Coupled Kinetic and Rheological Analysis of Lactic Acid Production from Whey by Kluyveromyces marxianus

Abstract

Whey, a high-volume by-product of the dairy industry, is a valuable substrate for biotechnological valorization that presents an environmental challenge. This study aimed to investigate the kinetic and rheological behavior of Kluyveromyces marxianus during lactic acid fermentation using deproteinized whey by varying initial lactose (35 and 50 g L^-1) and inoculum (3 × 10⁶ and 10 × 10⁶ cells mL^-1) concentrations and working volumes (0.6 and 0.9 L), through a full factorial (2³) experimental design. Growth kinetics, lactose consumption, and lactic acid production were modeled using the Gompertz, Luedeking-Piret, and Pirt models. Rheological behavior was obtained using the Ostwald-de Waele power-law model. A rheokinetic model that integrates biomass growth with fluid properties was proposed. The results showed that high lactose and low inoculum levels significantly improved lactic acid yield (0.285 g g^-1) and productivity (0.116 g L^-1 h^-1). Strong statistical interactions were identified between inoculum and substrate levels. The rheokinetic model demonstrated strong correlations between the consistency coefficient and the rheological behavior index, as well as microbial growth (correlation coefficient (R²) > 0.92), providing a predictive tool that eliminates the need for destructive measurements.

Keywords:
whey to biochemical conversion; lactose valorization; non-Newtonian fermentation; biomass prediction; rheokinetic modeling

Introduction

Whey, an abundant by-product of the dairy industry, poses a significant environmental challenge due to its high organic matter content and its massive global production volume, estimated at 180-200 million metric tons annually.¹,² However, this effluent is also a valuable source of nutrients, as it retains about 55% of the total nutrients from milk, including lactose, soluble proteins, minerals, and vitamins,³ which positions it as a promising substrate for the production of high-added-value bioproducts through fermentation processes; around 1 m³ of whey contains more than 9 kg of high-biological-value proteins, 50 kg of lactose and 3 kg of fat milk.⁴ This amount of nutrients characterizes whey with a high biochemical oxygen demand (BOD) and a significantly elevated chemical oxygen demand (COD), making it a significant source of environmental pollutants. When discharged into aquatic environments, whey contributes to the reduction of dissolved oxygen levels, posing ecological risks to marine organisms and threatening environmental and human health. Whey exhibits BOD and COD levels of approximately 50 and 80 g L^-1, respectively.⁵ The primary contributor to this elevated COD is lactose, present in concentrations ranging from 35 to 50 g L^-1, which accounts for more than 70% of the total COD. In comparison, municipal wastewater typically contains BOD and COD values around 0.20 and 0.41 g L^-1, respectively. This puts into perspective the fact that the organic load present in whey is approximately 150 times greater than that of urban effluents.⁶ Therefore, its biological transformation not only mitigates the environmental impact but also offers a way to recover waste economically.

Among the bioproducts of interest, lactic acid stands out for its versatility, finding applications in the food, pharmaceutical, and cosmetics industries and, crucially, as a monomer for the production of polylactic acid (PLA), a biodegradable bioplastic with growing demand.⁷-⁹ The biotechnological production of lactic acid, using fermenting microorganisms, is a sustainable alternative to conventional chemical methods, with the advantage of specifically generating the desired isomer (L- or D-lactic acid).¹⁰

The yeast Kluyveromyces marxianus (K. marxianus) is emerging as a particularly promising microorganism for the production of lactic acid from whey and other substrates.¹¹ Unlike many lactic acid bacteria, K. marxianus has a remarkable ability to directly assimilate lactose without the need for prior hydrolysis, which simplifies the process and reduces costs.¹²-¹⁴ Furthermore, it is known for its high growth rate, thermotolerance, and the ability to produce a variety of metabolites (bioethanol, lactic acid, enzymes, xylitol), making it a robust biotechnological platform for the bioconversion of dairy by-products.¹⁵,¹⁶

To efficiently optimize and scale lactic acid production in bioreactors, a thorough understanding of microbial growth kinetics and the rheological behavior of fermentation broths is imperative. Growth kinetics, substrate consumption, and product yield provide the key parameters (maximum growth rate (µ_max), saturation constant (Ks), which is the substrate concentration for which the specific growth rate is half of the maximum value, product yield coefficient on substrate (Y_P/S), biomass yield coefficient on substrate (Y_X/S), volumetric productivity (Q_P)) required for bioreactor design and modelling, enabling the prediction of process performance under different conditions.¹⁷-¹⁹ In parallel, the rheological properties of the fermentation broth (apparent viscosity, yield stress, Newtonian/non-Newtonian behavior) directly influence critical unit operations such as mixing, mass transfer (particularly oxygen), and heat transfer within the bioreactor.²⁰-²² Unfavorable rheological behavior, such as high viscosity or complex non-Newtonian profile, can lead to limitations in oxygen transfer, concentration gradients, and mechanical stress on cells, compromising productivity at on industrial scale.²³ The correlation between biomass concentration and rheological parameters, as well as the identification of models that describe these changes, is fundamental for process engineering.

Despite increasing attention to whey valorization with K. marxianus, few studies have comprehensively integrated detailed kinetic analysis of growth and lactic acid production with coupled rheological characterization and modelling of fermentation broths, especially under the influence of critical process variables such as initial substrate, inoculum, and oxygen concentrations. Understanding the interaction between these factors and system behavior is crucial for transitioning from laboratory scale to industrial production.

