Dynamic Modeling of Bacterial Cellulose Production Using Combined Substrate- and Biomass-Dependent Kinetics

Abstract

In this work, kinetic models are assessed to describe bacterial cellulose (BC) production, substrate consumption, and biomass growth by K. xylinus in a batch-stirred tank bioreactor, under 700 rpm and 500 rpm agitation rates. The kinetic models commonly used for Acetobacter or Gluconacetobacter were fitted to published data and compared using the Akaike Information Criterion (AIC). A stepwise fitting procedure was proposed for model selection to reduce computation effort, including a first calibration in which only the biomass and substrate were simulated, a selection of the three most effective models in terms of AIC, and a calibration of the three selected models with the simulation of biomass, substrate, and product. Also, an uncoupled product equation involving a modified Monod substrate function is proposed for a 500 rpm agitation rate, leading to an improved prediction of BC productivity. The M2c and M1c models were the most efficient for biomass growth and substrate consumption for the combined AIC, under 700 rpm and 500 rpm agitation rates, respectively. The average coefficients of determination for biomass, substrate, and product predictions were 0.981, 0.994, and 0.946 for the 700 rpm agitation rate, and 0.984, 0.991, and 0.847 for the 500 rpm agitation rate. It is shown that the prediction of BC productivity is improved through the proposed substrate function, whereas the computation effort is reduced through the proposed model fitting procedure.

1. Introduction

Bacterial cellulose (BC) has important applications in diverse fields, including tissue engineering, drug delivery, and food applications [1,2]. Commercial applications of BC include tissue engineering, biodegradable materials, foods, cosmetics, and the paper industry [3,4,5,6]. BC has several advantages over plant cellulose, including its high purity, crystallinity, mechanical strength, degree of polymerization, and water-holding capacity [2,7].

In addition, BC is biodegradable, renewable, and biocompatible [2]; it can be modified with in situ or ex situ treatments to obtain improved properties. Different carbon source mixtures can be used for BC production, for instance, glucose, glycerol, fructose, sucrose, and also mixtures of these [7,8]. Gluconacetobacter xylinus is the most used microorganism for BC production studies because of its high productivity [1]. Komagataeibacter xylinus is an aerobic Gram-negative bacterium with high cellulose production capability [9]. It was formerly classified as Acetobacter xylinum, then classified as Gluconacetobacter xylinus, and then reclassified as K. xylinus [10,11,12,13]. It can produce cellulose extracellularly under a temperature range of 25–30 °C and pH range 3–7, using glucose, fructose, sucrose, mannitol, and other carbohydrates as carbon sources [9,14]. It exhibits efficient cellulose production from fructose [14,15]. It has been observed that BC synthesis from fructose and glucose by K. xylinus is high compared to biomass growth and proliferation [16]. Still, this behavior depends on the dissolved oxygen concentration and the static/agitated mode [17]. BC synthesis using fructose as a carbon source by Acetobacter species comprises fructose metabolism to glucose-6-phosphate (G6P) by fructose hexokinase, flow of G6P into the pentose phosphate pathway, and the generation of glucose-1-phosphate (G1P) from G6P, as detailed in [14,18,19].

Batch fermentation by Acetobacter, Gluconacetobacter, and Komagataeibacter species, and bacterial fermentation with the production of exopolysaccharides, has been modeled using unstructured models for biomass, substrate, and product concentrations. Process kinetic modeling is useful for the study, design, optimization, and control of industrial fermentation [20,21,22,23], as it leads to lower experimental effort and accelerated bioprocess development [21]. Bioprocess modeling is useful for designing large-scale processes, based on small-scale process studies that include measuring and simulation [24].

The combined use of mathematical models with properly designed experiments facilitates the evaluation of the system behavior, compared to only laboratory experiments [20]. Mathematical models allow for the prediction of the rates of biomass growth, substrate consumption, and product formation [22], thus facilitating the understanding of the mechanisms of biomass growth and product formation [21,22]. In the fermentation processes, the yield coefficient relates the concentration of biomass to the concentration of substrate, product, and other nutrients, thus allowing for a comparison of the system efficiency [25].

The biomass growth rate models can be classified as structured or unstructured. In structured models, the definition of biomass includes the genetic, morphological, or biochemical features, so that the contents of intracellular components, such as RNA, enzymes, reactants, and products, are involved. In contrast, in unstructured models, the definition of biomass accounts for the interaction of microorganisms with the environment, but the changes that can occur in the inner cells are not considered [24,26]. Therefore, a complete representation of the true complexity of biological processes is not achieved by using unstructured models. However, unstructured models are more suitable for various cases, because training structured models requires obtaining large sets of experimental data for the intracellular components. Unstructured models have been widely used for laboratory and field settings, in applications including the microbial degradation of contaminants and treatment system design [24,27]. A comparison of models is commonly based on Akaike’s Information Criterion (AIC). The AIC is widely used to estimate the model that achieves the best approximation of the true process. It may apply to unrelated (non-nested) models [28,29]. It accounts for the number of parameters and goodness of fit, and it penalizes a considerable number of parameters and poor fitting. The best approximation model is the one with the lowest AIC value [28].

Some studies on modeling Gluconacetobacter/Acetobacter fermentations and exopolysaccharide production by Pantoea sp. are discussed as follows. In [20], vinegar production from vegetable waste by Acetobacter aceti in an agitated batch culture was modeled. A logistic model was used for biomass concentration, whereas Luedeking–Piret equations were used for substrate and product concentrations. The biomass, substrate, and product models were fitted to measurements. However, only the logistic model was used for the specific growth rate (SGR). The results indicate that acetic acid production and substrate consumption rates are proportional to the biomass growth rate and biomass concentration, and acetic acid production is a strong growth-associated process. In [30], the growth of Gluconacetobacter japonicus in an agitated batch fermentation tank with strawberry puree was modeled. A constant specific growth rate was used for the model of biomass growth during the exponential growth phase so that the resulting substrate model was a simple exponential in terms of time. The substrate model was fitted to measurements. The results show that the produced gluconic acid is proportional to the glucose consumed; the glucose consumption rate is proportional to the biomass growth rate; and the glucose consumption is growth-associated. However, model fitting was not performed for biomass, as only a constant specific growth rate was considered in the substrate model, and the gluconic acid concentration was modeled as a linear function of the glucose consumed.

