Dynamic flux balance analysis for predicting biomass growth and ethanol production in yeast fed-batch cultures

Proceedings Proceedings of of the the 9th 9th Vienna Vienna International International Conference Conference on on Proceedings ofModelling the 9th Vienna International Conference on Mathematical Available online Mathematical Modelling Proceedings of the 9th Vienna International Conference on at www.sciencedirect.com Mathematical Modelling Vienna, February Vienna, Austria, Austria, February 21-23, 21-23, 2018 2018 Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018

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IFAC PapersOnLine 51-2 (2018) 631–636

Dynamic flux balance analysis for predicting biomass growth and ethanol production in yeast fed-batch cultures Dynamic Dynamic flux flux balance balance analysis analysis for for predicting predicting biomass biomass growth growth and and ethanol ethanol production production in in yeast yeast fed-batch fed-batch cultures cultures Dynamic flux balance analysis for predicting biomass growth and ethanol production in yeast fed-batch cultures J. Plaza*, Ph. Bogaerts* J. Plaza*, Ph. Ph. Bogaerts* Bogaerts* J. Plaza*, ** 3BIO-BioControl, Université Libre de Bruxelles, J. Plaza*, Ph. Bogaerts* 3BIO-BioControl, Université Libre de de Bruxelles, Bruxelles, * 3BIO-BioControl, UniversitéB-1050 Libre CP 165/61, 50, Av. F.-D. Roosevelt, Brussels, Belgium * 3BIO-BioControl, Université Libre de Bruxelles, CP 165/61, 50, Av. F.-D. Roosevelt, B-1050 Brussels, Belgium CP 165/61, 50, Av. F.-D. Roosevelt, B-1050 Brussels, Belgium (E-mail: [email protected]) CP 165/61, 50, Av. F.-D. Roosevelt, B-1050 Brussels, Belgium (E-mail: [email protected]) (E-mail: [email protected]) (E-mail: [email protected]) Abstract: In this study, a metabolic flux analysis (MFA) (MFA) is firstly applied, at each time instant, for Abstract: In study, metabolic flux is firstly applied, each time instant, for Abstract: In this thisrange study,ofaaadmissible metabolic fluxes flux analysis analysis (MFA) is and firstly applied, at at each rates time and instant, for determining based on glucose ammonium uptake ethanol Abstract: In the thisrange study,ofaadmissible metabolic fluxes flux analysis (MFA) is and firstly applied, at each rates time and instant, for determining the based on glucose ammonium uptake ethanol determining the range of admissible fluxes based on glucose and ammonium uptake rates and ethanol production rate measured concentrations in Saccharomyces cerevisiae determining theestimated range of from admissible fluxes based on glucose and ammonium uptakefed-batch rates andcultures. ethanol production rate estimated from measured concentrations in Saccharomyces cerevisiae fed-batch cultures. production rate estimated fromthe measured concentrations in Saccharomyces cerevisiae fed-batch cultures. Given the similarity between upper bound of the biomass production flux and its estimation through production rate estimated fromthe measured concentrations in Saccharomyces cerevisiae fed-batch cultures. Given the similarity between upper bound of the biomass production flux and its estimation through Given the similarity between the upper bound of the biomass production flux and its estimation through concentration measurements, a flux balance analysis (FBA), aiming at maximizing the biomass growth, is Given the similarity between athe upper bound of the(FBA), biomass production flux and the its estimation through concentration measurements, flux balance analysis aiming at maximizing biomass growth, is concentration measurements, a flux balance analysis (FBA), aiming at maximizing the biomass growth, is proposed and allows narrowing the admissible flux ranges within the metabolic network. Finally, after concentration measurements, a flux balance analysis (FBA), aiming at maximizing the biomass growth, is proposed and allows narrowing the admissible flux ranges within the metabolic network. Finally, after proposed and allows narrowing the admissible flux ranges within the metabolic network. Finally, after withdrawing the information corresponding to the ethanol production rate, original inequality constraints proposed and allows narrowing the admissible flux ranges within the metabolic network. Finally, after withdrawing the corresponding to ethanol production rate, original constraints withdrawing the information information corresponding to the the ethanol production rate, originalininequality inequality constraints describing overflow metabolism of glucose are added to the FBA linear program order to compensate withdrawing the information corresponding to the ethanol production rate, originalininequality constraints describing overflow metabolism of glucose are added to the FBA linear program order to compensate describing overflow metabolism of glucose are added to the FBA linear program in order to compensate for the lack of information regarding ethanol production. final FBA model allows predicting biomass describing overflow metabolism of glucose are added to This the FBA linear program in order to compensate for the lack of information regarding ethanol production. This final FBA model allows predicting for the lack of