model for S. cerevisiae (refer for the specific details on the implementation to Chapter 2, Extended Methods, which has been tested and validated), we illustrate the steps (I-III) with the concrete example of converting a recent ge-nome-scale model of E. coli into a thermodynamic metabolic model.

Step I. Adding biochemical information to the stoichiometric network

Typically, genome-scale metabolic reconstructions contain a-priori defined proton balanc-es and - in order to obtain biological meaningful results - constraints on the directionality of certain metabolic processes (57). In a first step, we remove the a-priori proton balances from the metabolic network, because in our thermodynamic metabolic network models we use pH-dependent proton balances (Chapter 2, methods 1). Further, in order to remove any heuristic assumptions, in our model, we constrain the directionality of all metabolic processes only by the 2nd law of thermodynamics, and therefore we also remove in a preprocessing step, the directionality assignments from the metabolic network. Also, we merge the periplasmic and the extracellular space into one compartment, since the selectivity of the outer membrane is rather low, and chemical conditions (namely pH and ionic strength) are similar (135), and therefore the transport over the outer membrane can be neglected.

After these preprocessing steps, we then (i) determine for each metabolite the chemical spe-cies that are present at the compartments’ pH value, (ii) add metabolite transport processes to allow translocation of individual chemical species, and (iii) add charge and pH-dependent proton balances to enforce pH- and charge homeostasis in each compartment, where these balances are influenced by the chemical conversions and transport of chemical species. De-tails of these steps are illustrated in the following paragraphs.

Note, we differentiate between the reactants, i.e. the equilibrated mixture of chemical species at a specific pH, and chemical species, because this differentiation allows us to model the transport of chemical species between cellular compartments at different pH values (Fig. 2A) (35). The differentiation between chemical species and reactants in the charge balances and pH-dependent proton balances and metabolite transport allows to correctly model energy transfers between different metabolic processes.

(i) The chemical species of each metabolite (reactant) are determined from the acid dissocia-tion constants (pKa) of the metabolite. The pKa values are estimated from the chemical struc-ture of the metabolites, using the Marvin Suit from ChemAxon. This estimation is based on

67

Workflow

the partial charge distribution of each molecule and a set of training molecules. The chemical structure of the metabolites is downloaded from the KEGG database (136) and and encoded as IUPAC International Chemical Identifier (InChI) (137). Based on the pKa values and the compartmental pH values, we then calculate the abundance fraction of each chemical spe-cies of a metabolite (reactant) (138). In order to keep the model compact, we only consider chemical species, which have a fractional abundance of at least 10%. Charge and amount of protons for species of metabolites, whose structures cannot be properly encoded as InChI, such as molecular clusters, e.g. Fe-S clusters, or complex organometallics, were manually de-fined based on structural knowledge and databases such as EcoCyc (139) or ecmdb (140). For metabolites with unknown or ambiguous molecule structures, such as proteins or tRNA, where pKA values cannot be determined, we only considered one species with neutral charge.

In this step, it is advisable to check the elemental balances (for all elements except hydrogen) of all chemical transformations in the metabolic network. This consistency check detects po-tential discrepancies between the metabolites’ elemental compositions specified by (a) the InChI and (b) the original genome-scale metabolic network. If there is a discrepancy, the chemical transformations should be corrected to account for the elemental composition spec-ified by the InChI.

A0.2 H+

Succinate transport

20 % 80 % 100%

Proton symport(ΔrG’t 13.6 KJ mol-1)

Uniport (ΔrG’t -21.3 KJ mol-1)

Reactant(pH 7)

Reactant ( pH 5) Membrane potential

60 mV

0.2 H+ 2 H+ 2 H+

0

40

0

Frac

tion

of

chem

ical

con

vers

ions

(%)

Correlation between pH and standard Gibbs energies of reactions

Correlation between pH and stoichiometry of proton changes

B C

20

-1 10-1 1

Figure 2. Adding biochemical information to the metabolic network model. (A) Example of a transport process and its modelling (transport of unprotonated succinate), where we included two different variants (uniport and proton symport) of this transport process to take into account differences in the charge balance, proton balances, and the Gibbs energy of this metabolite transport (For the mathematical model description of the transport processes refer to Chapter 2, methods 1 & 3) (B) Histogram of Spearman rank correlation coefficients between pH values and stoi-chiometric coefficients describing the changes in protons of 1473 reactions shows the pH-dependence of the proton balances. (C) Histogram of Spearman rank correlation coefficients between pH values and standard transformed Gibbs energies of 1473 reactions shows the sensitivity of these Gibbs energies with respect to changes in pH.

68

Chapter 3

(ii) As for most metabolite transporters the exactly transported species is not known, we introduce for every metabolite transport multiple variants in order to allow for transport of different chemical species:

For metabolites that are transported by diffusion, proton symporter, proton antiporter, or unknown transport mechanism, we allow the transport of the species (with fractional abun-dance greater than 10%) or species combinations. Additionally, in case of transport processes, for which at the given pH value no charge neutral transport variant exists, we introduce an additional transport reaction, in which protons—balancing the charge—are co-translocated together with the respective species, i.e. adding a proton symporter or antiporter. This addi-tional transport variant ensures that for every metabolite a transport variant exists that does not translocate net charge. For metabolites that are transported by active transport mecha-nisms, we include transport processes for all considered species (with fractional abundance greater than 10%), where we accurately model the transport mechanism, e.g. for the phospho-transferase system, first the unphosphorylated molecule is transported over the membrane and subsequently phosphorylated in the cytoplasmic space.

For redox reactions, where half reactions are taking place on both sides of a membrane, e.g. complexes of the respiratory chain, we also model the transfer of electrons across the mem-brane, since this electron transfer influences the charge and Gibbs energy changes of the transport process (for details refer to Chapter 2, methods 1, 3). To model these redox process-es, one needs to determine, on which side of the membrane the redox cofactors are located and needs to add a chemical species for the electrons. The location of the redox cofactors are determined from the mechanisms of this membrane-bound redox reactions. The necessary information has to be taken from biochemical textbooks and the scientific literature. For an example please refer to the implementation of the respiratory chain of the S. cerevisiae model (Chapter 2, fig. S1, table S5-S7).

(iii) To specify the charge and proton balances, we calculate the stoichiometric coefficients of the charge and proton changes associated with the metabolic processes. The stoichiometric coefficients of the charge changes are based on the charges of the translocated chemical species and can be calculated according to Chapter 2, methods 1. Note, in case the metabolic network does not account for ions, e.g. potassium, calcium or magnesium, we introduce a generic ion species, for which we also include different transport variants, i.e. a uniport, a symport, and an ATP-driven active transport. Because the number of hydrogen atoms of reactants depends on the compartmental pH values, the stoichiometric coefficients of the proton changes are pH-dependent (Fig 2, A and B). To take into account this pH-dependency, the stoichiometric coefficients of the proton changes are calculated according to Chapter 2, methods 1, where we need the fractional abundances of the chemical species in the reactants (see (i)).

For building the thermodynamic metabolic network model of E. coli, the stoichiometry of the metabolic network was gathered from the most recent genome-scale metabolic recon-struction network (126), comprising 1136 unique metabolites located in the cytosol, peri-plasmatic and the extracellular space, 1471 chemical conversions, 782 metabolite transporter and 330 exchange processes. By merging the periplasmatic and the extracellular space, 325 metabolite transport processes between both compartments were removed from the mod-el and 193 periplasmatic chemical conversions were relocated into the extracellular space.

69

Workflow

Subsequently, the remaining set of 457 metabolite transport processes were enlarged to 604 transport variants by accounting for the different species (with fractional abundance greater than 10%). The resulting model contains in total 1132 metabolites and 2075 metabolic pro-cesses (1471 chemical conversions, 604 metabolite transport) and 330 exchange processes.

Step II. Determining Gibbs energies changes of metabolic processes

Next, we need to gather the necessary data to calculate the Gibbs energies of the metabolic processes (Chapter 2, methods 3). These Gibbs energies are used in the thermodynamic met-abolic network model to constrain the directionality of the metabolic processes through the second law of thermodynamics and to calculate the entropy production rates. To calculate the Gibbs energies of metabolic processes, we need (i) standard Gibbs energies of reactions, which describe the differences in the reactants’ Gibbs energies of formations, (ii) Gibbs en-ergies of metabolite transports associated with pH gradients, with translocation of protons by proton pumps, and with transport of charged metabolites over membrane potentials (Fig. 2A), and (iii) default ranges for the metabolite concentrations spanning typical physiological concentration levels. In the following paragraphs, we further illustrate these three different kinds of input data.

(i) We estimate the standard Gibbs energies of formations and reactions using the component contribution method (CCM) (27). CCM infers the standard Gibbs energies based on mea-sured equilibrium constants (31), and approximations for the Gibbs energies by the group contribution (29). The CCM estimates the means and errors for the standard Gibbs energies of reactions. Both estimates are later used to determine a thermodynamically consistent set of Gibbs reaction energies (cf. section ‘train the thermodynamic metabolic model’ (Step IV)). The Gibbs energy of formation of the biomass is estimated based on the elemental biomass composition using an empirical formula (69). Because metabolic processes take place in cel-lular compartments with different pH values and ionic strengths, we take into account the effect of pH and ionic strengths on the standard Gibbs energies of reactions. We do this, because the results of a sensitivity analysis for all 1473 chemical conversion show that this is indeed necessary (Fig. 2C). Therefore, the standard Gibbs energies are transformed to the compartmental pH values and ionic strengths (21). The Gibbs energy of formation of the biomass is transformed to the cytosolic pH.

(ii) The Gibbs energies of metabolite transports are calculated using the formula described in Chapter 2, methods 3. Here, we need the fractional abundances of the chemical species in the reactants (see Step I), and electrical potentials across cellular membranes, which can be found in the scientific literature.

(iii) Typically, intracellular metabolite concentrations vary between 1 µM and 10 mM (36-39), which we use as default range. For certain metabolites, such as central carbon metabolism intermediates, amino acids, redox cofactors, oxygen or carbon dioxide, we use ranges, which are based on minimal and maximum values reported in scientific literature or their maximum solubility (141,142). These data were reprocessed using a condition independent dry weight specific cellular volume of 0.0023 L gDW

-1 (141).

For building the thermodynamic metabolic network model of E. coli, we determined the Gibbs energies based on a pH of 7.6 (143) in the cytosol and 7.0 in the extracellular space

70

Chapter 3

(corresponding to M9 medium). The ionic strength was assumed to be 0.15 M (144) in the cytosol and 0.2 M in the extracellular space (corresponding to M9 medium). Based on the component contribution method, we were able to determine the Gibbs energy of reactions for 73% of the chemical transformations. The Gibbs energy of reactions, for which the CCM was not able to make an estimate, was allowed to vary by ± 105 KJ/mol.

Step III. Reducing the stoichiometric network

Third, we reduce the metabolic network (example given in Fig 3A) without compromising its predictive power. Because in the model every metabolic processes is related to a binary vari-able (by the second law of thermodynamics) and a non-linear equation (by the entropy pro-duction rate), a reduction in the number of metabolic processes speeds up the computational analyses of the model. Such reduction step is advisable to facilitate the later computational analysis of the thermodynamic metabolic model, but is not mandatory to proceed with step III. A reduction can be performed by specifying the scope of the model via defining possible growth substrates and products by constraining the extracellular rates of these compounds accordingly. After such a specification, a number of different model reductions are possi-ble: (i) identifying inactive metabolic processes, i.e. metabolic processes that can never carry metabolic flux at the specified conditions, (ii) merging of reactions within a linear pathway, and (iii) identifying linear dependent processes, i.e. reactions that have fully coupled fluxes (Note, step (iii) is not a reduction in the model size, but a reduction in the model complexity to facilitate later computational analyses).

(i) First, we identify inactive metabolic processes (Fig. 3B), using flux variability analysis (16), where we minimize and maximize the flux through each metabolic process using the mass balances as constraints. If the minimum and the maximum flux of a metabolic process is zero, then the metabolic process is inactive for the selected scope and can be removed from the network.

(ii) Second, the metabolic processes of linear pathways can be merged into one lumped met-abolic process (Fig. 3C). This is possible, since - under steady-state conditions—the metabol-ic fluxes through reactions of a linear pathway are equal. Here, we identify linear pathways from the structure of the stoichiometric matrix including the charge and proton balances, by identifying rows in the stoichiometric matrix that only contain two non-zero stoichiometric coefficients, which correspond to metabolites with only one production and consumption process. We can then eliminate this metabolite by merging its producing and consuming met-abolic process into one lumped reaction. This procedure should be repeated until all rows in the stoichiometric matrix have more than two non-zero stoichiometric coefficients, i.e. all reactions of linear pathways are merged. However, attention needs to be paid that metabolic processes, which include metabolite transports, are not lumped, since this lumping would disturb the charge and proton balances. Note, metabolic processes, for which standard Gibbs energies of reactions can be estimated from component-contribution method, can be exclud-ed from being lumped. The exclusion of these metabolic processes ensures that the second law is also considered locally for these metabolic processes, and therefore metabolites being part of these metabolic processes can be additionally constrained.

(iii) Third, linear dependent metabolic processes are identified by flux coupling analysis (96)

71

Workflow

(Fig. 3D), because such coupling allows the reduction of the model complexity (i.e. reduction in the binary variables), and accomplish a reduction in the number of optimizations that need to be performed in the later flux variability analyses. To this end, we determine the minimum and maximum flux ratios between any pair of two metabolic or exchange processes. Two pro-cesses are identified as linear dependent fully coupled, when their minimum and maximum flux ratios are identical. These linear dependent fully coupled processes are then grouped into subsets of the metabolic processes, where the each of this subsets is a degree of freedom that is later analyzed.

For building the thermodynamic metabolic network model of E. coli, we reduced the model considering the following substrates: acetate, fructose, galactose, glycerol, glucose, mannose, pyruvate, succinate and xylose. Using flux variability analysis 544 metabolic processes (294 chemical conversions, 250 metabolite transport) and 290 exchange processes were identified as inactive under the considered conditions and thus removed from the network. By lumping reactions of linear pathways, we further reduced the network by 339 chemical conversions. Therefore we reduced the model from 2075 to 1192 metabolic processes and from 330 to 40 exchange processes (total reduction of 48 %). By flux coupling analysis, we further reduced the 1232 processes (1192 metabolic- & 40 exchange-processes) to 1140 processes.

A

F

NAD+

NADP+B

C D E

Initial stoichiometricnetwork

2430 processes2100 metabolic processes330 exchange processes

(i) Removing inactive metabolic processes

1596 processes1556 metabolic processes40 exchange processes

A

F

NAD+

NADP+B

C D E

(ii) Merging metabolicprocesses of linear pathways

1232 processes1192 metabolic processes40 exchange processes

A

F

NAD+

NADP+

C D E

A

F

NAD+

NADP+

D E

(iii) Identify linear dependent metabolic processes

1140 processes

A

D

C

B

Figure 3. The automated model ex-traction and reduction is performed in multiple steps. (A) Initial stoichiomet-ric network. (B) Inactive processes are removed by flux variability analysis. (C) Processes of linear pathways are detected and merged. (D) Fully coupled processes are identified by flux coupling analysis.

72

Chapter 3

Step IV. Training the model

Fourth, unknown parameters of the thermodynamic metabolic model are estimated from experimental training data. These model parameters are (i) a thermodynamic consistent set of standard Gibbs energies for the chemical conversions (incl. chemical conversions for which no estimate can be obtained from the component contribution method), (ii) narrower phys-iological ranges for the concentrations and for the standard Gibbs energies of reactions, and (iii) the maximal cellular rate of entropy production.

(i) The training is done by regression analysis according to Chapter 2, methods 6, which also includes parametric bootstrap to determine the accuracy of the parameter estimates. The training data consists of measurements, i.e. mean and standard errors, of extracellular rates, the estimates for the standard Gibbs energies from the component contribution method (in-cluding mean and standard errors, see step II), and measurements of intracellular metabolite concentrations. The extracellular rates and metabolite concentrations should be measured from exponentially growing cultures at different conditions. The cultures should be selected in such a way that the cells grow at different growth rates (incl. unlimited growth) and have different metabolic operations, i.e. purely respiratory and respiro-fermentative metabolism, and therefore the model parameters, e.g. the maximum entropy production rate, contain the differences in the metabolic operations. To obtain a wide range of different growth rates, che-mostat cultures are ideally used.

(ii) After the regression analysis, we validate the estimated parameters by predicting meta-bolic operations for different growth conditions. These predictions are made by flux balance analysis with a cellular objective of maximizing growth according to Chapter 2, methods 8. The predicted metabolic operations are then evaluated by comparison with test data, i.e. ex-tracellular rates obtained from conditions that were not used for the training.

(iii) The solution space of the validated model is then further statistically analyzed using vari-ability analysis and Markov Chain Monte Carlo (MCMC) sampling according to Chapter 2, methods 8. The variability analysis and MCMC sampling allows to exactly predict different intracellular quantities, e.g. intracellular fluxes, concentrations and entropy production rates.

DiscussionThe developed workflow allows us to convert genome-scale metabolic reconstructions, which exist for a large range of different organisms, into a thermodynamic metabolic network mod-el. Such a thermodynamic metabolic network model allows to predict and analyze intracellu-lar fluxes without relying on heuristic assumptions. To build such a model, one requires only a carefully selected set of easily obtainable training data, i.e. extracellular rates, standard Gibbs energies of reactions, and intracellular metabolite concentrations. Key for the development of such models are (i) a correct thermodynamic description and (ii) a manageable size because the computational analyses are computationally very demanding. Here, we illustrated how such a model can be developed.

The limitation of the thermodynamic metabolic network models is the computational com-plexity introduced by the nonlinearity of these models. Therefore, in order to utilize these models, proficiency with mathematical optimization and high performance computing is re-

73

Methods

quired. This complexity along with the size of the genome scale metabolic network model is the reason, why we only exemplify in this chapter the necessary steps to build such a model.

MethodsThermodynamic metabolic model

The thermodynamic metabolic model (Chapter 2, methods 5), M(v,ln c) ≤ 0, consists of a set of linear and non-linear equations, which are based on: the mass balances including charge and proton balances; the cellular Gibbs energy balance; an equation to calculate the cellular entropy production rate σcell; equations to calculate the entropy production rates σ; equations to calculate the Gibbs energy exchange rates g; equations to calculate the Gibbs energies of reactions ∆rG’; equations to calculate the Gibbs energies of formation ∆fG’ of the metabolites i that are transferred across the system boundary; the second law of thermodynamics for the metabolic processes and the growth process,

M v c

S v v i

T g

ij jj MET i EXG

cellii EXG

celljj

( , ln ) ≤

= ∀

− =

=

∈ ∈

∈

∑∑

0

σ

σ σ∈∈∑

= − ∀ ∈

∀ ∈

+

=

=

MET

jG v

T

f i i

r jo

r j

i

r j

r j j j MET

G v i EXG

G G

g

G

σ∆

∆

∆ ∆∆

’

’

’ ’’ tt

f i f io

i

j r j

ij ii hRT S c j MET

G G RT c i EXG

G

+ ∀ ∈

= + ∈

> ≥

∉ +∑ ln

’ ’ ln

’

∆ ∆

∆σ 0 0.. \ , 5 2

0

∀ ∈

+ ≥

j MET BMSYN H Ot

BMSYN ATPHσ σ

, (Eq.1)

where: i are the metabolites, j ∈ MET are the metabolic processes, i ∈ EXG are the exchange processes, S is the stoichiometric matrix, v are rates through metabolic and exchange pro-cesses, T is temperature, ∆rG’o are standard Gibbs energies of the chemical conversions, ∆rG’t are the Gibbs energies of metabolite transports, R is the universal gas constant, ln c are the metabolite concentrations on the logarithmic scale, BMSYN is the biomass synthesis reaction, H2Ot are the water transporters across the membranes, ATPH is the ATP hydrolysis reaction.

Determining chemical species and Gibbs free energies of formation using the component contribution method

A list of formulas of all metabolic processes, in which the metabolites were encod-ed by their KEGG ID’s, was used as input for the component contribution method. Metabolites, for which no unique KEGG ID exists, were given a consecutive ‘fake KEGG ID’ starting from C99999 backwards.

In a first step, the CCM determines for each metabolite, calling ChemAxon, the pKa values

74

Chapter 3

and subsequently all at pH 7 present species with charge and number of protons. For each species, the Gibbs free energy of formation is then estimated using a combination of group and reactant contribution method. Finally, the standard deviation for each Gibbs free energy of reaction is calculated using a cross-validation benchmark and a set of measured Gibbs free energies of reactions.