As a hypothesis, it is believed that the initial concentrations of lactose, inoculum and broth volume significantly impact the growth kinetics of K. marxianus, the conversion efficiency of lactose to lactic acid, and that the biomass generated, together with other components of the fermentation broth, will predictably determine the rheological behavior of the system, directly affecting the design and operation considerations of bioreactors.

The objective of this study is to determine and model the kinetic profiles of K. marxianus growth, lactose consumption and lactic acid production from whey under different conditions of initial substrate, inoculum, and broth volume, as well as to rigorously correlate the rheological changes of the fermentation broth with the evolution of the biomass, to calculate fermentation yields and generate essential engineering parameters for the scaling of the process.

Experimental

Whey characterization

Characterization of the fresh whey was performed. The parameters analyzed included pH, measured using a Jenway potentiometer, model 3310. The density of the whey samples was measured using an Anton Paar DMA 4500 densitometer. The total amount of nitrogen was measured using the Kjeldahl method. The protein content of whey was calculated by multiplying its nitrogen content by 6.38. The concentration of reducing sugars (lactose) was calculated using the phenol-sulfuric method proposed by Dubois et al.²⁴ The fat, total solids, ash and percentage of titratable acidity (lactic acid %) were determined using the official methods of analysis from Association of Official Analytical Chemists²⁵ (AOAC 16.059 (Roese-Gottlieb), 16.032, 16.035 and 16.023, respectively). Chemical oxygen demand was measured using the closed reflux, colorimetric method,²⁶ and the color of the whey samples was determined by the colorimetric method.

Raw materials and preparation of the culture medium

Fresh liquid whey from a local cheese factory (Orizaba, Veracruz, Mexico) was used. For pretreatment, the whey was initially deproteinized through a thermo-acid treatment, adjusting the pH to 4.5 with 2 M HCl and heating at 90 °C to precipitate residual proteins. After cooling, the solid phase was removed by vacuum filtration through a filter paper Whatman No. 40. The clarified whey was adjusted to the desired lactose concentrations (30 and 50 g L^-1) by either dilution with distilled water or concentration by low temperature evaporation. Finally, the medium was sterilized by wet-steam autoclaving at 121 °C and 1.03 bar (Yamato SM 300) for 15 min.

Microorganism and inoculum preparation

A wild strain of K. marxianus, isolated from agave musts and identified using the polimerase chain reaction-restriction fragment length polymorphism (PCR-RFLP) method of the ITS-5.8S region, was employed. The strain was maintained on YPD agar slant (10 g L^-1 yeast extract, 20 g L^-1 peptone, 20 g L^-1 dextrose, 20 g L^-1 agar) at 4 °C with monthly refills. For long-term storage, it was stored in 20% (v/v) glycerol at -80 °C. For inoculum preparation, an aliquot of the preserved strain was transferred to a 250 mL Erlenmeyer flask containing 50 mL of sterile YPD medium. This culture was incubated with constant shaking (150 rpm) at 36 °C for 18 h, until it reached the exponential growth phase. Subsequently, the cells were harvested by centrifugation at 5000 rpm for 10 min, washed twice with sterile saline (0.9% NaCl) to remove traces of culture medium, and resuspended in a small amount of sterile saline. Cell concentration was determined by measuring the optical density at 600 nm (OD600) with a Genesys 10S UV-Vis spectrophotometer (Thermo Scientific) and correlated with biomass dry weight using a previously established calibration curve. Inoculum concentrations were adjusted for fermentation experiments (3 × 10⁶ and 10 × 10⁶ cells mL^-1).

Experimental conditions and experimental design

A 2³ factorial experimental design was employed to evaluate the effects of initial lactose concentration (35 and 50 g L^-1), initial fermentation volume (0.6 and 0.9 L), and inoculum concentration (3 × 10⁶ and 10 × 10⁶ cells mL ¹) on lactic acid production and rheological properties. Fermentations were carried out in 1 L Erlenmeyer flasks with working volumes of 0.6 and 0.9 L. The fermentation broth was maintained at 36 ± 0.5 °C with constant shaking at 150 rpm. Fermentations were monitored for 72 h. The samples were aseptically collected at 8-h intervals (0, 8, 16, 24, 32, 40, 48, 56, 64, 72 h) for quantification of biomass, lactose, and lactic acid. Biomass was determined by dry weight. A 10 mL aliquot of the fermentation broth was centrifuged at 5000 rpm for 10 min. The cell pellet was washed twice with sterile distilled water and dried to constant weight in an oven at 105 °C for 24 h. Lactose concentration was measured in the supernatant (obtained after centrifugation of the biomass) using the phenol-sulfuric acid method. Lactic acid concentration was determined in the supernatant using the ferric chloride colorimetric method.²⁷

Rheological analysis

The rheological properties of the fermentation broths were measured at each sampling point (every 8 h). A rotational rheometer (Anton Paar MCR 301, Austria) equipped with a vane geometry (ST22-4V-40) was used. Measurements were performed at 36 °C to replicate fermentation conditions. A shear rate range of 0.1 to 1000 s^-1 was applied to obtain flow curves (shear stress vs. shear rate). The flow curve data were fitted to the power law rheological model (equation 1):

(1) $${\eta }_{\mathrm{a}}=\mathrm{K}{\dot{\gamma }}^{\mathrm{n}}$$

where η_a is the apparent viscosity (Pa s), K is the consistency coefficient (Pa sⁿ), γ$\dot{\gamma }$ is the shear rate (s^-1), and n is the rheological behavior index (dimensionless). If (n > 1), the fluid exhibits a dilatant behavior, i.e., its apparent viscosity increases as the shear rate increases. If (n < 1), the fluid exhibits pseudo-plastic behavior, i.e., the apparent viscosity decreases as the shear rate increases. If (n = 1), the model reduces to the Newton model. The fitting rheological model was obtained by the RheoPlus/32 V2 software 8.1 (Anton Paar, Austria).