information regarding ethanol production. This final FBA model allows predicting biomass biomass growth and ethanol production rates, which are consistent with the experimental measurements. for the lack of information regarding ethanol production. This final FBA model allows predicting biomass growth and ethanol production rates, which are consistent with the experimental measurements. growth and ethanol production rates, which are consistent with the experimental measurements. growth ethanol production rates, which are consistent with the experimental measurements. © 2018,and IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: metabolic network, network, metabolic Keywords: metabolic metabolic flux flux analysis, analysis, flux flux balance balance analysis, analysis, underdetermined underdetermined systems, systems, Keywords: metabolic network, metabolic flux analysis, flux balance analysis, underdetermined systems, Saccharomyces cerevisiae, overflow metabolism. Keywords: metabolic network, metabolic flux analysis, flux balance analysis, underdetermined systems, Saccharomyces cerevisiae, overflow metabolism. Saccharomyces cerevisiae, overflow metabolism. Saccharomyces cerevisiae, overflow metabolism.   1. INTRODUCTION The goal of this study is to propose aa metabolic network-based  The goal this study is network-based 1. INTRODUCTION 1. INTRODUCTION The goal of of this study growth is to to propose propose a metabolic metabolic network-based predictor of biomass and ethanol production rates in 1. INTRODUCTION The goal of this study is to propose a metabolic network-based predictor of biomass growth and ethanol production rates in predictor of biomass growth and ethanol production rates of in Biotechnological processes require dynamical models for their fed-batch S. cerevisiae cultures, based on the measurements predictor of biomass growth and ethanol production rates in fed-batch S. cerevisiae cultures, based on the measurements of Biotechnological processes require dynamical models for their Biotechnological processes require dynamical models for their fed-batch S. cerevisiae cultures, based on the measurements of monitoring, optimization and control. Cultures of the glucose and ammonium uptake rates and appropriate linear Biotechnological processes require dynamical models for their fed-batch S. cerevisiae cultures, based on the measurements of glucose and uptake and linear monitoring, optimization optimization and and control. control. Cultures Cultures of of the monitoring, the glucose accounting and ammonium ammonium uptake rates rates and appropriate appropriate linear microorganisms, e.g. the yeast Saccharomyces cerevisiae, of in constraints for glucose glucose overflow metabolism. To monitoring, optimization and control. Cultures the glucose accounting and ammonium uptake rates and appropriate linear constraints for overflow metabolism. To microorganisms, e.g. the yeast Saccharomyces cerevisiae, in microorganisms, e.g. the yeast Saccharomyces cerevisiae, in constraints accounting for glucose The overflow metabolism. To bioreactors are typical examples of such processes which are this end, three steps are considered. first one uses available microorganisms, e.g. the yeast Saccharomyces cerevisiae, in this constraints accounting for glucose The overflow metabolism. To end, three steps are considered. first one uses available bioreactors are typical examples of such processes which are bioreactors are typical examples of as such processes which are this end, three steps are considered. The first one uses available often used in biopharmaceutical well as in agro-food measurements of both input (glucose and ammonium) and bioreactors are typical examples of as such processes which are measurements this end, three steps are considered. The first one uses available of both input (glucose and ammonium) and often used in biopharmaceutical well as in agro-food often used Although in biopharmaceutical as well as in agro-food measurements both input (glucose and industries. macroscopic models are to output (ethanol)of fluxes for determining determining theammonium) distributionsand of often used Although in biopharmaceutical well as agro-food measurements of fluxes both input (glucose and ammonium) and (ethanol) for the distributions of industries. macroscopic as models are inavailable available to output industries. Although macroscopic models are available to output (ethanol) fluxes for determining the distributions of predict the concentration time profiles of the main species admissible metabolic flux values. The second step introduces industries. Although macroscopic models are main available to admissible output (ethanol) fluxes forvalues. determining the distributions of metabolic flux The second step introduces predict the concentration time profiles of the species predict theglucose, concentration time ethanol) profiles (Richelle, of the main species admissible metabolic flux values. that Thethe second introduces (biomass, ammonium, Fickers and a FBA FBA based based on the the observation observation upperstep bound of the the predict theglucose, concentration time ethanol) profiles (Richelle, of the main species metabolic flux values. that