Flux variability analysis and flux coupling analysis

For the reduction of the stoichiometric network (Step III), we formulated a solution space ΩMR based on the linear constraints of M(v,ln c) ≤ 0 (Eq. 1),

ΩMRij jj MET i EXG i

loi i

upv S v v i v v v i EXG= = ∀( ) ∧ ≤ ≤ ∀ ∈( ) ∈ ∈∑| , (Eq.2)

where we constrained the extracellular rates, i ∈ EXG according to the medium conditions. Note: Because the solution space ΩMR only consists of linear constrains, we use linear pro-gramming algorithms (CPLEX) to solve the optimizations. To determine the flux variability (16), we solved the following optimization problems,

v v v j MET i EXGj i j iMR

( )min/max

( )min/ max := ∈ ∀ ∈ ∨ ∈Ω , (Eq.3)

where we minimized and maximized v of every metabolic and exchange process. A metabolic or exchange process, for which the minimum and maximum rate was zero, was identified as inactive and subsequently removed from the model. To determine linear dependent reac-tions, we determined the minimum and maximum flux ratios,

R v v v j j METj i j j i j iMR

( ) (i)min/max

( ) ( )min/ max / : ,1 2 1 2 1 2= ∈ ∀ ∈ ∨Ω ii EXG1 2,i ∈ , (Eq.4)

for each pair j1,j2 (i1,i2) of two metabolic or exchange processes. We then identified two pro-cesses as linearly dependent when the minimum equaled the maximum flux ratio according to (96).

75

Chapter 4

Accurate estimation of metabolic fluxes based on a thermodynamic metabolic network model and

isotopomer balancing

Bastian Niebel, Georg Hubmann, Gesa Behrends, Matthias Heinemann

(Manuscript in preparation)

Metabolic fluxes are a quantitative description of metabolic operation. Since metabolic fluxes cannot be measured directly, model-based methods have been developed to sta-tistically infer metabolic fluxes from data, i.e. from isotopomer patterns obtained from 13C-labelling experiments. However, the current analysis methods exploit only small met-abolic network models with numerous heuristic assumptions, which can lead to bias and false model predictions. Aiming to circumvent heuristic assumptions, here, we combined a recently developed thermodynamic metabolic network model and isotopomer balancing. Specifically, we used experimental data on extracellular rates, metabolite concentrations, standard Gibbs energies of reactions, and 13C labeling patterns, and estimated the metabol-ic fluxes for the yeast Saccharomyces cerevisiae with a minimal set of assumptions, thereby delivering an exact view of the thermodynamically feasible flux space. Further, the method also allowed to estimate the ratio of the forward and backward fluxes through enzymatic reactions, opening the door towards identifying kinetic rates laws in vivo.

BN carried out the computational simulations, analyzed the computational data, made the figures, and wrote the manuscript. GH developed the isotopomer model. GH & GB performed the labeling experiments and analyzed the mass spectroscopy data. GH and GB commented on the manuscript. BN, GH and MH designed the study. MH supervised the research and contributed to the writing.

76

Chapter 4

IntroductionMetabolic fluxes emerge as a result of environmental conditions, enzyme expression and reg-ulation at all cellular levels. Knowing metabolic fluxes can thus provide insights into metabol-ic operation and allows to evaluate cellular responses to environmental, genetic or chemical perturbations (145, 146). Metabolic fluxes cannot be measured directly. Instead, metabolic fluxes need to be inferred from experimental data analyzed with metabolic models (145). Be-cause physiological measurements alone provide limited information about intracellular flux-es, isotope tracer experiments are typically performed. In these experiments, cells are grown on 13C-labeled substrates, and then the steady-state isotopomer distribution of the 13C atoms in intracellular metabolites (typically protein-bound amino acids) is assessed with mass-spec-trometry or nuclear magnetic resonance. The isotopomer distribution patterns together with measured extracellular rates are then used to infer the intracellular metabolic fluxes (145-147).

Two different statistical approaches exist for inferring metabolic fluxes: metabolic flux ratio analysis (147-149) and whole isotopomer balancing (146, 147, 150). In metabolic flux ratios analysis, flux ratios between converging pathways are inferred from manually derived ana-lytical equations. In a second step, metabolic fluxes are inferred by fitting the estimated flux ratios and extracellular rates to a stoichiometric metabolic network (147). In whole isoto-pomer balancing, metabolic fluxes are determined by fitting isotopomer distribution patterns and extracellular rates to a metabolic network model, which contains mass- and isotopomer balances (150), with the later balances describing all possible isotopomer distributions in the metabolic network model. To efficiently solve the isotopomer balances, different mathemat-ical representations have been developed, e.g. cumomer balances (151) and elementary met-abolic units (152). Typically, the models used for whole isotopomer balancing also include – next to the net fluxes – also backward fluxes (so-called ‘exchange fluxes’) to allow exchange of labeling pattern also against the net flux direction. These backward fluxes are usually not considered in flux ratio analysis.

Because metabolic networks form underdetermined systems (153), metabolic network mod-els used in both of these approaches generally cover only a small number of metabolic pro-cesses, i.e. chemical transformations and metabolite transporters, being in the order of 40-50, and are constrained by additional assumptions. For example, pathways are assumed to be inactive and reactions are assumed to be unidirectional (147, 150). While constraining the models decreases the variance of the estimated fluxes (99, 154), this however may lead to inaccurate and biased flux estimates (155), if assumptions were incorrect. Because there is typically no possibility to check the validity of such assumptions, there is a strong need for new methods to determine intracellular fluxes in a more unbiased way, i.e. by using larger models and by replacing heuristic assumptions with physical principles.

Recently, we developed a novel approach to predict metabolic fluxes with an FBA-ap-proach with remarkable accuracy (Chapter 2, main text). This approach uses a thermody-namic metabolic network model M(v,ln c) ≤ 0 for Saccharomyces cerevisiae with variables v (metabolic fluxes) and ln c (natural logarithm of the intracellular metabolite concentra-tions). This thermodynamic metabolic network model contains a comprehensive description of the biochemical thermodynamics, mass-, charge-, and pH-dependent proton balances,

77

Results and discussion

a cellular Gibbs energy balance, and the second law of thermodynamics applied on the meta-bolic fluxes. The solution space of M(v,ln c) ≤ 0 defines the thermodynamically feasible space of metabolic operations, i.e. flux distributions and metabolite concentration levels.

Here, we exploit this thermodynamic metabolic network model, comprising 242 metabolic processes and without any a-priori reaction irreversibility, and combine it with isotopomer balancing in order to develop a new method to infer intracellular metabolic fluxes from ex-perimental data with unprecedented accuracy and without any heuristic assumptions. First, we generate samples of estimates of fluxes based on measured extracellular rates, intracellular metabolite concentrations, and standard Gibbs energies of reactions, which we then - in a second step - evaluate against measured isotopomer distribution patterns. Note, that in con-trast to previous methods, we also exploit metabolomics data to infer metabolic fluxes. We demonstrate the power of this method with S. cerevisiae grown in glucose batch cultures using extracellular rate measurements, metabolome data and isotopomer distribution patterns ob-tained from protein-bound amino acids.

Results and discussionEstimating metabolic fluxes using the thermodynamic metabolic model and isotopomer balancing

Our method for the accurate estimation of metabolic fluxes consists of three steps (Fig. 1): In the first step, experimental data, i.e. extracellular rates, standard Gibbs energies of reactions, and intracellular metabolite concentrations, are fitted by a regression analysis to the thermo-dynamic metabolic network model M(v,ln c) ≤ 0 (Chapter 2, methods 5). M(v,ln c) ≤ 0 de-scribes the metabolism of S. cerevisiae with 242 metabolic processes (which were considered to be fully reversible). Further, the model includes a detailed thermodynamic description of all metabolic processes, i.e. Gibbs energy changes ∆rG’ including the effect of pH and ion-ic strength in the different cellular compartments. We trained the model with experimental trained data from S. cerevisiae KOY.WT (cf. for description of the strains (156)), where six extracellular rates (this work), 30 intracellular concentrations of metabolites from central and amino acid metabolism (from (157)) were determined in shake flask cultures with exponen-tially growing cells. Further, we included estimates for the standard Gibbs energies of reac-tions from component contribution (cf. Chapter 3) in the training data. Regression analysis with this model was carried out as described in (Chapter 2, methods 6). From the regression analysis, we then obtained a thermodynamically consistent set of standard Gibbs energies of reactions, a consolidated set of extracellular rates (table S1) and intracellular metabolite con-centrations (table S2). Note, a consolidated data set consists of model estimates for statistical values (medians, confidence intervals) of the different measurements. Here, we found that the model describes the data well with a reduced chi square of 0.96.

In a next step, we constrained the extracellular rates and intracellular metabolite concentra-tion by the 97.5% confidence intervals of the consolidated data, which have been determined from the regression analysis (see above), and determined ~1000 randomly sampled flux dis-tributions from the solution space of the thermodynamic metabolic network model (now constrained by the ‘consolidated data’) by Markov Chain Monte Carlo sampling as described in Chapter 2, methods 8. From these sampled flux distributions, we then determined different

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statistical values of the metabolic fluxes, e.g. averages, standard deviations, medians, quan-tiles.

In the last step, we evaluated how well every sampled flux distribution could be fitted to the measured isotopomer distribution patterns and how the goodness of this fit influenced the estimates for the different statistical values of the metabolic fluxes. Here, we obtained labeling patterns from experiments where either 100% 1-13C labelled glucose (in the following referred to as [1-13C]-labeled glucose; for the patterns see table S3) or 80% C12 glucose & 20% uniform-ly 13C labelled glucose (in the following referred to as [U-13C] labelled glucose; for the patterns see table S4) was used. To determine the fit of the measured isotopomer pattern to a sampled flux distribution (see previous step), we minimized the residual between the measured and simulated isotopomer patterns using iterative isotopomer balancing (150) (cf. tables S5-S6 for the isotopomer model), while keeping the net fluxes fixed but with treating the so-called ‘exchange fluxes’ (i.e. the backward fluxes) as free variables. Note, that in our model, all meta-bolic processes had a corresponding exchange flux, which we constrained between 0 and 100 mmol gCDW-1 h-1.

By performing this fitting for every of the ~1000 sampled flux distributions, we confirmed that the isotopomer model describes the measured isotopomer patterns well (Fig. 2A-B; ta-bles S3 -S4). Then, we used each sampled flux distributions’ residual sum of squares, which describes how well this sampled flux distribution fitted the measured labeling patterns, to build a quality measure (or weight), which quantifies this flux distribution’s goodness of fit to the 13C-labeling patterns. Here, we determined for each sampled flux distribution weights that reflect the goodness of fit to the [1-13C] labeling patterns (Fig. 2C) or the [U-13C] labeling patterns (Fig. 2D). Also, we determined for each flux distribution a weight that quantifies how well the sampled flux distributions described both labeling patterns—resulting from the [1-13C]- or [U-13C]-labeled glucose—simultaneously (Fig. 2E). Then, with these weights based

Figure 1. Illustration of the workflow to estimate metabolic fluxes with the thermody-namic metabolic network model and isotopomer balancing.

I. Regression with thermodynamic metabolic networkmodel, M(v,ln c) ≤ 0

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Color indicates the weight determined from the fitted residual of the isotopomer balancing

Point in the solution space describing:

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on the labeling patterns, we estimated—from the sampled distributions—the different sta-tistical values of the metabolic fluxes, e.g. averages, standard deviations, medians, quantiles. In this way, we included the labeling patterns’ information about the flux distribution in the estimation of the metabolic fluxes.13C labeling patterns validate the flux estimates of the thermodynamic metabolic network model

Next, we investigated how far the measured labeling patterns would result in a shift of the estimated fluxes. Therefore, we determined for each metabolic flux the median and 97.5% confidence range using the ~1000 sampled flux distributions and the weights representing the goodness of fit of the sampled flux distributions to the different labeling patterns (table S7). We compared the shifts in the estimates for the medians (Fig. 3A-C) and 97.5% confidence ranges (Fig. 3D-F), when we determined these statistical values without using weights from the labeling patterns (i.e. only sampling of the regression solution space of the thermodynam-ic metabolic network model), and when we used the weights from the fits to labeling patterns. Here we found that the average absolute change in the median is between 0.3 and 1.8 mmol gCDW-1 h-1 and the average relative change in the confidence range is between 0.1 and 0.3 % (cf. Table 1). Note, in order not to bias this analysis by linear dependencies between metabolic processes, we determined the average changes in the medians confidence ranges using a set of 144 linear-independent metabolic processes determined by flux coupling analysis (96). Because the observed shifts in the fluxes’ median and the confidence ranges of the fluxes by the isotopomer labels are small, this small shifts could have the following reasons: The esti-mated statistics of the fluxes by the thermodynamic metabolic network model are already suf-ficiently good that the labeling patterns do not improve the estimates. Or there is not enough information in the labeling data to infer intracellular fluxes with our large scale model.

To test whether there is flux information in the labeling patterns, we repeated the workflow

Figure 2. The fitted labeling patterns, determined by isotopomer balancing with the sampled flux distributions from the thermodynamic metabol-ic network model, were in accordance with the measured labeling patterns from (A) [1-13C]-la-beled and (B) [U-13C]–labeled glucose (see tables S3-S4 for individual data). The black points rep-resent the averages and the error bars represent the measured and fitted standard deviation of the mass fragments of protein bound amino-ac-ids. The average and the standard deviation was determined from all sampled flux distributions using weight representing the goodness of fit of the individual sampled flux distribution. This weight was based on the fit of the sampled flux distribution with respect to the measured label-ing patterns (i.e. the residual sum of squares of the isotopomer balancing), i.e. (C) patterns from [1-13C]-glucose, (D) from [U-13C]-glucose, and (E) by combining the patterns from [1-13C]- & [U-13C]-glucose. Note, the plots (C-E) were sort-ed according to the [1-13C]&[U-13C] weights (E).

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(cf. Fig. 1) with a stoichiometric metabolic network model, describing the same metabol-ic processes as the thermodynamic metabolic network model, but lacking the exact ther-modynamic description of the physical principles operating within the cell. Here, we build such a model, where we assumed every reaction to be reversible. (Note, every intracellular rate was constrained between -100 and +100 mmol gCDW-1 h-1). After having constrained the extracellular rates by their consolidated 97.5% confidence intervals (table S1), we sampled ~1000 flux distributions from this solution space (for details on the MCMC sampling refer to Chapter 2, methods 8), fitted each of these flux distributions to the [1-13C] and [U-13C] label-ing patterns and determined the weights representing the goodness of fit with respect to label-ing data, as described in the previous section (figure S1). Here, we found—in contrast to what we found with the thermodynamic network model—substantial shifts in the medians (Fig. 3G-I) and confidence ranges (Fig. 3J-L) of the intracellular fluxes (cf. Table 1), when these statistical quantities were determined with and without using the weights from the labeling data. We concluded that there is information in the labeling patterns, which can be extracted

Figure 3. Comparison of the estimates of different statistical quantities (medians and 97.5% confidence ranges) of a set of 145 linear independent reactions, when the estimation was performed with and with-out the weights that describe the fit to the different 13C labeling patterns. The statistical quantities were determined from ~1000 sampled flux distributions of (A-F) the thermodynamic metabolic network model and of (G-L) a stoichiometric metabolic network model. Note both networks described the stoi-chiometry of the same metabolic processes, where all processes were assumed to be fully reversible. For the stoichiometry network model, we defined an a-priori range for the rates through the processes between -100 and +100 mmol gCDW-1.

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even by such a large scale stoichiometric metabolic network model. In turn, this means that the flux estimates obtained with the thermodynamic metabolic network model were already so good, that the labeling data did globally not provide any substantial improvements.

Based on this, we investigated, how including the labeling patterns would locally influence the estimated fluxes of single key metabolic processes located at the branch points of central metabolic pathways. To this end, we approximated the probability density function (PDF) of these fluxes from the sampled flux distributions using kernel density estimation using the weights representing the goodness of fit to the combined [1-13C] & [U-13C] labeling patterns. By comparing PDFs estimated without weights to PDFs estimated with the weights, we found only small differences between these PDFs (Fig. 4). Further, we compared these estimates with earlier determined values of these fluxes, which had been obtained by flux ratio analysis and isotopomer balancing from glucose limited chemostat or batch cultures for S. cerevisi-ae CEN.PK113-7D, FY4, ATCC 32167 (100-102). Here, we found that the estimates for the fluxes through upper glycolysis, indicated by glucose-6-phosphate isomerase (Fig. 4A), the oxidative branch of the pentose phosphate pathway, indicated by phosphogluconate dehydro-genase (Fig. 4B), through the pyruvate dehydrogenase (Fig. 4C), and the lower TCA cycle, succinate-coenzymeA ligase (Fig. 4D), were comparable to previously estimated fluxes. Still there was substantial scatter in these earlier determined fluxes’ values, which can be explained by biological variance, but also by the use of different assumptions in these methods.

As mentioned above, intracellular metabolic fluxes cannot be measured directly, but need to be inferred from models. Thus, general statements about the “correctness” of estimated fluxes are not possible. However, estimated fluxes are likely to be more correct, if a model has as little assumptions (=potential bias) as possible. In contrast to earlier models, the ther-modynamic metabolic network model that we used replaced any heuristic assumptions by a carefully curated description on the underlying thermodynamic processes, and additional data, i.e. metabolomics and standard Gibbs energies of reaction. Further, the large size of the metabolic network used with 242 metabolic processes makes less assumptions on active metabolic pathways than earlier models, which typically consist of 40-50 metabolic process-es. This large size together with the detailed description of metabolite transports, allows to

Labeling patterns

Thermodynamic metabolic network model Stoichiometric metabolic network modelAbsolute median

changeRelative variability

changeAbsolute median

changeRelative variability

changeMean SD Mean SD Mean SD Mean SD

[1-13C] 0.3 1.3 0.1 0.2 6.7 12.4 5.2 4.8

[U-13C] 1.6 5.6 0.3 0.9 1.5 2.9 1.7 1.4

[1-13C] & [1-13C] 1.8 6.2 0.3 0.9 9.2 14.7 7.8 7.6

Table 1. Statistics of the shift in the estimated median and the variability (indicated by the 97.5% confidence ranges) when using the weights indicating the fit to the labeling patterns in comparison to not using these weights, i.e. with-out labeling data. Note: The statistics of the stoichiometric metabolic network model are biased by the a-priori default flux range, which was set to be between -100 and +100 mmol gCDW-1 h-1. The thermodynamic metabolic network model did not have such an a-priori flux range, thus these statistics do not have this bias.

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include charge, proton, and cofactor balances and therefore does not introduce any heuristic constraints. Therefore, the estimated fluxes from this model provide a much more bias-free view on the intracellular flux distribution of S. cerevisiae compared to earlier analyses. To conclude, we found that the use of the labeling patterns only slightly improved the flux esti-mations. The estimations done by the thermodynamic network model without using 13C were already sufficiently accurate.

The ratio of the forward over the backward enzymatic rates correlate with the Gibbs energy of reaction

As mentioned above, in the model that we used to evaluate the 13C labeling pattern, every net metabolic flux has a so-called exchange flux. These exchange fluxes specify the back fluxes in the metabolic process, as identified from 13C labelling patterns. With our new method we could estimate the average and standard deviation for a number of exchange fluxes with nar-row ranges from the measured 13C labelling patterns by determining the average and standard deviation of the exchange fluxes from the fits (and the weights) of the isotopomer balances to the [1-13C] and [U-13C] labeling patterns. Note: we excluded from this analysis lumped meta-

Figure 4. 13C labeling patterns slightly improve the accuracy of estimated fluxes. The accuracy was determined by estimating the probability density function for metabolic fluxes of (A) glucose-6-phosphate isomerase (PGI), (B) phosphoglu-conate dehydrogenase (GND), (C) pyruvate dehydrogenase (PDHm), and (D) succinate-coenzymeA ligase (SUCOAS1m). The probability density functions were estimated using kernel density estimation with a Gaussian kernel. The blue areas are the probability density functions estimated with the weights of the combined [U-13C] & [1-13C] labeling patterns, and the red areas are the prob-ability density functions estimated without weights, i.e. labeling patterns. The black vertical lines indicate flux estimates which have been determined using different methods, isotopomer balancing and flux ratio analysis (solid lines are from (100), dashed lines are from (101), dotted lines are from(102)).

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bolic processes and transport processes, because there the backward flux provides no kinetic information. If the estimated standard deviation of an exchange flux was less than 1% of the a-priori allowed range (here, we used a-priori ranges from 0 to 100 (mmol/gCDW/h), the exchange flux was denoted as being identifiable. In this way, we identified the exchange flux of 24 metabolic processes from the [1-13C] pattern and 20 from the [U-13C] patterns. Here, 10 exchange fluxes could be identified from both patterns (strikingly, both patterns gave the same values for these exchange fluxes), therefore, we could identify in total 34 exchange fluxes (Supplementary table 8).

