The Candida tropicalis (ATCC 96745) strain was stored at 4°C in agar of yeast extract, peptone and xylose (YPX) with a concentration of 20 g L^-1 of each compound. The strain was subcultured in the same agar at 30 °C for 48 h before carrying out the pre-inoculum. Final pH was 5.6 and the fermentation medium was non-detoxified OPEFB hydrolysate (Manjarres-Pinzon et al., 2017).

The pre-inoculum was performed in 100 mL Erlenmeyer flasks with 40 mL of medium and 20 g L^-1 of xylose substrate. They were incubated at 30 °C and 120 rpm for 18 h. At the end, the Erlenmeyer flasks were inoculated with non-detoxified OPEFB hydrolysate as a fermentation medium aiming at studying their kinetics and modeling their behavior as well. The fermentations were also carried out in 100 mL Erlenmeyer flasks with 40 mL of YPX synthetic medium. The inoculum concentration was 1.6 g L^-1, the ratio of medium volume and Erlenmeyer flask volume was 0.4, pH 5.6 and 120 rpm of agitation rate. For the kinetic study, the fermentations with non-detoxified OPEFB hydrolysates were performed for 96 h.

Cellular concentration was determined by optical density spectrophotometry at 620 nm (Genesys 20, Thermo Scientific, Waltham, MA) and correlated with the dry weight method (Manjarres et al., 2018). Xylose and xylitol quantification was performed using an HPLC system (Shimadzu Prominence, Canby, OR), with IR detector, equipped with an Aminex HPX-87H column (Biorad). Elution was carried out with aqueous H₂SO₄ (0.005 M) at a flow rate of 0.6 mL min^-1. The oven temperature was maintained at 65 °C. The injection volume was 20 μL (Manjarres-Pinzon et al., 2017). Samples were prepared in duplicate and filtered.

Cell growth, substrate consumption and xylitol production kinetics were studied in flask scale with 100 mL capacity. The models studied for cell growth were Monod (Equation 1), Contois (Equation 2) and Tessier (Equation 3); Luedeking-Piret model (Equation 4) was applied for xylitol production and the general mass balance equation of substrate consumption was used for xylose consumption (Equation 5) (Aguiar et al., 2002).

Where, the estimated parameters were μ_max, B, α, β, Y_X/S and Y_P/S, K_S and K_t, according to the model used. The experimental maximum specific growth rate (μ_max) was calculated from the slope of the regression line of the natural logarithm of the cell numbers (LnX) in the increasing phase as a time function (Mohamad et al., 2016). Y_X/S is the biomass yield based on substrate consumption and it was considered that the entire substrate was used for cell growth. K_S is the saturation or substrate utilization constant, numerically equal to the substrate concentration where μ=μ_max/2. In the Tessier model, growth inhibition by substrate depletion is included, thus, μ is nullified or tends to zero with low substrate concentrations. K_t is the substrate growth inhibition constant.

The exponential phase, where xylitol is produced, and the stationary phase were used for modeling. The substrate consumption was oriented to cell growth. In a batch system, the change in cell concentration depends solely on cell growth. The cell can consume the substrate in three ways: cell growth, cell maintenance and metabolite production. In the substrate equation (Equation 5), ms is the cellular maintenance proportional to the biomass present, Y_X/S ^-1 is the substrate fraction used for cell growth and Y_P/S ^-1 is the substrate fraction used to produce xylitol. In the Luedeking-Piret model (Equation 4), the first term is oriented to the ability of the cell during cell growth to produce the metabolite (growth or exponential phase); whereas, the second term indicates the metabolite production based on the biomass present in the environment (cellular maintenance or stationary phase). If the product is obtained mainly during cell growth, β must have a low value, and α a high value. Primary metabolites are produced in the growth phase, where xylitol is one of them; on the contrary, secondary metabolites are generally obtained in the stationary phase.