Thethe second step introduces on upper bound of (biomass, ammonium, Fickers and aadmissible (biomass, glucose, ammonium, ethanol) (Richelle, Fickers and a FBA based on the observation that the upper bound of the Bogaerts, 2014), they require the choice and validation of a biomass growth flux is very close to its available (biomass, glucose, ammonium, ethanol) (Richelle, Fickers and a FBA based on the observation that the upper bound of the growth flux is close to Bogaerts, 2014), 2014), they they require require the the choice choice and and validation validation of of aa biomass Bogaerts, biomass growth flux is very very close metabolic to its its available available macroscopic reaction scheme, including its stoichiometry, and measurement, hence exhibiting an optimal behavior Bogaerts, 2014), they require the choice and validation of a biomass growth flux is very close to its available hence exhibiting an metabolic behavior macroscopic reaction reaction scheme, scheme, including including its its stoichiometry, stoichiometry, and and measurement, macroscopic measurement, exhibiting an optimal optimal metabolic behavior the structural and parametric of the reaction rates. which aims aims at athence maximizing biomass growth. The third step macroscopic reaction scheme,estimation including its stoichiometry, and which measurement, hence exhibiting an optimal metabolic behavior maximizing biomass growth. The third step the structural and parametric estimation of the reaction rates. the structural andisparametric estimation of the reaction rates. which aims at maximizing biomass growth. TheFBA thirdlinear step An alternative to exploit the knowledge available in proposes additional constraints to be used in the the structural andisparametric estimation of the reaction rates. which aims at maximizing biomass growth. TheFBA thirdlinear step proposes additional constraints to be used in the An alternative to exploit the knowledge available in An alternative is toMetabolic exploit the knowledge available in proposes additional constraints be used description in the FBA of linear metabolic networks. Flux Analysis is programs, which consist consist in an anto original the An alternative is toMetabolic exploit the available in programs, proposes additional constraints tooriginal be used description in the FBA of linear which in the metabolic networks. Fluxknowledge Analysis (MFA) (MFA) is the the metabolic networks. Metabolic Flux Analysis (MFA) is the programs, which consist in an original description of the most common approach for estimating the intraand extraglucose overflow metabolism in S. cerevisiae’s metabolic metabolic networks. Metabolic Flux Analysis (MFA) is the glucose programs,overflow which consist in aninoriginal description of the metabolism S. cerevisiae’s metabolic most common common approach for estimating estimating the intraintraand extraextramost approach for the and glucose overflow metabolism in S. cerevisiae’s metabolic cellular metabolic flux based aa selected network, network. These constraints constraints compensate for the themetabolic lack of of most common approach for estimating intra- and extra- network. glucose overflow metabolismcompensate in S. cerevisiae’s These for lack cellular metabolic flux values values based on onthe selected network, cellular metabolic flux values based on a selected network, network. These constraints compensate for the lack of some measured fluxes and the assumption that internal information which appears when withdrawing the ethanol cellular metabolic flux values based on a selected network, network. These constraints compensate for the lack of which appears when withdrawing the ethanol some measured measured fluxes fluxes and and the the assumption assumption that that internal internal information some information which appears when withdrawing the ethanol metabolites do not accumulate in cells. The system of linear production flux measurements. The final FBA model is then some measured fluxes and the assumption that internal information which appears when withdrawing the ethanol flux measurements. The model is metabolites do do not not accumulate accumulate in in cells. cells. The The system system of of linear linear production metabolites production The final final FBA FBA is then then equations which has to solved is generally only based basedflux on measurements. the input input measurements, measurements, i.e. model glucose and metabolites do not accumulate inbe cells. The system of linear only production flux measurements. The final FBA model is then on the i.e. glucose and equations which has to be solved is generally equations which has to be solved is generally only based on the input measurements, i.e. glucose and underdetermined, be estimated than the ammonium uptake fluxes, and allows allows predicting predicting biomass equations whichwith hasmore tofluxes be to solved is generally only based uptake on the fluxes, input measurements, i.e. glucose and ammonium and biomass underdetermined, with more fluxes to be estimated than the underdetermined, with more fluxes to balances be estimated than the ammonium uptake production fluxes, and allows predicting biomass available equations