For elementary reactions that follow mass action kinetics, one can show that the natural loga-rithm of the ratio between the forward, v+, and backward rate, v-, through the elementary re-action is proportional to the Gibbs energy of the reactions, , i.e. ∆rG’ = -RT ln (v+/ v-) (77,78), where R is the Gas constant, and T the temperature. For certain other types of enzymatic mechanisms this relationship has also been shown, and it has been suggested that it is gener-ally valid for enzymatic reactions (19, 20). However, this relationship has—to the best of our knowledge—never been tested until now.

To test whether such relationship exists, we first determined the average of all ln (v+/ v-) and ∆rG’ across all sampled points using the weights representing the goodness of fit to the [1-13C] and [U-13C] labeling patterns. Because the backward rate is the identified exchange flux, and the forward rate the net flux plus the exchange flux, this ratio can be directly determined from the metabolic processes’ net fluxes v and the exchange fluxes vxch using v+/ v- = |v|/vxch + 1. Here, we found for reactions, where the forward rate, v+, was close to the backward rate, v-,

Figure 5. The natural logarithm of the ratio of the forward rate vs. the backward rate negative-ly correlated with the reaction’s Gibbs energy. The location of the labels indicates the mean, and the error bars the standard deviation of the Gibbs energies and ratios. Red labels indicate estimates based on [1-13C] la-beling patterns and blue labels refer to [U-13C] labeling pat-terns. ln(forward rate / backward rate) (-)

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i.e. ln (v+/ v-) = 0, the relationship between ln (v+/ v-) and ∆rG’ did not hold. For reactions with ln (v+/ v-) > 0, we saw a very weak correlation between the ln (v+/ v-) and the Gibbs energy with ∆rG’ (F-statistic of a linear regression gave a p-value of 0.15 with a R2 of 0.12). The esti-mated slope of -1.8 +/- 1.2 KJ mol-1 was still in line with the theoretical proportionality factor of elementary reactions, -RT, which is -2.52 KJ mol-1. The reason for this weak correlation is likely due to missing explanatory variables and nonlinear effects, which are likely introduced by the catalytic properties of the enzyme, i.e. the kinetics of the enzyme, suggesting that the linear relationship of elementary steps does not generally hold for enzymatic reactions. One metabolic process, where it is particularly obvious that the linear relationship does not fit, is the transketolase reaction (for stoichiometry see TKT2 and TKT3 in supplementary table 7). We found that this metabolic process has a high ln (v+/ v-) with a low ∆rG’, which suggests that the backward flux through the enzyme catalyzing the chemical conversion must be blocked by other properties than a thermodynamic barrier introduced by a high ∆rG’, e.g. kinetic or regulatory properties of the enzyme. With this preliminary analysis, where we exploited the estimated narrow ranges for exchange fluxes and ∆rG’, we have opened up a new possibility to identify active regulation happening on individual enzymes.

ConclusionsWe have developed a novel method that allows to estimate metabolic fluxes based on extracel-lular rates, intracellular metabolite concentrations, and isotopomer patterns from 13C labeling experiments. The method allowed us to successfully predict intracellular flux distributions with using a large scale metabolic network without any heuristic assumptions, relying pure-ly on measurement data and physical constraints. This accuracy and lack of heuristics were achieved by using a thermodynamic metabolic network model, which is parameterized by physical constants and measurable quantities. Because we only got a marginal gain in the accuracy of the estimated fluxes when using 13C isotopomer patterns, this means that in fact we can determine fluxes with the thermodynamic metabolic network model on the basis of measured physiological rates, metabolomics and equilibrium constants alone. Not necessarily requiring 13C isotopomer patterns for precise flux estimates not only reduces experimental costs, but also allows to exactly quantify fluxes for substrates where no stable isotopomers exist or for substrates where labeling would not provide sufficient information. If additional accuracy is needed or exchange fluxes are of interest, for examples for determining rate equa-tions, 13C isotopomer patterns will give the necessary information. Because the accuracy re-quired for the estimated fluxes depends on specific applications, practical implementations of this method will be needed to show its utility. Further, these implementations will also show whether additionally measured isotopomer patterns, e.g. isotopomer patterns of intracellular metabolites are able to improve the accuracy (158).

Further, because we were not only able to predict metabolic fluxes with narrow bounds, but also ∆rG’ and exchange fluxes for a number of reactions, we could test the earlier proposed correlation between the Gibbs energy of the reaction and the natural logarithm of the forward and backward rate. Here, we found a very weak similarity to the correlation coefficient ex-pected for elementary reactions, suggesting that also for complex kinetic reactions, the Gibbs energy of the reaction has a major contribution on the rate. But in contrast to earlier assump-tions (19, 20), one cannot generalize this correlation, because reactions can have a high ratio

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between forward and back flux (i.e. low backward flux) and a high Gibbs energy of reaction. The reason for these outliers could eventually be explained by kinetic blockades of the back-ward fluxes. One conclusion of this finding is that it is not advisable to derive constraints on the metabolic or exchange fluxes based on the Gibbs energy of reaction as suggested earlier (78). An even more important conclusion from this finding is the possibility to identify and quantify regulatory control of enzymes. However further work will be needed to fully evaluate the potential of this method.

Materials and MethodsYeast strains

The haploid prototrophic Saccharomyces cerevisiae strain KOY PK2-1C83 (relevant genotype: wild type) (156) and its derivatives, KOY HXT1P (relevant genotype: HXT1 glucose chimera transporter) (156) and KOY TM6*P (relevant genotype: HXT1 glucose chimera transporter) (156) were used in this study. The medium used for yeast strain maintenance was YPD, which contained 2 % [v/w] peptone, 1 % [v/w] yeast extract and 2 % [v/w] glucose.

Batch cultivation conditions in minimal medium

All strains were cultivated in 250 ml Erlenmeyer flasks containing 25 ml of glucose minimal medium inoculated with exponentially growing yeast cells to an initial OD600 of 0.05. The in-oculum was prepared in the identical minimal medium. All cultivations were performed at 30 ˚C and cultures were continuously shaken at 300 rpm. The minimal medium was composed as follows (159): (NH4)2SO4 5 g l-1, KH2PO4 3 g l-1, MgSO4 7H2O 0.5 g l-1, EDTA 1.5 mg l-1, ZnSO4 7H2O 4.5 mg l-1, CoCl2 6H2O 0.3 mg l-1, MnCl2 4H2O 1 mg l-1, CuSO4 5H2O 0.3 mg l-1, CaCl2 2H2O 4.5 mg l-1, FeSO4 7H2O 3 mg l-1, Na2MoO4 2H2O 0.4 mg l-1, H3BO3 1 mg l-1, KI 0.1 mg l-1, D-Biotin 0.05 mg l-1, Ca pantothenate 1 mg l-1, Nicotinic acid 1 mg l-1, m-Inositol 25 mg l-1, Pyridoxine HCl 1 mg l-1, p-Aminobenzoic acid 0.2 mg l-1, Thiamine HCl 1 mg l-1, and D-glucose 10 g l-1. The medium was buffered at pH 5 with 10 mM KH-phthalate.

Determination of biomass

OD600 was determined every hour after the start of the cultivation. At the end, the yeast dry mass was determined by filtering a certain volume of culture through pre-weighed nitrocel-lulose filters with a pore size of 0.2 µm. Filters were washed once with distilled water and kept at 80°C for two days. Afterwards, they were weighed again. The cell dry mass at every mea-surement point was re-calculated from OD600. The ratio of dry mass to OD600 was obtained at the end of the fermentation.

Determination of glucose and extracellular metabolites

During the cultivation, 0.3 ml culture broth samples were centrifuged at 13 rpm for 2 min to separate the cells from the supernatant. The supernatant was transferred to filter columns (SpinX, pore size 0.22 µm), short spun, and transferred into HPLC vials. Glucose, pyruvate, glycerol, acetate and ethanol concentrations in the cultivation supernatant were determined by HPLC (Agilent, 1290 LC HPLC system) using a Hi-Plex H column and 5 mM H2SO4 as eluent at a constant flow rate of 0.6 ml min-1. The column temperature was kept constant at 60 ˚C. A volume of 10 µl of standards and samples was injected for analysis. Substrate concentra-

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tions were detected with refractive index and UV (constant wave length of 210 nm) detection. The chromatogram integration was done with Agilent Open Lab CDS software. Substrate and metabolite concentration were calibrated prior to the analysis of the fermentation samples using an external standard which included all compounds in one standard. The external stan-dard covered the compounds’ concentration range which was observed from the start until the end of the fermentation.

Estimation of extracellular rates

In at least three independent biological replicates, the extracellular rates were estimated from measured concentration time-courses, e.g. glucose, ethanol, acetate, glycerol, pyruvate, and biomass, of the batch cultivation. Extracellular rates were estimated by fitting the concentra-tion time courses to a model assuming exponential growth and constant yields in the culture. The ethanol concentrations were corrected for evaporation of ethanol, using a first order reac-tion. The evaporation constant was determined from an independent experiment, where the ethanol concentration was followed in Erlenmeyer flasks with an initial ethanol concentration of 4 g l-1shaken under cultivation. The evaporation constant was estimated to be 0.021 h-1 by fitting an exponential function to the measured ethanol time course. gPROMS Model Builder v.4.0 (Process Systems Enterprise Ltd.) was used for parameter estimation.

Quantification of mass isotopomer patterns in 13C labeling experiments13C labeling experiments were carried out as earlier described (147). Mass spectrometric frag-ment data of the hydrolyzed proteinogenic amino acids were integrated using MZmin2 (160) for peak area integration. These peak areas were then used to determine the relative mass iso-topomer pattern of individual fragments, which were corrected for naturally occurring stable isotopes using the iMS2flux software tool (161).

Thermodynamic metabolic network model and regression analysis

The thermodynamic metabolic network model for the yeast S. cerevisiae was taken from Chapter 2, methods 1. The model included a stoichiometric network, which covered: cen-tral metabolism, amino acid synthesis, other biosynthetic processes, a detailed description of the transport processes, pH-dependent proton balances, and charge balances. Further, we added an additional transketolase reaction (abbreviation TKT3; reaction stoichiometry s7p + e4p = r5p + f6p), because this addition was necessary to realistically represent the enzymatic mechanism (162). The Gibbs energies were determined using the component-contribution method (27). To correctly describe the thermodynamic properties, we used: a pH of 5.0 in the extracellular space, 7.0 in the cytosol and 7.4 in the mitochondria; an ionic strength of 0.2 M for all compartments; of 60 mV across the cytoplasmic membrane, and a membrane potential of 160 mV across the inner mitochondrial membrane; and a temperature at 303 K. Every reaction was considered to be fully reversible. The model was implemented and solved in the mathematical programming software GAMS. For details to the regression analysis and the Markov chain Monte Carlo sampling refer to Chapter 2, methods 8.

Network for isotopomer balancing

Since we only measured isotope labels in the carbon atoms, we extracted from the stoichi-ometry of thermodynamic metabolic network model a reduced stoichiometric network, in

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which we only considered mass balances for the carbon atoms. Note, because the metabolic fluxes distribution are fixed when evaluating the isotopomer balances, the other elemental balances are still implicitly considered. To this end, we removed all metabolites that do not convert carbon (e.g. ATP, ADP and AMP are replaced by AXP), and therefore only the car-bon in the adenine group was considered. Thus, the resulting stoichiometric network is an exact extraction of the full network, only considering carbon atoms. Also here, all reactions of the metabolic network were assumed to be fully reversible and had an exchange flux. Atom transitions were implemented according to the computationally derived previously published atom transitions at MetaCYC (163). We manually checked the atom transitions, especially for complex and lumped reactions (e.g. large products) and those involving stereospecific reac-tions on heterotopic groups (e.g. aconitase). (Supplementary table 4).

Isotopomer balancing using 13Cflux2

Isotopomer balancing calculations were carried out using the 13C metabolic flux analysis soft-ware 13Cflux2 (164). All net fluxes were fixed to the estimated flux distribution resulting from the sampling of the thermodynamic metabolic network model leaving only the exchange flux-es variable. All exchange fluxes were constrained between 0 and 100 mmol/gCDW/h. Every fit of the isotopomer patterns was run with 500 iterations, where we randomly selected one ex-change flux distribution as a starting point using MCMC sampler of 13Cflux2. Since 13Cflux2 builds on local optimization solver, global optimality cannot be guaranteed. However, we checked the quality of the local optimization by refitting the same fixed net flux distribution with different exchange flux distributions as starting points for 13Cflux2. Here, we found that we converged to the same values for the residuals.

Weighted estimations of statistical values

Weighted estimates for different statistical values, e.g. averages, standard deviations, medi-ans, quantiles, of the metabolic fluxes were determined from all sampled flux distributions using the wtd.stats R package Hmisc (http://cran.r-project.org/web/packages/Hmisc/). In these estimations, weights, w(s), were used which represent the goodness of fit of a sam-pled flux distribution s to the labeling patterns. This weights were based on the residu-al sum of squares, RSS(s), describing the fit of the flux distribution to the labeling patterns. The weight was based on the relative residual sum of squares, rRSS(s) = (RSS (s)-RSSmin)/RSS-min, which described the residual sum of squares of a sampled flux distribution, RSS (s), rel-ative to the minimum residual sum of squares across all sampled flux distribution RSSmin. The weight, w(s), were then determined from the rRSS(s) using, ws) = exp(-rRSS), where taking the exponential of the residual sum of squares ensured that flux distributions, which fitted poorly to the isotopomer patterns received a very low weight. Further, we determined for each flux distribution a weight that quantified a flux distributions’ goodness of fit to both labeling patterns simultaneously. This weight, w([1-13C]&([U-13C])(s), was determined using, w([1-13C]&([U-13C])(s) = exp(-rRSS([1-13C])(s)+ rRSS([U-13C])(s)), where we combined the relative residual sum of squares rRSS(s) from the fit of a flux distribution to the [1-13C] or [U-13C] la-beling patterns. Note, we normalized the weights w(s) such that there sum equaled the number of sampled flux distributions, e.g. if 1000 flux distributions are sampled, then ∑w(s) = 1000.

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Supplementary figures

Figure S1. The fitted labeling patterns determined by isotopomer bal-ancing with the sampled flux distributions from a stoichiometric met-abolic network model, were in accordance with the measured labeling patterns from (a) [1-13C]-labeled and (b) [U-13C]–labeled glucose. Based on the fit of the sampled flux distribution with respect to the mea-sured labeling patterns (i.e. the residual sum of squares of the isotopomer balancing), we determined for each sampled flux distribution a weight that quantifies the goodness-of fit of this flux distribution with respect to the different labeling patterns, i.e. (c) patterns from [1-13C]-glucose, (d) from [U-13C]-glucose, and (e) by combining the patterns from [1-13C]- & [U-13C]-glucose. Note, the plots (c-e) were sorted according to [1-13C]&[U-13C] weights (e).

0

20

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eigh

t of s

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oint

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Sampled pointsof the solution space

(#points = 989)

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Measured mass isotopomer distribution patterns (-)

Fitte

d m

ass

isot

opom

er

dist

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patte

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b [U-13C]

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42

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[U-13C]

07550250

89

Supplementary tables

Supplementary tables

ReactionMeasured Model

Mean SD 1.25 % 50% 98.75%

ac_EX 0.302 0.027 0.279 0.302 0.325

biomass 0.323 0.011 0.313 0.323 0.332

co2tot_EX 30.724 1.892 25.676 26.185 26.683

etoh_EX 21.824 0.858 23.137 23.639 24.136

glc-D_EX -15.761 0.380 -15.698 -15.459 -15.217

glyc_EX 2.067 0.097 2.001 2.083 2.160

o2_EX -1.373 0.111 -1.500 -1.409 -1.316

pyr_EX 0.165 0.007 0.159 0.165 0.171

MetabolitesMeasured Model

Mean SD 1.25 % 50% 98.75%

13dpg 0.0654 0.0245 0.0427 0.0645 0.0855

6pgc 0.9727 0.1484 0.8512 0.9743 1.1128

adp 0.9422 0.1656 0.8003 0.9444 1.0823

akg 4.3607 0.3717 4.0332 4.3579 4.6691

ala-L 6.4149 2.2492 4.4281 6.4143 8.3767

amp 0.3105 0.0314 0.2814 0.3100 0.3358

asp-L 16.9146 1.1820 15.8813 16.8990 17.8893

atp 6.0475 0.9148 5.2478 6.0619 6.8113

dhap 2.0322 0.4634 1.6220 2.0350 2.4177

f6p 1.9226 0.5375 1.2873 1.7796 2.2244

fdp 5.7885 0.3863 5.4496 5.7942 6.1176

fumarate 0.3714 0.0266 0.3473 0.3711 0.3958

g1p 0.5667 0.0199 0.5446 0.5614 0.5777

g6p 5.3876 0.7213 5.7143 6.1858 6.7865

gln-L 37.5700 5.3241 24.8285 29.4602 29.4602

MetabolitesMeasured Model

Mean SD 1.25 % 50% 98.75%

glu-L 79.8908 18.1540 65.5281 80.3087 95.7601

glyc3p 0.6273 0.0980 0.5458 0.6270 0.7126

gmp 0.1405 0.0038 0.1368 0.1400 0.1437

hom-L 0.9202 0.4714 0.4819 0.9260 1.3277

icit 0.1425 0.0039 0.1383 0.1421 0.1453

mal-L 0.8010 0.2688 0.5530 0.8020 1.0460

pep 0.3099 0.0437 0.2715 0.3106 0.3491

phe-L 0.1551 0.0121 0.1444 0.1548 0.1652

r5p 1.0259 0.3379 0.7330 1.0269 1.3068

ru5p-D 0.6996 0.2718 0.4646 0.7002 0.9193

s7p 0.7865 0.1840 0.6289 0.7883 0.9485

succ 2.0429 0.8535 1.3124 2.0449 2.7496

trp-L 0.0075 0.0101 0.0010 0.0080 0.0168

tyr-L 0.1259 0.0067 0.1201 0.1259 0.1329

xu5p-D 0.5594 0.1409 0.4411 0.5564 0.6881

table continues next page

Table S1. Consolidated extracellular rates. Extracel-lular rates were estimated from the thermodynamic model using regression analysis and parametric boot-strap as described in (Chapter 2, methods 6). The per-centages represent the estimated quantiles for the rates determined by the regression.

Table S2. Consolidated intracellular metabolite concentrations. Intracellular metabolite concentrations were esti-mated from the thermodynamic model using regression analysis and parametric bootstrap as described in (Chapter 2, methods 6). The percentages represent the estimated quantiles for the metabolite concentrations determined by the regression.

Table S3. [1-13C]-based mass fragments of protein-bound amino acids separated by gas chromatography and detect-ed by mass spectrometry. The model column represents the average and the standard deviation of the ~1000 sampled flux distributions of the thermodynamic metabolic network model. The average and the standard deviation were determined using the weights that represent the goodness of fit of the individual sampled flux distribution.