Experimental data for kinetic parameters estimation were used. The modeling protocol consists of defining a mathematical model, making a numerical solution scheme, applying numerical solution methods and checking the numerical stability of the model. The latter implies that the model represents the real values of a process, or that the solution be consistent and real. The aim of the modeling protocol is to find a routine that is stable and consistent when applied in solving the differential equations that represent the phenomenology under study.

The parameter estimation process was carried out using the software Berkeley Madonna 8.0, where the parameters of both cell growth models, substrate consumption and xylitol production models must be introduced. The software Berkeley Madonna 8.0 implies a numerical solution scheme, generates alerts and makes the modeling protocol as well. After the model numerical stability is done, the study can start. Furthermore, this software has an internal routine that allows calibrating the model and its constants. Model calibration consists of selecting all the variables that are coupled and adjusting the numerical solution that this software gives to the observed data, in order to find the most suitable magnitudes of the constants included in the model. The software, based on the root mean square error (RMSE) and the coefficient of variation (CV), generates a comparison between the observed and estimated data, also begins to generate random values of all the variables, makes iterations and evaluations of RMSE and CV until gets the combination of factors that results in the lowest CV.

The estimation procedure involved the variation of different parameter values for minimizing the differences between the experimental or observed data and those predicted by the model. The applied numerical solution method was Runge-Kutta of order 4 (RK4), which takes the average of 4 slopes and that is the value used to perform the iteration. Parameters initial data were chosen according to the data reported in the literature for this type of fermentation. For instance, the maximum specific growth rate of yeast was taken with values in the range of 0.2-1 h and 0.5-1 h, yields between 0 and 1, parameter B, K_S and K_t were 1.88, 599 and 305, respectively (Aguiar et al., 2002).

Biomass concentration with respect to fermentation time in YPX synthetic medium and in non-detoxified OPEFB hydrolysate is featured in Figure 1. It is important to note that fermentation in synthetic medium took 33 h, while in non-detoxified OPEFB hydrolysate took 96 h given that this medium contained different inhibitors and nutrients, which prolonged the yeast adaptation phase. Furthermore, no tests were made between 7 to 25 h since in initial experiments, Candida tropicalis reached the exponential phase in the YPX synthetic medium at that time. Thus, the aim was to evaluate the different growth phases of this yeast in the medium, and to try to model the fermentation results. The Contois model adequately predicts the experimental values of cell growth in synthetic medium. However, the behavior of all models predicts cell growth in the first 24 h in non-detoxified OPEFB hydrolysate and after that time, the values of the models tend to be lower than the experimental ones. The considerations that would improve the prediction of the model could be: more precise methods for the cell growth determination such as colony forming unit (CFU) and adjust only two variables for the model instead of three variables (biomass, substrate and product), as was done in this study. Therefore, the type of culture medium was a factor that affected the parameters sensitivity of a kinetic growth model.

Figure 1

In another study, the application of the Contois model in the cell growth of C. guilliermondii using a synthetic medium was suitable with small deviations (Aguiar et al., 2002). On the other hand, the non-detoxified OPEFB hydrolysate, outside of xylose, has other sugars such as glucose that can contribute to the increase in biomass production. In addition, the non-detoxified OPEFB hydrolysate was supplemented with different salts, such as 4 g L^-1 yeast extract, 3 g L^-1 (NH₄) 2SO₄, 0.5 g L^-1 MgSO₄ 7H₂O and 0.1 g L^-1 CaCl₂ 2H₂O, which help to improve the Candida metabolism (Manjarres-Pinzon et al., 2016).

Xylose consumption during fermentation in synthetic medium and in non-detoxified OPEFB hydrolysate is depicted in Figure 2. Xylose concentration gradually decreased over time in both media. Xylose was consumed by 50% in YPX and 70% in non-detoxified OPEFB hydrolysate according to experimental data. Xylitol concentration increased mainly during the rapid xylose consumption period, which confirmed xylitol accumulation due to imbalance factors of the enzymatic activities that involved xylose consumption in the cells (Mohamad et al., 2016).