Data collection and analysis

Experimental data on biomass, lactose, and lactic acid concentrations over time were used to calculate key kinetic parameters: specific growth rate (μ) was calculated from the Gompertz model (equation 2):

(2) $$\mathrm{X}(\mathrm{t})=\mathrm{Ae}(-\mathrm{e}(\frac{{\mu }_{max}\mathrm{e}}{\mathrm{A}}(\lambda -\mathrm{t})+1))$$

where X (g L^-1) is the biomass as a function of time (t), A (g L^-1) is the maximum achievable biomass, μ_max (h^-1) is the maximum rate at which biomass grows, and λ (h) is the latency time before exponential growth begins.

The product production rate (q_P) by Luedeking-Piret model (equation 3):

(3) $$q_P=\alpha \mu +\beta$$

where q_P (g_product g_biomass^-1 h^-1) is the specific product formation rate, μ (h^-1) is the rate of biomass growth, α (g_product g_biomass^-1) is the associated growth coefficient: product generated by growth, and β (g_product g_biomass^-1 h^-1) is the maintenance coefficient: product generated independently of growth.

The substrate consumption rate (q_S) by Pirt model (equation 4).

(4) $${\mathrm{q}}_{\mathrm{s}}=\frac{\mu }{{\mathrm{Y}}_{\mathrm{x}/\mathrm{s}}}+{\mathrm{m}}_{\mathrm{s}}$$

where q_S (g_substrate g_biomass^-1 h^-1) is the specific substrate consumption rate, Y_X/S (g_biomass g_substrate^-1) biomass yield on substrate (conversion efficiency), and m_s (g_substrate g_biomass^-1 h^-1) maintenance rate: substrate consumption independent of growth.

The global yields were calculated at the end of fermentation: lactic acid from lactose (equation 5):

(5) $${\mathrm{Y}}_{\mathrm{P}/\mathrm{s}}=\frac{\mathrm{\Delta }\mathrm{P}}{\mathrm{\Delta }\mathrm{S}}$$

where Y_P/S (g_product g_substrate^-1) is the product yield on substrate, ∆P (g L^-1) is the change in product concentration, and ∆S (g L^-1) is the change in substrate concentration.

Biomass from lactose (equation 6):

(6) $$Y_{x/s}=\frac{\mathrm{\Delta }X}{\mathrm{\Delta }S}$$

where Y_X/S (g_biomass g_substrate^-1) is the biomass yield on substrate, ∆X (g L^-1) is the change in biomass concentration.

And lactic acid volumetric productivity (equation 7).

(7) $$Q_P=\frac{P_f}{\mathrm{\Delta }t}$$

where Q_P (g L^-1 h^-1) is the rate of lactic acid production per unit volume and time, P_f (g L^-1) is the final concentration of lactic acid, and ∆t (h) is the time elapsed during which Pf occurred.

Finally, the experimental data from fermentation were analyzed statistically in RStudio, version 2025.05.1+513 (RStudio Inc., USA) to determine the effects of independent variables on lactic acid production. The models fitting were performed using minpack.lm packages interface of RStudio.

Results and Discussion

Characterization of whey

Whey, a by-product of cheesemaking, is a complex and nutrient-rich matrix. Its composition varies considerably depending on the raw material, the cheesemaking process, and storage conditions. Table 1 shows that the whey used is sweet, with a slightly acidic to neutral pH. This whey is produced by coagulating milk with rennet, unlike the acidic whey (pH < 5.0) produced by acid coagulation. The density value is typical for whey, slightly higher than that of water due to dissolved and suspended solids, fat, proteins, and lactose. The difference between total and volatile solids (approximately 1.74 kg m^-3) represents the organic matter content, highlighting the potential of whey as a substrate for microbial cultivation in bioprocesses. Lactose is the primary source of solids in whey and plays a crucial role in the production of value-added products, such as lactic acid. Its high concentration also contributes to the high chemical oxygen demand of whey. The color values indicate that whey has a yellowish-green color, which is common due to the presence of riboflavin (vitamin B2).³,²⁸,²⁹

Table 1
Chemical composition of residual whey

Parameter	Value
pH	6.550 ± 0.119
Density / (kg m-3)	1028 ± 0.000
Total solids / (kg m-3)	64.48 ± 0.119
Volatile solids / (kg m-3)	62.74 ± 0.085
Total nitrogen Kjeldahl / %	0.160 ± 0.010
Proteins / %	1.016 ± 0.010
Lactose / (kg m-3)	35.46 ± 0.286
CODT / (kg m-3)	87.66 ± 1.802
Fat / (kg m-3)	3.790 ± 0.041
Color
L	54.56
a	-0.42
b	09.42

Fermentation experiments

To investigate the factors influencing lactic acid production, 0.6 and 0.9 volumes of thermo-denatured whey were fermented for 72 h at 36 ± 2 °C. Initial conditions included lactose concentrations of 35 and 50 g L^-1 and yeast inoculum densities of 3 × 10⁶ and 10 × 10⁶ cells mL^-1. A factorial, completely randomized experimental design was utilized to assess the individual and combined effects of initial lactose concentration, inoculum concentration, and broth volume. These factors were set at two levels (low and high), with maximum lactic acid production serving as the measured response. The experiment involved eight unique fermentation conditions, with three replicates each, for a total of 24 fermentations. All experimental data were analyzed using RStudio software, version 2025.05.1+513. The results obtained in the fermentation series indicated a higher lactic acid yield at a high lactose concentration (50 g L^-1) (treatment 6), with a maximum production of 8.37 ± 0.17 g L^-1. This value is comparable to those reported in the literature under similar conditions. For example, in a comparative study, Plessas et al.³⁰ reported a production of 8.8 g L^-1 of lactic acid using the yeast K. marxianus in an undiluted whey medium, which also contained a lactose concentration of 50 g L^-1. Figure 1 shows that lactic acid accumulation was highest at the highest lactose concentration (50 g L^-1) under most conditions.

