Kinetic Modeling of Dihydroxyacetone Production from Glycerol by Gluconobacter oxydans ATCC 621 Resting Cells: Effect of Fluid Dynamics Conditions

Dihydroxyacetone production from glycerol has been studied. Cultures of Gluconobacter oxydans ATCC 621, a promising microorganism that is able to convert glycerol into dihydroxyacetone, has been employed. In this work, the influence of oxygen transport rate and the fluid dynamic conditions have been studied working with resting cells cultures. Several experiments were carried out at two different scales: 250 mL Erlenmeyer flasks and a 2 L stirred tank bioreactor, varying the agitation speed. Product and substrate concentration were determined employing high-performance liquid chromatography. Additionally, oxygen concentration was measured in the runs carried out in stirred tank reactors. Taking into account the results obtained in these experiments, three different behaviors were observed, depending on the mass transfer and chemical reactions rates. For experiments with low stirring speed (below 200 rpm for shake flasks and 300 rpm for reactors), the oxygen transport rate is the controlling step, while at high stirring speed (over 300 rpm in shake flasks and 560 rpm in the bioreactor), the chemical reaction is controlling the overall process rate. In some runs conducted at medium agitation, a mix control was found. All the kinetic models were able to reproduce experimental data and fulfill thermodynamic and statistical criteria, highlighting the importance of the mass transfer rate upon this system.


Introduction
Glycerol has become a waste due to the enormous availability caused by the huge growth of the biodiesel production industry [1], which produces glycerol as a by-product; approximately 10 kg of glycerol are produced per 100 kg of biodiesel [2]. An estimation of OECD-FAO (Organization for Economic Co-operation and Development-Food and Agricultural Organization) assumes that the global production of biodiesel will grow 4.5% annually, reaching 41 Mm3 in 2022 [2]. This makes the development of different sustainable processes that are able to transform the growing amount of glycerol produced into valuable products peremptory.
Glycerol has been employed in many industries (food, pharmaceutical, cosmetic, chemical, etc.), due to its versatile physicochemical properties [3]. Amongst all these applications, one promising route is the bioconversion of glycerol into chemical products and precursors, such as 1,3-dihydroxyacetone (DHA). DHA is extensively used in the cosmetic industry due to its employment as a building block for the production of various fine chemicals such as 1,3-propanediol and citric acid (being an intermediate compound in this processes) [4], with microbial production being the preferred synthesis pathway, in contrast with the chemical processes, which have expensive safety requirements [5].

Shake Flask Experiments
Figures 1-4 show the evolution of both substrate (glycerol) and product (DHA) concentrations for the different experiments carried out at low-scale (shake flasks). A clear linear tendency with time for DHA production as well as for glycerol consumption can be observed in these figures, at least at short times. However, only at the lowest agitation (100 rpm), this linear tendency is maintained throughout the run; while in the rest of the runs, the linear trend is only observed at shorter times. Furthermore, it is important to note that at the lowest agitation speed, both evolutions-glycerol and DHA concentrations-are independent of the biomass concentration employed, which is a clear sign of the existence of a mass transfer limitation phenomenon affecting the DHA production rate. Finally, it is necessary to point out that the glycerol is totally consumed during the experimental time in all of the runs carried out in this work.  Table 1 On the other hand, as can be seen in Figures 2-4, the linear tendency commented does not last during the entirety of the experiments represented in these figures. Although the linear phase is the same for each of the stirring speeds over all the runs, both the slope and duration of this phase vary with agitation speed. Thus, stirring speed-and therefore, mass transfer rate-had an important impact on DHA production rate for the runs shown in  This effect can also be stated over the specific productivity values (P X ), which are shown in Figure 5. In this figure, it can be appreciated that the effect of the stirring speed is more impactful over P X for the amount of biomass employed in each case, which is in line with the observation previously described for the experimental results. In addition, these results are comparable to the work present in the literature. Jackson et al. reached a P X of 3.21 × 10 −2 mol·g X −1 ·h −1 employing an N of 180 rpm and a biomass concentration of 0.66 g X ·L −1 using commercial glycerol and G. oxydans resting cells [29], whereas in this work, we have reached a value of 2.25 × 10 −2 mol·g X −1 ·h −1 for similar conditions. However, as can be seen in Figure 5, this productivity can be increased substantially through incrementing stirring rate and, therefore, favoring OTR. time for runs 1 to 6 (see Table 1) carried out in shake flasks. All runs conducted at agitation speed of 100 rpm employing different biomass concentrations according to (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, (e) 0.5, and (f) 1.0 g·L −1 .  Table 1 Table 1 Table 1 Table 1) carried out in shake flasks. All runs conducted at agitation speed of 300 rpm employing different biomass concentrations according to (a) 0.    Table 1