Figure 2

In synthetic media, both Monod and Tessier models adequately predicted experimental values in the first 3 h of fermentation; but after 24 h, the models distanced themselves from the experimental values, being the Tessier model the closest in terms of the reported values. The values estimated by the models behave similarly in non-detoxified OPEFB hydrolysate, where it can be emphasized that the estimated results were similar between the models and completely away from the experimental ones. It is necessary to highlight that the modeling of substrate consumption was based on Equation 5, which is closely related to the growth kinetics variables of each model studied.

Xylitol production during fermentation in synthetic medium and in non-detoxified OPEFB hydrolysate is shown in Figure 3. All the models were similar to each other and adequately predicted the xylitol production, especially in synthetic medium. However, it should be kept in mind that the base equation in all models for the development of this kinetic was Luedeking-Piret equation, although they also depend on other parameters, which are found both in cell growth and in substrate consumption. The Luedeking-Piret equation is a function of cell growth and cell maintenance to describe product formation (Veeravalli and Mathews, 2018). The Luedeking-Piret equation was originally developed to explain the fermentation kinetics of glucose to lactic acid secretion based on instantaneous growth of bacterial rate and to the bacterial density. Nonetheless, it has been modified to relate the production rate of any metabolite with substrate consumption rate and substrate concentration (Mahdinia et al., 2019). Xylitol concentration was higher in non-detoxified OPEFB hydrolysate with a value around 6 g L^-1 compared to the synthetic medium that was 1.3 g L^-1. Studies have shown that xylitol production is affected by many factors such as the type of microorganism, the medium composition, such as initial xylose concentration, inhibitors, etc., and the environmental conditions such as pH, temperature, agitation and aeration (Mohamad et al., 2016; Pappu and Gummadi, 2016).

Figure 3

A kinetic model is helpful to understand and optimize a fermentation process (Zhu et al., 2016). Table 1 shows the estimated parameters of each model during fermentation of C. tropicalis in two culture media. Mathematical modeling of cell growth behavior is highly investigated to control, predict and design the processing, stability and safety of food and bioproducts (Guo et al., 2018; Pappu and Gummadi, 2016). The equations of Monod and Tessier are normally used to describe growth with a low cell population; while the Contois model usually works better in cultures with high cell density (Aguiar et al., 2002).

Table 1

Monod kinetics, expressed as substrate degradation resulting from microbial uptake and cell concentration, has a great effect on degradation rates. Contois kinetics considers the fact that a high concentration of cells adhered to the surface of the particles could inhibit substrate degradation. Tessier kinetics takes into account the maintenance energy for cellular activity, which means that the maximum growth rate will be reduced when the substrate concentration is lower and microorganisms compete for food sources (Wang and Witarsa, 2016).

It is necessary to highlight that the experimental value of μ_max in the synthetic medium was similar to that reported by the Tessier model, whereas it was closer to the Contois equation in non-detoxified OPEFB hydrolysate (Table 1). The experimental value of μ_max was higher in the synthetic medium than in the non-detoxified OPEFB hydrolysate, which indicates that the medium influences the cell growth potential. Generally, yeast extract provides good growth factors and organic nitrogen for microorganisms. In addition, the activities of enzymes involved in cell growth are inhibited by changes in the culture medium (Zhu et al., 2016).

The μ_max of yeast is usually between 0.2 and 0.5 h^-1. In C. guilliermondii growth using a synthetic medium for the production of xylitol, a value of μ_max was found for the Contois model of 0.25 h^-1 (Aguiar et al., 2002). Nevertheless, in other works, the value has been below than what was previously reported. In xylitol production with C. parapsilosis in synthetic medium, μ_max was between 0.1 to 0.13 h^-1 (Aranda-Barradas et al., 2000); value of 0.12 h^-1 was reported with C. tropicalis in synthetic medium (Mohamad et al., 2016) and with Debaryomyces nepalensis in synthetic medium were values from 0.029 to 0.078 h^-1 (Pappu and Gummadi, 2016). The values of this latter study were close to the experimental values reported in the present study. This indicates that μ_max experimentally depends on the fermentation conditions and the microorganisms used as well.