Figure 1
Lactic acid production by treatment.

When comparing the conditions (lactose 35 g L^-1, inoculum 3 × 10⁶, volume 900 mL) with (lactose 50 g L^-1, inoculum 3 × 10⁶, volume 900 mL), production increased from an average of 4.16 to 7.19 g L^-1. Similarly, in the 600 mL volume with 3 × 10⁶ inoculum, it increased from 5.17 to 8.35 g L^-1. This is because higher substrate availability typically results in greater product accumulation. Koukoumaki et al.,³¹ investigating the production of cellular protein and ethanol from whey using K. marxianus, demonstrated a clear relationship between increased lactose concentration and increased metabolite activity. That is, although the metabolites are distinct, the trend is analogous: a higher lactose concentration leads to greater accumulation of the product of interest, up to a point at which the system becomes saturated or inhibited.

On the contrary, at the initial inoculum concentration, there is lower lactic acid production at high inoculum concentrations (10 × 10⁶ cells mL^-1) than at low concentrations (3 × 10⁶ cells mL^-1). This may occur because a high inoculum concentration can lead to competition for nutrients; i.e., a high cell density can quickly deplete essential nutrients other than lactose, such as micronutrients present in the whey (e.g., nitrogen sources, mineral salts), which limits growth and metabolic activity. Another reason is that, if the process is not strictly aerobic, as in lactic acid fermentations, a high inoculum could quickly deplete dissolved oxygen, leading to an early anaerobic state that may not be optimal for overall productivity. Saini et al.,³² in a study of alcoholic fermentations using whey as a substrate and a K. marxianus strain, observed that the highest ethanol production was achieved with a low inoculum level of 6% v/v, rather than 8% v/v. This supports the idea that inoculum size plays a vital role in the development of whey-based fermentations.

The volume effect did not significantly affect other factors. However, a smaller volume is slightly more beneficial for lactic acid production. This could be explained by the fact that a lower volume in a similar fermentation vessel implies a higher surface area-to-volume ratio, which may improve the transfer of available oxygen. In lower volumes, mixing could be more efficient, ensuring a more homogeneous distribution of nutrients and cells, and reducing gradients. In a recent study³³ focusing on K. marxianus, the volumetric oxygen transfer coefficient (k_La) was evaluated in 250 mL shaker flasks containing 50 or 150 mL of culture. It was found that lower volumes (resulting in a greater relative surface area) yield higher k_La values. This confirms that lower volumes facilitate greater oxygen transfer, thereby favoring metabolic production.

The data obtained in the fermentation series were statistically analyzed to evaluate the effect of each factor and its interactions. Figure 2a shows the Pareto chart of standardized effects. It can be seen that factors beyond the critical value (12.71) have a statistically significant impact at the α = 0.05 confidence level. Lactose concentration is the most crucial factor affecting lactic acid production.

Figure 2
(a) Standardized effects of factors on lactic acid production, (b) individual influence of lactose, inoculum, and volume on lactic acid fermentation.

As mentioned above, high lactose concentrations lead to increased lactic acid production. Inoculum also exhibits a significant effect, suggesting that optimal inoculum size is essential to ensure a sufficient number of active cells for efficient substrate conversion. Volume has a significant effect, although less pronounced than either lactose or inoculum. This suggests that the fermentation volume independently influences lactic acid production, as nutrient distribution can vary with volume, thereby affecting microbial growth and metabolic activity. The analysis revealed two strong interactions: the first between lactose and the inoculum. This implies that the effect of lactose concentration on lactic acid production is not independent of the effect of inoculum concentration, and vice versa. There is an antagonistic relationship between these two factors, i.e., a particular lactose concentration produces different results depending on the inoculum level, indicating that optimizing both simultaneously is crucial; the second, the inoculum-volume interaction, shows a moderately significant effect, indicating that the effect of inoculum concentration is influenced by volume or vice versa. This could be related to limitations in nutrient availability due to the volume of the inoculum. Another hypothesis for this interaction is that at higher volumes, the available oxygen is lower. According to González and Fernández,³⁴ if anaerobic conditions begin too soon, the population density will not be high enough to achieve a reasonable conversion rate. Figure 2b presents the main effects plots. The figure illustrates how a change in the level of a single factor, averaged across the levels of the other factors, affects lactic acid production. The coded levels of -1 and 1 represent the low and high levels of each factor, respectively. Lactose shows a steep positive slope, indicating a strong positive main effect. As lactose concentration increases, average lactic acid production increases significantly because greater substrate availability, within non-inhibitory ranges, generally leads to greater product formation.³⁵ In contrast, the inoculum exhibits a steep negative slope. This indicates a strong adverse main effect: increasing the inoculum concentration decreases the average lactic acid production. However, an adequate inoculum is essential to initiate fermentation; nevertheless, high inoculum concentrations can lead to substrate depletion rates exceeding product formation or per-cell nutrient limitations, ultimately hampering overall lactic acid yield.³⁶ The volume graph also shows a negative slope, although less steep than that of the inoculum. This indicates an adverse main effect: increasing volume decreases average lactic acid production. This effect can be attributed to several factors, including reduced nutrient efficiency, limited oxygen availability at higher volumes, and less efficient mixing, all of which can negatively impact microbial metabolism.³⁷

Figure 3 shows the interactions between lactose, inoculum, and volume on lactic acid production. In the inoculum-volume interaction, both lines exhibit a negative slope, indicating that increasing the inoculum level generally decreases lactic acid production, regardless of volume. However, the lines are not parallel, suggesting a strong interaction. Specifically, at a low inoculum level, the difference in lactic acid production between a low volume (-1) and a high volume (1) is greater compared to a high inoculum level (1). This suggests that the adverse effect of increasing inoculum is slightly reduced at high volumes or exacerbated at low volumes.