In [31], Komagataeibacter xylinus growth in a batch-stirred tank bioreactor and the effect of agitation rates were studied. The integral form of the biomass growth rate model with Monod expression for the specific growth rate was fitted to data, considering neither product nor biomass inhibition terms. However, model fitting was performed for neither substrate nor product concentration. In [1], bacterial cellulose production by Gluconacetobacter xylinus in a static batch culture was studied, with biodiesel-derived crude glycerol and pure glycerol as carbon sources using magnetic functionalization. The specific production rate of bacterial cellulose production was modeled using several substrate inhibition models, including Haldane, Yano, Aiba, and Edward. The results indicate that Haldane, Yano, Aiba, and Edward models are best for representing the inhibition of the bacterial cellulose production rate for high substrate concentrations. Although the BC production rate was fitted, the models of biomass concentration, substrate concentration, and product concentration were not. In [32], the synthesis of exopolysaccharide (EPS) from waste molasses by Pantoea sp. in an agitated submerged batch culture was studied and modeled. The logistic model was used for the biomass growth rate, whereas the Luedeking–Piret model was used for product formation and substrate consumption, with fitting to the experimental data. The results indicate the following: (i) the stationary phase of biomass growth is accurately described by using the logistic SGR model; (ii) the exopolysaccharide formation and substrate consumption rates are proportional to the biomass growth rate and biomass concentration according to the Luedeking–Piret equations; and (iii) product formation is growth-associated, according to the coefficients of the Luedeking–Piret equation for product formation. However, only the logistic model is used for biomass growth. A comparison of the above modeling studies is given in

Table 1. Comparison of unstructured models for either synthesis of exopolysaccharides (EPS) or Gluconacetobacter/Komagataeibacter process. SGR: specific growth rate.

Microorganisms, Culture Medium, and Culture Conditions	Modeling Features	Reference
Microorganism: Pantoea sp. Culture medium: peptone, Na2HPO4, Triton X-100, H3BO3, ZnCl2, FeCl3, supplemented with sugar beet molasses (SBM). Culture conditions: agitated batch culture; 30 °C temperature; 6.5 initial pH, and 200 rpm agitation.	Features of the model used for training: algebraic equation for biomass, product, and substrate concentrations. It is deduced based on a logistic model for biomass growth, the Luedeking–Piret model for the product, and the Luedeking–Piret model for the substrate. SGR model: the logistic model is used for biomass growth, so that the SGR is a linear function of the biomass concentration with a negative slope. Fitted variables: biomass, product, and substrate concentrations. Main limitations: the logistic model is used for biomass growth, which is overly simple.	[32]
Microorganism: Gluconacetobacter xylinus. Culture medium: yeast extract, peptone, sodium phosphate dibasic, citric acid, and glycerol. Biodiesel-derived crude glycerol and puree glycerol are used as carbon sources. Culture conditions: static batch culture with magnetic functionalization of bacterial cellulose; 30 °C temperature; and 5.0 pH.	Features of the model used for training: algebraic specific production rate as a function of substrate concentration, using Haldane, Yano, and Aiba models. SGR model: --- Fitted variables: specific rate of bacterial cellulose production. Main limitations: the model used for training is the specific production rate, so concentrations of biomass and substrate are not modeled.	[1]
Microorganism: Gluconacetobacter japonicus. Culture medium: commercial strawberry purée. Culture conditions: agitated batch culture; 29 °C temperature; and 3.5 initial pH.	Features of the model used for training: algebraic equation for total glucose; linear equation for gluconic acid as a function of glucose concentration. SGR model: constant. Fitted variables: substrate (glucose) concentration and gluconic acid concentration. Main limitations: the SGR model used is overly simple.	[30]
Microorganism: K. xylinus. Culture medium: fructose, corn steep solid (CSS) solution, and (NH4)2SO4. Fructose is used as a carbon source. Culture conditions: stirred batch culture; 4.5 initial pH; and 30 °C temperature.	Features of the model used for training: Ordinary Differential Equations (ODE) with biomass, substrate, and product concentrations as state variables. SGR model: Monod, Haldane, third Edward’s functions, with and without biomass inhibition terms. Fitted variables: biomass, substrate, and product concentrations. Main limitations: dependence on pH and temperature is missing.	This study

.

In summary, the main limitations of the studies on unstructured models for either the synthesis of bacterial cellulose (BC) or Gluconacetobacter/Acetobacter fermentation are the following: (i) complete model training considering biomass, substrate, and product (bacterial cellulose) concentrations simultaneously and using ordinary differential equations (ODEs) has not been explored; and (ii) a comparison of different specific growth rate (SGR) models has not been explored. This work focuses on using unstructured kinetic models to describe the dynamics of bacterial cellulose production.

The objectives of this study are the following: (i) to determine the suitability of unstructured models for describing experimental data in the BC production process, using ordinary differential equations (ODEs), with biomass, substrate, and product concentrations as state variables; and (ii) to assess the suitability of different specific growth rate (SGR) models for the description of the BC production process, identifying the SGR models that lead to the highest fit quality in terms of AIC and considering different SGR models, including models with substrate inhibition effects, models with biomass inhibition effects, the Monod model, the Moser model, and models combining biomass and substrate inhibition effects. The contributions concerning closely related studies are as follows:

• A stepwise fitting procedure was proposed for model selection, leading to reduced computation effort. It includes a first calibration, in which only biomass and substrate concentrations are included in the cost function; a selection of the three most effective models in terms of AIC (lowest AIC); and a second calibration with only the three selected models, and biomass, substrate, and product concentrations are considered in the cost function.
• Different expressions of the specific growth rate are used, including combined substrate and biomass inhibition functions, and they are compared in terms of the Akaike Information Criterion (AIC).
• A modified product equation is proposed for a 500 rpm agitation rate, using a modified Monod substrate function, leading to improved prediction of BC productivity.

We considered agitated batch BC production as it implies simpler modeling. The model for BC production in static mode includes carbon source concentration in the reservoir zone and carbon source concentration in the aerobic zone. These concentrations are related to the formation of different zones and the diffusion of the carbon source between them [21]. Hence, measurements of both concentrations are required for proper modeling, but they are not provided in current experimental studies. Therefore, we consider agitated batch BC production and selected experimental data accordingly.

2. Materials and Methods

The bacterial cellulose production by K. xylinus in agitated batch culture reported by [31] is considered in this study. The culture medium is corn steep solid (CSS) solution, (NH₄)₂SO₄, and fructose as a carbon source, with 4.5 initial pH and 30 °C temperature. During the exponential growth phase, cells grew exponentially, fructose was consumed rapidly, and dissolved oxygen (DO) concentration declined rapidly. When fructose was completely depleted, cell growth and DO increase stopped, but bacterial cellulose was still produced until the end of cultivation. Bacterial cellulose was a growth-related product.

2.1. Experimental Data

The data used were obtained from [31], where BC was produced by K. xylinus, with fructose as a carbon source. A 3 L stirred bioreactor in batch mode was used, with the culture maintained at 30 °C, subject to 1 vvm air flow rate and impellers at 700 rpm and 500 rpm agitation. The culture medium includes 25 g/L fructose, centrifuged 35 g/L of corn steep solid (CSS) solution, 3.3 g/L (NH₄)₂SO₄, and 4.5 initial pH. Data were extracted through image digitization using WebPlotDigitizer software, WebPlotDigitizer Version 5.1 (https://automeris.io/WebPlotDigitizer/index.html, Accessed on 1 August 2024).