STBR Experiments
The results of the runs performed at STBR scale are given in Figure 6. As can be seen, both glycerol and DHA concentrations change with the stirring speed employed in each run. For the lowest agitation (300 rpm), both glycerol and DHA concentration maintained a linear time course

STBR Experiments
The results of the runs performed at STBR scale are given in Figure 6. As can be seen, both glycerol and DHA concentrations change with the stirring speed employed in each run. For the lowest agitation (300 rpm), both glycerol and DHA concentration maintained a linear time course trend.  Table 1 carried out in a stirred tank bioreactor (STBR). All runs conducted with a biomass concentration of 0.5 g·L −1 , employing several stirring speeds according to: (a) 300, (b) 360, (c) 420, (d) 560, (e) 650, and (f) 750 rpm.

Discussion
The results observed in both of the experimental devices employed leads to the following considerations, which have to be addressed in order to formulate an adequate kinetic model to describe DHA production:


There are three different observed patterns, which depend on the stirring speed employed. This is valid for both shake flasks and STBR.  For low agitation, DHA production is strongly related to the stirring speed and, therefore, to the mass transfer rate.  For high stirring speeds, biomass concentration exerts a strong influence on DHA evolution. When the amount of biocatalyst is maintained, DHA evolution is not affected by agitation speed. Thus, the production rate of DHA depends on the chemical reaction rate.  Table 1 carried out in a stirred tank bioreactor (STBR). All runs conducted with a biomass concentration of 0.5 g·L −1 , employing several stirring speeds according to: (a) 300, (b) 360, (c) 420, (d) 560, (e) 650, and (f) 750 rpm.
However, as stirring increases, the evolutions of both compound concentrations change into hyperbolic type, in which the first part of the time-course evolution for both concentrations resembles a linear evolution with a higher slope than the first experiment at 300 rpm, and it is the same tendency observed in the runs carried out in shake flasks.
For the runs with higher agitation, DHA evolution is the same, which is maintained despite the stirring speed increase. It is important to point out that for all the experiments shown in Figure 6, the biomass concentration employed remains the same (0.5 g·L −1 ).
As with the experiments carried out in shake flasks, P X was evaluated for each run, and the results are shown in Table 2. As can be seen in this table, the productivity reached for these experiments is lower than that obtained in flask experiments. However, it is clear that the same tendency can be observed in both cases, as the productivity increases with the stirring rate.

Discussion
The results observed in both of the experimental devices employed leads to the following considerations, which have to be addressed in order to formulate an adequate kinetic model to describe DHA production: • There are three different observed patterns, which depend on the stirring speed employed. This is valid for both shake flasks and STBR.

•
For low agitation, DHA production is strongly related to the stirring speed and, therefore, to the mass transfer rate.

•
For high stirring speeds, biomass concentration exerts a strong influence on DHA evolution. When the amount of biocatalyst is maintained, DHA evolution is not affected by agitation speed. Thus, the production rate of DHA depends on the chemical reaction rate.

•
There is a mixed behavior when the agitation takes a medium value; in this case, the process rate seems to depend on the oxygen availability, or the OTR from gas to the culture media, which is directly related to the stirring speed.
Taking into account these factors, several different kinetic models were proposed to describe each one of the situations described above.

Kinetic Model for Low Agitation: Mass Transfer Rate Evaluation
OTR can be evaluated by means of the oxygen mass balance [27,28], according to Equation (1): where C O2 is the oxygen concentration in the liquid phase (mol·L −1 ) and C * O2 is the saturation oxygen concentration (mol·L −1 ).
For the experiments carried out at low agitation, it can be assumed that the concentration of dissolved oxygen (DO) will be very low or negligible due to the slower income of oxygen compared to the oxygen consumption by the culture. Therefore, the DHA production rate will be strongly related to OTR. Therefore, the DHA production rate can be described by Equation (2): where C DHA is DHA concentration (mol·L −1 ) and k L a is the volumetric mass transfer coefficient (h −1 ). As predicted by Equation (2), a linear evolution of C DHA with time is expected. Thus, integrating Equation (2), assuming an initial DHA concentration = 0, Equation (3) is obtained: where t is time (h). This tendency can be observed in two different ways in the experiments carried out in this work: Thus, Equation (2) can be used to describe the DHA production rate for the cases in which this linear tendency is observed throughout the whole process, which is evident in runs 1 to 6 (in shake flasks, at 100 rpm stirring speed) and 19 (in STBR, at 300 rpm stirring speed).
In Figure 6, the experimental results of these experiments are represented versus the results achieved using Equation (3) in both cases. The kinetic and statistical parameters values are given in Table 3. Table 3. Kinetic and statistical parameters calculated for low agitation experiments with mass transfer control of the overall process rate. Runs 1 to 6 (shake flasks) and 19 (STBR).