Figure 3
Analysis of interactions between factors in lactic acid production: (a) inoculum-volume, (b) lactose-inoculum, and (c) lactose-volume.

Optimal volumetric productivity in fermentation systems is highly dependent on the balance between cell density and substrate availability. Excess inoculum in small volumes could lead to rapid substrate depletion, as observed in the steep decrease. The significant interaction between lactose and inoculum shows a relatively flat response to lactose, indicating that at a high inoculum level, increasing lactose has a minimal effect on lactic acid production. In contrast, a low inoculum (-1) shows a steep positive slope, meaning that at a low inoculum level, increasing lactose dramatically increases lactic acid production. This highlights that the effect of lactose concentration on lactic acid formation depends strongly on the initial cell density. When the inoculum is low, providing more lactose is very beneficial. However, at high inoculum levels, other limiting factors (e.g., nutrient availability per cell, oxygen transfer at high cell densities) may become dominant, making it less practical to increase lactose. The interaction between lactose and volume shows two lines with a positive slope, indicating that increasing lactose generally increases lactic acid production, regardless of volume. The lines are parallel, suggesting a weaker interaction effect than the other interactions. However, low volume consistently shows higher lactic acid production than high volume at both lactose levels. This indicates that a lower fermentation volume (-1) generally promotes lactic acid production. This could be related to improved mixing, resulting in a more even distribution of nutrients.

Finally, a regression analysis of lactic acid production was performed as part of the design evaluation. The linear regression model was found (equation 8) to have an adequate fit (correlation coefficient (R²) = 0.845) to predict lactic acid production based on the coded levels of lactose, inoculum, and volume, as follows:

(8) $$\text{Lactic acid}=0.57+1.77\times \text{lactose}-\mathrm{3.18\; \times }\text{inoculum}-0.28\times \text{volume}$$

Figure 4 shows that most points are close to the line, indicating that the predictions of the model are adequate for the range of experimental conditions evaluated.

Figure 4
Validation of the regression model for lactic acid production.

Lactic acid production via fermentation processes critically depends on the kinetic behavior of the microorganism under different culture conditions. The modified Gompertz model (equation 1) was used to describe microbial growth because it has proven to be a robust tool for estimating key parameters such as the specific growth rate (µ) and the lag phase (λ). The characterization of these parameters enables the identification of optimal operating conditions, comparison of treatments, and the selection of appropriate strategies to maximize yield in biotechnological processes. In this context, Figure 5 presents the individual analysis by treatment (1-8), facilitating the visual comparison of these kinetic indicators.

Figure 5
Distribution of growth kinetic parameters (µ, λ) and quality of fit by treatment.

The data obtained showed an adequate adjusted R-squared value (R² > 0.78; presented in Supplementary Information (SI) section, Table S1). A higher µ indicates faster growth, which is desirable for efficient lactic acid production. At the same time, a λ close to zero implies that the culture does not experience a significant delay in the onset of growth. As shown in the Figure 5, treatment 5 exhibits the highest µ (0.201 h^-1) and a considerable lag phase (λ = 0.234 h). This could indicate that, although growth is rapid, the initial delay limits the total production (5.17 g L^-1, Figure 1), possibly due to adaptation effects. Treatment 6 (µ = 0.142 h^-1, λ = 0) exhibits the highest production (8.36 g L^-1), confirming that a high growth rate without a lag phase promotes efficient lactic acid accumulation. Treatment 2 (µ = 0.1 h^-1, λ ca. 0.02 h) also showed high production (7.19 g L^-1), confirming that the slight delay somewhat compromises system efficiency. Treatments 7 and 3 showed the lowest growth rates (µ < 0.07 h^-1) and also had the lowest yields (2.55 and 3.10 g L^-1), reflecting a clear positive correlation between µ and yield. The values found for µ, although close to values reported in the literature,¹⁸,³⁸ are relatively low. One reason may be that no micronutrients (such as ammonium sulfate and yeast extract) were supplied for optimal growth and development.

Figure 6 presents data obtained from the Luedeking-Piret model (equation 2), showing how lactic acid production is coupled with cell growth, where α (g product g biomass^-1) is the growth-associated production, i.e., the amount of product generated per biomass increase, and β (g product g biomass^-1 h^-1) is the non growth associated production, or maintenance, i.e., the product generated per cell per hour, regardless of growth.³⁹

Figure 6
Distribution of the α and β coefficients: comparative analysis of the kinetic behavior in lactic acid production.

