1. Yeast

Two commercial oenological yeasts were used in this study: Torulaspora delbrueckii Zymaflore alpha® supplied by Laffort S.A.S., France, and Saccharomyces cerevisiae QA23® supplied by Lallemand S.A.S., France. The yeasts were maintained on sterile YPD agar slants [yeast extract 1 % (w/v), peptone 1 % (w/v), glucose 2 % (w/v) and agar 2 % (w/v)] at 4 °C. When the inoculation cultures had been prepared, each yeast was transferred from the agar slant to a flask containing 50 mL of YPD [yeast extract 1 % (w/v), peptone 1 % (w/v), glucose 2 % (w/v)]. The yeasts were incubated at 25 °C with agitation (130 rpm) for 11 h. Subsequently, they were transferred to an Erlenmeyer flask containing 300 mL of YPD to give an initial concentration of 5.10⁶ cells/mL, and then incubated with agitation (130 rpm) at 25 °C for 16 h.

2. Medium

Two synthetic media, MS170 and MS300, were used. They contained (per litre): 110 g glucose, 110 g fructose, 6 g L-malic acid, 6 g citric acid, mineral salts (750 mg KH₂PO₄, 500 mg K₂SO₄, 250 mg MgSO₄·7H₂O, 155 mg CaCl·2H₂O, 200 mg NaCl, 4 mg MnSO₄·H₂O, 4 mg ZnSO₄·7H₂O, 1 mg CuSO₄·5H₂O, 1 mg KI, 0.4 mg CoCl₂·6H₂O, 1 mg H₃BO₃, and 1 mg (NH₄)₆Mo₇O₂₄), vitamins (20 mg myo-inositol, 2 mg nicotinic acid, 1.5 mg calcium pantothenate, 0.25 mg thiamin-HCl, 0.25 mg pyridoxine–HCl, and 0.003 mg biotin), anaerobic growth factors (15 mg ergosterol, 443 μg oleic acid, and 1 mL of a Tween80/ethanol solution (1:1, v/v)). The assimilable nitrogen was in the form of ammonium ions, NH₄Cl, and a concentrated solution of 19 α-amino acids. The composition of the concentrated amino acid solution was (per litre of solution): 1.4 g tyrosine, 13.7 g tryptophan, 2.5 g isoleucine, 3.4 g aspartic acid, 9.2 g glutamic acid, 28.6 g arginine, 3.7 g leucine, 5.8 g threonine, 1.4 g glycine, 38.6 g glutamine, 11.1 g alanine, 3.4 g valine, 2.4 g methionine, 2.9 g phenylalanine, 6.0 g serine, 2.5 g histidine, 1.3 g lysine, 1.0 g cysteine, and 46.8 g proline. The ammonium salts and α-amino acids (all amino acids except proline) in the medium were considered to be assimilable nitrogen (Salmon and Barre, 1998). Medium MS300 contained an equivalent of 324 mg N/L that was assimilable by yeasts (204 mg N/L from amino acid solution corresponding to 13 mL/L of amino acid solution and 120 mg N/L from NH₄Cl corresponding to 0.46 g/L). Medium MS170 contained only 176 mg N/L of assimilable nitrogen (108 mg N/L from amino acid solution corresponding to 7.4 mL/L and 68 mg N/L from NH₄Cl corresponding to 0.26 g/L). The pH of each medium was adjusted to 3.3 before autoclaving for 15 min at 121 °C. Vitamins were filtered and added after the thermal treatment.

3. Analytical methods

4. Fermentation

Each fermentation was performed in a 2.5 L reactor filled with 1.5 L of medium. The reactor has a sampling probe and an opening at the top equipped with a filter (0.45 μm) to avoid overpressure. The inoculation volume was calculated to initially reach 5*10⁸ and 10¹⁰ cells/L, respectively, for S. cerevisiae and T. delbrueckii. Pure cultures of S. cerevisiae and T. delbrueckii were performed in MS170 and MS300. The duration of fermentation corresponded to the time needed to consume 97 % of the sugar initially present. All fermentations were carried out at 20 °C under magnetic agitation (250 rpm) and were performed in duplicate.

5. Statistical analysis

The data are given as mean values (± standard deviation). Statistical analyses were performed using the RStudio programme (Racine, 2012). Following verification of variance homogeneity (Fisher test, p > 0.05), one-way analysis of variance (ANOVA) (p < 0.05) was used to determine statistically significant differences between modalities. In the text, no significant differences are shown between the mean values compared (if p value is higher than 0.05) or significant differences (if p value is lower than 0.05).