representing mass and available growth and ethanol rates which are consistent with underdetermined, with more fluxes to balances be estimated than the growth ammonium uptake production fluxes, and allows predicting biomass and ethanol rates which are consistent with available equations representing mass and available available equations representing massfor balances and available growth and ethanolmeasurements. production rates which are consistent with measurements. Many solutions overcoming that experimental available equations representing massfor balances and available growth and ethanolmeasurements. production rates which are consistent with their experimental measurements. Many solutions overcoming that their measurements. Many solutions for overcoming that their experimental measurements. underdeterminacy have been proposed, them Flux measurements. Many solutions for among overcoming that their experimental measurements. underdeterminacy have been proposed, among them Flux underdeterminacy have (Orth, been Thiele proposed, among 2010), them Flux Balance and flux text is organized as follows: Section 2 describes underdeterminacy have (Orth, been Thiele proposed, among 2010), them Flux The text is as follows: 2 describes the the case case Balance Analysis Analysis (FBA) (FBA) and Palsson, Palsson, flux The Balance Analysis (FBA) (Orth, Thiele and Palsson, 2010), flux The text isS.organized organized asfed-batch follows: Section Section 2 and describes the case spectrum analysis (Llaneras and Pico, 2007), elementary flux study on cerevisiae cultures the metabolic Balance Analysis (FBA) (Orth, Thiele and Palsson, 2010), flux The text is organized as follows: Section 2 describes the case study on S. cerevisiae fed-batch cultures and the metabolic spectrum analysis (Llaneras and Pico, 2007), elementary spectrum analysis (Llaneras andetPico, 2007), elementary flux study on S. cerevisiae fed-batch cultures and metabolic modes de al., 2016), etc. network under consideration. Section 33 the explains the spectrum analysis (Llaneras andetPico, flux study on S. cerevisiae fed-batch cultures the metabolic network under consideration. Section and explains the modes (Fernandes (Fernandes de Sousa Sousa al., 2007), 2016),elementary etc. Recently, Recently, modes (Fernandes de Sousa et al., 2016), etc. Recently, network under consideration. Section 3 explains the solutions using additional constraints accounting for overflow methodology to estimate the external input (glucose and modes (Fernandes de Sousa et al., 2016), etc. Recently, network under consideration. Section 3 explains the methodology to estimate the external input (glucose and solutions using additional constraints accounting for overflow solutions using additional constraints accounting for overflow methodology to output estimate the external input fluxes. (glucose and metabolism phenomena have been in framework ammonium) (ethanol and Section solutions using additional constraints accounting overflow methodology to output estimate the external input fluxes. (glucose and ammonium) and and (ethanol and biomass) biomass) Section metabolism phenomena have been proposed proposed in the thefor framework metabolism phenomena have been proposed in the framework ammonium) and output (ethanol and biomass) fluxes. Section of mammalian cell cultures (Richelle, Mhallem Gziri and 4 presents the three above-mentioned steps: MFA-in-out for metabolism phenomena have been proposed in the framework ammonium) and output (ethanol and biomass) fluxes. Section 4 presents the three above-mentioned steps: MFA-in-out for of mammalian cell cultures (Richelle, Mhallem Gziri and of mammalian cell cultures (Richelle, Mhallem Gziri and 4 presents thethethree above-mentioned steps: MFA-in-out for Bogaerts, 2016; Bogaerts, Mhallem Gziri and Richelle, 2017). determining distributions of the admissible flux values of mammalian cell cultures (Richelle, Gziri and 4determining presents thethethree above-mentioned steps: MFA-in-out for distributions of the the admissible admissible flux values values Bogaerts, 2016; Bogaerts, Bogaerts, Mhallem Gziri Mhallem and Richelle, Richelle, 2017). Bogaerts, 2016; Mhallem Gziri and 2017). determining the distributions of flux When added to linear programs in MFA or FBA, they allow based on the input and output flux measurements, FBA-in-out Bogaerts, 2016; Bogaerts, Mhallem Gziri and Richelle, 2017). determining the distributions of the admissible FBA-in-out flux values based on the input and output flux measurements, When added to linear programs in MFA or FBA, they allow When added to linear programs in MFA or FBA, they allow based on the input and output flux measurements, FBA-in-out narrowing the of flux values. When added linear programs in MFA FBA, they allow based on the input and output flux measurements, FBA-in-out narrowing thetointervals intervals of admissible admissible flux or values. narrowing the intervals of admissible flux values. narrowing the intervals of admissible flux values. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 1 Copyright 2018 responsibility IFAC 1 Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 1 Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.03.107

