An extended model of wine fermentation including aromas and acids
Jan Bartsch, Alfio Borzì, Christina Schenk, Dominik Schmidt, Jonas Müller, Volker Schulz, Kai Velten
aa r X i v : . [ q - b i o . M N ] J a n An extended model of wine fermentation including aromas and acids
Jan Bartsch* , Alfio Borzì , Christina Schenk , Dominik Schmidt , Jonas Müller , Volker Schulz , andKai Velten University of Würzburg, Institute for Mathematics, 97974 Würzburg, Germany, [email protected] University of Würzburg, Institute for Mathematics, 97974 Würzburg, Germany, [email protected] Trier University, Department of Mathematics, 54286 Trier, Germany, [email protected] Geisenheim University, Modeling and Systems Analysis , 65366 Geisenheim, Germany, [email protected] Geisenheim University, Modeling and Systems Analysis , 65366 Geisenheim, Germany, [email protected] Trier University, Department of Mathematics, 54286 Trier, Germany, [email protected] Geisenheim University, Modeling and Systems Analysis, 65366 Geisenheim, Germany, [email protected] corresponding author:*January 14, 2019
Abstract
The art of viticulture and the quest for making wines has along tradition and it just started recently that mathematiciansentered this field with their main contribution of modellingalcoholic fermentation. These models consist of systems ofordinary differential equations that describe the kinetics ofthe bio-chemical reactions occurring in the fermentation pro-cess.The aim of this paper is to present a new model of wine fer-mentation that accurately describes the yeast dying compo-nent, the presence of glucose transporters, and the formationof aromas and acids. Therefore the new model could becomea valuable tool to predict the taste of the wine and providethe starting point for an emerging control technology thataims at improving the quality of the wine by steering a well-behaved fermentation process that is also energetically moreefficient. Results of numerical simulations are presented thatsuccessfully confirm the validity of the proposed model bycomparison with real data.
Keywords: wine fermentation; differential equations;aroma modelling; acid modelling; numerical simulation
The ambition for making excellent wines has a very longhistory [Pel06] and focuses on the crucial step of a ‘per-fect’ alcoholic fermentation. For this quest, also mathe-matics is contributing by investigating models consisting ofsystems of ordinary differential equations (ODEs) that de-scribe the kinetics of the bio-chemical reactions occurring inthe fermentation process and provide the starting point foran emerging control technology that aims at improving thequality of the wine by steering a well-behaved fermentationprocess that is also energetically more efficient.Among some of the most representative and recent modelsof wine fermentation, we refer to [D +
10] and [D + +
10, GCC + +
14, MB16, MBH17, MV15,SS15, SSRvW17, SV16]. In this work and any other im-provement in simulation and optimization of wine fermen-tation, a differential model of fermentation of increasing so-phistication seems indispensable.That is why in this paperwe aim at developing a new model that improves upon exist-ing ones by including and merging together different com-ponents like the yeast dying behavior, glucose transporters,and the presence of aromas and acids.Wine fermentation is a very complex bio-chemical pro-cess where each bio-chemical component plays an importantrole. In particular, oxygen plays a crucial role for yeast activ-ity, helping this to cope with the initial high concentrationsof sugar and nutrients. However, oxygen which is not con-sumed by the yeast leads to oxidation of the wine; see e.g.