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

ala_L__c_[1-3]#M0 0.5166 0.0106 0.5242 0.0026

ala_L__c_[1-3]#M1 0.4633 0.0076 0.4584 0.0027

ala_L__c_[1-3]#M2 0.0143 0.0026 0.0172 0.0027

ala_L__c_[1-3]#M3 0.0058 0.0058 0.0002 0.0001

ala_L__c_[1-2]#M0 0.5306 0.0046 0.5366 0.0023

ala_L__c_[1-2]#M1 0.4632 0.0046 0.4560 0.0026

ala_L__c_[1-2]#M2 0.0062 0.0030 0.0073 0.0024

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

gly__c_[1-2]#M0 0.9685 0.0373 0.9690 0.0065

gly__c_[1-2]#M1 0.0259 0.0424 0.0304 0.0062

gly__c_[1-2]#M2 0.0056 0.0129 0.0006 0.0004

gly__c_[1]#M0 0.9813 0.0080 0.9816 0.0057

gly__c_[1]#M1 0.0187 0.0080 0.0184 0.0057

val_L__c_[1-5]#M0 0.2723 0.0040 0.2776 0.0024

val_L__c_[1-5]#M1 0.4892 0.0028 0.4822 0.0038

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Chapter 4

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

val_L__c_[1-5]#M2 0.2294 0.0039 0.2257 0.0025

val_L__c_[1-5]#M3 0.0088 0.0020 0.0142 0.0035

val_L__c_[1-5]#M4 0.0002 0.0005 0.0003 0.0001

val_L__c_[1-5]#M5 0.0001 0.0003 0.0000 0.0000

val_L__c_[1-4]#M0 0.2762 0.0042 0.2880 0.0026

val_L__c_[1-4]#M1 0.4857 0.0048 0.4891 0.0027

val_L__c_[1-4]#M2 0.2274 0.0035 0.2159 0.0023

val_L__c_[1-4]#M3 0.0100 0.0025 0.0069 0.0024

val_L__c_[1-4]#M4 0.0007 0.0006 0.0001 0.0000

val_L__c_[4-5]#M0 0.8592 0.0031 0.8743 0.0080

val_L__c_[4-5]#M1 0.1169 0.0024 0.1214 0.0075

val_L__c_[4-5]#M2 0.0239 0.0042 0.0043 0.0011

leu_L__c_[1-6]#M0 0.1104 0.0202 0.1546 0.0020

leu_L__c_[1-6]#M1 0.2614 0.0351 0.3937 0.0031

leu_L__c_[1-6]#M2 0.2306 0.0246 0.3409 0.0020

leu_L__c_[1-6]#M3 0.3486 0.0637 0.1058 0.0030

leu_L__c_[1-6]#M4 0.0302 0.0117 0.0048 0.0016

leu_L__c_[1-6]#M5 0.0187 0.0029 0.0001 0.0001

leu_L__c_[1-6]#M6 0.0001 0.0002 0.0000 0.0000

leu_L__c_[1-5]#M0 0.1585 0.0076 0.1794 0.0054

leu_L__c_[1-5]#M1 0.3855 0.0058 0.4134 0.0046

leu_L__c_[1-5]#M2 0.3454 0.0051 0.3189 0.0050

leu_L__c_[1-5]#M3 0.1074 0.0028 0.0856 0.0047

leu_L__c_[1-5]#M4 0.0028 0.0016 0.0027 0.0010

leu_L__c_[1-5]#M5 0.0004 0.0004 0.0000 0.0000

ile_L__c_[1-6]#M0 0.2817 0.0227 0.2698 0.0029

ile_L__c_[1-6]#M1 0.4693 0.0163 0.4771 0.0025

ile_L__c_[1-6]#M2 0.2309 0.0111 0.2328 0.0026

ile_L__c_[1-6]#M3 0.0173 0.0118 0.0196 0.0023

ile_L__c_[1-6]#M4 0.0008 0.0021 0.0007 0.0003

ile_L__c_[1-5]#M0 0.2681 0.0066 0.2738 0.0029

ile_L__c_[1-5]#M1 0.4782 0.0076 0.4800 0.0024

ile_L__c_[1-5]#M2 0.2392 0.0050 0.2292 0.0026

ile_L__c_[1-5]#M3 0.0133 0.0034 0.0165 0.0021

ile_L__c_[1-5]#M4 0.0006 0.0009 0.0004 0.0001

ile_L__c_[1-5]#M5 0.0006 0.0006 0.0000 0.0000

pro_L__c_[1-5]#M0 0.2828 0.0233 0.2984 0.0065

pro_L__c_[1-5]#M1 0.4754 0.0302 0.4785 0.0027

pro_L__c_[1-5]#M2 0.2244 0.0149 0.2084 0.0055

pro_L__c_[1-5]#M3 0.0167 0.0158 0.0144 0.0022

pro_L__c_[1-5]#M4 0.0006 0.0016 0.0003 0.0001

pro_L__c_[1-5]#M5 0.0000 0.0002 0.0000 0.0000

pro_L__c_[1-4]#M0 0.2897 0.0141 0.3128 0.0067

pro_L__c_[1-4]#M1 0.4824 0.0089 0.4868 0.0021

pro_L__c_[1-4]#M2 0.2208 0.0079 0.1949 0.0054

pro_L__c_[1-4]#M3 0.0066 0.0035 0.0054 0.0011

pro_L__c_[1-4]#M4 0.0005 0.0008 0.0000 0.0000

met_L__c_[1-5]#M0 0.3183 0.0220 0.3084 0.0069

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

met_L__c_[1-5]#M1 0.4611 0.0100 0.4777 0.0022

met_L__c_[1-5]#M2 0.2021 0.0143 0.1995 0.0060

met_L__c_[1-5]#M3 0.0114 0.0107 0.0138 0.0020

met_L__c_[1-5]#M4 0.0025 0.0046 0.0004 0.0003

met_L__c_[1-5]#M5 0.0046 0.0061 0.0000 0.0001

met_L__c_[1-4]#M0 0.3129 0.0403 0.3130 0.0071

met_L__c_[1-4]#M1 0.4608 0.0254 0.4802 0.0022

met_L__c_[1-4]#M2 0.1952 0.0170 0.1955 0.0061

met_L__c_[1-4]#M3 0.0285 0.0489 0.0111 0.0019

met_L__c_[1-4]#M4 0.0026 0.0056 0.0002 0.0001

ser_L__c_[1-3]#M0 0.5491 0.0292 0.5471 0.0110

ser_L__c_[1-3]#M1 0.4407 0.0361 0.4419 0.0114

ser_L__c_[1-3]#M2 0.0095 0.0102 0.0109 0.0013

ser_L__c_[1-3]#M3 0.0007 0.0035 0.0001 0.0001

ser_L__c_[1-2]#M0 0.5616 0.0150 0.5539 0.0111

ser_L__c_[1-2]#M1 0.4348 0.0155 0.4405 0.0116

ser_L__c_[1-2]#M2 0.0036 0.0052 0.0056 0.0011

ser_L__c_[2-3]#M0 0.9661 0.0141 0.9732 0.0040

ser_L__c_[2-3]#M1 0.0297 0.0144 0.0265 0.0038

ser_L__c_[2-3]#M2 0.0042 0.0041 0.0004 0.0002

thr_L__c_[1-4]#M0 0.4979 0.0310 0.5029 0.0064

thr_L__c_[1-4]#M1 0.4755 0.0305 0.4620 0.0033

thr_L__c_[1-4]#M2 0.0252 0.0292 0.0341 0.0051

thr_L__c_[1-4]#M3 0.0011 0.0030 0.0009 0.0003

thr_L__c_[1-4]#M4 0.0002 0.0007 0.0001 0.0001

thr_L__c_[1-3]#M0 0.5116 0.0114 0.5104 0.0065

thr_L__c_[1-3]#M1 0.4712 0.0092 0.4612 0.0034

thr_L__c_[1-3]#M2 0.0149 0.0131 0.0280 0.0050

thr_L__c_[1-3]#M3 0.0023 0.0045 0.0004 0.0002

phe_L__c_[1-9]#M0 0.2129 0.0138 0.1455 0.0032

phe_L__c_[1-9]#M1 0.4224 0.0080 0.3848 0.0026

phe_L__c_[1-9]#M2 0.2876 0.0091 0.3477 0.0029

phe_L__c_[1-9]#M3 0.0661 0.0037 0.1148 0.0026

phe_L__c_[1-9]#M4 0.0030 0.0007 0.0069 0.0004

phe_L__c_[1-9]#M5 0.0046 0.0092 0.0002 0.0000

phe_L__c_[1-9]#M6 0.0009 0.0014 0.0000 0.0000

phe_L__c_[1-9]#M7 0.0022 0.0019 0.0000 0.0000

phe_L__c_[1-9]#M8 0.0002 0.0003 0.0000 0.0000

phe_L__c_[1-9]#M9 0.0002 0.0003 0.0000 0.0000

phe_L__c_[1-8]#M0 0.2143 0.0143 0.1475 0.0032

phe_L__c_[1-8]#M1 0.4242 0.0085 0.3880 0.0025

phe_L__c_[1-8]#M2 0.2865 0.0105 0.3471 0.0030

phe_L__c_[1-8]#M3 0.0656 0.0038 0.1117 0.0026

phe_L__c_[1-8]#M4 0.0044 0.0017 0.0056 0.0002

phe_L__c_[1-8]#M5 0.0024 0.0023 0.0001 0.0000

phe_L__c_[1-8]#M6 0.0003 0.0007 0.0000 0.0000

phe_L__c_[1-8]#M7 0.0019 0.0056 0.0000 0.0000

phe_L__c_[1-8]#M8 0.0003 0.0006 0.0000 0.0000

Table S3. [1-13C]-based mass fragments (continued)

table continues next page

91

Supplementary tables

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

phe_L__c_[8-9]#M0 0.9744 0.0055 0.9709 0.0047

phe_L__c_[8-9]#M1 0.0244 0.0043 0.0287 0.0045

phe_L__c_[8-9]#M2 0.0013 0.0019 0.0004 0.0002

asp_L__c_[1-4]#M0 0.5092 0.0248 0.5031 0.0076

asp_L__c_[1-4]#M1 0.4649 0.0314 0.4624 0.0033

asp_L__c_[1-4]#M2 0.0217 0.0094 0.0338 0.0060

asp_L__c_[1-4]#M3 0.0015 0.0042 0.0007 0.0002

asp_L__c_[1-4]#M4 0.0028 0.0041 0.0000 0.0000

asp_L__c_[1-3]#M0 0.5209 0.0170 0.5106 0.0077

asp_L__c_[1-3]#M1 0.4574 0.0222 0.4614 0.0035

asp_L__c_[1-3]#M2 0.0183 0.0047 0.0278 0.0059

asp_L__c_[1-3]#M3 0.0034 0.0021 0.0003 0.0001

asp_L__c_[2,4]#M0 0.9695 0.0026 0.9648 0.0051

asp_L__c_[2,4]#M1 0.0294 0.0024 0.0346 0.0049

asp_L__c_[2,4]#M2 0.0011 0.0010 0.0007 0.0003

glu_L__c_[1-5]#M0 0.2770 0.0145 0.2984 0.0065

glu_L__c_[1-5]#M1 0.4762 0.0199 0.4785 0.0027

glu_L__c_[1-5]#M2 0.2299 0.0077 0.2084 0.0055

glu_L__c_[1-5]#M3 0.0135 0.0078 0.0144 0.0022

glu_L__c_[1-5]#M4 0.0028 0.0050 0.0003 0.0001

glu_L__c_[1-5]#M5 0.0005 0.0016 0.0000 0.0000

glu_L__c_[1-4]#M0 0.3094 0.0329 0.3128 0.0067

glu_L__c_[1-4]#M1 0.4699 0.0194 0.4868 0.0021

glu_L__c_[1-4]#M2 0.2136 0.0159 0.1949 0.0054

glu_L__c_[1-4]#M3 0.0055 0.0053 0.0054 0.0011

glu_L__c_[1-4]#M4 0.0016 0.0018 0.0000 0.0000

glu_L__c_[3,5]#M0 0.6910 0.0296 0.5826 0.0236

glu_L__c_[3,5]#M1 0.2642 0.0252 0.3987 0.0222

glu_L__c_[3,5]#M2 0.0448 0.0098 0.0187 0.0050

lys_L__c_[1-6]#M0 0.1992 0.0271 0.1680 0.0037

Mass isotopomer pattern

Measurment Model

Mean SD Mean SD

lys_L__c_[1-6]#M1 0.3762 0.0250 0.4039 0.0027

lys_L__c_[1-6]#M2 0.3123 0.0183 0.3287 0.0038

lys_L__c_[1-6]#M3 0.1112 0.0145 0.0954 0.0027

lys_L__c_[1-6]#M4 0.0010 0.0020 0.0040 0.0009

lys_L__c_[1-6]#M5 0.0000 0.0001 0.0001 0.0000

lys_L__c_[1-6]#M6 0.0001 0.0005 0.0000 0.0000

lys_L__c_[1-5]#M0 0.2482 0.0762 0.1951 0.0076

lys_L__c_[1-5]#M1 0.3158 0.0690 0.4214 0.0042

lys_L__c_[1-5]#M2 0.3423 0.0357 0.3048 0.0071

lys_L__c_[1-5]#M3 0.0880 0.0242 0.0767 0.0045

lys_L__c_[1-5]#M4 0.0053 0.0068 0.0021 0.0005

lys_L__c_[1-5]#M5 0.0004 0.0011 0.0000 0.0000

tyr_L__c_[1-9]#M0 0.2283 0.0363 0.1455 0.0032

tyr_L__c_[1-9]#M1 0.4191 0.0206 0.3848 0.0026

tyr_L__c_[1-9]#M2 0.2805 0.0167 0.3477 0.0029

tyr_L__c_[1-9]#M3 0.0659 0.0032 0.1148 0.0026

tyr_L__c_[1-9]#M4 0.0051 0.0035 0.0069 0.0004

tyr_L__c_[1-9]#M5 0.0005 0.0009 0.0002 0.0000

tyr_L__c_[1-9]#M6 0.0004 0.0010 0.0000 0.0000

tyr_L__c_[1-9]#M7 0.0000 0.0001 0.0000 0.0000

tyr_L__c_[1-9]#M8 0.0001 0.0003 0.0000 0.0000

tyr_L__c_[1-8]#M0 0.2300 0.0404 0.1475 0.0032

tyr_L__c_[1-8]#M1 0.4100 0.0208 0.3880 0.0025

tyr_L__c_[1-8]#M2 0.2779 0.0175 0.3471 0.0030

tyr_L__c_[1-8]#M3 0.0657 0.0050 0.1117 0.0026

tyr_L__c_[1-8]#M4 0.0036 0.0015 0.0056 0.0002

tyr_L__c_[1-8]#M5 0.0040 0.0006 0.0001 0.0000

tyr_L__c_[1-8]#M6 0.0012 0.0023 0.0000 0.0000

tyr_L__c_[1-8]#M7 0.0011 0.0012 0.0000 0.0000

tyr_L__c_[1-8]#M8 0.0066 0.0033 0.0000 0.0000

Table S3. [1-13C]-based mass fragments (continued)

Table S4. [U-13C]-based mass fragments of protein-bound amino acids separated by gas chromatography and de-tected by mass spectrometry. The model column represents the average and the standard deviation of the ~1000 sampled flux distributions of the thermodynamic metabolic network model. The average and the standard deviation were determined using the weights that represent the goodness of fit of the individual sampled flux distribution to the different labeling patterns.

Mass fragmentMeasurment Model

Mean SD Mean SD

ala_L__c_[1-3]#M0 0.74960 0.00630 0.74139 0.00798

ala_L__c_[1-3]#M1 0.05170 0.00590 0.05959 0.00853

ala_L__c_[1-3]#M2 0.02800 0.00130 0.03729 0.00719

ala_L__c_[1-3]#M3 0.17080 0.00350 0.16174 0.00752

ala_L__c_[1-2]#M0 0.77700 0.00370 0.77671 0.00237

ala_L__c_[1-2]#M1 0.03870 0.00390 0.03366 0.00474

ala_L__c_[1-2]#M2 0.18430 0.00320 0.18963 0.00237

gly__c_[1-2]#M0 0.80180 0.04400 0.77409 0.00378

gly__c_[1-2]#M1 0.02960 0.02120 0.03889 0.00755

Mass fragmentMeasurment Model

Mean SD Mean SD

gly__c_[1-2]#M2 0.16860 0.03720 0.18702 0.00377

gly__c_[1]#M0 0.81300 0.01590 0.79353 0.00001

gly__c_[1]#M1 0.18700 0.01590 0.20647 0.00001

val_L__c_[1-5]#M0 0.60210 0.00480 0.57469 0.00772

val_L__c_[1-5]#M1 0.05120 0.00320 0.07252 0.00908

val_L__c_[1-5]#M2 0.16500 0.00230 0.17215 0.00486

val_L__c_[1-5]#M3 0.14080 0.00200 0.13749 0.00461

val_L__c_[1-5]#M4 0.00880 0.00100 0.01275 0.00153

val_L__c_[1-5]#M5 0.03210 0.00160 0.03041 0.00175

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92

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Mass fragmentMeasurment Model