The parameter values obtained showed an adequate adjusted R-squared (R² > 0.92, SI section). Treatments with high α (T3 = 4.97, T7 = 3.02) show a production strongly linked to growth. According to Wu et al.,⁴⁰ this behavior is characteristic of semi-growth-associated lactic acid fermentations. They also reported high α values, ranging from 1.21 to 9.11 g g^-1, which support the results obtained in our study. The α = 4.97 at T3 suggests that for every gram of new biomass, almost 5 g of lactic acid is produced, indicating the effect of a supersaturated inoculum density, consistent with the initial inoculum level (10 × 10⁶ cells mL ¹). A β close to zero, as at T2 (0.0035), denotes production almost exclusively linked to growth, while negative β values (T1, T4-T8) even indicate that substrate is consumed during the stationary phases, possibly for intracellular maintenance.

Figure 7a presents the Y_X/S and m_s values from the Pirt model (equation 3) for each treatment with an adequate fit (R² > 0.94, SI section). Figure 7b presents the overall yields Y_P/S, Y_X/S, and volumetric productivity (Q_P) by treatment.

Figure 7
(a) Integrated analysis of Y_X/S and m_s (maintenance rate) of the kinetic behavior in lactic acid production, (b) overall yields and productivity in lactic acid production.

Treatments with high biomass yield over substrate (Y_X/S) values such as 5 (0.22), 6 (0.18), and 2 (0.136) indicate greater substrate-to-biomass conversion. This is consistent with their moderate α values (5: 1.97, 6: 2.66, 2: 2.42) and with lactic acid production, confirming good production linked to growth. In contrast, treatments with low Y_X/S, such as 3, 7, and 8 (< 0.07), show lower yields, coinciding with high β values in T3 and T7 (4.97 and 3.02, respectively), indicating excess production per cell even with low biomass. This often occurs in inhibited fermentations, where the microorganism redirects its metabolism toward producing a product despite limited growth. High β values in 4 (0.165), 7 (0.127), and 8 (0.125) suggest high substrate consumption for maintenance, regardless of growth. These are correlated with β < 0 or close to 0, indicating minimal production in the stationary phase, which is typical under stress conditions, such as acidic environments. The values determined for the maintenance coefficient and the biomass-to-product coefficient are consistent with previous published studies⁴¹,⁴² that have used whey as a substrate and the yeast K. marxianus.

Figure 7b presents the overall yields Y_P/S, Y_X/S, and volumetric productivity (Q_P) by treatment. Treatments 5 (0.285), 6 (0.260), and 2 (0.225) stand out with the highest overall Y_P/S yields, meaning they convert a greater proportion of the substrate into product. This trend coincides with intermediate α and β values close to zero or negative, indicating growth-related production with efficient substrate use. Furthermore, their productivities (Q_P) are high (5: 0.072, 6: 0.116, 2: 0.104 g L^-1 h^-1), indicating their overall performance, in line with the descriptions of Palaniraj and Nagarajan,⁴³ for efficient lactic fermentations. Regarding Y_X/S yield, treatments 5 and 6 also show high yields (0.122 and 0.108, respectively), indicating high biomass accumulation and good conversion to product. This suggests that lactic acid production is a function of growth.⁴⁴ In contrast, treatment 7 exhibits a high Y_X/S (0.335), but a low overall Y_P/S (0.115) and low productivity (Q_P = 0.036). This indicates that a significant portion of the substrate is utilized for biomass and maintenance, resulting in low lactic acid production. The overall coefficients of product yield and biomass on substrate obtained are consistent with the existing scientific literature⁴²,⁴⁴,⁴⁵ on the use of whey and the yeast K. marxianus.

By comparison, the achieved productivity of 0.116 g L^-1 h^-1 is reasonable, given the conditions under which the experiment was conducted, considering published values. For example, Smets et al.⁴⁶ achieved 1.44 g L^-1 h^-1 using glucose as a substrate and a strain of K. marxianus specifically modified to produce more lactate by deleting the PDC1 (pyruvate decarboxylase 1) gene to direct metabolism toward this pathway. Similarly, Gosalawit et al.⁴⁷ reported productivities of 0.90 and 1.06 g L^-1 h^-1 with glucose and genetically optimized yeasts (deleted the PDC1 gene). In our case, we used unenriched whey (ammonium sulfate/ammonium nitrate as the nitrogen source) and a wild-type strain of K. marxianus isolated from agave must. In this context, the results obtained are entirely consistent and reinforce the idea that the highest yields are usually obtained when specific genetic modifications and efficiently metabolizable substrates, such as glucose, are combined.

Finally, based on the kinetic analysis performed, it is concluded that the proposed models (modified Gompertz, Luedeking-Piret, and Pirt) provide a satisfactory fit to the experimental data obtained during fermentation for lactic acid production. As shown in Figure 8a, the Gompertz and Luedeking-Piret models accurately describe both the growth of K. marxianus and lactic acid synthesis, adequately reproducing the exponential phase, the inflection point, and the stationary phase of the culture. This indicates that the estimated kinetic parameters accurately reflect the behavior of the system under the experimental conditions evaluated. Furthermore, the Pirt model applied to lactose consumption showed a remarkable fit (see Table S1, in the SI section), adequately describing the relationship between microbial growth rate and specific substrate consumption. Figure 8b visualizes the quality of the fit and confirms that the mathematical models adequately describe the production and consumption dynamics in the fermentation system. Supplementary figures (treatments 1-5, 7, and 8, Figures S1 S14) are presented in the SI section. Treatment 6, which showed the highest production, productivity, and yields, is presented in this central section to avoid overloading the main results section.

Figure 8
(a) Kinetic adjustment using growth, production, and consumption models in lactic acid production (treatment 6), (b) validation of the Luedeking-Piret and Pirt growth models associated with lactic acid production and lactose consumption (treatment 6).