Experimental values of xylitol yield (Y_P/S) and cellular yield (Y_X/S) were higher in the non-detoxified OPEFB hydrolysate than in the synthetic medium. Data of the models were closer to the experimental results of Y_X/S than of Y_P/S; the latter being similar among the models. This is probably due to the fact that the base equation used for product modeling was the same for all models as mentioned above. Y_X/S and Y_P/S values were found between 0.09 to 0.15 (g g^-1) and between 0.13 to 0.41 (g g^-1), respectively, for xylitol production with C. tropicalis in synthetic medium (Mohamad et al., 2016). Regarding the xylitol production with D. nepalensis in synthetic medium, Y_X/S and Y_P/S values were reported between 0.1 to 0.6 (g g^-1) and 0.1 to 0.5 (g g^-1), respectively (Pappu and Gummadi, 2016). The experimental values of the present study were slightly higher for Y_X/S, whereas the Y_P/S data were in the literature range.

Non-growing or slowly growing microbial cultures can recycle much of the metabolic energy derived from initial fermentation of a polysaccharide to boost the additional production of a secondary metabolite (Singh et al., 2019). In order to determine if this could be happening in this study, the amount of substrate used for biomass in the non-growth phase and the maintenance coefficient (ms) for C. tropicalis in synthetic medium and in non-detoxified OPEFB hydrolysate were determined (Table 1). The ms indicates the portion of substrate consumed for the maintenance of cellular function and is used as a correction for the microbial growth kinetics (Wang and Witarsa, 2016). In synthetic media, the estimated values of the ms showed that in the Tessier model more xylose was used for its non-growing components, which contributed to xylitol formation; while in the non-detoxified OPEFB hydrolysate, the Monod model showed greater use of xylose for its non-growing components. The estimated values of the ms were low compared to the literature (Pappu and Gummadi, 2016; Singh et al., 2019), which could assume that xylitol is mainly produced during the cell growth phase.

The estimated product formation coefficient (α), which represents xylitol production associated with growth, was always higher in all models in both media, except for the Tessier model in non-detoxified OPEFB hydrolysate. This indicates that xylitol formation depends on cell growth. The xylitol production in the stationary phase is represented by β, which is the coefficient of non-growth product. A comparatively low β value suggests that xylitol production remains low in non-growing conditions (Ghosh et al., 2012). These results agreed with what was previously reported for the ms.

Statistical criteria to validate fit of data to models, CV and RSME of the models are portrayed in Table 2. CV and RSME were lower in synthetic medium than in non-detoxified OPEFB hydrolysate, except for CV and RSME of Contois model substrate that reported a value of 48% and 5.684, respectively. Lower CV and RSME values represent the best fit of the model (Pappu and Gummadi, 2016). If we focus only on these values, the Monod model would be the most representative in synthetic medium and the Contois model in the non-detoxified OPEFB hydrolysate. Nonetheless, taking into account the experimental values of μ_max and Y_X/S, the models that are closest to reality were Tessier in synthetic medium and Contois in non-detoxified OPEFB hydrolysate. The latter is essential to clarify, since a model is the mathematical representation of a phenomenon or process, and apparently, it can have a good predictive capacity represented in different statistical comparisons and experimental errors; but first of all, the model must be in accordance with the adequate prediction of what happens in reality. The Tessier model in synthetic medium and Contois in non-detoxified OPEFB hydrolysate had a CV in the growth kinetics of 32 and 33%, respectively. This means that the kinetic behavior of the experimental values can be predicted in the conditions posed in this study by around 67%. This may be due to the way the cell concentration was determined and the different assumptions used to apply the models.

Table 2

Since fermentation is a complex bioprocess, it is difficult to obtain a complete profile to reveal what really happens during the entire process and predict the cell growth kinetics and metabolism under certain conditions. Although a huge amount of research on mathematical modeling of microbial processes has been carried out, the applicability of biological concepts into the models remained limited (Tian et al., 2018). It is necessary to keep in mind that these types of processes are complex and depend on various variables.