Mean SD Mean SD

val_L__c_[1-4]#M0 0.61200 0.00340 0.60291 0.00377

val_L__c_[1-4]#M1 0.04500 0.00300 0.05300 0.00737

val_L__c_[1-4]#M2 0.29220 0.00530 0.29531 0.00434

val_L__c_[1-4]#M3 0.01520 0.00500 0.01291 0.00166

val_L__c_[1-4]#M4 0.03560 0.00180 0.03587 0.00092

val_L__c_[4-5]#M0 0.76240 0.01060 0.75468 0.00820

val_L__c_[4-5]#M1 0.05390 0.00700 0.07773 0.01639

val_L__c_[4-5]#M2 0.18370 0.00620 0.16760 0.00819

leu_L__c_[1-6]#M0 0.43210 0.04700 0.46642 0.00545

leu_L__c_[1-6]#M1 0.04630 0.00640 0.06501 0.01037

leu_L__c_[1-6]#M2 0.29610 0.03940 0.34308 0.00704

leu_L__c_[1-6]#M3 0.18260 0.10960 0.03160 0.00464

leu_L__c_[1-6]#M4 0.03190 0.03270 0.08337 0.00272

leu_L__c_[1-6]#M5 0.00900 0.00900 0.00383 0.00051

leu_L__c_[1-6]#M6 0.00200 0.00340 0.00669 0.00032

leu_L__c_[1-5]#M0 0.50330 0.00800 0.47844 0.00299

leu_L__c_[1-5]#M1 0.14650 0.00460 0.16653 0.00507

leu_L__c_[1-5]#M2 0.24720 0.00510 0.24528 0.00192

leu_L__c_[1-5]#M3 0.06430 0.00230 0.07121 0.00042

leu_L__c_[1-5]#M4 0.03150 0.00170 0.03113 0.00039

leu_L__c_[1-5]#M5 0.00720 0.00070 0.00741 0.00019

ile_L__c_[1-6]#M0 0.54530 0.01890 0.47116 0.00298

ile_L__c_[1-6]#M1 0.11690 0.01530 0.16488 0.00345

ile_L__c_[1-6]#M2 0.13240 0.00850 0.13704 0.00231

ile_L__c_[1-6]#M3 0.15510 0.01710 0.15168 0.00151

ile_L__c_[1-6]#M4 0.02140 0.01350 0.03848 0.00061

ile_L__c_[1-6]#M5 0.02870 0.01270 0.02955 0.00050

ile_L__c_[1-6]#M6 0.00010 0.00060 0.00721 0.00018

ile_L__c_[1-5]#M0 0.54390 0.01430 0.48229 0.00213

ile_L__c_[1-5]#M1 0.10850 0.00960 0.16032 0.00326

ile_L__c_[1-5]#M2 0.26300 0.00480 0.24704 0.00145

ile_L__c_[1-5]#M3 0.04680 0.00530 0.07113 0.00081

ile_L__c_[1-5]#M4 0.03200 0.00200 0.03147 0.00031

ile_L__c_[1-5]#M5 0.00580 0.00150 0.00775 0.00030

pro_L__c_[1-5]#M0 0.55630 0.01810 0.47990 0.00892

pro_L__c_[1-5]#M1 0.11260 0.02270 0.16701 0.01248

pro_L__c_[1-5]#M2 0.25030 0.01720 0.24057 0.00573

pro_L__c_[1-5]#M3 0.05040 0.01960 0.07395 0.00388

pro_L__c_[1-5]#M4 0.02720 0.01770 0.03052 0.00135

pro_L__c_[1-5]#M5 0.00330 0.00480 0.00805 0.00149

pro_L__c_[1-4]#M0 0.62320 0.00830 0.59531 0.00736

pro_L__c_[1-4]#M1 0.03920 0.00530 0.06904 0.01379

pro_L__c_[1-4]#M2 0.29360 0.00490 0.28419 0.00817

pro_L__c_[1-4]#M3 0.00950 0.00270 0.01743 0.00325

pro_L__c_[1-4]#M4 0.03450 0.00200 0.03404 0.00178

met_L__c_[1-5]#M0 0.54420 0.01250 0.48150 0.00223

met_L__c_[1-5]#M1 0.22760 0.01010 0.27263 0.00128

met_L__c_[1-5]#M2 0.03050 0.00620 0.05446 0.00322

Mass fragmentMeasurment Model

Mean SD Mean SD

met_L__c_[1-5]#M3 0.13010 0.00650 0.12257 0.00095

met_L__c_[1-5]#M4 0.05610 0.00290 0.06099 0.00098

met_L__c_[1-5]#M5 0.01160 0.00440 0.00786 0.00014

met_L__c_[1-4]#M0 0.55910 0.01620 0.49287 0.00119

met_L__c_[1-4]#M1 0.21740 0.01420 0.27042 0.00104

met_L__c_[1-4]#M2 0.13910 0.00820 0.16299 0.00140

met_L__c_[1-4]#M3 0.07420 0.01840 0.06527 0.00084

met_L__c_[1-4]#M4 0.01020 0.00530 0.00845 0.00029

ser_L__c_[1-3]#M0 0.73900 0.02080 0.74607 0.02358

ser_L__c_[1-3]#M1 0.06920 0.01670 0.05465 0.02437

ser_L__c_[1-3]#M2 0.05250 0.01170 0.03314 0.02202

ser_L__c_[1-3]#M3 0.13930 0.00980 0.16614 0.02279

ser_L__c_[1-2]#M0 0.74530 0.01050 0.75972 0.02361

ser_L__c_[1-2]#M1 0.11210 0.00870 0.06765 0.04722

ser_L__c_[1-2]#M2 0.14260 0.00560 0.17262 0.02361

ser_L__c_[2-3]#M0 0.78650 0.00760 0.77749 0.00131

ser_L__c_[2-3]#M1 0.03450 0.00750 0.03212 0.00261

ser_L__c_[2-3]#M2 0.17910 0.00710 0.19039 0.00131

thr_L__c_[1-4]#M0 0.67870 0.02470 0.60680 0.00282

thr_L__c_[1-4]#M1 0.12030 0.02020 0.18567 0.00231

thr_L__c_[1-4]#M2 0.01270 0.01190 0.02031 0.00359

thr_L__c_[1-4]#M3 0.16540 0.01830 0.14917 0.00213

thr_L__c_[1-4]#M4 0.02280 0.01030 0.03805 0.00070

thr_L__c_[1-3]#M0 0.68440 0.01060 0.62114 0.00150

thr_L__c_[1-3]#M1 0.12170 0.01380 0.17916 0.00169

thr_L__c_[1-3]#M2 0.16630 0.00900 0.15878 0.00143

thr_L__c_[1-3]#M3 0.02760 0.00320 0.04093 0.00141

phe_L__c_[1-9]#M0 0.43420 0.00320 0.41534 0.00151

phe_L__c_[1-9]#M1 0.09060 0.00620 0.09260 0.00153

phe_L__c_[1-9]#M2 0.11480 0.00530 0.12001 0.00109

phe_L__c_[1-9]#M3 0.16840 0.00580 0.16968 0.00132

phe_L__c_[1-9]#M4 0.08940 0.00380 0.09615 0.00045

phe_L__c_[1-9]#M5 0.04260 0.00140 0.04461 0.00007

phe_L__c_[1-9]#M6 0.03200 0.00130 0.03481 0.00010

phe_L__c_[1-9]#M7 0.01900 0.00270 0.01884 0.00011

phe_L__c_[1-9]#M8 0.00440 0.00110 0.00365 0.00002

phe_L__c_[1-9]#M9 0.00450 0.00100 0.00431 0.00005

phe_L__c_[1-8]#M0 0.43900 0.01010 0.42343 0.00062

phe_L__c_[1-8]#M1 0.09140 0.00460 0.08856 0.00102

phe_L__c_[1-8]#M2 0.21250 0.00460 0.21962 0.00042

phe_L__c_[1-8]#M3 0.08400 0.00360 0.08615 0.00059

phe_L__c_[1-8]#M4 0.09740 0.00300 0.10452 0.00040

phe_L__c_[1-8]#M5 0.03150 0.00370 0.03208 0.00019

phe_L__c_[1-8]#M6 0.03330 0.00210 0.03712 0.00018

phe_L__c_[1-8]#M7 0.00600 0.00350 0.00401 0.00002

phe_L__c_[1-8]#M8 0.00490 0.00200 0.00450 0.00003

phe_L__c_[8-9]#M0 0.79280 0.00490 0.77645 0.00197

phe_L__c_[8-9]#M1 0.02750 0.00370 0.03420 0.00393

Table S4. [U-13C]-based mass fragments (continued)

93

Supplementary tables

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Mass fragmentMeasurment Model

Mean SD Mean SD

phe_L__c_[8-9]#M2 0.17970 0.00370 0.18935 0.00197

asp_L__c_[1-4]#M0 0.67520 0.02790 0.60684 0.00282

asp_L__c_[1-4]#M1 0.12140 0.02110 0.18568 0.00231

asp_L__c_[1-4]#M2 0.01240 0.01490 0.02027 0.00356

asp_L__c_[1-4]#M3 0.16920 0.02160 0.14917 0.00211

asp_L__c_[1-4]#M4 0.02180 0.01190 0.03804 0.00070

asp_L__c_[1-3]#M0 0.68880 0.01710 0.62117 0.00150

asp_L__c_[1-3]#M1 0.11890 0.01520 0.17914 0.00169

asp_L__c_[1-3]#M2 0.16260 0.00510 0.15876 0.00144

asp_L__c_[1-3]#M3 0.02970 0.00490 0.04092 0.00141

asp_L__c_[2,4]#M0 0.78210 0.00330 0.77347 0.00344

asp_L__c_[2,4]#M1 0.03120 0.00320 0.04013 0.00687

asp_L__c_[2,4]#M2 0.18670 0.00410 0.18640 0.00344

glu_L__c_[1-5]#M0 0.54160 0.02100 0.47990 0.00892

glu_L__c_[1-5]#M1 0.11360 0.01590 0.16701 0.01248

glu_L__c_[1-5]#M2 0.25580 0.01890 0.24057 0.00573

glu_L__c_[1-5]#M3 0.05430 0.01810 0.07395 0.00388

glu_L__c_[1-5]#M4 0.03110 0.00870 0.03052 0.00135

glu_L__c_[1-5]#M5 0.00340 0.00760 0.00805 0.00149

glu_L__c_[1-4]#M0 0.61630 0.01310 0.59531 0.00736

glu_L__c_[1-4]#M1 0.04710 0.02030 0.06904 0.01379

glu_L__c_[1-4]#M2 0.29280 0.00870 0.28419 0.00817

glu_L__c_[1-4]#M3 0.00880 0.00580 0.01743 0.00325

glu_L__c_[1-4]#M4 0.03500 0.00530 0.03404 0.00178

glu_L__c_[3,5]#M0 0.66230 0.09710 0.64088 0.01374

Mass fragmentMeasurment Model

Mean SD Mean SD

glu_L__c_[3,5]#M1 0.25510 0.10350 0.30528 0.02748

glu_L__c_[3,5]#M2 0.08250 0.02080 0.05384 0.01374

lys_L__c_[1-6]#M0 0.50360 0.01900 0.46147 0.00667

lys_L__c_[1-6]#M1 0.06790 0.02820 0.07538 0.01232

lys_L__c_[1-6]#M2 0.32630 0.01530 0.33484 0.00835

lys_L__c_[1-6]#M3 0.02330 0.01600 0.03690 0.00545

lys_L__c_[1-6]#M4 0.07180 0.00730 0.08049 0.00317

lys_L__c_[1-6]#M5 0.00210 0.00490 0.00452 0.00065

lys_L__c_[1-6]#M6 0.00490 0.00700 0.00640 0.00039

lys_L__c_[1-5]#M0 0.51710 0.05820 0.47239 0.00584

lys_L__c_[1-5]#M1 0.13300 0.02250 0.17773 0.00945

lys_L__c_[1-5]#M2 0.25440 0.05740 0.23975 0.00403

lys_L__c_[1-5]#M3 0.06220 0.01980 0.07251 0.00096

lys_L__c_[1-5]#M4 0.02960 0.01040 0.03061 0.00101

lys_L__c_[1-5]#M5 0.00360 0.00610 0.00703 0.00037

tyr_L__c_[1-8]#M0 0.43580 0.34460 0.42343 0.00062

tyr_L__c_[1-8]#M1 0.18200 0.22560 0.08856 0.00102

tyr_L__c_[1-8]#M2 0.13040 0.20870 0.21962 0.00042

tyr_L__c_[1-8]#M3 0.47710 0.51920 0.08615 0.00059

tyr_L__c_[1-8]#M4 0.06810 0.08460 0.10452 0.00040

tyr_L__c_[1-8]#M5 0.07960 0.14120 0.03208 0.00019

tyr_L__c_[1-8]#M6 0.08220 0.12430 0.03712 0.00018

tyr_L__c_[1-8]#M7 0.02200 0.04510 0.00401 0.00002

tyr_L__c_[1-8]#M8 0.02200 0.03760 0.00450 0.00003

Table S4. [U-13C]-based mass fragments (continued)

Table S5. Atom transition network which served as a basis for the isotopomer balances. The network was based on the stoichiometric network of Chapter 2, methods 1. The network was extended for a third transketolase (TKT3). Since we only considered the mass balance for the carbon atoms we excluded the following reactions: GLCS2, BPNT, SO4SO3, SULR, ADK1, ADK1m, ATPHYD, NDPK2, NADH2-u6i, PPA, PPAm, NADH2-u6m, ComplexIII, Com-plexIV, ATPS3m, ATPASE, POSQt, POSQtm, Ht, Htm, IONPUMP, IONPUMPm, H2Ot+0, NH3t+1, NH3t+1-H, O2t, PIt-1, PIt-1+H, SO4t-2+2H, SO4t-2+H, SO4t-2, ADPATPtm-4-2, ADPATPtm-4-3, NH3tm+1, NH3tm+1-H, O2tm+0, PItm-2+2H, PItm-2+H, PItm-2.

Abbreviation Stoichiometry Atom mappings

ac_EX ac[e] = C#1@1 C#2@1 =

co2tot_EX co2tot[e] = C#1@1 =

etoh_EX etoh[e] = C#1@1 C#2@1 =

glc-D_EX glc-D[e] = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 =

glyc_EX glyc[e] = C#1@1 C#2@1 C#3@1 =

pyr_EX pyr[e] = C#1@1 C#2@1 C#3@1 =

succ_EX succ[e] = C#1@1 C#2@1 C#3@1 C#4@1 =

GHMT2 ser-L[c] = gly[c] + mlthf[c] C#1@1 C#2@1 C#3@1 = C#2@1 C#3@1 + C#1@1

HSDy aspsa[c] = hom-L[c] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

HSK hom-L[c] = phom[c] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

PGPS 3pg[c] + glu-L[c] = ser-L[c] + akg[c]C#1@1 C#2@1 C#3@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@1 C#2@1

C#3@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2

ENO 2pg[c] = pep[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

94

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Abbreviation Stoichiometry Atom mappings

FBA fdp[c] = dhap[c] + g3p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#4@1 C#1@1 C#3@1 + C#5@1

C#2@1 C#6@1

PGI g6p[c] = f6p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#6@1 C#2@1 C#3@1

C#4@1 C#5@1

GAPD g3p[c] = 13dpg[c] C#1@1 C#2@1 C#3@1 = C#2@1 C#3@1 C#1@1

PDHm pyr[m] = accoa[m] + co2tot[m] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 + C#3@1

HEX1 glc-D[c] = g6p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

PFK f6p[c] = fdp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

PGK 3pg[c] = 13dpg[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

PGM 2pg[c] = 3pg[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

PYK pep[c] = pyr[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

TPI dhap[c] = g3p[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

PRMICIi prfp[c] = prlp[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 C#13@1 C#14@1 C#15@1 = C#14@1 C#1@1 C#2@1 C#3@1

C#4@1 C#10@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#11@1 C#12@1 C#13@1 C#15@1

ATPPRT prpp[c] = prbatp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@2 C#1@1 C#3@1 C#2@1 C#2@2 C#4@1 C#5@1 C#3@2 C#6@1 C#4@2 C#7@1 C#8@1 C#9@1 C#5@2 C#10@1

HISTD hist[c] = his-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#4@1 C#6@1

C#5@1 C#3@1

HISTP hisp[c] = hist[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

HSTPT glu-L[c] + imacp[c] = akg[c] + hisp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2

C#6@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#6@2 C#5@2

IGPDH eig3p[c] = imacp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#6@1 C#1@1 C#2@1 C#3@1

C#4@1 C#5@1

PRATPAMPC prbatp[c] = prfp[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 C#13@1 C#14@1 C#15@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 C#13@1 C#14@1 C#15@1

PRPPS r5p[c] = prpp[c] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

FDH for[c] = co2tot[c] C#1@1 = C#1@1

METS 5mthf[c] + hcys-L[c] = met-L[c] C#1@1 + C#1@2 C#2@2 C#3@2 C#4@2 = C#1@1 C#1@2 C#2@2 C#3@2 C#4@2

CYSTGL cyst-L[c] = 2obut[c] + cys-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#2@1 C#1@1 C#4@1

C#6@1 + C#3@1 C#5@1 C#7@1

HSERTA accoa[c] + hom-L[c] = achms[c]C#1@1 C#2@1 + C#1@2 C#2@2 C#3@2 C#4@2 = C#1@1 C#1@2 C#2@2 C#2@1

C#3@2 C#4@2

AHSERL2 achms[c] = ac[c] + hcys-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#4@1 + C#2@1 C#3@1

C#5@1 C#6@1

FUM mal-L[c] = fumarate[c] Var1: C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

Var2: C#2@1 C#1@1 C#4@1 C#3@1 = C#1@1 C#2@1 C#3@1 C#4@1

FUMm mal-L[m] = fumarate[m] Var1: C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@2

Var2: C#2@1 C#1@1 C#4@1 C#3@1 = C#1@1 C#2@1 C#3@1 C#4@2

MDH mal-L[c] = oaa[c] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

MDHm mal-L[m] = oaa[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@2

PGL 6pgl[c] = 6pgc[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

G6PDH2 g6p[c] = 6pgl[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

Table S5. Atom transition network (continued)

95

Supplementary tables

table continues next page

Abbreviation Stoichiometry Atom mappings

GND 6pgc[c] = co2tot[c] + ru5p-D[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#6@1 + C#5@1 C#1@1 C#4@1

C#2@1 C#3@1

RPI r5p[c] = ru5p-D[c] C#2@1 C#4@1 C#5@1 C#3@1 C#1@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

RPE ru5p-D[c] = xu5p-D[c] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

TALA g3p[c] + s7p[c] = e4p[c] + f6p[c]C#1@1 C#2@1 C#3@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 C#7@2 = C#7@2 C#2@2 C#6@2 C#4@2 + C#2@1 C#1@2 C#3@1 C#1@1 C#5@2 C#3@2

TKT3 s7p[c] + e4p[c] = r5p[c] + f6p[c]C#1@1C#2@1C#3@1C#4@1C#5@1C#6@1C#7@1 +

C#1@2C#2@2C#3@2C#4@2 = C#2@1C#4@1C#6@1C#7@1C#5@1 + C#2@2C#1@1C#4@2C#3@2C#1@2C#3@1

TKT2 e4p[c] + xu5p-D[c] = f6p[c] + g3p[c]C#1@1 C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#2@1

C#1@2 C#4@1 C#3@1 C#1@1 C#3@2 + C#5@2 C#2@2 C#4@2

TKT1 r5p[c] + xu5p-D[c] = g3p[c] + s7p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 =

C#5@2 C#2@2 C#4@2 + C#1@2 C#1@1 C#3@2 C#2@1 C#5@1 C#3@1 C#4@1

ACOAH ac[c] = accoa[c] C#1@1 C#2@1 = C#1@1 C#2@1

ACS ac[c] = accoa[c] C#1@1 C#2@1 = C#1@1 C#2@1

ACSm ac[m] = accoa[m] C#1@1 C#2@1 = C#1@1 C#2@1

ALCD2x etoh[c] = acald[c] C#1@1 C#2@1 = C#1@1 C#2@1

HCITSm accoa[m] + akg[m] = hcit[m]C#1@1 C#2@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#2@2 C#1@2 C#1@1

C#4@2 C#2@1 C#5@2 C#3@2

PYRDC pyr[c] = acald[c] + co2tot[c] C#1@2 C#2@2 C#3@2 = C#1@2 C#2@2 + C#3@2

AATA2oxoadp[c] + glu-L[c] = akg[c] +

L2aadp[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 + C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

MCITDm hcit[m] = b124tc[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#2@1 C#1@1 C#3@1

C#7@1 C#4@1 C#5@1 C#6@1

HACNHm b124tc[m] = hicit[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#4@1

C#5@1 C#3@1 C#7@1 C#6@1

HICITDm hicit[m] = oxag[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#3@1

C#4@1 C#5@1 C#6@1 C#7@1

AASAD1 L2aadp[c] = L2aadp6sa[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#3@1 C#2@1 C#5@1

C#4@1 C#6@1

THRD_Lm thr-L[m] = 2obut[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

OXAGm oxag[m] = 2oxoadp[m] + co2tot[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#3@1 C#2@1

C#5@1 C#4@1 C#7@1 + C#6@1

SACCD2 saccrp-L[c] = akg[c] + lys-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 = C#4@1 C#5@1 C#8@1 C#9@1 C#11@1 + C#1@1 C#2@1 C#3@1

C#6@1 C#7@1 C#10@1

SACCD1 glu-L[c] + L2aadp6sa[c] = saccrp-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 = C#1@2 C#2@2 C#3@2 C#1@1 C#2@1 C#4@2 C#5@2 C#3@1 C#4@1

C#6@2 C#5@1

THRA acald[c] + gly[c] = thr-L[c] C#1@1 C#2@1 + C#1@2 C#2@2 = C#1@1 C#2@1 C#1@2 C#2@2

THRS phom[c] = thr-L[c] C#1@1 C#2@1 C#3@1 C#4@1 = C#2@1 C#1@1 C#3@1 C#4@1

DHQT 3dhq[c] = 3dhsk[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#2@1 C#1@1 C#7@1

C#4@1 C#3@1 C#5@1 C#6@1

DHQS 2dda7p[c] = 3dhq[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#3@1

C#5@1 C#6@1 C#7@1 C#4@1

DDPA e4p[c] + pep[c] = 2dda7p[c]C#1@1 C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 = C#1@2 C#2@1 C#1@1

C#2@2 C#4@1 C#3@1 C#3@2

PSCVTi pep[c] + skm5p[c] = 3psme[c]C#1@1 C#2@1 C#3@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 C#7@2 =

C#1@1 C#1@2 C#2@2 C#2@1 C#3@2 C#4@2 C#5@2 C#6@2 C#3@1 C#7@2

ALDD2y acald[c] = ac[c] C#1@1 C#2@1 = C#1@1 C#2@1

ALDD2xm acald[m] = ac[m] C#1@1 C#2@1 = C#1@1 C#2@1

ALDD2ym acald[m] = ac[m] C#1@1 C#2@1 = C#1@1 C#2@1

Table S5. Atom transition network (continued)

96

Chapter 4

Table S5. Atom transition network (continued)

table continues next page

Abbreviation Stoichiometry Atom mappings

ANPRT anth[c] + prpp[c] = pran[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#1@2 C#5@1 C#6@1 C#2@2

C#3@2 C#4@2 C#5@2 C#7@1

CHORM chor[c] = pphn[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 =

C#3@1 C#8@1 C#2@1 C#4@1 C#1@1 C#7@1 C#5@1 C#9@1 C#10@1 C#6@1

CHORS 3psme[c] = chor[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 = C#1@1 C#3@1 C#7@1 C#2@1 C#4@1 C#5@1 C#8@1 C#6@1 C#9@1 C#10@1

IGPS 2cpr5p[c] = 3ig3p[c] + co2tot[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1

C#11@1 C#12@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#9@1 C#8@1 C#10@1 C#11@1 + C#12@1

PHETA1akg[c] + phe-L[c] = glu-L[c] +

phpyr[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 C#7@2 C#8@2 C#9@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2

C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 C#7@2 C#8@2 C#9@2

PRAIi pran[c] = 2cpr5p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#11@1 C#5@1 C#6@1

C#7@1 C#10@1 C#8@1 C#9@1 C#12@1

PPNDH pphn[c] = co2tot[c] + phpyr[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 = C#9@1 + C#6@1 C#2@1 C#1@1 C#4@1 C#3@1 C#5@1 C#10@1 C#7@1 C#8@1

PPND2 pphn[c] = 34hpp[c] + co2tot[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 = C#3@1 C#4@1 C#1@1 C#2@1 C#5@1 C#10@1 C#6@1 C#7@1 C#8@1 + C#9@1

SHK3D 3dhsk[c] = skm[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#3@1

C#4@1 C#5@1 C#6@1 C#7@1

SHKK skm[c] = skm5p[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#2@1 C#1@1 C#3@1

C#5@1 C#4@1 C#6@1 C#7@1

TRPS13ig3p[c] + ser-L[c] = g3p[c] +

trp-L[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 + C#1@2 C#2@2 C#3@2 = C#11@1 C#6@1 C#10@1 + C#1@1 C#2@1

C#3@1 C#4@1 C#1@2 C#5@1 C#8@1 C#7@1 C#2@2 C#9@1 C#3@2

TYRTAakg[c] + tyr-L[c] = 34hpp[c] +

glu-L[c]

C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 + C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