Based on the kinetic parameters obtained from the modeling of treatment 6, a Shiny application in RStudio was developed to graph the growth curves of K. marxianus, lactose consumption, and lactic acid production (Figure S15, SI section). This approach allowed us to evaluate the effects of the maximum growth rate (μ_max), the initial lactose concentration, and the maximum biomass reached (X_max) on the kinetic behavior of these curves. In Figure S16 (SI section), maintaining the conditions of treatment 6 (50 g L^-1 of lactose and X_max = 3.6 g L^-1) but increasing μ_max to a relatively high value (0.35 h^-1), an accelerated lactose consumption is observed, which leads to the rapid achievement of both maximum biomass and maximum lactic acid production (around 15 h). On the other hand, maintaining the conditions of treatment 6 (μ_max = 0.147 h^-1, lactose = 50 g L^-1) and reducing the maximum biomass reached (X_max) to 2 g L^-1 (Figure S17, SI section), substrate consumption was limited (ca. 12 g L^-1), which negatively impacted lactic acid production (ca. 3.7 g L^-1) and the growth of K. marxianus, reaching its maximum at 20 h. When the initial lactose concentration (35 g L^-1) and the maximum biomass reached (2 g L^-1) were reduced, but μ_max = 0.147 h^-1 was maintained (Figure S18, SI section), a reduction in substrate consumption (ca. 17 g L^-1), delayed cell growth (more than 20 h to reach maximum), and low lactic acid production (ca. 3.1 g L^-1) were observed. Finally, rapid and high lactic acid production (Figure S19, SI section) requires high substrate availability (50 g L^-1), a relatively high growth rate (μ_max = 0.35 h^-1), and high biomass accumulation in the culture medium. This is because lactic acid synthesis in K. marxianus is closely linked to cell growth.⁴⁰

Measuring the rheological behavior of fermentation broths is crucial for enhancing mixing efficiency, predicting potential scale-up issues, and maintaining a homogeneous environment that supports microbial growth and lactic acid production. Figure 9 shows the apparent viscosity vs. shear rate for the fermentation of treatment 6 (the best kinetic parameters), fitted with the power law model.

Figure 9
Apparent viscosity curves fitted to the power law model for treatment 6, evolution during fermentation.

The experimentally determined models are found to fit adequately, with a coefficient of determination R² > 0.98 (Tables S2-S9, SI section). Figure 9 shows that the fluid behaves as a non-Newtonian fluid with dilatant characteristics, exhibiting a rheological behavior coefficient n > 1 (Tables S2-S9) and displaying consistency coefficient (K) characteristics, which are commonly observed in microbial cultures such as K. marxianus, due to the presence of biomass that promotes particle-particle interactions (stretching and deformation between particles, hindering fluid flow).

Under these conditions, scale-up is known to require special attention, as increasing impeller speed in dilatant broths can further increase apparent viscosity, affecting mixing and heat and oxygen transfer. However, previous studies on non-Newtonian fermentation systems have shown that effective scale-up is possible without relying solely on higher stirring speeds. For example, Hubbard⁴⁸ demonstrated that bioreactors processing non-Newtonian broths can be successfully scaled up by optimizing energy dissipation and impeller configuration rather than increasing rotational speed.⁴⁹ Similarly, improved aeration strategies have been reported to optimize turbulence, bubble dispersion, and k_La (volumetric oxygen mass transfer coefficient) even in viscous media, thereby reducing the need for high shear.⁴⁹-⁵¹ Coaxial and hybrid stirring systems have also been shown to maintain adequate mass transfer at lower shear rates, preventing excessive viscosity buildup during scale-up.⁵²

Supplementary figures (treatments 1-5, 7, and 8, Figures S20-S25, and S26) are presented in the SI section. The rheological behavior figures show that there was no significant change in the apparent viscosity of the fermentation broths over time. This could be attributed to insufficient biomass growth, which did not significantly influence the rheological properties of the broths. To support this, we have identified that the consistency coefficient (K) correlates with biomass concentration. Furthermore, the value of this parameter varies depending on the morphology and aggregate structure of the microorganism. On the other hand, the viscous behavior index (n) also varies, but more stably,⁵³ as observed in the values obtained in this study.

Finally, a mathematical equation was developed that quantifies biomass growth over time, using rheological parameters obtained from the fermentation broths. This modified Gompertz rheokinetic model describes microbial growth, and its key lies in integrating rheological parameters (K and n) into the growth kinetics. This incorporation provides the model with two significant advantages: it enables the prediction of biomass concentration based on the rheological properties of the culture medium. It includes information on how the viscosity and flow behavior of the system directly affect microbial proliferation. The model was as follows:

(9) $$\mathrm{X}={\mathrm{X}}_{max}\mathrm{e}\{-\mathrm{e}[\frac{\mathrm{B}-\mathrm{t}}{\mathrm{C}}+\alpha (\mathrm{K}-\bar{\mathrm{K}})+\beta (\mathrm{n}-\bar{\mathrm{n}})]\}\mathrm{e}(-\mathrm{D}(\mathrm{t}-{\mathrm{t}}_{\mathrm{d}}))$$

where X (g L^-1) is the biomass concentration as a function of time, X_max (g L^-1) is the maximum biomass reached. It represents the maximum growth capacity of the system, and t (h) is the fermentation time. K and n are rheological parameters of the power law model, where K (Pa sⁿ) is the consistency coefficient, n (dimensionless) is the rheological behavior index. $\bar{\mathrm{K}}$ and $\bar{\mathrm{n}}$ are the average values of K and n, respectively. B (h) is the phase change time, i.e., where the maximum growth rate occurs (inflection point of the curve, associated with phase lag), C (h) is the growth scale and indicates the duration of the logarithmic phase (curve width or curvature), i.e., how fast growth occurs after the lag phase; the higher the value of C, the more extended the curve, D is the decay rate constant (h^-1), α (Pa s^-n) indicates the sensitivity of growth to K and expresses how much the consistency of the medium influences the growth rate, β (dimensionless) indicates the sensitivity of growth to n. It describes how much the type of flow (pseudoplastic or dilatant) influences growth, t_d (h) is the time from which the decay begins.