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Fig. 1. Metabolic network used for describing the metabolism of Saccharomyces cerevisiae. for introducing biomass growth maximization and FBA-in for introducing additional constraints, describing glucose overflow metabolism, to be used when only the input flux measurements are available. Finally, Section 4 proposes conclusions and perspectives.

purely aerobic conditions. The pH was maintained at 5 with KOH 5 M. The metabolic network of S. cerevisiae used in this work (Fig. 1) is based on the reaction networks presented by (Nissen et al., 1997; Franzén, 2003; Sainz et al., 2003; Lequeux et al., 2010; Quirós et al., 2013). The network includes the central carbon metabolism, nitrogen and amino acids metabolism, pentose phosphate pathways, fermentative metabolism with glycerol and ethanol production under aerobic conditions, TCA cycle, transport reactions (including external transport to the cell and cytosomal/mitochondrial transport) and precursors for biomass production (protein, lipids, RNA, carbohydrates).

2. CASE STUDY: Saccharomyces cerevisiae FED-BATCH CULTURES AND METABOLIC NETWORK This study uses experimental data from (Richelle, Fickers and Bogaerts, 2014). Fed-batch cultures of S. cerevisiae commercial strain (Bruggeman) were carried out during 21 h in a 20 L bioreactor (Biostat C-DCU3, Sartorius B. Braun Biotech International) with an initial biomass concentration of 0.1 g/L dry weight and initial volume of 6.5 L of a medium having the following composition (per liter of solution): yeast extract (Sigma), 13.5 g; KH2PO4, 3.5 g; MgSO4·7H2O, 1.7 g; CaCl2·2H2O 1.7 g. The feeding time profile was selected to simulate the different regimes of an industrial baker’s production: an initial phase where sugar concentration is such that the respirative phase is prominent and ethanol production is low; a second phase with high glucose concentration to ensure fermentation with high ethanol production; and a last phase with accumulated ethanol consumption. The feeding medium contained 300 g/L of glucose and different concentrations of (NH4)2SO4 depending on the experiment: 16.5 g/L (experiment 1), 33 g/L (experiment 2) and 33 g/L during the first 15 h followed by a feeding without (NH4)2SO4 (experiment 3). The cultures were performed at 30°C at a stirrer speed of 750 rpm and an air flow of 20 slpm to ensure

The synthesis of biomass is calculated from the fluxes producing biomass precursors using the following reaction: 𝛼𝛼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 𝛽𝛽 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 + 𝛾𝛾 𝑅𝑅𝑅𝑅𝑅𝑅 + 𝛿𝛿 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 → 1 𝑔𝑔 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵

(1)

where the coefficients α, β, γ and δ are the necessary weights for biomass formation (Sainz et al., 2003; Hjersted, Henson and Mahadevan, 2007). In this study, the calculated weights implemented in Quirós et al. (2013) are used and assumed constant under different regimes in the experiments (Simeonidis et al., 2010). The metabolic network includes: • •

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91 reaction pathways; 63 internal metabolites that are assumed to be balanced (i.e., they do no not accumulate in the cell);

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633

The concentration measurements of experiment 1 are represented in Fig. 2 with the smoothing splines (function spaps in MATLAB) which were used for estimating the concentration time derivatives.

3 measured compounds: glucose and ammonium (inputs) as external substrates entering the cell and ethanol as a metabolite produced by the cell (output).

S. cerevisiae’s network can be described by a stoichiometric matrix N ϵ ℝ63x91 with 63 rows corresponding to the internal metabolites and 91 columns corresponding to the metabolic fluxes. The mass balance of each internal metabolite in the metabolic network is: (2) Ċ𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑁𝑁 ∙ 𝑣𝑣 (𝑡𝑡) − µ ∙ 𝐶𝐶𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡) where µ·Cint met is the dilution term accounting for biomass growth, µ being the specific growth rate. Based on the assumptions that the internal metabolites are at quasi steady-state – given that intracellular dynamics are much faster than extracellular dynamics – and the dilution term µ·Cint met may be neglected in comparison with the reaction term N·v(t), the system of mass balances of the internal metabolites becomes: 𝑁𝑁 ∙ 𝑣𝑣 (𝑡𝑡) = 0 (3) This system of equations is underdetermined with 91 (fluxes) - 63 (equations) = 28 degrees of freedom. 3. FROM MEASURED MEASURED FLUXES

CONCENTRATIONS

Fig. 2. Measured concentrations of glucose, biomass, ammonium, ethanol (with 95 % confidence intervals) and their corresponding smoothing splines (in blue). Experiment 1. The time derivatives are computed with the analytical expressions of the corresponding smoothing splines (function fnder in MATLAB). Based on these latter and the mass balances (9), (10), (11), (12), the external fluxes are given in Fig. 3.

TO

S. cerevisiae’s network can be related to the external concentration measurements through transport reaction fluxes that can describe either consumption of an external substrate or production of a metabolite or of the biomass itself. Accordingly, the time profiles of the external input fluxes (glucose and ammonium) and output fluxes (ethanol and biomass) are linked to the internal fluxes of vector v by the following relations: (4) 𝑣𝑣𝐺𝐺 = 𝑣𝑣1 (𝑡𝑡) 𝑣𝑣𝑁𝑁 = 𝑣𝑣48 (𝑡𝑡) (5) (6) 𝑣𝑣𝐸𝐸 = 𝑣𝑣68 (𝑡𝑡) (7) 𝑣𝑣𝑋𝑋 = 𝑣𝑣79 (𝑡𝑡) Note that, in the sequel, the measurements (4), (5), (6) used in MFA and FBA will be written in matrix form as follows: (8) 𝑁𝑁𝑒𝑒 𝑣𝑣(𝑡𝑡) = 𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡) 𝑇𝑇 with 𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 = [ 𝑣𝑣𝐺𝐺 𝑣𝑣𝑁𝑁 𝑣𝑣𝐸𝐸 ] and Ne ϵ ℝ3x91 the corresponding stoichiometric matrix.