Run
C X (g·L −1 ) C * O2 (mol·L− 1 )·10 4 k L a·10 3 ·(h −1 ) r 2 As can be seen in Figure 7, the kinetic model (Equation (3)) is able to reproduce the experimental results of all the runs carried out at low agitation speeds in a suitable way. The results presented in Table 3 show a very small deviation of the kinetic parameter, k L a, and a value of r 2 very close to 1. Furthermore, as can be seen in Figure 7a, the biomass concentration does not exert any influence on DHA production rate. Hence, it is possible to say that for this stirring speed, the mass transfer rate is controlling the overall process rate.
On the other hand, it is possible to estimate the value of k L a for the rest of experiments. However, it is necessary to employ only the initial linear stage, since the DHA concentration time evolves with time from a linear to a hyperbolic trend, as has been shown in Figures 2-4 and Figure 6.
The results achieved for k L a calculation are presented in Table 4. With the results shown in this table, k L a values were fitted to agitation speed in order to correlate the stirring speed with the mass transfer rate. These correlations are given by Equations (4) and (5).  the effect of stirring speed depending on a function that is almost linear in both cases. While Equation (4) is a reasonable relationship [16,30], Equation (5) is not, which is surely due to the kLa values having been calculated under conditions where mass transfer is not the limiting step; thus, these values are not realistic [16]. Therefore, this method is useful to evaluate the impact of the mass transfer rate over the DHA production rate in a rapid and simple way. As can be seen in Table 4, all of the calculated k L a values are directly related to the agitation speed. Furthermore, as given by Equations (4) and (5), these relationships are similar in both systems, the effect of stirring speed depending on a function that is almost linear in both cases. While Equation (4) is a reasonable relationship [16,30], Equation (5) is not, which is surely due to the k L a values having been calculated under conditions where mass transfer is not the limiting step; thus, these values are not realistic [16]. Therefore, this method is useful to evaluate the impact of the mass transfer rate over the DHA production rate in a rapid and simple way.

Kinetic Model for High Agitation: Chemical Reaction Rate Calculation
As the reaction of the transformation of glycerol into DHA is carried out by a membrane-embedded enzyme [18], it can be described by a Michaelis-Menten-type kinetic model. Accordingly, DHA production model is stated by Equation (6): where k C is the Michaelis-Menten (M-M) kinetic constant (L·g X −1 ·h −1 ), C G is glycerol concentration (mol·L −1 ); and K G is the M-M affinity constant (mol·L −1 ). This kinetic model assumes that the enzymes on the membrane have enough oxygen available, which is an essential co-factor for the DHA production reaction. Therefore, this kinetic model only can be applied for high stirring speeds, and will be used only in runs 16 to 18 (agitation of 400 rpm in shake flasks) and runs 20 to 24 (stirring speed higher than 560 rpm in STBR).
The results of the kinetic model fitting to experimental results are shown in Figure 8, and the kinetic and statistical parameters are enclosed in Table 5. As can be appreciated in Figure 8, the model is able to reproduce the experimental results very well, being able to describe the results of different runs carried out at several stirring speeds and employing different biomass concentrations in shake flasks and STBR, respectively. On the other hand, for both scales, the kinetic model satisfies statistical criteria, with very low sum of residuals (SQR), standard error of estimates (SEE), and residual mean square error (RMSE) values, while the values of VE reaches very high values (more than 98%).

Kinetic Model for Intermediate Cases: Oxygen Concentration Effect
Although the kinetic models formulated for the previously commented fluid dynamic conditions are able to describe experiments conducted at both low and high stirring speeds, in the experimental program, there are experiments performed under intermediate conditions, and the results of these runs cannot be described using the previous kinetic models. In these experiments, the agitation and stirring speeds are neither high nor low. Consequently, in these runs, the OTR and chemical reaction rate will have a similar impact on the DHA production rate. Thus, it is necessary to propose another different kinetic model that is able to reflect the impact of oxygen availability on the DHA production rate.
For this purpose, it was mandatory to evaluate the DO concentration in the culture media during each run. Unfortunately, this was not possible for shake flasks experiments due to a lack of the required equipment. Thus, the next model was obtained only for the results corresponding to STBR experiments. The evolution of DO concentration is given in Figure 9, in which some interesting trends can be observed:

•
In the run with less agitation (300 rpm), DO concentration reduces to 0 shortly after the start of the experiments. This reinforces the idea that in this case, there is a shortage in oxygen availability caused by the low mass transfer rate.