2.2. Statistical Analysis: Performance Metrics

The model performance is evaluated by using the following performance metrics:

• The root mean squared error (RMSE) is obtained as follows [33,34,35]:

  $$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_1^N{\left(Y_{sim}-Y_{exp}\right)}^2}$$
• The determination coefficient is calculated as follows [34,36]:

  $${\mathrm{R}}^2=1-{\displaystyle \frac{\sum _1^N{\left(Y_{sim}-Y_{exp}\right)}^2}{\sum _1^N{\left(Y_{exp}-\overline{Y}\right)}^2}}$$

  where $Y_{exp}$ represents measurements of either biomass, substrate, or product concentrations; $Y_{sim}$ represents simulated values; and $\overline{Y}$ represents the average of measurements. The RMSE and the R² are criteria for qualifying parameter estimation [34]. The RMSE is a standard metric used in model evaluation. RMSE is the square root of the mean squared error (MSE). The MSE is the averaged form of the L2 norm, which is the Euclidean distance. For normal errors, minimizing either MSE or RMSE yields the most likely model. RMSE is a reasonable first choice for normally distributed errors [37]. The coefficient of determination (R²) is a measure of the ‘fit’ of a model to the experimental data. It estimates the capability of a model to represent the relationship between the dependent and independent variables [28,36].

In addition, the Akaike Information Criteria (AIC) allows one to select the best model [38,39,40]:

$$\mathrm{A}\mathrm{I}\mathrm{C}=\left\{\begin{array}{c}N\ln \left({\displaystyle \frac{SSE}{N}}\right)+2\left(N_p+1\right)for{N}_p\le N/40\\ N\ln \left({\displaystyle \frac{SSE}{N}}\right)+{\displaystyle \frac{2N\left(N_p+1\right)}{{N-N}_p-2}}for{N}_p>N/40\end{array}\right.$$

where $SSE$ is the sum of squared errors, $N$ is the number of observations, and $N_p$ is the number of parameters of a model. The best model is the one with the lowest AIC value [38,40]. The AIC is widely used to estimate the model that achieves the best approximation of the true process. It applies to unrelated (non-nested) models. It accounts for the number of parameters and goodness of fit, and it penalizes various parameters and poor fitting [28,29].

2.3. Model Formulation: Mass Balance Models

The biomass formation rate in agitated batch fermentation is [20,41]:

$${\displaystyle \frac{dX}{dt}}=\mu X$$

(1)

where $X$ is the biomass concentration and $\mu$ is the specific growth rate. The Luedeking–Piret equation with constant yield coefficients is used for bacterial cellulose (BC) production rate in batch fermentation [20,41,42,43,44]:

$${\displaystyle \frac{dP}{dt}}=\alpha {\displaystyle \frac{dX}{dt}}+\beta X$$

where $P$ is the BC concentration, $\alpha$ is the synthesis constant of the product associated with microbial growth, $\beta$ is the synthesis constant of product associated with biomass, and $\alpha$ and $\beta$ are constants. A high $\alpha /\beta$ relation indicates that product formation is growth-associated [20]. Substituting the $dX/dt$ expression (1) gives:

$${\displaystyle \frac{dP}{dt}}=\alpha \mu X+\beta X$$

(2)

The substrate consumption rate in batch fermentation includes substrate conversion to biomass, substrate conversion to product, and substrate consumption for maintenance, with constant coefficients [22,41,42,43]:

$${\displaystyle \frac{dS}{dt}}=-\left({\displaystyle \frac{1}{Y_{x/s}}}{\displaystyle \frac{dX}{dt}}+{\displaystyle \frac{1}{Y_{p/s}}}{\displaystyle \frac{dP}{dt}}+m_{s}X\right)$$

where $S$ is the substrate concentration, $Y_{x/s}$ is the yield coefficient of biomass per substrate uptake, $Y_{p/s}$ is the yield coefficient of product per substrate uptake, and $m_s$ is the maintenance coefficient. The terms $Y_{x/s}$, $Y_{p/s}$, and $m_s$ are constants. Substituting the $dX/dt$ Equation (1) and the $dP/dt$ Equation (2) into the above $dS/dt$ equation gives Equation (3):

$${\displaystyle \frac{dS}{dt}}=-\left(k_{11}\mu +k_{12}\right)X$$

(3)

where $k_{11}$ and $k_{12}$ are constants.

2.4. Model Formulation: Specific Growth Rate Expressions

In some unstructured modeling studies, the specific growth rate comprises substrate inhibition function with neither product nor pH inhibition [22,44,45]. Sometimes, the existence of product inhibition effect is certain, but substrate inhibition or substrate monotonically increasing term is used without product inhibition term [44,45]. In addition, some argue that when the specific growth rate ($\mu$) is subject to product inhibition but the measurement of that product is not available, a biomass inhibition term can be used instead [20,46].

In [20], the biomass concentration model is Verhulst logistic, so that the specific growth rate is ${\mu }_{max}\left(1-X/X_m\right),$ where $X$ is the biomass concentration, ${\mu }_{max}$ is the maximum growth rate, and $X_m$ is the maximum biomass concentration, which is achieved at the stationary phase. Both ${\mu }_{max}$ and $X_m$ are constants. The Verhulst logistic model is substrate-independent and leads to a sigmoid curve of the biomass growth [26]. In [46], the specific growth rate involves a substrate-dependent term and the function $1/\left(1+K_{x}X\right)$, where $K_x$ is a positive constant. In [47], the specific growth rate is:

$$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S}}\left(1-{\displaystyle \frac{X}{X_m}}\right).$$

where $S$ is the substrate concentration, $X$ is the biomass concentration, and $K_s$ is a positive constant. Commonly used substrate-dependent-type SGR models with monotonically increasing shapes include the Monod and Moser models [45,48,49]. The Monod model is a function of the limiting nutrient, and its main features are the following: (i) it considers only one limiting substrate; (ii) it does not account for the inhibitory effect of high substrate concentrations; and (iii) it becomes almost constant regarding substrate concentration for high substrate concentrations [26,50]. Monod model leads to a representation of biomass stationary phase as a result of nutrient exhaustion [51]. The Moser model is a modified form of the Monod model involving a power parameter for substrate concentration [45]. Common substrate inhibition expressions include the Haldane and the third function of Edwards [52]:

$${\mu }_{max}{\displaystyle \frac{S}{K_s+S+{\left(1/K_i\right)S}^2}}$$

$${\mu }_{max}{\displaystyle \frac{S}{K_s+S+\frac{S^2}{K_i}\left(1+\frac{S}{K_s}\right)}}$$

where $S$ is the substrate concentration and ${\mu }_{max}$, $K_s$, and $K_i$ are constants. These models increase for low substrate values and decrease for high values [45,52,53,54].