Fig. 3. External input (glucose, ammonium) and output (ethanol, biomass) fluxes based on smoothing splines and mass balances. Experiment 1. 4. METABOLIC FLUX ANALYSIS (MFA) AND FLUX BALANCE ANALYSIS (FBA) 4.1. MFA based on glucose and ammonium input fluxes and ethanol output flux: MFA-in-out

The external fluxes vX, vG, vN and vE can be deduced from smoothing splines of the corresponding concentrations and the following mass balances: 𝑋𝑋̇ = 𝑣𝑣𝑋𝑋 𝑋𝑋(𝑡𝑡) − 𝐷𝐷(𝑡𝑡)𝑋𝑋 (9) 𝐺𝐺̇ = −𝑣𝑣𝐺𝐺 𝑋𝑋(𝑡𝑡) + 𝐷𝐷(𝑡𝑡)( 𝐺𝐺𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝐺𝐺(𝑡𝑡) ) (10) 𝑁𝑁̇ = −𝑣𝑣𝑁𝑁 𝑋𝑋(𝑡𝑡) + 𝐷𝐷(𝑡𝑡)( 𝑁𝑁𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝑁𝑁(𝑡𝑡) ) (11) 𝐸𝐸̇ = 𝑣𝑣𝐸𝐸 𝑋𝑋(𝑡𝑡) − 𝐷𝐷(𝑡𝑡)𝐸𝐸(𝑡𝑡) (12) where: • X, G, N, E are the concentrations of biomass, glucose, ammonium and ethanol; • vX, vG, vN and vE are the external fluxes; • D is the dilution rate; • Gfeed and Nfeed are glucose and ammonium feeding concentrations.

A first MFA is applied which aims at determining the fluxes v which satisfy the equality constraints (3) corresponding to the internal metabolites mass balances, the measurement equality constraints (8) which include all the measured fluxes (glucose, glutamine and ethanol) and the positivity inequality constraints 𝑣𝑣𝑗𝑗 ≥ 0 (j being the indices of fluxes which are assumed to be positive, i.e. 𝑗𝑗 ∈ {1, . . . ,91} except fluxes 𝑗𝑗 ∈ {11,12,68} involving ethanol and fluxes 𝑗𝑗 ∈ {30, … ,46,48,80, … ,85} involved in amino acid metabolism). Given the large underdeterminacy (28 degrees of freedom), the intervals of admissible values are computed, at each time instant tk and for each flux vi, by solving 2 linear programs (LP), with the use of the function linprog of Matlab, which

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allows determining the minimum and maximum values of vi(tk): 𝑣𝑣𝑖𝑖𝑀𝑀𝑀𝑀𝑀𝑀,𝑀𝑀𝑀𝑀𝑀𝑀 (𝑡𝑡𝑘𝑘 ) = Min, Max 𝑣𝑣𝑖𝑖 ∀𝑖𝑖 ∈ [1, 𝑛𝑛], ∀𝑡𝑡𝑘𝑘 𝑣𝑣

Large intervals remain among the amino acids transport fluxes due to the lack of any information about those reactions.

(13)

4.3 FBA based on glucose and ammonium input fluxes: FBA-in with additional constraints accounting for overflow metabolism

under the constraints, 𝑁𝑁𝑁𝑁 = 0; 𝑁𝑁𝑒𝑒 𝑣𝑣 ≤ (1 + 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); 𝑁𝑁𝑒𝑒 𝑣𝑣 ≥ (1 − 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); 𝑣𝑣𝑗𝑗 ≥ 0; 𝑣𝑣16 + 𝑣𝑣76 < 𝑣𝑣2

(14)

Instead of using equality constraints (8), inequality constraints are considered in (14) involving the smallest variation coefficient ev (1% in this work) which allows feasible solutions to the LP at each time instant. An extra inequality constraint in the LP ensures that the glycolysis (v2) remains the main use of glucose in central carbon metabolism in comparison with its use for carbohydrate formation (v76) and for the pentose phosphate pathways (v16) (Franzén, 2003; Gonzalez et al., 2003; Quirós et al., 2013).

The main goal of this paper being to predict biomass growth and ethanol production in yeast cultures, the ethanol output flux measurements have to be withdrawn from the former LPs {(15),(16),(17),(18)}. However, as they were the only “witnesses” of the different culture phases (respiration, fermentation), new constraints have to be added in the LPs to account for the glucose overflow metabolism phenomenon. Two main operating regimes are depicted in Fig. 5:

Due to the system underdeterminacy, most of the different pathways (TCA, fermentative pathways, nitrogen and amino acids pathways, biomass precursors) show wide admissible flux intervals (results not shown).