1. Minimal nitrogen and stored nitrogen

A structured model was developed based on nitrogen-based biomass growth. In this model, the nitrogen in the medium (N) is assimilated by a yeast cell and distributed between two compartments: the constitutive compartment and the storage compartment. The constitutive compartment contains the minimal quantity of nitrogen (N_min) required to ensure the vital functions of a cell. Apart from the vacuoles, the constitutive compartment corresponds to all the nitrogenous elements of the cytoplasm, the membranes and the contents of the organelles, including free amino acids. Like the constitutive compartment, the storage compartment contributes to cell activity; however, only the storage compartment contains the nitrogen reserves (N_s) available for growth. The storage compartment corresponds to the nitrogen content of the vacuoles. During fermentation, the size and composition of amino acids in the cytosolic pool is fairly constant, in contrast to those in the vacuolar pool (Huber-Wälchli and Wiemken, 1979; Kitamoto et al., 1988). Therefore, it can be assumed that the amount of nitrogen in the constitutive compartment (N_min) is invariable and the amount of nitrogen in the storage compartment (N_s) varies during fermentation. As the cells multiply, the amount of nitrogen required for the formation of the additional constitutive compartment is drawn from the storage compartment of the mother cell. The remaining nitrogen stock is equitably distributed between the daughter cells (Figure 1. ).

Figure 1. Nitrogen partition during cell multiplication.

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N_S t=i represents the amount of nitrogen in the storage compartment before a cell multiplication occurs at time t=i.

2. Additional assumptions

Some assumptions were made to describe the evolution of the fermentation. It was assumed that the reactors used were perfectly agitated and that the volumes sampled over time did not disturb the evolution of the fermentation. The model describes the evolution of total sugars and does not differentiate between glucose and fructose. Similarly, the model considers total assimilable nitrogen without discerning between ammonia and amino acids. Ethanol and glycerol production are two reactions which are considered independent and instantaneous and which use sugar as a substrate. There is therefore no accumulation of sugars in the cells. For each yeast, molar mass (M) and chemical formula (CH_xO_yN_z) vary during fermentation. It is assumed that molar mass (M') and chemical formula (CH_xO_yN_z') of a cell containing an empty storage compartment do not vary. The minimum amount of nitrogen contained in the constituent compartment is the only nitrogen contribution to this molar mass. This molar mass is called the constitutive molar mass.

3. Stoichiometry

The model is based on five reactions: growth (r1), ethanol production (r2), YAN absorption (r3), yeast death (r4), and glycerol production (r5).

$\left(r1\right) 1 C_6{H}_{12}{O}_6+g1 N_S\to g2 {CH}_x{O}_y{N}_{z^{\text{'}}}+g3 {CO}_2+g4 H_{2}O$


$\left(r2\right) 1 C_6{H}_{12}{O}_6\to 2 C_2{H}_{5}OH+2 {CO}_2$


$\left(r3\right) 1 N\to 1 N_s$


$\left(r4\right) 1 {CH}_x{O}_y{N}_{z^{\text{'}}} \left(alive\right)\to 1 {CH}_x{O}_y{N}_{z^{\text{'}}} \left(dead\right)$


$\left(r5\right) 1 C_6{H}_{12}{O}_6+4H^{+}+4é \to 2 C_3{H}_8{O}_3$

Nitrogen mass balance in growth reaction (r1) allows us to establish a relationship between stoichiometric coefficients g1 and g2 (Eq. (6)):

$z^{\text{'}}=\frac{g1}{g2}$
    

Therefore, the growth reaction rate is written as follows (Eq. (7)):

$\left(r1\right) 1 C_6{H}_{12}{O}_6+g1{ N}_S\to g2 {CH}_x{O}_y{N}_{(g1/g2)}+g3 {CO}_2+g4 H_{2}O$

The molar mass containing the lowest fraction of nitrogen experimentally observed was CH_1.741O_0.698N_0.097 and CH_1.710O_0.632N_0.113 for S. cerevisiae and T. delbrueckii respectively. These values were used to determine the growth reaction stoichiometric coefficients for each yeast (8-9).

S. cerevisiae (Eq. (8)):

$\left(r1\right) 1 C_6{H}_{12}{O}_6+g1{ N}_S\to 5.525 {CH}_{1.741}{O}_{0.698}{N}_{(g1/5.525)}+0.475 {CO}_2+1.191 H_{2}O$
    

T. delbrueckii (Eq. (9)):

$\left(r1\right) 1 C_6{H}_{12}{O}_6+g1{ N}_S\to 5.398 {CH}_{1.710}{O}_{0.632}{N}_{(g1/5.398)}+0.602 {CO}_2+1.386 H_{2}O$
    

The stoichiometric coefficient g1 is identified alongside parameters described in the following section.