[Com09]. During fermentation yeast grows by metabolizingsugar in the presence of nutrients such as nitrogen. The con-sumed sugar is converted into ethanol. On the other hand,alcohol inhibits yeast growth because it is toxic for the yeast,see [JMG89, LaVU82, SPPCA03] for further references andattempts to model the influence of ethanol on the fermenta-tion rate. In fact, the alcohol which is produced consists ofmany components of which ethanol is the most relevant one.Further, as discussed in [D + transporters , for the assimilation of sugar andnitrogen. Our work starts with the basic model proposed in [D + X , assimil-able nitrogen N , ethanol E and sugar S . In [BMM + O ; 3) the impact of oxygen and sugar on the yeastproduction. For clarity of illustration see the terms in bold in (1) and compare with [D + d X d t = µ max ( T ) NK N + N SK S + SO K O + O X − ϕ ( E ) X d N d t = − k µ max ( T ) NK N + N SK S + SO K O + O X d E d t = β max ( T ) SK S + S K E ( T ) K E ( T ) + E X d S d t = − k d E d t − k µ max ( T ) NK N + N SK S + SO K O + O X d O d t = − k µ max ( T ) NK N + N SK S + SO K O + O X . (1)Since the yeast is also active even if no oxygen is left, weintroduce a parameter ε > , to model this fact. However,in order to ensure that the yeast reaches a stationary state,we add an additional dying term in the equation for the yeastgiven by − k d X ; see [SSRvW17]. These additions result inthe model (A) .In the following, the parameters k i represent the stoichio-metric coefficients , which describe the substances amountration in a chemical reaction, e.g. k describes, in whichratio nitrogen ( N ) is used in the conversion to yeast ( X ).To have mathematically more compact equations, we oftencombine several stoichiometric coefficients in to one.In every equation, the Michaelis-Menten-kinetics withMichaelis-Menten constants are used to model the conver-sion of the substances into each other, a well known modelfor enzyme kinetics. The Michaelis-Menten constants aregiven by the substrate concentration for which the reactionrate is the half of the maximal reaction rate. Their values are2resented in Table 3.The time dependent functions µ max ( · ) , β max ( · ) describethe specific maximal reaction rate for the given substrate,depending on the temperature. ϕ ( · ) : R +0 → R + isthe function, which is used to model the toxic influ-ence of alcohol on yeast. We define Σ( t ) as the sum ofall different alcohols contained in wine; in our model Σ( t ) := E ( t ) + A ( t ) + B ( t ) + P ( t ) . Where A (isoamylalcohol), B (isobutanol), P (propanol) are additionalalcohols which are described in detail in Section 2.The toxicity function ϕ (Σ) was proposed in [BMM + ϕ appropriately de-scribes the effect of alcohol toxicity in the evolution of theyeast. We have ϕ (Σ) = (cid:18) . π arctan( k d (Σ − tol )) (cid:19) k d (Σ − tol ) , (2) where tol describes the tolerance bound of the alcohol con-centration and k d , k d are parameters, which correlate withthe dying of the yeast cells related to the exceeding of the tol threshold. We follow the work in [D +
11] and consider the presenceof glucose transporters that are essential for the assimilationof sugar (here glucose) and nitrogen. In fact, these trans-porters are responsible for the glucose passing the yeast cellmembrane. Therefore we distinguish two nitrogen compo-nents: there exists a part of nitrogen N x , which is convertedinto yeast X , and one part N T r , which is responsible for thesynthesis of the transporters
T r . These components are in-cluded in our model in the way proposed by [D +
11] and asillustrated in the system (B) as follows (B) ( d T r d t = η max ( T ) N tr N tr + N tr SK S + S O K O + O X d N tr d t = − k ′ η max ( T ) N tr K tr + N tr SK S + S O K O + O X , where η max is given in (3). Hence, we arrive at the com-bined system (A) - (B) , where (A) also contains the glucosetransporters that influence the ethanol production throughthe conversion of sugar. Specifically, this influence is imple-mented by the additional factor (1 + Φ( T ) T r ) , where Φ( T ) is a function of temperature T and is given in (3). (A) d X d t = µ max ( T ) N x K x + N x SK S + S (cid:16) O K O + O + ε (cid:17) X − k d X − ϕ (Σ) X d N x d t = − k µ max ( T ) N x K x + N x SK S + S (cid:16) O K O + O + ε (cid:17) X d S d t = − k d E d t − k µ max ( T ) N x K x X + N x SK S + S (cid:16) O K O + O + ε (cid:17) X d O d t − k µ max ( T ) N x K x X + N x SK S + S O K O + O X d E d t = β max ( T ) SK S + S K E ( T ) K E ( T )+ E X (1 + Φ( T ) T r ) Where the sugar-related Michaelis-Menten constant K , K are associated to part of sugar utilized for nutrition of yeast,respectively to part required for the metabolization into al-cohol. The next important step in the development of our modelis to take into account the presence of aroma compounds,which are responsible for the taste of wine; see e.g. [Pis01].We consider three secondary aromas as in [Ran67, Zho14]:propanol ( P ) , isoamyl alcohol ( A ) and isobutanol ( B ) . TheODE equations that describe the dynamics of these sub-stances are given in (C) as follows (C) d P d t = k µ max ( T ) N x K x X + N x SK S + S (cid:16) O K O + O + ε (cid:17) X d A d t = k β max ( T ) SK S + S K E ( T ) K E ( T )+ E X (1 + Φ( T ) T r )+ k µ max ( T ) N x K x X + N x SK S + S (cid:16) O K O + O + ε (cid:17) X d B d t = k β max ( T ) SK S + S K E ( T ) K E ( T )+ E X (1 + Φ( T ) T r )+ k µ max ( T ) N x K x X + N x SK S + S (cid:16) O K O + O + ε (cid:17) X .
These equations represent an extension of the model dis-cussed in [Zho14] to take into account the coupling with theother components of our model. Since the propanol pro-duction stops if the assimilable nitrogen part is depleted, inthe related equation only the nitrogen part N x appears; see[M +
16] for a related discussion. In contrast to that, thereis an additional influence of sugar to isoamyl alcohol andisobutanol. Therefore, a supplementary summand has to betaken into account. The additional Michaelis-Menten con-stants related to sugar K S , K S describe the saturation con-stants associated to the corresponding alcohols.Now, to conclude the discussion on our new model and be-fore we address the presence of acids, we give in Table 1and Table 3 the values of the parameters appearing in our3 ymbol value symbol value tol ( g/l ) β (( d ◦ C ) − ) β ( d − ) µ d ◦ C ) − ) µ ( d − ) Φ ( l/ ( g ◦ C )) Φ ( l/g ) η (( d ◦ C ) − ) η ( d − ) (A) - (B) - (C) Values for temperature dependence;units are specified in brackets symbol values symbol values K X K S K S K E ( g/ ( l · min )) K E K O K S K S K T r K AA K MA K T A K S g/l unlessnoted otherwise (A) - (B) - (C) model. In Table 1, the stoichiometric coeffi-cients are given in terms of relative ratios; in Table 2, param-eters for the functions that represent the time dependenceof the reaction time are presented; in Table 3, we give theMichaelis-Menten constants in g/l . We take the parametervalues as they are given in [BMM +
14, Zho14, D + symbol value symbol value k k ′ k k k k k k k k k d k d (A) - (B) - (C) Stoichiometric coefficientsThese values also enter in the functional dependence onthe temperature of some constants appearing in our model.This functional dependence is given in (3). β max ( T ) = β T − β µ max ( T ) = µ T − µ Φ( T ) = Φ T − Φ (3) η max ( T ) = η T − η K E ( T ) = − K E T + K E Notice that in (3), the relation between temperature and max- substance values (g/l) substance values (g/l) X N x S O E P A B T r N T r AA M A
T A R + . We consider linear de-pendence in order to be consistent with the references, andmention that it is also common to model the temperature de-pendence in chemical reactions with the so called Arrhenius equation, see [Lai84]. Note that all kinetic constants dependon temperature, but we focused on the for our study mostimportant ones.In the simulation results presented in the following Section2.1, we consider that the working temperature is 18°C in thefirst half of the simulation and 30°C in the second one; see[D + In this section, we present results of simulation with our (A) - (B) - (C) model. For this purpose, we use the initial valuesgiven in Table 4.For the numerical solution of our non-linear ODE sys-tem we use the Runge-Kutta method (as implemented by theMATLAB function ode45 ). We chose a time interval of 40days, which is typical for a complete fermentation process[D + y -axis always repre-sents this concentration in g/l , while on the x -axis the timeevolution in days is given.In Figure 1, we show the time behaviour of propanol ( ),isoamyl alcohol ( ) and isobutanol ( ). Propanolreaches the highest