C#7@1 C#8@1 C#9@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2

ACHBSm2obut[m] + pyr[m] = 2ahbut[m] +

co2tot[m]C#1@1 C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 = C#1@1 C#1@2 C#2@1

C#2@2 C#4@1 C#3@1 + C#3@2

IPPMIb 2ippm[c] = 3c3hmp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#6@1 C#7@1 C#5@1 = C#1@1 C#2@1 C#3@1

C#4@1 C#5@1 C#6@1 C#7@1

IPPS 3mob[c] + accoa[c]= 3c3hmp[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 = C#1@1 C#2@1 C#1@2

C#3@1 C#2@2 C#5@1 C#4@1

OMCDC 3c4mop[c] = 4mop[c] + co2tot[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#4@1

C#3@1 C#5@1 C#7@1 + C#6@1

IPPMIa 3c2hmp[c] = 2ippm[c]C#1@1 C#2@1 C#5@1 C#3@1 C#4@1 C#7@1 C#6@1 = C#1@1 C#2@1 C#3@1

C#4@1 C#5@1 C#6@1 C#7@1

IPMD 3c2hmp[c] = 3c4mop[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 = C#1@1 C#2@1 C#3@1

C#4@1 C#5@1 C#6@1 C#7@1

KARA1m alac-S[m] = 23dhmb[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#2@1 C#1@1 C#5@1 C#4@1 C#3@1

ACLSm (2) pyr[m] = alac-S[m] + co2tot[m]C#1@1 C#2@1 C#3@1 + C#1@2 C#2@2 C#3@2 = C#1@2 C#1@1 C#2@2 C#3@1

C#2@1 + C#3@2

DHAD1m 23dhmb[m] = 3mob[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#5@1 C#3@1 C#4@1

DHAD2m 23dhmp[m] = 3mop[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#6@1

C#4@1 C#5@1

ILETA akg[c] + ile-L[c] = 3mop[c] + glu-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2

C#6@2 = C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 + C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

KARA2m 2ahbut[m] = 23dhmp[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#6@1

C#5@1 C#4@1

97

Supplementary tables

Table S5. Atom transition network (continued)

Abbreviation Stoichiometry Atom mappings

LEUTAakg[c] + leu-L[c] = 4mop[c] +

glu-L[c]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 = C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2 + C#1@1 C#2@1 C#3@1

C#4@1 C#5@1

VALTAakg[c] + val-L[c] = 3mob[c] +

glu-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 + C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

ComplexII succ[m] = fumarate[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

ACt-1 & ACt-1+H ac[e] = ac[c] C#1@1 C#2@1 = C#1@1 C#2@1

CO2t-1 & CO2t-1+H co2tot[e] = co2tot[c] C#1@1 = C#1@1

ETOHt+0 etoh[e] = etoh[c] C#1@1 C#2@1 = C#1@1 C#2@1

GLCt+0 glc-D[e] = glc-D[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

GLYCt+0 glyc[c] = glyc[e] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

PYRt-1 & PYRt-1+H pyr[e]= pyr[c] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

SUCCt-2, SUCCt-2+H & SUCCt-2+2H

succ[e] = succ[c] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

2OBUTtm-1+H and 2OBUTtm-1

2obut[c] = 2obut[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@2

2OXOADPAKGtm-2-22oxoadp[m] + akg[c] = 2oxoadp[c]

+ akg[m]

C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2

C#4@2 C#5@2

3MOBtm-1 & 3MOBtm-1+H

3mob[c] = 3mob[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

3MOPtm-1 & 3MOPtm-1+H

3mop[c] = 3mop[m]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 = C#1@1 C#2@1 C#3@1 C#4@1

C#5@1 C#6@1

ACtm-1 and ACtm-1+H ac[c] = ac[m] C#1@1 C#2@1 = C#1@1 C#2@1

ACALDtm+0 acald[c] = acald[m] C#1@1 C#2@1 = C#1@1 C#2@2

ALAtm+0 ala-L[c] = ala-L[m] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

AKGMALtm-2-2akg[c] + mal-L[m] = akg[m] +

mal-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2 = C#1@1

C#2@1 C#3@1 C#4@1 C#5@1 + C#1@2 C#2@2 C#3@2 C#4@2

ASPGLUtm-1-1asp-L[c] + glu-L[m]= asp-L[m] +

glu-L[c]C#1@1 C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 = C#1@1

C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2

CITICITtm-3-3 cit[c] + icit[m] = cit[m] + icit[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2

C#5@2 C#6@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2 C#5@2 C#6@2

CITMALtm-3-2 & CITMALtm-3+H-2

cit[c] + mal-L[m] = cit[m] + mal-L[c]C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 + C#1@2 C#2@2 C#3@2 C#4@2

CO2tm-1 & CO2tm-1+H co2tot[c] = co2tot[m] C#1@1 = C#1@1

ETOHtm+0 etoh[c] = etoh[m] C#1@1 C#2@1 = C#1@1 C#2@1

GLNtm+0 gln-L[c] = gln-L[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

GLUtm-1 & GLUtm-1+H glu-L[c] = glu-L[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@2

GLYtm+0 gly[c] = gly[m] C#1@1 C#2@1 = C#1@1 C#2@1

GLXtm-1 & GLXtm-1+H glx[c]= glx[m] C#1@1 C#2@1 = C#1@1 C#2@1

MALPItm-2-2 mal-L[c] = mal-L[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

OAAtm-2 and OAAtm-2+H and OAAtm-2+2H

oaa[c] = oaa[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@2

ORNtm+1 & ORNtm+1-H

orn[c] = orn[m] C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 = C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

PYRtm-1 and PYRtm-1+H

pyr[c]= pyr[m] C#1@1 C#2@1 C#3@1 = C#1@1 C#2@1 C#3@1

SUCCPItm-2-2 succ[c]= succ[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

SUCFUMtm-2-2fumarate[m] + succ[c] = fumarate[c]

+ succ[m]C#1@1 C#2@1 C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 C#4@2 = C#1@1 C#2@1

C#3@1 C#4@1 + C#1@2 C#2@2 C#3@2 C#4@2

THRtm+0 thr-L[c] = thr-L[m] C#1@1 C#2@1 C#3@1 C#4@1 = C#1@1 C#2@1 C#3@1 C#4@1

98

Chapter 4

Table S6. Stoichiometry of the biomass synthesis reaction of the atom transition network, which was based on the biomass composition described in Chapter 2, methods 1. Also here, we only considered compounds which contrib-uted to the carbon balances.

MetaboliteStoichiometric

coefficientAtom mappings

ala-L -0.4588 C#1@1 C#2@1 C#3@1

amp -0.046 C#1@1 C#1@2 C#1@5 C#1@8 C#2@8 C#1@9 C#2@9 C#3@9 C#4@9 C#5@9

arg-L -0.1607 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

asn-L -0.1017 C#1@1 C#2@1 C#3@1 C#4@1

asp-L -0.2975 C#1@1 C#2@1 C#3@1 C#4@1

accoa -0.4536 C#1@1 C#2@1

cmp -0.0447 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1

co2tot 0.0062 C#1@1

cys-L -0.0066 C#1@1 C#2@1 C#3@1

damp -0.0036 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1

dcmp -0.0024 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1

dgmp -0.0024 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1

dtmp -0.0036 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1

ergst -0.0007C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 C#13@1 C#14@1

C#15@1 C#16@1 C#17@1 C#18@1 C#19@1 C#20@1 C#21@1 C#22@1 C#23@1 C#24@1 C#25@1 C#26@1 C#27@1 C#28@1

f6p -0.8079 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

g6p -1.1401 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

gln-L -0.1054 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

glu-L -0.3018 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

gly -0.2904 C#1@1 C#2@1

glyc3p -0.0247 C#1@1 C#2@1 C#3@1

glycogen -0.129625C#1@2 C#1@4 C#1@5 C#1@3 C#2@2 C#2@4 C#2@5 C#2@3 C#3@2 C#3@4 C#4@2 C#4@4 C#4@5 C#4@3 C#5@5

C#5@2 C#5@4 C#5@3 C#3@5 C#3@3 C#6@5 C#6@2 C#6@4 C#6@3

gmp -0.046 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1

hcys-L 0.0051 C#1@1 C#2@1 C#3@1 C#4@1

his-L -0.0663 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

ile-L -0.1927 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

leu-L -0.2964 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

lys-L -0.2862 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1

met-L -0.0558 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

phe-L -0.1339 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1

pro-L -0.1647 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

ser-L -0.1976 C#1@1 C#2@1 C#3@1

thr-L -0.1914 C#1@1 C#2@1 C#3@1 C#4@1

tre -0.0234 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1

trp-L -0.0284 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1

tyr-L -0.102 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1

ump -0.0599 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1

val-L -0.2646 C#1@1 C#2@1 C#3@1 C#4@1 C#5@1

zymst -0.0015C#1@1 C#2@1 C#3@1 C#4@1 C#5@1 C#6@1 C#7@1 C#8@1 C#9@1 C#10@1 C#11@1 C#12@1 C#13@1 C#14@1

C#15@1 C#16@1 C#17@1 C#18@1 C#19@1 C#20@1 C#21@1 C#22@1 C#23@1 C#24@1 C#25@1 C#26@1 C#27@1

99

Supplementary tables

Table S7. Estimates of the metabolic fluxes, which have been determined from ~1000 randomly sampled flux distri-butions and the weights representing the goodness of fits to the different labeling patterns. The percentages represent the estimated quantiles for the fluxes.

table continues next page

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

2OBUTtm_1 -9.208 -3.474 -0.703 -9.136 -3.487 -0.703 -9.136 -3.503 -0.675 -9.119 -3.507 -0.679

2OBUTtm_1+H 0.705 3.476 9.210 0.740 3.491 9.210 0.682 3.509 9.210 0.690 3.509 9.201

2OXOADPAKGtm_2-2 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

3MOBtm_1 -11.182 -4.124 -0.373 -11.182 -4.118 -0.373 -11.435 -4.046 -0.414 -11.548 -4.018 -0.417

3MOBtm_1+H 0.194 3.941 11.003 0.213 3.938 11.214 0.237 3.881 11.391 0.237 3.857 11.398

3MOPtm_1 -11.897 -3.989 -0.460 -11.897 -3.989 -0.471 -11.233 -4.026 -0.547 -11.233 -4.029 -0.555

3MOPtm_1+H 0.398 3.927 11.836 0.451 3.930 12.085 0.494 3.971 11.446 0.496 3.975 11.446

AASAD1 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

AATA 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

ac_EX 0.283 0.305 0.323 0.283 0.305 0.323 0.283 0.303 0.324 0.283 0.303 0.324

ACALDtm_0 11.711 18.555 26.698 11.711 18.555 26.593 11.314 18.647 26.959 11.308 18.647 26.914

ACGKm 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

ACHBSm 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063

ACLSm 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183

ACOAH -1.476 -0.424 -0.039 -1.476 -0.419 -0.039 -1.493 -0.426 -0.045 -1.493 -0.424 -0.047

ACONT -3.793 1.899 8.023 -3.733 1.905 8.023 -3.444 1.964 8.023 -3.444 1.982 8.016

ACONTm -7.617 -1.557 4.124 -7.560 -1.577 4.124 -7.544 -1.586 3.900 -7.544 -1.610 3.900

ACOTAm 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

ACS 0.826 1.880 3.274 0.840 1.879 3.274 0.827 1.748 3.163 0.851 1.748 3.157

ACSm -2.085 -0.572 0.212 -2.071 -0.572 0.212 -1.756 -0.484 0.212 -1.754 -0.484 0.212

ACt_1 -2.555 -0.824 -0.148 -2.555 -0.824 -0.141 -2.539 -0.823 -0.141 -2.547 -0.823 -0.141

ACt_1+H -0.161 0.526 2.261 -0.161 0.527 2.277 -0.165 0.527 2.244 -0.168 0.527 2.249

ACtm_1 -9.065 -3.456 0.324 -9.065 -3.497 0.324 -8.803 -3.771 0.269 -9.065 -3.797 0.261

ACtm_1+H -1.838 2.166 7.849 -1.807 2.233 7.862 -1.679 2.453 8.024 -1.663 2.514 8.075

ADK1 1.155 2.211 3.604 1.170 2.210 3.604 1.156 2.079 3.493 1.180 2.079 3.489

ADK1m -2.085 -0.572 0.212 -2.071 -0.572 0.212 -1.756 -0.484 0.212 -1.754 -0.484 0.212

ADPATPtm_4-2 0.130 1.070 3.477 0.132 1.073 3.488 0.147 1.106 3.488 0.147 1.113 3.488

ADPATPtm_4-3 1.571 6.268 11.567 1.571 6.176 11.540 1.408 5.654 11.246 1.381 5.592 11.223

AGPRm 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

AGTm -1.910 -0.782 -0.224 -1.849 -0.778 -0.224 -1.783 -0.778 -0.249 -1.765 -0.766 -0.251

AHSERL2 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019

AKGDm -1.430 -0.109 1.146 -1.418 -0.119 1.125 -1.174 0.010 1.059 -1.174 0.003 1.057

AKGMALtm_2-2 3.681 26.535 46.635 4.340 26.421 46.306 4.340 25.060 43.689 4.340 24.757 42.911

ALATA_L -5.970 14.067 28.621 -5.970 13.858 28.385 -6.740 13.264 28.063 -7.096 12.995 28.063

ALATA_Lm -29.099 -14.614 5.786 -28.623 -14.440 5.948 -28.438 -13.873 6.461 -28.417 -13.717 6.713

ALAtm_0 -6.114 13.919 28.475 -6.114 13.711 28.239 -6.887 13.118 27.916 -7.242 12.849 27.916

ALCD2m -25.897 -17.844 -11.146 -25.748 -17.844 -11.088 -26.031 -17.978 -10.762 -26.031 -17.991 -10.762

ALCD2x -12.774 -6.010 2.068 -12.857 -6.007 2.068 -12.931 -5.873 2.302 -12.931 -5.868 2.273

ALDD2xm 0.039 0.309 0.921 0.039 0.306 0.921 0.037 0.300 0.865 0.037 0.300 0.871

ALDD2y 0.033 0.311 1.413 0.035 0.319 1.452 0.042 0.307 1.252 0.043 0.311 1.252

ALDD2ym 0.032 0.309 0.829 0.032 0.308 0.844 0.027 0.287 0.806 0.028 0.286 0.806

ampSYN1 0.010 0.023 0.032 0.010 0.023 0.032 0.011 0.023 0.032 0.012 0.023 0.032

ampSYN2 0.006 0.014 0.027 0.006 0.014 0.027 0.006 0.014 0.026 0.006 0.014 0.026

ANPRT 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

ANS 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

ARGSL 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

100

Chapter 4

Table S7. Estimates of the metabolic fluxes (continued)

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

ARGSS 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

ASAD 0.200 0.710 1.771 0.203 0.704 1.771 0.200 0.700 1.678 0.200 0.687 1.670

ASNS1 0.032 0.033 0.033 0.032 0.033 0.033 0.032 0.033 0.033 0.032 0.033 0.033

ASPGLUtm_1-1 -5.050 11.232 27.149 -4.873 11.340 27.149 -5.134 10.574 26.920 -5.134 10.574 26.920

ASPK 0.200 0.710 1.771 0.203 0.704 1.771 0.200 0.700 1.678 0.200 0.687 1.670

ASPTA -28.097 -12.299 4.377 -27.955 -12.396 4.377 -27.479 -11.711 4.398 -27.479 -11.778 4.398

ASPTAm -5.050 11.232 27.149 -4.873 11.340 27.149 -5.134 10.574 26.920 -5.134 10.574 26.920

ATPASE 6.130 9.010 12.383 6.142 8.989 12.383 6.157 8.984 12.254 6.157 8.969 12.241

ATPHYD 0.100 0.636 2.156 0.110 0.638 2.156 0.115 0.640 2.095 0.116 0.640 2.095

ATPPRTPRATPAMPC 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

ATPS3m 0.966 2.294 3.925 0.966 2.274 3.906 0.993 2.170 3.875 0.993 2.144 3.811

biomass 0.316 0.320 0.326 0.316 0.320 0.326 0.316 0.320 0.326 0.316 0.321 0.326

BPNT 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019

CBPS 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

CHORM 0.075 0.076 0.077 0.075 0.076 0.077 0.075 0.076 0.077 0.075 0.076 0.077

CHORS 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

CITICITtm_3-3 -6.583 -1.466 4.703 -6.533 -1.441 4.775 -6.533 -1.342 4.635 -6.525 -1.274 4.636

CITMALtm_3-2 -8.646 -0.821 3.378 -8.603 -0.822 3.431 -8.387 -0.763 3.378 -8.313 -0.780 3.378

CITMALtm_3+H-2 -5.514 0.571 9.235 -5.514 0.548 9.235 -5.612 0.458 8.630 -5.619 0.438 8.571

cmpSYN 0.016 0.016 0.017 0.016 0.016 0.017 0.016 0.016 0.017 0.016 0.016 0.017

CO2t_1 -2.225 -0.766 -0.096 -2.132 -0.761 -0.096 -2.061 -0.758 -0.092 -2.040 -0.758 -0.092

CO2t_1+H -26.410 -25.725 -24.214 -26.408 -25.727 -24.232 -26.409 -25.727 -24.429 -26.408 -25.726 -24.431

CO2tm_1 -4.385 -1.387 -0.122 -4.385 -1.378 -0.122 -4.397 -1.331 -0.117 -4.412 -1.329 -0.109

CO2tm_1+H -29.508 -21.452 -13.698 -29.057 -21.369 -13.656 -28.740 -21.220 -13.503 -28.587 -21.202 -13.454

co2tot_EX 26.302 26.493 26.668 26.306 26.494 26.668 26.313 26.496 26.666 26.315 26.496 26.666

ComplexII -0.126 0.039 0.232 -0.126 0.039 0.232 -0.126 0.031 0.228 -0.126 0.031 0.228

ComplexIII 2.728 2.821 2.900 2.728 2.821 2.900 2.726 2.819 2.899 2.725 2.819 2.899

ComplexIV 2.728 2.821 2.900 2.728 2.821 2.900 2.726 2.819 2.899 2.725 2.819 2.899

CS 0.015 0.147 0.322 0.016 0.147 0.322 0.016 0.149 0.319 0.016 0.149 0.319

CSm 0.044 0.215 0.399 0.046 0.214 0.399 0.044 0.210 0.399 0.044 0.209 0.399

CYSTGL 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

CYSTS 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

dampSYN 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

dcmpSYN 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

DDPA 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

dgmpSYN 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

DHAD1m 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183

DHAD2m 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063

DHQS 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

DHQT 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

dtmpSYN 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

ENO 26.162 26.394 26.640 26.164 26.394 26.637 26.164 26.390 26.637 26.166 26.390 26.637

ergstSYN 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

etoh_EX 23.672 23.859 24.044 23.673 23.860 24.044 23.674 23.864 24.040 23.674 23.864 24.035

ETOHt_0 -24.044 -23.859 -23.672 -24.043 -23.859 -23.672 -24.035 -23.864 -23.673 -24.034 -23.864 -23.673

ETOHtm_0 -25.897 -17.844 -11.146 -25.748 -17.844 -11.088 -26.031 -17.978 -10.762 -26.031 -17.991 -10.762

FBA 14.208 14.374 14.511 14.208 14.374 14.511 14.213 14.377 14.511 14.217 14.377 14.511

FBP 0.072 0.704 2.843 0.078 0.706 2.883 0.086 0.709 2.689 0.091 0.715 2.689

table continues next page

101

Supplementary tables

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

FDH 0.082 0.237 0.390 0.082 0.238 0.390 0.083 0.244 0.394 0.083 0.245 0.394

FTHFL -0.389 -0.237 -0.082 -0.389 -0.237 -0.082 -0.392 -0.244 -0.082 -0.393 -0.244 -0.082

FUM -37.474 -15.254 6.899 -36.865 -14.029 7.629 -35.435 -10.617 9.535 -34.933 -10.304 9.535

FUMm -6.805 15.436 37.686 -6.810 14.188 37.154 -8.952 10.737 35.888 -9.360 10.359 35.217