Under the above formulations of specific growth rates, the expressions used in this work combine substrate-dependent functions and biomass inhibition functions. The substrate-dependent functions are Monod, Moser, Haldane, and the third function of Edwards [52]. Since acid concentration measurements are not available in [31], biomass inhibition terms are used instead of the product inhibition term, namely:

$${\displaystyle \frac{1}{1+K_{x}X}},{\left(1-{\displaystyle \frac{X}{X_m}}\right)}^m,\mathrm{and}\left(1-{\left({\displaystyle \frac{X}{X_m}}\right)}^m\right)$$

where $X_m$, $K_x$, and $m$ are positive constants, and the last two terms are modifications of $\left(1-X/X_m\right)$ [55]. As batch cultivation in [31] is under constant temperature, its effect is not considered in the specific growth rate.

The specific growth rate expressions are:

• [M0a]

  $$\mu ={\mu }_{max}{\left(1-{\displaystyle \frac{X}{X_{max}}}\right)}^m$$

  (4)
• [M0b]

  $$\mathsf{\mu }={\mathsf{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{\mathrm{S}}{{\mathrm{K}}_{\mathrm{s}}+\mathrm{S}}}$$

  (5)
• [M0c]

  $$\mathsf{\mu }={\mathsf{\mu }}_{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{\mathrm{S}}{{\mathrm{K}}_{\mathrm{s}}+\mathrm{S}+{\left(1/{\mathrm{K}}_{\mathrm{i}}\right)\mathrm{S}}^2}}$$

  (6)
• [M0d]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S+\frac{S^2}{K_i}\left(1+\frac{S}{K_{s2}}\right)}}$$

  (7)
• [M0e]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S^n}{K_s+S^n}}$$

  (8)
• [M1a]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S}}{\displaystyle \frac{K_p}{K_p+X}}$$

  (9)
• [M1b]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S}}{\left(1-{\displaystyle \frac{X}{X_{max}}}\right)}^m$$

  (10)
• [M1c]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S}}\left(1-{\displaystyle \frac{X}{X_{max}}}\right)$$

  (11)
• [M2a]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S^n}{K_s+S^n}}{\left(1-{\displaystyle \frac{X}{X_{max}}}\right)}^m$$

  (12)
• [M2b]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S^n}{K_s+S^n}}\left(1-{\left({\displaystyle \frac{X}{X_{max}}}\right)}^m\right)$$

  (13)
• [M2c]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S^n}{K_s+S^n}}\left(1-{\displaystyle \frac{X}{X_{max}}}\right)$$

  (14)
• [M3a]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S+{\left(1/K_i\right)S}^2}}{\left(1-{\displaystyle \frac{X}{X_{max}}}\right)}^m$$

  (15)
• [M3b]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S+{\left(1/K_i\right)S}^2}}{\displaystyle \frac{K_p}{K_p+X}}$$

  (16)
• [M4b]

  $$\mu ={\mu }_{max}{\displaystyle \frac{S}{K_s+S+\frac{S^2}{K_i}\left(1+\frac{S}{K_{s2}}\right)}}{\left(1-{\displaystyle \frac{X}{X_{max}}}\right)}^m$$

  (17)

The terms ${\mu }_{max}$, $K_s$, $n$, $X_{max}$, $K_i$, $m$, $K_p$, and $K_{s2}$ are constants. The numbers of parameters of the specific growth rate expressions are: 3 (M0a), 2 (M0b), 3 (M0c), 4 (M0d), 3 (M0e), 3 (M1a), 4 (M1b), 3 (M1c), 5 (M2a), 5 (M2b), 4 (M2c), 5 (M3a), 4 (M3b), and 6 (M4b).

2.5. Parameter Estimation

The coefficients of models M0a (4), M0b (5), M0c (6), M0d (7), M0e (8), M1a (9), M1b (10), M1c (11), M2a (12), M2b (13), M2c (14), M3a (15), M3b (16), and M4b (17) are estimated via model fitting using MATLAB 2014 (Natick, MA, USA). This was performed to obtain a close match between simulation and measurements of biomass, substrate, and product concentrations. The objective function $J$ is minimized using the MATLAB command fmincon, being $J$ defined as:

• For calibration of $X$ and $S$ models

  $$J={\sum }_1^N{\displaystyle \frac{{\left(X_{sim}-X_{exp}\right)}^2}{X_{max}^2}}+{\sum }_1^N{\displaystyle \frac{{\left(S_{sim}-S_{exp}\right)}^2}{S_{max}^2}}$$
• For calibration of $X$, $S$, and $P$ models

  $$\begin{gathered} J={\sum }_1^N{\displaystyle \frac{{\left(X_{sim}-X_{exp}\right)}^2}{X_{max}^2}}+{\sum }_1^N{\displaystyle \frac{{\left(S_{sim}-S_{exp}\right)}^2}{S_{max}^2}} \\ +{\sum }_1^N{\displaystyle \frac{{\left(P_{sim}-P_{exp}\right)}^2}{P_{max}^2}} \end{gathered}$$

  where $X_{exp}$, $S_{exp}$, and $P_{exp}$ are the measurements of biomass, substrate, and product concentrations. The terms $X_{max}$, $S_{max}$, and $P_{max}$ are the maximum of the biomass, substrate, and product concentration measurements. $X_{sim}$, $S_{sim}$, and $P_{sim}$ are the simulations of biomass, substrate, and product concentrations, respectively. Finally, $N$ is the number of measurements.

Simulation values $X_{sim}$, $S_{sim}$, and $P_{sim}$ are obtained by numerical solution of models defined in Equations (1)–(3), with specific growth rates in Equations (4)–(17), using MATLAB ode45 solver. The definition of the cost function $J$ can be found in [39,44,56]. The above definitions of $J$ are the product of $N$ and the sum of the mean squared errors (MSE)

$${\sum }_1^N{\left(X_{sim}-X_{exp}\right)}^2/N,{\sum }_1^N{\left(S_{sim}-S_{exp}\right)}^2/N,{\sum }_1^N{\left(P_{sim}-P_{exp}\right)}^2/N$$

with weights $1/X_{max}^2$, $1/S_{max}^2$, $1/P_{max}^2$, respectively, as can be noticed from [39,44]. The above weights compensate for the different magnitudes of $X$, $S$, and $P$. In addition, they allow one to obtain proper parameter estimation for $X$, $S$, and $P$ models simultaneously, using a single objective function $J$ instead of three objective functions. A flowchart for parameter estimation, simulation, and assessment of model performance is given in Figure 1.