• Respiratory regime is observed when there is no saturation of the glucose respiratory capacity, hence glucose is oxidized trough the TCA cycle (respiration). If the glucose uptake flux is close to zero, the remaining respiratory capacity is used for the oxidation of ethanol present in the culture medium (ethanol consumption). • Respiro-fermentative regime is observed when the glucose uptake rate is greater than the maximum respiratory capacity of glucose, consequently the surplus is going through the fermentative pathways producing ethanol.

4.2 FBA based on glucose and ammonium input fluxes and ethanol output flux: FBA-in-out For narrowing the admissible flux intervals obtained with MFA-in-out, a Flux Balance Analysis (FBA) is proposed. The observation that the upper bound of the biomass growth flux (v79) is very close to its available experimental measurement allows assuming that the yeast cell metabolism works for biomass growth maximization. Hence, the following FBA problem is considered: 𝑜𝑜𝑜𝑜𝑜𝑜 𝑣𝑣79 (𝑡𝑡𝑘𝑘 ) = max 𝑣𝑣79 ( 𝑡𝑡𝑘𝑘 ) ∀𝑡𝑡𝑘𝑘 (15) 𝑣𝑣 under the constraints, 𝑁𝑁𝑁𝑁 = 0; 𝑁𝑁𝑒𝑒 𝑣𝑣 ≤ (1 + 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); 𝑁𝑁𝑒𝑒 𝑣𝑣 ≥ (1 − 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); (16) 𝑣𝑣𝑗𝑗 ≥ 0; 𝑣𝑣16 + 𝑣𝑣76 < 𝑣𝑣2

followed by 2 LP’s, which allow obtaining the admissible flux intervals corresponding to biomass maximization: 𝑣𝑣𝑖𝑖𝑀𝑀𝑀𝑀𝑀𝑀,𝑀𝑀𝑀𝑀𝑀𝑀 (𝑡𝑡𝑘𝑘 ) = Min, Max 𝑣𝑣𝑖𝑖 ∀𝑖𝑖 ∈ [1, 𝑛𝑛], ∀𝑡𝑡𝑘𝑘 𝑣𝑣

(17)

under the constraints, 𝑁𝑁𝑁𝑁 = 0; 𝑁𝑁𝑒𝑒 𝑣𝑣 ≤ (1 + 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); 𝑁𝑁𝑒𝑒 𝑣𝑣 ≥ (1 − 𝑒𝑒𝑣𝑣 )𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑘𝑘 ); 𝑜𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑗𝑗 ≥ 0; 𝑣𝑣16 + 𝑣𝑣76 < 𝑣𝑣2 ; 𝑣𝑣79 (𝑡𝑡𝑘𝑘 ) = 𝑣𝑣79 (𝑡𝑡𝑘𝑘 ) ∀𝑡𝑡𝑘𝑘

(18) Fig. 5. Representation of glucose overflow metabolism. Ethanol consumption (blue lines): ethanol and nearly zero glucose are oxidized. Respiration (red lines): glucose is fully oxidized. Respiro-fermentation (green lines): overflow metabolism with glucose excess and production of ethanol.

Fig. 4 shows the similarity between the measured specific 𝑜𝑜𝑜𝑜𝑜𝑜 . growth rate vX (in green) and the maximized value 𝑣𝑣79

The overflow metabolism can be taken into account within FBA through some simple additional constraints: • A threshold value has to be set for the maximum specific respiratory capacity of glucose vGsat. Based on the available measurements for experiment 1 (Fig. 3), it can be observed that ethanol production, i.e. the fermentation phase, occurs when vG exceeds 0.003 c-mol g-1 h-1. • In case of fermentation, the transformation of acetaldehyde is split into ethanol and acetate. An upper bound on this latter is deduced from the estimates of v11, v12, v13 provided by FBA-in-out of experiment 1 and corresponds to (19) 𝑣𝑣13 < 0.15(𝑣𝑣11 + 𝑣𝑣12 )

Fig. 4. Biomass flux maximization (v79) corresponding to the FBA-in-out problem in blue and growth rate from the measurements in green (vX). Experiment 1. The admissible flux intervals corresponding to this FBA-inout problem (results not shown) are significantly narrowed in comparison with the ones obtained based on MFA-in-out. 4

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experiment 1 (16.5 g/L nitrogen feeding). Middle: experiment 2 (33 g/L nitrogen feeding). Bottom: experiment 3 (nitrogen starvation after 15h).

which indicates that produced acetate does not exceed 15% of produced ethanol. • When glucose uptake flux vG is lesser than 0.003 c-mol g-1 h-1, respiration of glucose and/or ethanol takes place. An upper bound on ethanol consumption (or lower bound considering the negative sign) can be deduced from the experimental data of experiment 1 (Fig. 3): (20) −0.003 < 𝑣𝑣68 < 0 The constraints (19) and (20) can be lumped in: If vG ≥ 0,003 (21) 𝑣𝑣13 < 0.15(𝑣𝑣11 + 𝑣𝑣12 ) else vG < 0,003 -0.003 < 𝑣𝑣68 < 0