G3PD1 2.279 3.216 3.974 2.279 3.216 3.976 2.266 3.246 3.979 2.264 3.248 3.979

G3PD2 0.201 1.114 1.873 0.201 1.116 1.873 0.188 1.152 1.887 0.188 1.157 1.894

G3PT 2.207 3.045 4.724 2.209 3.045 4.761 2.203 3.078 4.791 2.204 3.083 4.791

G5P5 0.052 0.053 0.054 0.052 0.053 0.054 0.052 0.053 0.054 0.052 0.053 0.054

G6PDH2 0.406 0.732 1.159 0.413 0.732 1.160 0.413 0.730 1.159 0.407 0.730 1.160

GALU 0.171 0.174 0.177 0.171 0.174 0.177 0.171 0.174 0.177 0.171 0.174 0.177

GAPD 26.598 26.816 27.026 26.600 26.816 27.026 26.604 26.820 27.022 26.604 26.820 27.022

GHMT2 0.184 0.340 0.493 0.185 0.340 0.494 0.185 0.347 0.497 0.185 0.348 0.497

glc_D_EX -15.603 -15.517 -15.418 -15.603 -15.517 -15.418 -15.600 -15.520 -15.418 -15.600 -15.520 -15.418

GLCS2 0.041 0.042 0.042 0.041 0.042 0.042 0.041 0.042 0.042 0.041 0.042 0.042

GLCt_0 15.418 15.517 15.603 15.418 15.517 15.604 15.418 15.520 15.600 15.418 15.520 15.600

GLNS 0.131 0.990 3.121 0.141 0.988 3.145 0.159 0.930 3.029 0.160 0.923 3.029

GLNtm_0 -0.131 0.729 2.860 -0.123 0.725 2.883 -0.103 0.668 2.769 -0.103 0.659 2.769

GLU5K 0.052 0.053 0.054 0.052 0.053 0.054 0.052 0.053 0.054 0.052 0.053 0.054

GLUDy -3.571 -0.476 3.477 -3.571 -0.532 3.477 -3.666 -0.822 3.373 -3.620 -0.864 3.334

GLUDym -4.691 -0.308 3.199 -4.609 -0.263 3.263 -4.609 -0.057 3.102 -4.579 -0.014 3.101

GLUSxm -0.131 0.729 2.860 -0.123 0.725 2.883 -0.103 0.668 2.769 -0.103 0.659 2.769

GLUtm_1 -13.949 3.998 15.275 -13.586 4.126 15.358 -12.262 4.372 15.275 -11.872 4.415 15.282

GLUtm_1+H -31.703 -19.832 -2.401 -31.647 -19.590 -1.955 -31.340 -19.086 -1.652 -31.339 -18.907 -1.597

GLXtm_1 -15.692 -4.531 -0.248 -15.692 -4.503 -0.248 -15.624 -4.694 -0.226 -15.624 -4.672 -0.233

GLXtm_1+H -0.705 3.684 14.732 -0.677 3.682 14.908 -0.648 3.883 14.732 -0.647 3.872 14.862

glyc_EX 2.020 2.090 2.143 2.021 2.090 2.143 2.022 2.087 2.143 2.023 2.087 2.143

GLYCt_0 2.020 2.090 2.143 2.021 2.090 2.143 2.022 2.087 2.143 2.023 2.087 2.143

GLYK 0.112 0.966 2.628 0.115 0.966 2.650 0.112 1.004 2.685 0.113 1.007 2.685

GLYtm_0 0.224 0.782 1.910 0.226 0.779 1.910 0.251 0.779 1.795 0.251 0.766 1.783

gmpSYN1 0.000 0.008 0.022 0.000 0.008 0.022 0.000 0.008 0.020 0.000 0.008 0.020

gmpSYN2 -0.006 0.007 0.016 -0.006 0.007 0.016 -0.004 0.008 0.016 -0.004 0.007 0.016

GND 0.406 0.732 1.159 0.413 0.732 1.160 0.413 0.730 1.159 0.407 0.730 1.160

h_EX 6.983 7.025 7.066 6.983 7.026 7.066 6.982 7.027 7.065 6.981 7.027 7.065

h2o_EX -22.205 -22.002 -21.787 -22.205 -22.002 -21.787 -22.204 -22.002 -21.786 -22.204 -22.002 -21.786

H2Ot_0 21.787 22.002 22.205 21.787 22.002 22.205 21.787 22.002 22.205 21.787 22.002 22.204

H2Otm_0 3.860 28.507 52.842 3.593 27.666 51.330 3.331 24.879 49.712 3.130 24.479 48.532

HACNHm 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

HCITSm 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

HEX1 15.418 15.517 15.603 15.418 15.517 15.604 15.418 15.520 15.600 15.418 15.520 15.600

HICITDm 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

HISTD 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

HISTP 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

HSDy 0.200 0.710 1.771 0.203 0.704 1.771 0.200 0.700 1.678 0.200 0.687 1.670

HSERTA 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019

HSK 0.182 0.692 1.753 0.185 0.686 1.753 0.182 0.682 1.660 0.182 0.668 1.652

HSTPT 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

Ht 0.003 0.231 1.418 0.004 0.231 1.418 0.002 0.223 1.439 0.002 0.223 1.451

Htm 0.011 0.760 4.472 0.013 0.759 4.504 0.013 0.782 6.346 0.015 0.779 6.625

Table S7. Estimates of the metabolic fluxes (continued)

table continues next page

102

Chapter 4

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

ICDHxm 14.559 21.641 27.711 14.559 21.590 27.711 14.531 21.646 27.564 14.531 21.641 27.532

ICDHy -3.716 0.483 4.059 -3.682 0.522 4.059 -3.542 0.766 4.002 -3.508 0.769 3.970

ICDHym -29.012 -21.951 -12.856 -29.012 -21.910 -12.821 -28.946 -21.962 -12.856 -28.996 -21.997 -12.856

ICL -1.153 0.152 1.477 -1.089 0.176 1.477 -1.057 0.034 1.224 -1.053 0.044 1.224

IGPDH 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

IGPS 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

ILETA -0.063 -0.062 -0.061 -0.063 -0.062 -0.061 -0.063 -0.062 -0.061 -0.063 -0.062 -0.061

IONPUMP 0.838 1.284 2.508 0.838 1.285 2.508 0.838 1.278 2.491 0.838 1.277 2.464

IONPUMPm 0.039 0.470 1.530 0.044 0.470 1.540 0.054 0.499 1.606 0.056 0.499 1.649

IPMD 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097

IPPMIa -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094

IPPMIb -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094

IPPS 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097

KARA1m 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183 0.177 0.180 0.183

KARA2m 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063 0.061 0.062 0.063

LEUTA -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094 -0.097 -0.095 -0.094

MALPItm_2-2 -1.800 31.059 48.702 -1.044 31.404 48.782 3.889 33.208 49.750 4.442 33.426 49.735

MALS 0.123 0.967 2.179 0.160 0.969 2.179 0.150 0.811 2.006 0.160 0.811 1.999

MCITDm 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

MDH -25.192 -17.541 -11.358 -25.127 -17.541 -11.114 -25.747 -17.643 -10.387 -25.747 -17.664 -10.387

MDHm -11.238 -3.049 5.264 -10.500 -3.023 5.127 -10.194 -2.917 5.631 -10.143 -2.904 5.561

MEm 15.340 22.251 27.770 15.340 22.193 27.747 15.269 22.086 27.428 15.269 22.078 27.380

METS 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018

MTHFC 0.165 0.321 0.474 0.165 0.321 0.475 0.166 0.328 0.478 0.166 0.328 0.478

MTHFD 0.165 0.321 0.474 0.165 0.321 0.475 0.166 0.328 0.478 0.166 0.328 0.478

MTHFR3 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018

NADH2_u6i 0.239 0.927 1.953 0.239 0.927 1.953 0.216 0.907 1.880 0.216 0.901 1.880

NADH2_u6m 0.104 0.686 1.526 0.110 0.685 1.528 0.121 0.689 1.492 0.122 0.689 1.492

NDPK2 0.171 0.174 0.177 0.171 0.174 0.177 0.171 0.174 0.177 0.171 0.174 0.177

nh3_EX -1.825 -1.792 -1.769 -1.825 -1.792 -1.769 -1.825 -1.792 -1.769 -1.825 -1.792 -1.769

NH3t_1 1.904 2.517 3.682 1.920 2.517 3.835 1.935 2.487 3.950 1.935 2.486 3.950

NH3t_1-H -1.873 -0.722 -0.103 -1.873 -0.721 -0.122 -2.116 -0.691 -0.133 -2.116 -0.690 -0.139

NH3tm_1 0.117 1.194 4.513 0.130 1.196 4.513 0.133 1.207 4.415 0.133 1.198 4.400

NH3tm_1-H -5.807 -1.191 3.844 -5.807 -1.213 3.800 -5.757 -1.424 3.800 -5.757 -1.484 3.689

o2_EX -1.486 -1.446 -1.400 -1.486 -1.446 -1.399 -1.485 -1.445 -1.399 -1.485 -1.445 -1.399

O2t 1.400 1.446 1.486 1.400 1.446 1.486 1.399 1.445 1.485 1.399 1.445 1.485

O2tm_0 1.364 1.411 1.450 1.364 1.410 1.450 1.363 1.409 1.449 1.363 1.409 1.449

OAAtm_2 -3.212 -0.936 -0.090 -3.143 -0.933 -0.090 -2.982 -0.905 -0.090 -2.953 -0.899 -0.090

OAAtm_2+2H -12.587 -3.437 10.386 -12.587 -3.562 10.423 -12.396 -3.105 10.824 -12.396 -3.134 10.824

OAAtm_2+H -10.946 -3.037 -0.222 -10.856 -3.048 -0.222 -10.905 -2.984 -0.300 -10.856 -3.030 -0.301

OCBT 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

OMCDC 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097 0.094 0.095 0.097

ORNTACm 0.051 0.051 0.052 0.051 0.051 0.052 0.051 0.052 0.052 0.051 0.052 0.052

ORNtm_1 0.678 4.537 11.195 0.678 4.529 11.012 0.701 4.587 10.959 0.704 4.583 10.847

ORNtm_1-H -11.247 -4.589 -0.729 -11.052 -4.575 -0.717 -10.957 -4.635 -0.729 -10.739 -4.635 -0.729

OXAGm 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

PC 0.098 1.261 4.941 0.099 1.245 4.918 0.095 1.169 4.844 0.095 1.156 4.844

PDHm 0.110 0.877 2.349 0.117 0.877 2.349 0.110 0.769 2.070 0.117 0.772 2.070

Table S7. Estimates of the metabolic fluxes (continued)

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103

Supplementary tables

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

PFK 14.432 15.077 17.313 14.435 15.085 17.338 14.437 15.088 17.053 14.437 15.090 17.053

PGI 13.811 14.238 14.570 13.818 14.239 14.570 13.824 14.246 14.570 13.824 14.246 14.570

PGK -27.026 -26.816 -26.598 -27.024 -26.816 -26.598 -27.020 -26.820 -26.602 -27.018 -26.820 -26.601

PGL 0.406 0.732 1.159 0.413 0.732 1.160 0.413 0.730 1.159 0.407 0.730 1.160

PGM -26.640 -26.394 -26.162 -26.637 -26.394 -26.162 -26.637 -26.390 -26.162 -26.637 -26.390 -26.164

PGMT -0.177 -0.174 -0.171 -0.177 -0.174 -0.171 -0.177 -0.174 -0.171 -0.177 -0.174 -0.171

PGPS 0.258 0.414 0.568 0.259 0.415 0.570 0.259 0.422 0.572 0.259 0.422 0.572

PHETA1 -0.044 -0.043 -0.042 -0.044 -0.043 -0.042 -0.044 -0.043 -0.042 -0.044 -0.043 -0.042

pi_EX -0.065 -0.063 -0.063 -0.065 -0.063 -0.063 -0.065 -0.063 -0.063 -0.065 -0.063 -0.063

PIt_1 -2.286 -0.836 -0.200 -2.270 -0.830 -0.187 -2.146 -0.840 -0.173 -2.123 -0.835 -0.169

PIt_1+H 0.265 0.900 2.349 0.265 0.893 2.349 0.244 0.904 2.235 0.236 0.902 2.209

PItm_2 -11.023 -3.364 -0.315 -11.023 -3.311 -0.315 -11.320 -3.435 -0.324 -11.320 -3.426 -0.330

PItm_2+2H 0.491 5.214 17.358 0.510 5.251 17.453 0.531 5.105 17.305 0.610 5.144 17.305

PItm_2+H 25.834 51.504 70.265 25.834 50.899 70.238 25.151 49.639 67.682 25.134 49.187 67.560

poscharge_EX 0.401 0.407 0.414 0.402 0.407 0.414 0.402 0.407 0.414 0.402 0.407 0.414

POSQt 0.034 0.480 1.709 0.034 0.481 1.709 0.029 0.478 1.694 0.029 0.477 1.665

POSQtm 0.039 0.470 1.530 0.044 0.470 1.540 0.054 0.499 1.606 0.056 0.499 1.649

PPA 1.958 3.013 4.412 1.977 3.013 4.412 1.959 2.882 4.299 1.985 2.884 4.299

PPAm -2.085 -0.572 0.212 -2.071 -0.572 0.212 -1.756 -0.484 0.212 -1.754 -0.484 0.212

PPCK -25.470 -20.929 -14.255 -25.398 -20.887 -14.152 -25.395 -20.960 -14.103 -25.398 -20.929 -14.103

PPND2 0.032 0.033 0.033 0.032 0.033 0.033 0.032 0.033 0.033 0.032 0.033 0.033

PPNDH 0.042 0.043 0.044 0.042 0.043 0.044 0.042 0.043 0.044 0.042 0.043 0.044

PRAIi 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

PRMICIi 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022 0.021 0.021 0.022

PRPPS 0.030 0.030 0.031 0.030 0.030 0.031 0.030 0.030 0.031 0.030 0.030 0.031

PSCVTi 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

PYK 0.772 5.309 11.993 0.790 5.339 12.094 0.839 5.236 12.134 0.839 5.265 12.134

pyr_EX 0.160 0.167 0.171 0.160 0.167 0.171 0.160 0.167 0.171 0.160 0.167 0.171

PYRDC 23.065 24.313 25.655 23.084 24.321 25.655 23.146 24.319 25.488 23.152 24.326 25.488

PYRt_1 -2.390 -0.805 -0.150 -2.359 -0.799 -0.150 -2.349 -0.804 -0.153 -2.343 -0.799 -0.153

PYRt_1+H -0.018 0.635 2.224 -0.015 0.633 2.224 -0.013 0.633 2.179 -0.013 0.633 2.179

PYRtm_1 -4.779 6.869 17.258 -4.749 6.930 17.315 -4.795 6.928 17.621 -4.795 6.953 17.715

PYRtm_1+H -58.920 -40.782 -20.980 -58.920 -40.501 -20.538 -58.785 -39.982 -19.660 -58.757 -39.813 -19.660

RPE 0.177 0.395 0.680 0.182 0.395 0.681 0.182 0.393 0.680 0.177 0.393 0.681

RPI -0.479 -0.337 -0.229 -0.479 -0.337 -0.229 -0.478 -0.336 -0.229 -0.479 -0.336 -0.229

SACCD1 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

SACCD2 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093 0.091 0.092 0.093

SHK3D 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

SHKK 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086 0.084 0.085 0.086

so4_EX -0.025 -0.025 -0.024 -0.025 -0.025 -0.024 -0.025 -0.025 -0.024 -0.025 -0.025 -0.024

SO4SO3 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019 0.018 0.018 0.019

SO4t_2 -2.690 -1.045 -0.242 -2.690 -1.045 -0.237 -2.690 -1.080 -0.213 -2.690 -1.080 -0.213

SO4t_2+2H 0.034 0.320 1.241 0.036 0.320 1.248 0.032 0.319 1.248 0.034 0.319 1.248

SO4t_2+H 0.078 0.694 1.986 0.080 0.697 1.994 0.079 0.728 1.986 0.081 0.728 1.986

SUCCPItm_2-2 -6.533 15.705 38.286 -6.821 14.542 37.027 -8.581 10.769 36.086 -8.838 10.354 35.104

SUCCt_2 -2.161 -0.634 -0.062 -2.161 -0.623 -0.062 -2.161 -0.618 -0.059 -2.161 -0.609 -0.059

SUCCt_2+2H 0.060 0.622 2.506 0.061 0.621 2.447 0.062 0.636 2.541 0.065 0.636 2.546

SUCCt_2+H -1.898 0.023 1.683 -1.833 0.023 1.683 -1.910 -0.007 1.683 -1.910 -0.010 1.671

Table S7. Estimates of the metabolic fluxes (continued)

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104

Chapter 4

Metabolic processwithout labeling with [1-13C] with [U-13C] with [1-13C] & [U-13C]

1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75% 1.25% 50% 98.75%

SUCFUMtm_2-2 -37.595 -15.373 6.779 -36.985 -14.149 7.509 -35.555 -10.739 9.417 -35.052 -10.425 9.417

SUCOAS1m -1.146 0.109 1.430 -1.108 0.119 1.430 -1.058 -0.008 1.182 -1.046 -0.001 1.180

SULR -0.019 -0.018 -0.018 -0.019 -0.018 -0.018 -0.019 -0.018 -0.018 -0.019 -0.018 -0.018

TALA 0.131 0.240 0.382 0.133 0.240 0.383 0.133 0.239 0.382 0.131 0.239 0.382

THRA -1.633 -0.571 -0.061 -1.611 -0.565 -0.061 -1.531 -0.556 -0.057 -1.519 -0.545 -0.057

THRD_Lm 0.059 0.060 0.061 0.059 0.060 0.061 0.059 0.060 0.061 0.059 0.060 0.061

THRS 0.182 0.692 1.753 0.185 0.686 1.753 0.182 0.682 1.660 0.182 0.668 1.652

THRtm_0 0.059 0.060 0.061 0.059 0.060 0.061 0.059 0.060 0.061 0.059 0.060 0.061

TKT1 0.027 0.158 0.432 0.030 0.158 0.432 0.030 0.158 0.402 0.030 0.159 0.400

TKT2 0.012 0.216 0.502 0.015 0.215 0.505 0.018 0.213 0.506 0.018 0.212 0.510

TKT3 -0.242 -0.072 0.125 -0.242 -0.071 0.125 -0.242 -0.071 0.116 -0.242 -0.069 0.116

TPI 12.104 12.279 12.447 12.109 12.280 12.447 12.124 12.283 12.439 12.124 12.284 12.438

TRESYN 0.007 0.007 0.008 0.007 0.007 0.008 0.007 0.007 0.008 0.007 0.008 0.008

TRPS1 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

TYRTA -0.033 -0.033 -0.032 -0.033 -0.033 -0.032 -0.033 -0.033 -0.032 -0.033 -0.033 -0.032

umpSYN 0.035 0.035 0.036 0.035 0.035 0.036 0.035 0.035 0.036 0.035 0.035 0.036

VALTA -0.086 -0.085 -0.084 -0.086 -0.085 -0.084 -0.086 -0.085 -0.084 -0.086 -0.085 -0.084

zymstSYN 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Table S7. Estimates of the metabolic fluxes (continued)

Table S8. Estimates of the exchange fluxes, which have been determined from ~1000 randomly sampled flux distri-butions and the weights representing the goodness of fits to the different labeling patterns. Only exchange fluxes that could be identified are shown (n.i. means not identifiable).

Metabolic processwith [1-13C] with [U-13C]

Mean SD Mean SD

AASAD1 50.00 0.38 n.i. n.i.

ACGKm 50.00 0.34 n.i. n.i.

ACHBSm n.i. n.i. 0.00 0.02

ACLSm n.i. n.i. 0.00 0.04

AGPRm 50.00 0.38 n.i. n.i.

DHAD1m n.i. n.i. 50.00 0.89

DHAD2m 50.00 0.38 50.00 0.54

FUM 0.00 0.28 0.00 0.07

FUMm 0.00 0.52 0.00 0.06

G3PD1 50.00 0.41 n.i. n.i.

G3PD2 50.00 0.43 n.i. n.i.

G3PT 50.00 0.43 n.i. n.i.

G5P5 50.00 0.43 50.00 0.91

GHMT2 n.i. n.i. 0.00 0.10

GLU5K 50.00 0.99 n.i. n.i.

GLYK 50.00 0.41 50.00 0.94

HEX1 50.00 0.49 n.i. n.i.

HISTD 50.00 0.37 50.00 0.90

HISTP 50.00 0.39 n.i. n.i.

IGPDH 50.00 0.32 n.i. n.i.

KARA1m n.i. n.i. 50.00 0.58

KARA2m 50.00 0.39 n.i. n.i.

MALS n.i. n.i. 0.00 0.00

MDH n.i. n.i. 0.00 0.06

METS 50.00 0.40 n.i. n.i.

Metabolic processwith [1-13C] with [U-13C]

Mean SD Mean SD

MTHFR3 50.00 0.33 50.00 0.66

OCBT 50.00 0.38 50.00 0.63

ORNTACm 50.00 0.50 n.i. n.i.

PDHm n.i. n.i. 0.00 0.07

PGMT 50.00 0.44 n.i. n.i.