3. Results

The model performance for the 700 rpm agitation rate is presented in

Table 2. Quality-of-fit scores for 700 rpm agitation rate at the first calibration. RMSEx is the RMSE value for biomass concentration ($X$), and RMSEs is the RMSE value for substrate concentration ($S$). R2x is the coefficient of determination for biomass concentration ($X$) and R2s is the coefficient of determination for substrate concentration ($S$).

Model	RMSEx	RMSEs	R2x	R2s	AIC
M0a	0.053761	0.051327	0.97792	0.98378	−185.206
M0b	0.061041	0.042049	0.97153	0.98911	−188.3623
M0c	0.061363	0.044923	0.97123	0.98758	−183.6492
M0d	0.064275	0.043261	0.96844	0.98848	−179.1874
M0e	0.060673	0.038143	0.97188	0.99104	−187.6856
M1a	0.058479	0.038479	0.97387	0.99088	−189.2826
M1b	0.055625	0.0411	0.97636	0.9896	−186.9084
M1c	0.05228	0.041629	0.97912	0.98933	−192.4374
M2a	0.050104	0.03359	0.98082	0.99305	−192.7555
M2b	0.072258	0.059127	0.96011	0.97848	−163.0504
M2c	0.049386	0.03195	0.98137	0.99372	−197.9226
M3a	0.048482	0.031961	0.98204	0.99371	−195.3432
M3b	0.057165	0.034364	0.97503	0.99273	−189.3736
M4b	0.055737	0.037616	0.97627	0.99129	−181.6289

, and a boxplot of the residuals is shown in Figure 2.

The order of the models from the highest to the lowest performance is presented as follows, from the lowest to the highest AIC, from the lowest to the highest RMSE, and from the highest to the lowest R2:

• AIC: M2c, M3a, M2a, M1c, M3b, M1a, M0b, M0e, M1b, M0a, M0c, M4b, M0d, and M2b.
• RMSEx: M3a, M2c, M2a, M1c, M0a, M1b, M4b, M3b, M1a, M0e, M0b, M0c, M0d, and M2b.
• RMSEs: M2c, M3a, M2a, M3b, M4b, M0e, M1a, M1b, M1c, M0b, M0d, M0c, M0a, and M2b.
• R2x: M3a, M2c, M2a, M1c, M0a, M1b, M4b, M3b, M1a, M0e, M0b, M0c, M0d, and M2b.
• R2s: M2c, M3a, M2a, M3b, M4b, M0e, M1a, M1b, M1c, M0b, M0d, M0c, M0a, and M2b.

Therefore, the M2c model has the highest performance in terms of AIC, whereas the M3a and M2c models exhibit the highest performance in terms of R2x and R2s, respectively. The M2b model has the lowest performance in terms of AIC, R2x, and R2s. The R2 value (R2x and R2s) for the M2b model is the lowest. From the boxplot for biomass concentration (Figure 2a), it follows that: (i) M0a has the median (Q2) closest to the zero error and with better symmetry when comparing distances between Q1-Q2 and Q2-Q3; however, this model does not correspond to the lowest interquartile range (IQR). Other models with the median close to zero are M2a, M2c, M3a, M0e, M1b, M1c, M2b and M3b. (ii) The models with a lower IQR and with the median closest to zero are M2b and M0e, followed by the M2a, M2c, M3b, and M3a models. From the boxplot for the substrate concentration (Figure 2b), it follows that (i) the model with the median (Q2) closest to zero is M0d, followed by the M2c, M3a, M2a, and M1c models, and (ii) the models with the lowest IQR and with the median closest to zero are M0d and M2c, followed by M3a and M2a.

Therefore, the three models with the highest performance in terms of AIC (lowest AIC) are M2c, M3a, and M2a. The results of the second calibration include the model parameters given in

Table 3. Estimated parameters and quality-of-fit scores for the M2c, M3a, and M2a models, 700 rpm agitation rate at the second calibration. RMSEx is the RMSE value for biomass concentration ($X$); RMSEs is the RMSE value for substrate concentration ($S$); and RMSEp is the RMSE value for product (bacterial cellulose) concentration ($P$). R2x is the coefficient of determination for biomass concentration ($X$); R2s is the coefficient of determination for substrate concentration ($S$); and R2p is the coefficient of determination for product concentration ($P$).

Model	Estimated Parameters and Confidence Intervals	RMSE	R2
M2c	${\mu }_{max}=0.71205\pm 3.9997$; $K_s=37.1744\pm 6.101$; $n=0.91871\pm 2.5036$; $X_{mx}=7.0587\pm 7.7363$; $k_{11}=2.3147\pm 0.7552$; $k_{12}=0.2371\pm 0.0750$; $\alpha =0.10195\pm 0.02059$; $\beta =0.0017471\pm 0.0005668$.	RMSEx = 0.049568; RMSEs = 0.031743; RMSEp = 0.074324.	R2x = 0.98123; R2s = 0.9938; R2p = 0.94627.
M3a	${\mu }_{max}=0.71275\pm 194.9067$; $K_s=42.5962\pm 617.8332$; $K_i=69.9847\pm 169539.1866$; $X_{mx}=7.4095\pm 452.0619$; $m=0.9477\pm 398.2741$; $k_{11}=1.2746\pm 0.5600;k_{12}=0.37306\pm 0.07769;$ $\alpha =0.098546\pm 0.02074$; $\beta =0.0018538\pm 0.00057564.$	RMSEx = 0.048459; RMSEs = 0.032281; RMSEp = 0.074962;	R2x = 0.98206; R2s = 0.99358; R2p = 0.94534.
M2a	${\mu }_{max}=0.45807\pm 88.3202$; $K_s=25.6801\pm 8127.2933$; $n=1.0437\pm 29.0828$; $X_{mx}=7.8132\pm 494.3498;$ $m=1.211\pm 28.447$; $k_{11}=2.51\pm 0.46$; $k_{12}=0.2197\pm 0.0535;$ $\alpha =0.10129\pm 0.02057$; $\beta =0.0017568\pm 0.0005662.$	RMSEx = 0.049837; RMSEs = 0.033548; RMSEp = 0.074281.	R2x = 0.98102; R2s = 0.99307; R2p = 0.94633.

. The confidence intervals (CIs) were calculated using the mean squared error (MSE) method of confidence intervals. The MSE method leads to large CIs, and several values are more than 100% with respect to parameter estimates [57,58]. In [58], the MSE confidence intervals, including 208% values, confirm local identifiability; a $7.3\times {10}^3$% confidence interval implies that the corresponding parameter is not locally identifiable; parameter accuracy is poor when the CIs pass over 100%. According to these criteria, the accuracy and parameter identifiability of parameters in Table 3 are as follows: parameters ${\mu }_{max}$, $n$, $X_{mx}$, for the M2c model, have low accuracy, as their confidence intervals are more than 100%; parameters ${\mu }_{max}$, $K_s$, $K_i$, $X_{mx}$, $m$, for the M3a model, and ${\mu }_{max}$, $K_s$, $n$, $X_{mx}$, $m$, for the M2a model, are not locally identifiable as their confidence intervals are more than $1.0\times {10}^3$%.