and isobutanol the lowest concentrationat the final time. We also remark that the propanol pro-duction reaches the static stage earlier than the productionof isoamyl alcohol and isobutanol due to a different depen-dence of propanol to sugar and nitrogen.In Figure 2, we depict the evolution of nitrogen N X (solid-dotted) and N T r ( ) and oxygen. In this case the solidthick line ( ) represents the oxygen concentration, and theright y -axis gives the concentration amount of it. We seethat the depletion of oxygen is very much related to the con-4igure 1: Time evolution of aromassumption of nitrogen.Figure 2: Time evolution of nitrogen and oxygenIn Figure 3, we plot the concentration of the yeast cells,showing the toxic influence of ethanol and of the other al-cohols contributing to the dying phase of yeast. In Figure 4,we show the relation between ethanol ( ) and sugar ( ).One can see how ethanol is produced (by yeast) as far assugar is available; the production of ethanol stops at the timewhen the sugar is depleted. In Figure 5, we present the timeevolution of the concentration of the glucose transporters inthe wine. Figure 3: Time evolution of yeastFigure 4: Time evolution of sugar and ethanol5igure 5: Time evolution of glucose transporters The results presented in the previous section meet the ex-pectation and experience of practitioners and encourage usto make a further step of sophistication of our wine fermen-tation model considering the production of acids. This ad-dition is also useful for validating our model with results ofmeasurements focusing on the particular acid content of thewine.Now, we discuss the functional dependence of the mostrelevant acids on the remaining components participating thewine fermentation process. We start discussing the aceticacid (AA) . This is a by-product of alcoholic fermentation,which is produced by the yeast in the order of several hun-dreds of milligrams per liter; see [V + > g/l ) in the final product might indi-cate bacterial activity during or after fermentation. This acidbelongs to the family of volatile acids and is responsible foracescence if its concentration reaches more than 1 g/l. Fur-thermore, for a good wine it must not exceed a certain con-centration due to an EU Regulation, which differs depend-ing on the particular wine produced (see Appendix I C in[EUV]). Another two important acids contained in wine are malic acid (MA) and tartaric acid (TA) , which we also dis-cuss below. For detailed information concerning these acidssee [Rob03]. During fermentation, sugar conversion by lactic acid bacte-ria [EE05] (otherwise called wild yeast) can lead to an in- crease of acetic acid concentration, while post-fermentationspoilage can be caused by the presence of acetic acidbacteria that form acetic acid by oxidation of ethanol([VMSMF + + K S describing the saturating sugar concentration forthis acid. These considerations are modelled in the followingequation (4). We have d AA d t = k θ max ( T ) µ max ( T ) N x K x + N x SK S + S O K O + O X + k ζ max ( T ) d E d t − k a AA. < wh (4)
Notice that malic and tartaric acids are present in the grape.The more matured the grape is, the less is the concentra-tion of malic acid, which is responsible for the sour tasteof fruits. During the fermentation process, the concentra-tions of these acids decreases. In the case of malic acid, thishappens through malolactic fermentation , that is, the con-version of malic acid into the softer lactic acid and CO ; see[Rob03]. We remark that the bacteria that are responsible forthis process need a low concentration of oxygen to act opti-mally. Furthermore we consider a linear precipitation rate k m of the acid. Hence, we arrive at the following equation(5) describing the dynamics of malic acid: d MA d t = − k γ max ( T ) MAK MA + MA (cid:18) O K O + O + k m (cid:19) (5) The concentration of tartaric acid is reduced through precip-itation of wine scale. This happens if the acid is no moreresolvable in the developing wine; [Rob03]. Factors thatcontribute to the reduction of tartaric acid are low temper-ature and rising alcohol concentration. Therefore we modelthe tartaric acid dynamics by the following equation, where k t is the rate of precipitation: d T A d t = − k γ max ( T ) T AK TA + T A (cid:18) K E ( T ) K E ( T ) + E + k t (cid:19) , (6) where the temperature dependent factors are again approxi-mated by linear functions as follows γ max ( T ) = γ T − γ , ˜ γ max ( T ) = ˜ γ T − ˜ γ θ max ( T ) = θ T + θ , ζ max ( T ) = ζ T + ζ . time [d] t e m pe r a t u r ( ° C ) Temperatur function