The FBA-in problem is defined through the LPs {(15),(16),(17),(18),(21)}, the rows corresponding to ethanol being withdrawn from Ne and vmes in (16) and (18). The admissible flux intervals (results not shown) are very similar to the ones obtained with the FBA-in-out problem, showing that the additional constraints (21) accounting for glucose overflow metabolism compensate efficiently for the lack of measurement about ethanol production. We focus here on the results about the prediction of biomass growth and ethanol production. Fig. 6 shows that optimal biomass growth flux predicted by FBA-in fits the experimental measurements. Note that experiment 1 was the only one used for tuning the parameters involved in constraints (21), so that results with experiments 2 and 3 can be considered as cross-validation tests.

Fig. 7. Maximum (in blue) and minimum (in red) admissible values of ethanol production flux (v68) predicted with FBA-in and measured ethanol production rate (vE, in green). Top: experiment 1 (16.5 g/L nitrogen feeding). Middle: experiment 2 (33 g/L nitrogen feeding). Bottom: experiment 3 (nitrogen starvation after 15h). Similarly, the admissible values of ethanol production flux v68 predicted with FBA-in fits the experimental measurements and reproduce the culture behaviour in respiration as well as in respiro-fermentative phases, and this in direct (experiment 1) and in cross-validation (experiments 2 and 3). 5.

CONCLUSION

In this study, a three-step procedure is applied for building a FBA-based predictor of biomass growth and ethanol production rates. In step 1, intervals of admissible flux values are determined through a MFA based on the input (glucose and ammonium uptake) fluxes and on the output (ethanol production) flux measurements. Based on the observation that the upper bound of biomass growth rate fits the experimental data, step 2 turns to a FBA problem considering that the yeast cells maximize biomass growth rate. Finally, in step 3, the measurements concerning the output ethanol flux are withdrawn as it has to be predicted. This loss of information is compensated with additional inequality constraints in the LPs which account for glucose overflow metabolism description.

Fig. 6. Optimal biomass growth flux (v79, in blue) predicted with FBA-in and measured growth rate (vX, in green). Top: 5

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The obtained FBA LPs allow predicting biomass growth and ethanol production, which is illustrated with experimental S. cerevisiae fed-batch cultures in direct and cross-validation.

10.1155/2010/621645. Llaneras, F. and Pico, J. (2007) ‘A procedure for the estimation over time of metabolic fluxes in scenarios where measurements are uncertain and/or insufficient’, BMC Bioinformatics, 8(1), p. 421. doi: 10.1186/1471-2105-8-421.

In future work, this FBA-based flux predictor could be coupled with dynamical mass balances for predicting concentration time profiles of the main extracellular species, provided that the glucose and ammonium specific uptake rates measurements are replaced with appropriate kinetic models. Generalized kinetic models could be considered to this end (Grosfils, Vande Wouwer and Bogaerts, 2007; Richelle and Bogaerts, 2015). Although the proposed inequality constraints for describing glucose overflow metabolism allow reproducing the different culture phases, their fundamental justification should be deeper analysed. Especially, the agreement with well-known macroscopic models of overflow metabolism should be checked (Sonnleitner and Käppeli, 1986). Finally, systematic approaches for identifying limiting constraints could be tested and compared to our results (Nikdel and Budman, 2017). 6.

Nikdel, A. and Budman, H. (2017) ‘Identification of active constraints in dynamic flux balance analysis’, Biotechnlogy Progress, 33, pp. 26–36. doi: 10.1002/btpr.2388. Nissen, T. L., Schulze, U., Nielsen, J. and Villadsen, J. (1997) ‘Flux Distributions in Anaerobic, Glucose-Limited Continuous Cultures of Saccharomyces Cerevisiae’, Microbiology, 143(1), pp. 203–218. doi: 10.1099/00221287143-1-203. Orth, J. D., Thiele, I. and Palsson, B. Ø. (2010) ‘What is flux balance analysis?’, Nature biotechnology. NIH Public Access, 28(3), pp. 245–8. doi: 10.1038/nbt.1614. Quirós, M., Martínez-Moreno, R., Albiol, J., Morales, P., Vázquez-Lima, F., Barreiro-Vázquez, A., Ferrer, P. and Gonzalez, R. (2013) ‘Metabolic Flux Analysis during the Exponential Growth Phase of Saccharomyces cerevisiae in Wine Fermentations’, PLoS ONE. Edited by M. Polymenis, 8(8), p. e71909. doi: 10.1371/journal.pone.0071909.

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