The simulations and measurements of biomass production, substrate consumption, and BC production during batch fermentation assays using the M2c, M3a, and M2a models are shown in Figure 3. Figure 4 presents a boxplot of the residuals. From the boxplot for biomass concentration (Figure 4a), it follows that: (i) the M2c and M2a models have a median closer to zero compared to the M3a model; (ii) the M2c and M2a models have a similar IQR and range and are better than the M3a model. From the boxplot for substrate concentration (Figure 4b), it follows that: (i) M2c has the median closest to zero; (ii) M2c and M2a have similar IQR, but M2c presents better symmetry when comparing distances between Q1-Q2 and Q2-Q3. From the boxplot for BC concentration (Figure 4c), it follows that: (i) the three models have medians near zero; (ii) the M2c and M3a models have a similar IQR and range and are better than the M2a model. Figure 5 shows a plot of observed versus simulated values of $P$, $X$, and $S$, for the M2c, M3a, and M2a models, with a 700 rpm agitation rate in the second calibration.

The performance of models for the 500 rpm agitation rate is presented in

Table 4. Quality-of-fit scores for 500 rpm agitation rate, first calibration. RMSEx is the RMSE value for biomass concentration ($X$); RMSEs is the RMSE value for substrate concentration ($S$); and RMSEp is the RMSE value for product (bacterial cellulose) concentration ($P$). R2x is the coefficient of determination for biomass concentration ($X$); R2s is the coefficient of determination for substrate concentration ($S$); and R2p is the coefficient of determination for product concentration ($P$).

Model	RMSEx	RMSEs	R2x	R2s	AIC
M0a	0.044434	0.056655	0.98589	0.98189	−187.3691
M0b	0.052438	0.038735	0.98034	0.99154	−197.0915
M0c	0.052651	0.039572	0.98018	0.99117	−193.4266
M0d	0.052076	0.038253	0.98062	0.99174	−191.5316
M0e	0.052697	0.03796	0.98015	0.99187	−194.3819
M1a	0.12744	0.079194	0.88392	0.96462	−137.4426
M1b	0.04614	0.038197	0.98478	0.99177	−196.6857
M1c	0.046914	0.037597	0.98427	0.99203	−199.6311
M2a	0.046447	0.037757	0.98458	0.99196	−193.2806
M2b	0.06077	0.070234	0.9736	0.97217	−163.4092
M2c	0.05592	0.031866	0.97611	0.99375	−191.7991
M3a	0.04902	0.03774	0.98282	0.99196	−191.0376
M3b	0.055973	0.039585	0.97761	0.99116	−187.5064
M4b	0.049466	0.040792	0.98251	0.98251	−184.8671

, and a boxplot of the residuals is shown in Figure 6.

The order of the models from the highest to the lowest performance is as follows, from the lowest to the highest AIC, from the lowest to the highest RMSE, and from the highest to the lowest R2:

• AIC: M1c, M0b, M1b, M0e, M0c, M2a, M2c, M0d, M3a, M3b, M0a, M4b, M2b, and M1a.
• RMSEx: M0a, M1b, M2a, M1c, M3a, M4b, M0d, M0b, M0c, M0e, M2c, M3b, M2b, and M1a.
• RMSEs: M2c, M1c, M3a, M2a, M0e, M1b, M0d, M0b, M0c, M3b, M4b, M0a, M2b, and M1a.
• R2x: M0a, M1b, M2a, M1c, M3a, M4b, M0d, M0b, M0c, M0e, M3b, M2c, M2b, and M1a.
• R2s: M2c, M1c, M3a y M2a; M0e, M1b, M0d, M0b, M0c, M3b, M4b, M0a, M2b, and M1a.

Therefore, the M1c model has the highest performance in terms of AIC, whereas the M0a and M2c models have the highest performance in terms of R2x and R2s, respectively. The M1a model exhibits the lowest performance in terms of AIC, R2x, and R2s. The R2 value (R2x and R2s) for the M1a model is the lowest. From the boxplot for biomass concentration (Figure 6a), (i) the M0b, M0c, M0d, M0e, M1b, M1c, M2a, M3a, and M3b models have medians near zero; (ii) the M0a model has the lowest IQR range, but the M0b, M0c, M0d, M0e, M1b, M1c, M2a, M2b, M3a, and M3b models have a lower range compared to other models; and (iii) M2b presents better symmetry when comparing distances between Q1-Q2 and Q2-Q3, considering the low error identified by the IQR.

From the boxplot for substrate concentration (Figure 6b), (i) the M1a and M2c models have medians near zero; (ii) M0e, M1b, M1c, M2a, and M2c have low IQR, but M2c has better symmetry when comparing distances between Q1-Q2 and Q2-Q3 with a median close to zero; and (iii) other models with lower ranges, M0b, M0c, M0d, M3a, and M3b, than the rest of the models have a median error far from the zero value.

The three models with the lowest AIC are M1c, M0b, and M1b. In the first attempt at second calibration, the fitting of the product model in Equation (2) resulted in overlarge residuals. Therefore, BC production in Equation (2) is modified by incorporating a $f_s$ function to the $\alpha$ term:

$${\displaystyle \frac{dP}{dt}}=\alpha f_s\mu X+\beta X$$

(18)

$$f_s={\displaystyle \frac{c_3}{1+e^{-c_1\left(S-c_2\right)}}}$$

(19)

where $c_1$, $c_2$, $c_3$ are constants. The formulation of the above product model is as follows. From the measurements of $P$, $X$, and $S$, it follows that: (i) the BC production rate ($dP/dt$) is higher for $t\le 7.5$ h and is lower for $t\ge 12$ h; (ii) the biomass growth rate ($dX/dt$) is lower for $t\le 7.5$ and is higher for $t\ge 12$ h; and (iii) the substrate concentration ($S$) is higher ($24.04–26.07$) g/L for $t\le 7.5$ h and is lower ($0–21.1$) g/L for $t\ge 12$ h. The above behavior of $dP/dt$ can be expressed in terms of substrate concentration: i) the BC production rate is higher for substrate concentration in the range $\left[24.04–26.07\right]$ g/L and is lower for substrate concentration in the range $\left[0–21.1\right]$ g/L. To achieve a proper representation of this $dP/dt$ behavior, the $f_s$ function is chosen to increase with $S$, fulfilling $f_s\approx 0forS\in [0–21.1]$ g/L and $f_s>0forS\in [24.04–26.07]$ g/L, such that $f_s\mu \approx 0forS\in [0–21.1]$ g/L and $f_s\mu >0forS\in [24.04–26.07]$ g/L, which implies $dP/dt\approx \beta X$ for $S\in [0–21.1]$ g/L. To this end, $f_s$ is defined through a commonly used sigmoid expression, as defined in Equation (19). An illustration of $f_s$, ${\mu }_s$, and $f_s{\mu }_s$ versus substrate concentration $S$ is presented in Figure 7, where ${\mu }_s$ is the substrate-dependent term of $\mu$:

$${\mu }_s={\mu }_{max}{\displaystyle \frac{S}{K_s+S}}$$

for the M1c and M1b models.