PRMICIi 50.00 0.34 50.00 0.71

PYRDC n.i. n.i. 0.00 0.19

TKT2 n.i. n.i. 0.00 0.02

TKT3 0.00 0.19 0.00 0.03

105

Conclusions and future perspectives

Bastian Niebel

106

Conclusions and future perspective

This thesis investigated how thermodynamic principles govern metabolic operations. To this end, we developed a thermodynamic metabolic network model for the yeast Saccharomyces cerevisiae, which encompasses ~250 metabolic processes, and a comprehensive thermody-namic description for metabolic and growth processes (Chapter 2). Notably, we did not assign any a-priori directionalities on these metabolic processes, but rather constrained their direc-tionalities by the second law of thermodynamics. Then we analyzed with this model experi-mental data of S. cerevisiae, which has been determined from chemostat cultures at different glucose uptake rates (83). Here we inferred the cellular entropy production rate with the ther-modynamic metabolic network model from this data (Chapter 2, Fig. 1). We identified that the cells reach a maximum entropy production rate, which, to our surprise, happened at the same critical glucose uptake rate as the onset fermentation.

Based on this finding, we speculated that this limitation might be the cause for the coun-terintuitive reshuffling in the metabolic operation from respiration at low (with a high ATP yield) and fermentation at high glucose uptake rates (with a low ATP yield). To test this hy-pothesis, we performed, for different glucose uptake rates, flux balance analysis (FBA)—using the growth rate as an cellular objective—with the thermodynamic metabolic network model and a maximum limit in the cellular entropy production rate. FBA with our model and the maximum limit in the entropy production rate correctly predicted the physiology of S. cer-evisiae including aerobic-fermentation at high glucose uptake rates, and also predicted the maximum growth rate (Chapter 2, Fig. 2). When we repeated the same optimizations without the limitation in the entropy production rate, the model did not predict aerobic-fermentation and also did not predict a maximum growth rate (Chapter 2, Fig. 2). Besides, exploring the physiological impact of the maximum limit in the cellular entropy production rate, we were also interested in its constraining effect on the intracellular behavior. To this end, we deter-mined the entropy fluxes produced by the different metabolic processes. We found a major reshuffling of the entropy production from respiratory to fermentative pathways, when the maximum limit in the cellular entropy production rate was reached (Chapter 2, Fig. 3). From these analyses, we concluded that the maximum entropy production rate is indeed the cause for the switch between respiration and fermentation and explains maximum growth rates. Further, we concluded that S. cerevisiae indeed maximizes its growth rate.

In the future, we plan to show the existence of this limitation and its constraining effect on a multitude of different organisms. To this end, we developed a computational workflow (Chapter 3, Fig. 1) which allows to construct thermodynamic metabolic network models, similar to the one developed for S. cerevisiae in Chapter 2. This workflow builds on existing genome scale metabolic reconstructions that are publicly available for a wide array of different organisms (131,132), ranging from bacteria, fungi, mammalian cells, to plants.

Another open question is the underlying mechanistic reason for the maximum cellular entro-py production rate. We speculate that the respective cause is linked to the cellular morphol-ogy. More specifically the surface to volume ratio of the cell might serve as a limiting factor. We will test this in future by using thermodynamic network models that will be developed for different organisms and determine their maximum entropy production rates. Then we determine for each of these organism the cellular surface to volume ratios. From this compar-ison, we will be a step further to gain fully insight into the mechanistic principles behind the

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limitation in the entropy production rate.

Further, we used the thermodynamic metabolic network model to investigate the relationship between thermodynamic principles and enzyme kinetics. To investigate this relationship, we inferred intracellular metabolic fluxes and backward rates by combining the thermodynamic network model with isotopomer balances in a novel statistical method (Chapter 4, Fig. 1). This method allows to infer metabolic fluxes based on a multitude of different experimental data, i.e. extracellular rates, metabolomics data, standard Gibbs energies of reactions, and isotopomer patterns from 13C-labeling. Besides infering intracellular metabolic rates for S. cerevisiae, we used our method and inferred for 34 reactions their Gibbs energies and their ratio of the forward over the backward enzymatic rates (Chapter 4. Fig. 4). By comparing the experimentally inferred correlation between the Gibbs energies and these ratios with a ther-modynamic theory, we could identify possible enzymatic regulatory control.

To test this enzymatic control, future research should aim at identifying additional ratios be-tween the forward and the backward rate by increasing the quantity of the measure isoto-pomer patterns using advanced analytical techniques (158). Thereby we will establish a cor-relation between these ratios and the Gibbs energy of reactions for a large amount of different enzymes. Then, we will determine these correlations for the different yeast strains at different glucose uptake rates (156) and integrating proteome data. From this multi-strain comparison, we will be able to identify the in vivo enzymatic control (by inferring kinetic parameters from the complete data, i.e. forward/backward rates, metabolite and protein levels).

Overall, in this thesis we investigated and showed the importance of thermodynamic prin-ciples on metabolic operations. We developed statistical methods to infer model parameters and intracellular quantities from experimental data, to analyze intracellular behaviors using these statistical methods, and to predict new metabolic behaviors with high unprecedented accuracy. This excellent predictive capabilities will not only facilitate fundamental research of metabolism, but also impact the development of new cell factories in biotechnology indus-tries.

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In biological cells, metabolism describes the chemical conversions of substrates into biomass and by-products. These conversions are carried out by enzymes in different cellular compartments, between which the metabolites are transported via transporters. Metabolism is organized in a complex reaction network, i.e. the metabolic network. Although the topology of this network is well known, the main principles governing the fluxes through metabolism are only poorly understood.

Because distinct flux patterns occur across organisms, we hypothesized that fundamental thermodynamic constraints might govern metabolism. To test this, we developed a constraint-based model for Saccharomyces cerevisiae with a comprehensive description of the biochemical thermodynamics and including a cellular Gibbs energy balance. Using nonlinear regression analyses on quantitative metabolome and physiology data, we found that an upper limit exists in the cellular entropy transfer rate. Applying this limit in flux balance analyses with growth maximization as the objective, our model correctly predicted physiology, intracellular metabolic fluxes and maximal growth rates for different glucose uptake rates and carbon sources. Thus, reaction stoichiometry, fundamental thermodynamic constraints and the objective of growth maximization shape metabolic fluxes in yeast. This finding has enormous consequences for our understanding of how metabolic fluxes are regulated, distributed, and controlled: Here, without considering any type of regulation (neither on the genetic level nor on the enzymatic level), we can now—with the uncovered principles—predict intracellular fluxes under all tested conditions. This means that all regulatory mechanisms must have evolved to accomplish these identified principles.

In order to apply these fundamental thermodynamic constraints on other metabolic networks, we developed a workflow to build thermodynamic metabolic network models, starting from existing metabolic reconstructions. Our workflow scheme consists of the four steps, adding biochemical information to the existing stoichiometric network, determining Gibbs energies for the metabolic processes, reducing the stoichiometric network, and training the model on experimental data. We used this workflow to develop the model for Saccharomyces cerevisiae and illustrate the development of a thermodynamic metabolic network model for Escherichia coli.

Finally, we combined our thermodynamic metabolic network model with isotopomer balancing in order to quantify metabolic fluxes in S. cerevisiae. This statistical method allowed us to combine a wide range of different experimental data, i.e. extracellular rates, metabolite concentrations, standard Gibbs energies of reactions, and 13C based isotopomer patterns, and estimated with a minimal set of assumptions the metabolic fluxes along with their confidence intervals, thereby delivering a precise view on the true flux space. Further, this method also enabled us to estimate the ratio of the forward and backward fluxes through enzymatic

Summary

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reactions, opening the door towards identifying kinetic rates laws in vivo.

In conclusion, we identified the fundamental thermodynamic principle that governs metabolism, which is a limitation in the cellular entropy transfer. Using this principle as a constraint in metabolic network models, we inferred, analyzed and predicted metabolic operations in S. cerevisiae. With this work, we are now able to exactly quantify and predict metabolism, this will have major impact on fundamental research and on the biotech industry, which needs this capabilities for the development of new microbial cell factories.

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Metabolisme omschrijft de chemische omzettingen van substraten in biomassa en bijproduc-ten in cellen. Deze omzettingen worden uitgevoerd door enzymen in verschillende cellulaire compartimenten waartussen de metabolieten via transporteurs worden getransporteerd. Me-tabolisme is georganiseerd in een complex netwerk van reacties genaamd het metabolisch netwerk. Hoewel de topologie van dit netwerk bekend is, zijn de belangrijkste principes, die de fluxen door het netwerk bepalen, slecht begrepen.

Omdat vergelijkbare flux patronen voorkomen bij verschillende organismen, veronderstelden we dat fundamentele thermodynamische principes metabolisme zouden kunnen reguleren. Om dit te testen, ontwikkelden we een constraint-based model voor Saccharomyces cerevi-siae met een uitvoerige beschrijving van de biochemische thermodynamica en een cellulaire Gibbs energiebalans. Met behulp van niet-lineaire regressie analyse en kwantitatieve meta-boloom en fysiologie data, vonden we dat er een maximale overdrachtssnelheid bestaat van cellulaire entropie. Als deze limiet wordt gebruikt in flux balans analyse in combinatie met groei maximalisatie als doelstelling, voorspelt ons model correct de fysiologie, intracellulaire metabolische fluxen en maximale groeisnelheden bij verschillende glucose opname snelhe-den en koolstofbronnen. Klaarblijkelijk bepalen reactiestoichiometrie, fundamentele thermo-dynamische beperkingen en groei maximalisering de metabolische fluxen in gist.

Deze ontdekking heeft een enorme consequentie voor ons begrijp hoe de metabolische fluxen zijn gereguleerd, gedistribueerd en gecontroleerd. We kunnen nu zonder rekening te houden met enig type regulatie (noch op genetisch niveau, noch op enzymatisch niveau) intracellu-laire fluxen voorspellen onder alle geteste condities met behulp van de ontdekte thermodyna-mische beperking. Dit betekent dat alle regulatiemechanismen ontstaan moeten zijn om de geïdentificeerde principes te bereiken.

Om deze fundamentele thermodynamische beperkingen toe te kunnen passen op andere metabolische netwerken, ontwikkelden we een workflow om thermodynamisch metaboli-sche netwerk modellen te genereren, uitgaande van bestaande metabolische reconstructies. Onze methode bestaat uit vier stappen; het toevoegen van biochemische informatie aan het bestaande stoichiometrische netwerk, het bepalen van de Gibbs energie voor de metabo-lische processen, vermindering van het stoichiometrische netwerk, en het trainen van het model met experimentele data. Wij hebben deze workflow ontwikkeld voor het model van Saccharomyces cerevisiae en demonstreren de ontwikkeling van een thermodynamisch meta-bolisch netwerk model voor Escherichia coli gebruikt.

Tot slot combineerden we ons thermodynamisch metabolisch netwerk model met isotopo-meer balanceren om metabolische fluxen te kwantificeren in S. cerevisiae. Deze statistische methode maakte het voor ons mogelijk om veel verschillende experimentele gegevens te com-bineren, zoals extracellulaire snelheden, metabolietconcentraties, standaard Gibbs energieën

Samenvatting

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van reacties, en op 13C gebaseerde isotopomeer patronen, en met een minimale set van aanna-mes de metabolische fluxen samen met hun kansdichtheid te bepalen, waardoor er een exacte weergave ontstaat van de thermodynamisch mogelijke flux ruimte. Verder konden we met deze methode ook een schatting maken van de verhouding tussen de voorwaartse en achter-waartse flux in enzymatische reacties, wat de deur opent naar het identificeren van regels voor kinetische snelheden in vivo.

Concluderend, identificeerden we de fundamentele thermodynamische beginsel dat meta-bolisme reguleert, een beperking in de overdrachtssnelheid van cellulaire entropie. Door dit principe als een beperking in metabolische netwerk modellen op te nemen, konden wij me-tabolische operaties begrijpen, analyseren en voorspellen. Met dit werk, kunnen we nu exact metabolisme kwantificeren en voorspellen. Dit zal een enorme invloed hebben op fundamen-teel onderzoek en op de biotechnologische industrie, die deze kennis nodig hebben voor de ontwikkeling van nieuwe microbiologische cel fabrieken.

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Der Stoffwechel einer Zelle (Metabolismus) besteht aus chemischen Umwandlungen von Substraten in Biomasse und Nebenprodukten. Diese Umwandlungen werden durch Enzyme in verschiedenen Zellkompartimenten, die durch Metabolittransporter miteinander verbunden sind, katalysiert. Insgesamt sind metabolische Wege in einem komplexen Reaktionsnetzwerk, dem metabolischen Netzwerk, organisiert. Obwohl die Topologie dieses Netzwerks bekannt ist, liegen die grundsätzlichen Prinzipien, die metabolische Flüsse bestimmen, im Dunkeln.

Da bestimmte Flussverteilungen evolutionär konserviert sind, vermuteten wir, dass fundamentale thermodynamische Prinizipien den Metabolismus regulieren. Um dies zu testen, entwickelten wir ein Modell für Saccharomyces cerevisiae unter Einbezug einer umfassenden Beschreibung der biochemischen Thermodynamik sowie einer Gibbs-Energie Bilanz. Durch nicht-lineare Regressionsanalyse quantitativer Metabolom- und Physiologie-Daten konnten wir ein oberes Limit der zellulären Entropie-Transferrate definieren. Indem wir dieses Limit in Fluss-Bilanz-Analysen mit der Zielfunktion der Wachstumsmaximierung verwendeten, konnten wir die zelluläre Physiologie, intrazelluläre metabolische Flüsse und maximale Wachstumsraten für verschiedene Glukosekonzentrationen und Kohlenstoff-Quellen korrekt prädizieren. Damit können wir zeigen, dass die Reaktionsstöchiometrie, die thermodynamischen Randbedingungen sowie die Zielfunktion der Wachstumsmaximierung die metabolischen Flüsse in Hefe bestimmen. Diese Erkenntnis hat enorme Konsequenzen für unsere Verständnis der metabolischen Flussregulation. Ohne etwaige genetische oder enzymatische Regulationen annehmen zu müssen, können wir nun intrazelluläre Flüsse unter allen getesteten Bedingungen vorhersagen. Dies bedeutet, dass alle regulatorischen Mechanismen so evolutionär entstanden sein müssen, dass sie den identifizierten Prinzipien genügen.

Um diese fundamentalen thermodynamischen Bedingungen in anderen metabolischen Netzwerken anzuwenden, entwickelten wir ein Protokoll zur Herstellung thermodynamischer metabolischer Netzwerk-Modelle, aufbauend auf existierenden metabolischen Rekonstruktionen. Unser Protokoll besteht aus vier Schritten: (1) Einbeziehung biochemischer Information in das existierende stöchiometrische Netzwerk, (2) Bestimmung der Gibbs-Energien für die metabolischen Prozesse, (3) Reduktion des stöchiometrischen Netzwerks, (4) Modell-Trainierung auf experimentelle Daten. Wir verwendeten dieses Protokoll, um ein Modell für S. cerevisiae zu erstellen und die Entwicklung eines thermodynamischen metabolischen Netzwerk-Modells für Escherichia coli zu illustrieren.

Schließlich kombinierten wir unser thermodynamisches metabolisches Netzwerk-Modell mit Isotopen-Bilanzierungen, um metabolische Flüsse in S. cerevisiae zu quantifizieren. Diese statistische Methode erlaubte uns, einen weiten Bereich an verschiedenen experimentellen Daten (extrazelluläre Raten, Metabolitkonzentrationen, Standard-Gibbsenergien von

Zusammenfassung

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Reaktionen, 13C-basierte Isotopenmuster) zu kombinieren. Mit einem minimalen Anteil an Annahmen konnten wir die metabolischen Flüsse in ihren Konfidenzintervallen schätzen und somit ein genaues Bild des tatsächlichen Flussraumes erstellen. Außerdem erlaubte uns diese Methode, das Verhältnis von Flussrichtungen durch enzymatische Reaktionen zu schätzen, was die Bestimmung von kinetischen Raten in vivo ermöglichen wird.

In dieser Doktorarbeit haben wir das fundamentale thermodynamische Prinzip gefunden, das den Metabolismus steuert: eine Limitation in dem zellulären Entropie-Transfer. Mit diesem Limit konnten wir metabolische Operationen erklären, analysieren und prädizieren. Wir sind somit in der Lage, den Metabolismus exakt zu quantifizieren und vorherzusagen, was von großer Bedeutung für die Grundlagenforschung und Biotech-Industrie, einschließlich der Entwickung mikrobieller Zell-Fabriken, ist.

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In December 2009, I contacted a young scientist in Zurich, Switzerland, because I was interested in his work and wanted to do my PhD in his group. Besides the interesting research, Zurich would be close to my family, I would be able to go to the mountains. But after the first telephone call the Zurich plans were shattered, since the young group leader, Matthias Heinemann, was moving to Groningen, The Netherlands, and start his research group there. My first reaction was to open a map and find Groningen on it. But actually it did not matter, since I was extremely enthusiastic about his research vision and I knew, I want to do my PhD in his group…

…December 2014, after spending a lot of ‘blood and sweat’, I wrote this small book that covers the achievements of these years. Something that was only possible, because of the help of many people:

Matthias, thanks for all the input, advice and criticism. Working with you in the last years not only improved my professional working spirit, but also developed me personally. Looking back, I can answer full of confidence that it was the right decision to move to Groningen and join your group.

Ernst, thanks for helping me to understand statistics. I still profit from your courses on statistics. Your critical comments on my work were essential and allowed me to see a lot of my research from a new perspective.

Guille, thanks for the Mate tea supply and also for getting me through the early stages of my PhD. Your guidance and friendship were essential in this critical phase.

Georg, Gesa & Simeon, my German crew, thanks for all the input and help in my work. I always enjoyed working with you, and not only because we could talk in German. Georg & Gesa, danke für den Einblick in die “richtige” Laborarbeit. Simeon, viel Spass beim debuggen ;-).

Daphne & Kuba, do you also remember the times, when we were sitting alone in the office with the group being super small? A lot changed since then, but it was always great to be your colleague and friend. Kuba, your cuisine is legendary, I am already looking for the next dinner. Daphne, good luck across the channel.

Alex, happy cell imaging! It is a pleasure to work with you together; and thanks for giving me shelter :-).

Martijn, I was really happy to advise you during your Master thesis. I wish you good luck with your PhD in Japan.

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All current members, Alvaro, Brenda, Renate, Robson, Silke, Thanasis, Tom, Yonathan, Zheng, and former members, Anne, Bas, Chetak, Ignacio, Mareike, of MSB, it was a pleasure to be your co-worker. It was amazing to see how we evolved from a small conglomerate of people from all backgrounds to a driving group of well-connected co-workers and friends.

Prof. Bakker, Prof. Driessen, Prof. Sauer, thanks for being on the assessment committee for my thesis. With your busy schedules, I greatly appreciate that you took the time and critically evaluated my thesis.

Mama & Papa, danke für all eure Unterstützung, ohne euch hätte ich es nie geschafft. Ohne das Freiheitsverständnis, das ihr mir vermittelt habt, wäre ich nie in der Lage gewesen diese Arbeit zu bewältigen.

Oma Getrud, danke, dass du und Opa Karl mir schon früh die richtigen Werte vermittelt habt und mir gezeigt habt, was es heisst, gewissenhaft zu sein. Es ist schade das Opa heute nicht hier ist!

Oma Inge & Opa Helmut, ihr habt mir euer grosses Interesse an allem und jedem weitergegeben, das viel dazu beigetragen hat, dass ich heute hier bin. Danke!

Anna & Jonas, meine lieben Geschwister, Blut ist dicker als Wasser, das stimmt bei uns auf alle Fälle. Auch wenn wir uns manchmal nerven, am Ende halten wir wieder zusammen. Danke für all die Unterstützung!

Sarah, ich weiss, nicht wie ich diesen Endspurt ohne dich geschafft hätte! Vielen Dank für die Aufmunterungen, die Ablenkungen, die Motivationen, die Korrekturen, und das exzellente Coverdesign.

Martin B., wir kennen uns jetzt schon eine ganze Ewigkeit! Du stehst mir immer mit Rat und Tat zu Seite! Danke!

All my friends from my studies and from home, Alexander, Christian, Henning, Martin T., Philipp, you influenced me a lot in my development, thanks a lot!

All my friends from Groningen, Ana, Audrey, Evelyn, Indiana, Gosia, Maxi, Sole, Ralph, Robin, thanks for all the parties, borrels, and other entertainments in Holland!

Also I like to thank everyone, I did not mention here!

Die Tat wird vergessen, doch das Ergebnis bleibt bestehen. Ovid