4. Kinetics

Reaction rates are expressed in mol/L/h. $X$ and a represent the concentration of living yeast cells and the correlation coefficient linking the concentration of living cells to mass concentration respectively. The value of a is 37.6*10⁹ and 71.6*10⁹ cells/g for S. cerevisiae and T. delbrueckii respectively. $X_n$, the constitutive yeast molar concentration, is defined by (Eq. (10)):

$X_n=\frac{X}{a*M\text{'}}$

Microbial growth rate (r1) is assumed to be inhibited by ethanol (E) and limited by the nitrogen stock (N_s). It is expressed using the following logistic-like equation (Eq. (11)):

$r1=X_n.k_{\mu }.\left(1-\frac{X}{X_{max}}\right).\left(\frac{K_E}{K_E+E}\right).\left(\frac{\frac{N_s}{X}}{{\frac{N_s}{X}+K}_X}\right)$

where, $k_{\mu }$ is the growth kinetics constant, $X_{max}$ is the maximum living cells concentration, $K_E$ is the constant of growth inhibition by ethanol and $K_X$ is the constant of growth limitation by stored nitrogen. Ethanol production rate (r2) is assumed to be inhibited by ethanol (E) and limited by sugar (S) and nitrogen stock (N_s) (Eq. (12)).

$r2=k_s.X_n.\left(\frac{S}{{S+K}_S}\right).\left(\frac{K_{ES}}{K_{ES}+E}\right)\text{.}\left(\frac{\frac{N_s}{X}}{\frac{N_s}{X}+K_{NS}}\right)$

The terms $k_s$, $K_S$, $K_{ES}$ and $K_{NS}$ represent the constants of ethanol production kinetics, ethanol production limitation by sugar, ethanol production inhibition by ethanol and ethanol production limitation by stored nitrogen respectively.

Nitrogen uptake rate is related to the concentration of assimilable nitrogen and the number of cells present in the medium and is described as follows (Eq. (13)):

$r3=k_N.X_n.N$

Where $k_N$ is the constant of nitrogen assimilation kinetics.

In the death reaction (r4), $E_{lim}$ is the mortality coefficient (Eq. (14)).

$r4=X_n.e^{\left(\frac{E-E_{lim}}{10}\right)}$

Glycerol production (r5) is limited by low sugar concentration. The production of glycerol (G) is coupled to the oxidation of NADH to NAD⁺. Regarding the surplus of NADH produced during the growth of biomass, the production of glycerol contributes to maintaining the redox equilibrium (Nordstöm, 1968). In addition, the production of ethanol is coupled with the oxidation of NADH to NAD^+: the more ethanol produced the less glycerol is needed to maintain the redox equilibrium. The rate of glycerol production is described as follows (Eq. (15)):

$r5=k_G.X_n.\left(\frac{S}{{S+K}_{SG}}\right).\left(\frac{K_{EG}}{K_{EG}+E}\right)$

where $k_G$ is the constant of glycerol production kinetics, $K_{SG}$ is the constant of glycerol production limitation by sugar, and $K_{EG}$ is the constant of glycerol production inhibition while ethanol is being produced. The dynamic evolution of each of the six variables of the model is given by the system (A):

$\begin{array}{cc}\left(A\right)& \left\{\begin{array}{c}\frac{dX}{dt}=\left(g2*r1-r4\right)\\ \begin{array}{c}\frac{dS}{dt}=-\left(r1+r2+r5\right)\\ \frac{dN}{dt}=-r3\\ \begin{array}{c}\frac{d{N}_s}{dt}=\left(r3-g1*r1\right)\\ \frac{dE}{dt}=2*r2\\ \frac{dG}{dt}=r5\end{array}\end{array}\end{array}\right.\end{array}$

5. Calculation methods

All calculations were performed using Matlab® 2014a. Integration of the ordinary differential equations system (A) uses the function ode15 of Shampine and Reichelt (1997). Parameter estimation is based on a constrained non-linear multivariable function minimisation. The function to be minimised is the least-square sum expressing the difference between experimental values and calculated values of the five measured variables: concentration of living yeast cells (X), sugar (S), YAN (N), ethanol (E) and glycerol (G). A Sequential Quadratic Programming (SQP) method was used to minimise the least-square sum. The confidence intervals given for the estimated parameters correspond to a 10 % quasi-invariance region of the least-square sum. The initial value of the specific nitrogen in storage compartment ${\left(\frac{N_s}{X}\right)}_0$was 0.1 mg/10⁹ cells for both yeasts.