Then, product Equation (18) is used instead of Equation (2) in the second calibration, with the estimation of parameters $c_1$, $c_2$, $c_3$. The results include the model parameters given in

Table 5. Estimated parameters and quality-of-fit scores for the M1c, M0b and M1b models for 500 rpm agitation rate, at the second calibration. RMSEx is the RMSE value for biomass concentration ($X$); RMSEs is the RMSE value for substrate concentration ($S$); and RMSEp is the RMSE value for product (bacterial cellulose) concentration ($P$). R2x is the coefficient of determination for biomass concentration ($X$); R2s is the coefficient of determination for substrate concentration ($S$); and R2p is the coefficient of determination for product concentration ($P$).

Model	Estimated Parameters	RMSE	R2
M1c	${\mu }_{max}=0.43596$; $K_s=37.6763$; $X_{mx}=15.0202;$ $k_{11}=1.725$; $k_{12}=0.3859;$ $\alpha =0.19561$; $\beta =0.0010854;$ $c_1=1.4262$; $c_2=24.7182;$ $c_3=4.7823$.	RMSEx = 0.04766; RMSEs = 0.039016; RMSEp = 0.094667.	R2x = 0.984; R2s = 0.99097; R2p = 0.84697.
M0b	${\mu }_{max}=0.45646$; $K_s=45.7344;$ $k_{11}=2.2663$; $k_{12}=0.35638;$ $\alpha =0.20644$; $\beta =0.00098285;$ $c_1=1.1261$; $c_2=25.0636;$ $c_3=5.4439$.	RMSEx = 0.054207; RMSEs = 0.039804; RMSEp = 0.10556.	R2x = 0.9793; R2s = 0.9906; R2p = 0.7928.
M1b	${\mu }_{max}=0.44013$; $K_s=37.3568$; $X_{mx}=14.9667;$ $m=1.1389$; $k_{11}=1.9455;$ $k_{12}=0.3577$; $\alpha =0.30138$; $\beta =0.00098133;$ $c_1=0.95578$; $c_2=25.9594$; $c_3=5.3961$.	RMSEx = 0.04645; RMSEs = 0.039577; RMSEp = 0.10599.	R2x = 0.9848; R2s = 0.99071; R2p = 0.79111.

, and the predictions of biomass, substrate, and product concentrations are shown in Figure 7.

The simulations and measurements of biomass production, substrate consumption, and BC production during batch fermentation assays using the M1c, M0b, and M1b models are shown in Figure 8. In addition, a boxplot of the residuals is shown in Figure 9.

From the boxplot for biomass concentration (Figure 9a), it follows that (i) the three models have medians near zero; (ii) the three models have a similar IQR and range. From the boxplot for substrate concentration (Figure 9b), (i) M1b is the one with the median closest to the zero error; (ii) M1c and M1b have a similar IQR range, but M1b presents better symmetry when comparing distances between Q1-Q2 and Q2-Q3. From the boxplot for BC concentration (Figure 9c), (i) the M0b model exhibits the median closest to zero; (ii) the three models have a similar IQR; and (iii) M1c presents better symmetry when comparing distances between Q1-Q2 and Q2-Q3, but it is not the model with the median closest to zero. Figure 10 shows a plot of observed versus simulated values of X, S, and P for the M1c, M0b, and M1b models, with a 500 rpm agitation rate in the second calibration.

4. Discussion

The biomass mass balance model, the Luedeking–Piret equation for the substrate, and the Luedeking–Piret equation for the product and the SGR models described the dynamic behavior of BC production by K. xylinus. Despite the simplicity of the used mass balance models and the complexity of the fermentation process by K. xylinus, a high R2 is obtained. The R2 and AIC values indicate that the biomass mass balance, the Luedeking–Piret equation for substrate and product equations with the M2c and M1c as SGR models, have a high capability for the description of biomass formation, substrate consumption, and bacterial cellulose formation, for 700 and 500 rpm, respectively. The AIC and R2 values indicate that the M2c and M1c models are the most suitable SGR models for 700 and 500 rpm, respectively. For 500 rpm, it was necessary to modify the specific growth rate in the Luedeking–Piret equation for the product.

The AIC criterion allows for identifying the models with a proper balance between the fitting accuracy and the number of parameters. Since the AIC value depends on both the SSE and the number of parameters, the order of models is different for the AIC and R2 indices, and a higher R2 value does not guarantee a lower AIC for a given model, whereas a high number of parameters may worsen the AIC value. For instance, the highest R2 value for the 500 rpm agitation rate was obtained using the the M2c model, but the lowest AIC was obtained with the M1c model.

The M4b model achieved a low performance in terms of AIC despite its high R2x and R2s values, for both the 700 and 500 rpm cases, which is related to its higher number of parameters. In addition, a high number of parameters does not guarantee a higher R2 value for a given model (see Table 2 and Table 4). For instance, the M4b model has the largest number of parameters, but the highest R2s and R2x are achieved by the M2c and M3a models for a 700 rpm agitation rate and by the M2c and M0a models for a 500 rpm agitation rate.

The higher performance of the M2c and M1c models over more complex SGR models (M2a, M2b, M3a, M3b, and M4b), for 700 and 500 rpm, respectively, in terms of the AIC value, implies that the $\left(1-X/X_m\right)$ biomass function has a higher capability than the ${\left(1-X/X_m\right)}^m$ and $\left(1-{\left(X/X_m\right)}^m\right)$ functions. This is related to the AIC metrics, which depends on the number of parameters and fit quality, and it penalizes a higher number of parameters and poor fitting quality. Hence, the lower number of parameters and higher fitting capability of $\left(1-X/X_m\right)S^n/\left(K_s+S^n\right)$ and $\left(1-X/X_m\right)S/\left(K_s+S\right)$ terms appearing in the M2c and M1c models for 700 and 500 rpm, respectively, led to higher performance in terms of AIC. The higher performance in terms of AIC achieved by the M2c and M1c models, for 700 and 500 rpm, respectively, implies the following:

• A higher performance of monotonically increasing substrate models (Monod and Moser) compared to substrate inhibition models.
• A higher performance of the $\left(1-X/X_m\right)$ biomass function compared to ${\left(1-X/X_m\right)}^m$, $\left(1-{\left(X/X_m\right)}^m\right)$, and $K_p/\left(K_p+X\right)$.