Figure 6: Temperature profile in the experiment.Summarizing, we can augment our model with the follow-ing equations (D) d AA d t = k θ max ( T ) µ max ( T ) N x K x + N x SK S + S O K O + O X + k ζ max ( T ) d E d t − k a AA d MA d t = − k γ max ( T ) MAK MA + MA (cid:16) O K O + O + k m (cid:17) d T A d t = − k ˜ γ max ( T ) T AK TA + T A (cid:16) O K O + O + k t (cid:17) . In this section, we present results of simulation and com-parison to real data to validate our new model (A) - (B) - (C) - (D) . We refer to measurements performed in an experimentas part of the ROENOBIO project in collaboration withthe Dienstleistungszentrum Ländlicher Raum (DLR) Mosel(Germany). In a time interval of almost 30 days, periodicmeasurements of concentrations of various important sub-stances were taken (for more information see [SSRvW17]).Notice that in this experiment, the temperature was con-trolled in a way suggested by an independent control strategy[SSRvW17]. This temperature profile is depicted in Figure6. In the following plots, we compare the results of simu-lation with our model with the measured data. The initialvalues for the simulation are given in Table 4 and the valuesof the parameters for the acetic model are given in Table 5and Table 6. symbol values symbol values k k k a k k m k k t (D) . symbol values symbol values γ γ ˜ γ ˜ γ ζ ζ θ θ (D) in g/l .We choose the parameters of Table 5 and Table 6 based onexperience. Another possibility would be to solve a parame-ter fitting problem using for example MATLAB built-in func-tions as fminsearch in combination with an ODE solver.In Figure 7, we see measured concentration of acetic acidin g/l over a time period of about 30 days. The solid lineshows the result of our simulation with our extended model.In Figure 8, one can see the results of the measurements ofthe other two chosen acids collected in the same experiment.One can see that the time evolution of all three acids is repro-duced in our simulation with a striking accuracy. The aceticacid concentration raises in both experiment and simulationuntil a specific time and then diminishes with the same rate.In the evolution of malic ( ) and tartaric ( ) acid, wesee different stages of decrease in the measured data, as wellas in our model.7igure 7: Time development of acetic acid.Figure 8: Time development of malic and tartaric acid.
In this paper, a new extended model of wine fermentationhas been presented. It extends previous models by includingthe time development of aromas and acids and is augmentedby the presence of glucose transporters. Results of numeri-cal experiments and comparison to real data have been illus-trated that successfully validate the proposed model.This work was done in the context of the project ROENO-BIO aiming at the development of multi-dimensional simu- lation and control tools for an improved production and qual-ity of white wines. The proposed model represents the corecomponent (kinetics) of bio-chemical reaction fluid modelsfor simulation of wine fermentation in tanks and in relatedcontrol problems where an optimal fermentation pattern isdriven by temperature regulation.
This work was supported by the BMBF (German FederalMinistry of Education and Research) project ROENOBIO(Robust energy optimization of fermentation processes forthe production of biogas and wine) with contract number . We would like to thank Achim Rosch andChristian von Walbrunn for providing the measurements bywhich we validated our results and Peter Fürst and Peter Pet-ter (fp-sensor systems) for equipping those experiments withtheir technology.
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