1. Yeast growth

Passing from MS170 medium to MS300 medium induced a 77 % increase in initial YAN concentration, engendering a slight increase in concentration of S. cerevisiae maximum living cells (from 240 (±10) *10⁹ to 281 (±13) *10⁹ cells/L). The increase in initial YAN concentration also induced an increase in T. delbrueckii maximum living cells (from 420 (±20) *10⁹ to 542 (±53) *10⁹ cells/L), which corresponds to an 29 % increase, whereas that of S. cerevisiae was 17 %.

In terms of population evolution, the model faithfully describes the latency, growth, stationary and decline phases of each culture of S. cerevisiae and T. delbrueckii (Figures 2 and 3). In each fermentation, the profile of the nitrogen in the storage compartment showed an accumulation phase followed by a depletion phase (Figures 2 and 3). In medium MS170, yeast growth stopped when there was no longer enough stored nitrogen for the production of new cells (Figures 2a and 3a). The model suggests that not enough nitrogen was assimilated from this medium to satisfy requirements for reaching the maximal concentration of living cells ($X_{max}$). This is in accordance with the hypothesis of a limiting nitrogen concentration in MS170 for this specific strain of S. cerevisiae and another strain of T. delbrueckii (Taillandier et al., 2014). In medium MS300, nitrogen is not limiting and yeast growth stops when $X_{max}$ is reached.

Table 1. Parameters of the model.

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The confidence intervals given for the estimated parameters corresponding to a 10 % quasi-invariance region of the least-square sum.

Figure 2. S. cerevisiae pure culture fermentations in synthetic media: MS170 (a) and in MS300 (b).

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Data points are experimental measurements, while solid lines are dynamic model predictions. Vertical bars represent the standard deviations.

Figure 3. T. delbrueckii pure culture fermentations in synthetic media: MS170 (a) and MS300 (b).

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Data points are experimental measurements, while solid lines are dynamic model predictions. Vertical bars represent the standard deviations.

2. Nitrogen partition

Early on in the performed fermentations, the yeasts assimilated nitrogen and the content of the storage compartment increased and rapidly became higher than that of the constitutive compartment (Figure 4). In the latency and growth phases, the storage compartment of S. cerevisiae contained a nitrogen pool which reached a concentration 5 to 20 times higher than that of the constitutive compartment (Figure 4a). Amino acids were accumulated in vacuoles at levels 5 to 40-fold higher than the corresponding cytosolic concentration of S. cerevisiae (Ohsumi and Anraku, 1981; Sato et al., 1984; Klionsky, 1990). The hypothesis regarding the association of the storage compartment with vacuoles can therefore be confirmed.

The model showed that the ratio N_s/N_min was always under 2.5 in T. delbrueckii fermentations (Figure 4b). This is much lower than ratios calculated in S. cerevisiae fermentations. As far as T. delbrueckii is concerned, we have not found data about nitrogen partitioning during fermentation in the literature. The modelling therefore gives an estimation of non-accessed data and additional experiments are necessary to determine the reliability of such an estimation.

The transport of amino acids from cytosol to vacuole is performed through an energy-requiring process (Okorokov et al., 1985). The higher the N_s/N_min ratio, the higher the energy required to maintain the transport of amino acids against the concentration gradient. The N_s/N_min ratios for S. cerevisiae and T. delbrueckii were revealed to be different. Our model therefore allows us to conclude that the energy requirements for amino acid transport from cytosol to vacuole in T. delbrueckii may be different to those of S. cerevisiae.

Figure 4. Nitrogen partition in S. cerevisiae (a) and T. delbrueckii (b) pure cultures.

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3. Nitrogen requirements

In both media, S. cerevisiae (Figure 2) and T. delbrueckii (Figure 3) completely consumed the YAN initially present in ≈55 h. In each fermentation, the stored nitrogen profile showed an accumulation phase followed by a depletion phase (Figures 2 and 3). In medium MS170, yeast growth stopped when there was no longer enough stored nitrogen for the production of new cells (Figures 2a and 3a). The model suggests that there is not enough nitrogen assimilated in this medium to satisfy requirements for reaching the maximal concentration of living cells ($X_{max}$). This is in accordance with the hypothesis of a limiting nitrogen concentration in MS170 for this specific strain of S. cerevisiae and another strain of T. delbrueckii (Taillandier et al., 2014). In medium MS300, nitrogen is not limiting and yeast growth stops when $X_{max}$ is reached.