The M2b and M1a models achieved the lowest performance (highest AIC) for 700 rpm and 500 rpm, respectively, which implies a lower performance of $K_p/\left(K_p+X\right)$, $\left(1-{\left(X/X_m\right)}^m\right)$ biomass terms compared to $\left(1-X/X_m\right)$ and ${\left(1-X/X_m\right)}^m$. Comparing the AIC performance of substrate models (M0b, M0c, M0d, and M0e) for both 500 and 700 rpm (Table 4 and Table 5), one obtains the following: (i) the higher performance of the M0b and M0e models over M0c and M0d models implies a higher effectiveness of monotonically increasing substrate models (Monod and Moser) compared to substrate inhibition models (Haldane and Edwards III); (ii) the higher performance of model M0b over the M0c, M0d, and M0e models implies a higher effectiveness of the Monod model over the Haldane, Edwards III, and Moser models; and (iii) the higher performance of the M0e model over the M0c and M0d models implies a higher effectiveness of the Moser model over the Haldane and Edwards III models. For the biomass, substrate, and product differential equations with the M2c and M1c models, the biomass growth is described through the $\left(1-X/X_m\right)S^n/\left(K_s+S^n\right)$ and $\left(1-X/X_m\right)S/\left(K_s+S\right)$ terms appearing in the M2c and M1c models, so that the zero SGR (which corresponds to the stationary phase of biomass) is generated by biomass concentration at its maximum value ($X_{max}$) and zero substrate concentration. In contrast, for substrate inhibition SGR models (M0c, M0d, and M3b), but also for the M0b, M0e, and M1a models, the zero SGR (stationary phase) is generated by the zero substrate concentration. This is related to the metabolic pathway of K. xylinus. Further experimental data at 700 and 500 rpm, given in [31], indicate that, (i) after fructose is exhausted, there is a quite short stationary phase and a large death phase, so that there is no secondary growth phase; (ii) the times for glucose exhaustion and the onset of the stationary phase are quite similar. Therefore, biomass growth has a strong dependence on the carbon source (fructose), whereas the utilization of subproducts for growth is negligible. This agrees with the terms $S^n/\left(K_s+S^n\right)$ and $S/\left(K_s+S\right)$ in the M2c and M1c models.

For the 700 rpm agitation rate, the high performance of the differential equations for product and substrate concentrations implies that the bacterial cellulose production and fructose consumption rates are proportional to the biomass growth rate and biomass concentration through constant coefficients, according to the Luedeking–Piret models. For the M2c model, the $\alpha$ obtained is more than 50-times as large as $\beta$, which indicates that the bacterial cellulose formation is a strong growth-associated fermentation process. In contrast, for the 500 rpm agitation rate, the bacterial cellulose production and fructose consumption rates are not linearly proportional to the biomass growth rate. It is related to the strong uncoupling action that weak acids can exert on various microorganisms, including bacteria [20].

The procedure shown in Figure 1 can be applied to other fermentation assays. However, the application of the models to other Acetobacter or Gluconacetobacter batch fermentation assays requires new calibration using biomass, substrate, and product measurements, followed by model performance comparisons, using the AIC and R2 criteria. Comparing model performance for different studies is not convenient because model performance depends on various factors, including measurement uncertainty and model structure.

Modeling studies of either the synthesis of exopolysaccharides (EPS) or Gluconacetobacter/Komagataeibacter process are minimal. Niknezhad et al. 2022 and Cañete et al. 2016 used an overly simplified specific growth rate (SGR) model, thus leading to algebraic equations for the substrate or product concentration [30,32]. In contrast, in this study, (i) several SGR models are used, including Monod, Haldane, third Edward’s functions, with and without biomass inhibition terms; (ii) ordinary differential equations (ODEs) are used for concentrations of biomass, substrate, and bacterial cellulose; and (iii) the performance of the models is compared by using the AIC.

5. Conclusions

From the results obtained, the M2c and M1c models’ specific growth rates were satisfactory for the representation of the biomass formation, substrate consumption, and bacterial cellulose formation in Acetobacter fermentation, for the 700 rpm agitation rate and 500 rpm agitation rate, respectively, as they achieved the lowest AIC values. In contrast, the M2b and M1a models achieved the lowest performance in terms of AIC value, for the 700 rpm and 500 rpm agitation rates, respectively. These results imply that more complex SGR models with combined substrate and biomass effects, with either an exponential biomass term or inhibition substrate term (M2a, M2b, M3a, M3b, and M4b models), lead to lower performance than the Monod or Moser substrate terms with a linear non-exponential biomass term. The Luedeking–Piret model was suitable for describing substrate consumption and product formation. However, for the 500 rpm agitation rate, uncoupled product formation with a modified Monod substrate function was necessary for obtaining proper product prediction.

The biomass, substrate, and product differential equations with the M2c and M1c models agree with the biomass growth dynamics, which exhibit a strong dependence on the carbon source (fructose), with negligible utilization of subproducts. For the 700 rpm agitation rate, the bacterial cellulose production and fructose consumption rates are proportional to the biomass growth rate and biomass concentration through constant coefficients, and the bacterial cellulose formation is a strong growth-associated fermentation process.

The main limitations of the fitted models are as follows: (i) the effects of pH and temperature are not considered in the SGR models; (ii) the consumption of multiple nutrients is not considered in the specific growth rate; and (iii) the concentration of a subproduct (e.g., a produced acid) is not considered in the ODE. The results (the modeling strategy and the identified models) are a first step in the design of a large-scale bacterial cellulose production process through studies that include process simulation and process optimization for enhanced productivity. Process simulation has the following benefits in large-scale processes: (i) it allows for assessing the system performance under different operation conditions; (ii) it allows for the optimization of operating parameters for feedstock changes; and (iii) it allows better planning and optimization decisions.

The results are a first step for improving large-scale bacterial cellulose production and specifically for (i) optimizing operating parameters, for instance, temperature and pH, including the case that the feedstock is changing, and (ii) assessing certain feeding strategies, for instance, intermittent feeding of a carbon source. Further studies must incorporate the effects of pH, temperature, and acid concentration on biomass growth, substrate consumption, and BC formation. The models identified with the highest goodness of fit were M2c and M1c, for 700 and 500 rpm, respectively. The effect of pH and temperature can be included through pH and temperature functions in the coefficients (${\mu }_{max}$ and $K_s$). In case of further studies with acid concentration measurements, the inhibitory effect of the acid concentration can be included through an inhibition term with structure ${\left(1-A/A_{max}\right)}^m$ instead of the biomass term ${\left(1-X/X_{max}\right)}^m$.