Figure 10 shows the fit of the rheokinetic model to the experimental biomass growth data obtained during the fermentation process of experimental run 6.

Figure 10
Biomass growth kinetics modeled from rheological parameters for treatment 6.

Figure 10 illustrates biomass growth, which the proposed model accurately captures. This model incorporates the rheological parameters of the power law model as indirect explanatory variables for yeast growth, describing the fermentation system with non-Newtonian characteristics. To avoid overloading the main body of the document, the individual graphs corresponding to the model fits for each fermentation treatment are included in the SI section (treatments 1-5, 7, and 8, Figures S27-S33). These graphs exhibit behavior similar to that shown here, reinforcing the robustness of the proposed model and its applicability to the simultaneous analysis of kinetic and rheological data in biotechnological processes.

The fit of the experimentally determined models is adequate, showing a high degree of correlation between the experimental data and the model prediction. This is supported by the coefficients of determination (R²) reported in Table 2, along with the fitted kinetic parameters for each fermentation run.

Table 2
Rheokinetic parameters for biomass growth in different treatments

Treatment	Parameter							R2
	Xmax	B	C	α	β	D	td
1	1.839	0.669	21.877	-5851.057	-16.395	0.002	71.974	0.947
2	2.549	8.853	9.011	-5446.788	-4.568	0.006	69.593	0.999
3	1.414	3.390	8.208	8254.293	-12.917	0.000	62.543	0.998
4	2.742	6.733	7.313	-4275.122	-10.253	0.015	43.810	0.995
5	2.343	3.491	5.477	-334.747	7.361	0.006	28.928	0.987
6	3.562	6.458	8.783	-7700.238	-17.865	0.004	40.486	0.927
7	1.221	3.449	11.307	-8451.790	-10.526	0.083	71.638	0.995
8	1.802	3.275	10.679	7011.102	18.059	0.011	71.487	0.999

Table 2 shows the parameters obtained from modeling microbial growth as a function of rheological parameters (K and n). It can be seen that negative values of α (α(K ‒ K-)) indicate a consistency coefficient below average; that is, when K increases above its average (positive values of α), the viscosity of the medium increases, resulting in a decrease in growth. The same behavior is reflected in β. Negative values of β (β(n - $\bar{\mathrm{n}}$)) indicate an index of rheological behavior below average; that is, as n increases (resulting in positive values of β), the dilatant behavior of the medium increases, meaning the medium begins to thicken. In general, if the combined value of α(K - $\bar{\mathrm{K}}$) + β (n - $\bar{\mathrm{n}}$) is negative, the value of the exponent decreases, which increases biomass.

Multiple kinetic models have been developed for fermentation, allowing the description of microbial growth dynamics, as reported in recent studies with K. marxianus⁵⁴ and other yeasts and bacteria.⁵⁵-⁵⁸ However, these models frequently omit rheological parameters, even though microbial biomass directly impacts properties such as apparent viscosity (η_a), consistency coefficient (K), and rheological performance index (n), thereby affecting process efficiency. In contrast, some studies⁵³,⁵⁹,⁶⁰ have shown strong correlations between biomass and rheological parameters; however, they have not formally integrated these into a dynamic kinetic framework. In this context, the model proposed in this work represents a significant contribution, as it combines a rheokinetic approach into fermentations with K. marxianus grown in whey. This approach allows microbial growth to be estimated from experimentally measured rheological parameters. This integration provides a predictive tool for evaluating system behavior without the need for destructive measurements, thereby opening up new possibilities for designing and monitoring fermentation processes in complex media.

Conclusions

It was observed that the initial concentrations of lactose and inoculum exhibited statistically significant and antagonistic effects on lactic acid production by K. marxianus, with the high lactose level (50 g L^-1) and the low inoculum level (3 × 10⁶ cells mL^-1) being the most optimal, with yields of up to 0.285 g g^-1 and productivities of 0.116 g L^-1 h^-1. The proposed kinetic models (Gompertz, Luedeking-Piret, and Pirt) adequately described the dynamics of microbial growth, substrate consumption, and lactic acid production, confirming growth-associated kinetics. Rheological analysis revealed non-Newtonian behavior with dilatant characteristics in the fermentation broths. The proposed rheokinetic model enabled the estimation of microbial growth from rheological parameters (K and n) obtained from the Ostwald-de Waele power-law model, establishing a non-destructive predictive tool for biomass growth.

Supplementary Information

Supplementary information is available free of charge at http://jbcs.sbq.org.br as file.

Acknowledgments

The authors thank the Tecnológico Nacional de México/Instituto Tecnológico Superior de Coatzacoalcos, Veracruz, México, for providing space for this study.

During the preparation of this work, the authors used ChatGPT-5 mini (OpenAI, USA, free version) to improve the readability and language of the manuscript. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

Data Availability Statement

All data supporting the findings of this study are available within the article and its supplementary information.

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Edited by

• Editor handled this article:
  Adriana Nunes Correia (Associate)

Publication Dates

• Publication in this collection
  06 Feb 2026
• Date of issue
  2026

History

• Received
  20 Sept 2025
• Accepted
  09 Jan 2026