The stoichiometric coefficient g1 links the consumption of nitrogen in the storage compartment to cell production. Its value is 0.41 ± 0.06 and 0.42 ± 0.03 respectively for S. cerevisiae and T. delbrueckii. However, the yeast stoichiometric coefficients g1 are not significantly different and the maximal concentration of T. delbrueckii living cells ($X_{max}$) is twice as high as that of S. cerevisiae. This means that twice as much nitrogen is required to produce one S. cerevisiae cell than to produce one T. delbrueckii cell. However, the maximum biomass concentration was close to 8 g/L for all fermentations of S. cerevisiae and T. delbrueckii. Despite both yeasts having a similar maximal dry weight, T. delbrueckii performs fermentations with a higher maximal concentration of living cells. The latter observation, alongside the nitrogen requirements suggested by the model, are coherent with the fact that S. cerevisiae cells size is higher than that of T. delbrueckii (Hernández-López et al., 2007).

The determined yeast constitutive molar mass was CH_1.741O_0.698N_0.074 and CH_1.710O_0.632N_0.078 for S. cerevisiae and T. delbrueckii respectively. For both yeasts, nitrogen fractions experimentally measured during the fermentations were higher than the estimated nitrogen fraction of yeast with an empty storage compartment (data not shown). The minimum amount of nitrogen required to ensure the vital functions of cells was 38 and 40 mg N per gram of yeast for S. cerevisiae and T. delbrueckii respectively.

4. Ethanol production

Sugar had been completely consumed by the end of all performed fermentations. For both yeasts, final ethanol concentration was ≈105 g/L and it was not significantly affected by initial YAN concentration. By increasing the initial concentration of YAN, the fermentation time decreased from 197 ±11 to 117 ±7 h and from 245 ±11 to 218 ±7 h for S. cerevisiae and T. delbrueckii respectively. The ethanol production reaction (r2) is limited by sugar and stored nitrogen and is inhibited by ethanol. The evolution of sugar limitation $\left(\frac{S}{{S+K}_S}\right)$, stored nitrogen limitation $\left(\frac{\frac{N_s}{X}}{\frac{N_s}{X}+K_{NS}}\right)$ and ethanol inhibition $\left(\frac{K_{ES}}{K_{ES}+E}\right)$ are represented as a function of the consumed sugar in figure 5. For both yeasts, sugar limitation and ethanol inhibition continuously occurred while sugar was being consumed. The identified value of $K_{NS}$ was 0 mg N/10⁹ cells for T. delbrueckii. In both media the production of ethanol by T. delbrueckii was not limited by stored nitrogen. Therefore, the lower ethanol production rate observed in T delbrueckii fermentation in MS170 was caused by the lower concentration of living cells rather than a limitation of stored nitrogen. The model enabled us to rank the effect of limitations and inhibition on ethanol productivity, and therefore to rank these phenomena.

The identified value of $K_{NS}$ was (5 ± 3) *10^-6 mg N/10⁹ cells for S. cerevisiae. In MS300, ethanol production of S. cerevisiae was not limited by stored nitrogen. In MS170, S. cerevisiae consumed the stored nitrogen until there was not enough for the production of new cells. Subsequently, the stored nitrogen suddenly became limiting for ethanol production and the ethanol production rate sharply decreased by 55 % (Figure 5b). Unlike that of T. delbrueckii, the ethanol production rate of S. cerevisiae was impacted by the limited stored nitrogen in MS170.

Figure 5. Fermentation limitations and inhibition in T. delbrueckii (a) and S. cerevisiae (b) pure cultures.

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T. delbrueckii stored nitrogen limitation has the same profile in MS170 and MS300 media.

1. Conclusion

The present comprehensive model of wine-making fermentation is based on the partition of assimilated nitrogen between the constitutive and the storage compartments and includes the following variables: yeast living cells, sugar, YAN, ethanol, glycerol and stored nitrogen. It contains a set of five ordinary differential equations in order to efficiently predict the evolution of S. cerevisiae and T. delbrueckii pure culture fermentations. This mass-balance model provides a stoichiometric approach to biomass production, unlike nitrogen backboned models used in winemaking. The developed model gives an estimation of non-accessed data, such as nitrogen partition between vacuole and cytosol during fermentation. Although this estimation is coherent with the partition of amino acids observed in S. cerevisiae fermentation, additional experiments are necessary to determine its reliability.

2. Future work

The developed model is robust enough to give a precise description of the fermentation evolution of the pure cultures of two yeasts. The next step will be to validate the model with more initial concentrations of yeast assimilable nitrogen. The model will be further developed so that it can be used for modelling the mixed culture fermentation of S. cerevisiae and T. delbrueckii.