•
In the experiments performed with more than 560 rpm, it can be seen that the DO concentration never reaches 0. Therefore, it is safe to say that the OTR is far greater than the DHA production rate (chemical reaction).

•
There are two runs, corresponding to those carried out at 360 and 420 rpm, in which the DO concentration reaches 0 in the first 3 h. However, in both cases, this oxygen shortage only lasts for a short time period. • Therefore, it is necessary to take into account the DO concentration in order to explain the DHA production rate. Thus, Equation (7) was formulated with these premises: where C O2 is DO concentration (mol·L −1 ) and n is an empirical exponent.   The results achieved by fitting Equation (7) to the experimental results of runs 20 and 21 are shown in Figure 10, and the kinetic and statistical parameters obtained are given in Table 6.

Microorganism
The strain employed in this work is Gluconobacter oxydans ATCC 621, which was provided by the Spanish Type Culture Collection (CECT 360). Bacteria were stored in a glycerol solution stock at −80 °C. As the experiments were carried out at resting cell conditions, in which the bacteria cells are able to metabolize glycerol into DHA but can not replicate due to the lack of nitrogen source, it was necessary to grow the cells previously in two successive growth steps, employing the same methodology described in previous works [16,31].

Experimental Procedure
Two different experimental setups were employed in this work. In the first one, resting cells  As can be seen in Figure 10, the experimental results of both runs are perfectly described by the model shown in Equation (7). This is also reflected by the values of the kinetic and statistical parameters, which can be seen in Table 6. This result highlights even more the importance of the OTR on DHA production, especially in the cases in which the oxygen availability in the culture is not oversaturated. Furthermore, it has been demonstrated that in order to describe and to maximize the DHA production rate, it is critical to choose OTR values that meet the oxygen requirements of the enzyme to perform the glycerol transformation at the maximum reaction rate.

Microorganism
The strain employed in this work is Gluconobacter oxydans ATCC 621, which was provided by the Spanish Type Culture Collection (CECT 360). Bacteria were stored in a glycerol solution stock at −80 • C. As the experiments were carried out at resting cell conditions, in which the bacteria cells are able to metabolize glycerol into DHA but can not replicate due to the lack of nitrogen source, it was necessary to grow the cells previously in two successive growth steps, employing the same methodology described in previous works [16,31].

Experimental Procedure
Two different experimental setups were employed in this work. In the first one, resting cells were transferred to 250 mL Erlenmeyer-type (shake flasks), with a working volume of 50 mL. The medium used in these experiments contains 20 mM of acetate buffer (initial pH of 5.5) and 25 g·L −1 (0.27 mol·L −1 ) of glycerol (supplied by Panreac AppliChem, Barcelona). The temperature was fixed at 30 • C, and the agitation was varied from 100 to 400 rpm. In the second experimental setup, a 1 L STBR with a working volume of 0.5 L was employed. The medium used in this setup was the same as the one previously described, as well as the temperature. The stirring speed was changed from 360 to 750 rpm, and the aeration was kept at 0.5 L·min -1 of the air flow rate. All of the experiments described in this work have been performed in triplicate, and only the mean value has been presented. The experimental error for all of the data was below 5%.

Analytical Methods
The concentration of biomass was measured employing spectrophotometry, as the relationship between the biomass concentration and the absorbance at 600 nm is linear. Both the equation employed to relate the biomass concentration to the absorbance and the methodology used to this purpose were stated elsewhere [15,16].
DHA, glycerol, and glyceric acid were determined by HPLC, with an Agilent 1100 series and a Rezex RHM Monosaccharide H+ column. Temperature was maintained at 80 • C, with an 0.5 mL·min −1 of elution rate using 1.2 mM H2SO4 acid water dilution as the mobile phase and 15 µL of injection volume. For DHA concentration determination, a diode array detector (DAD) was employed at 270 nm to measure DHA concentration, with a retention time of 16.8 min, and a refractive index detector (RID) was employed to measure the other compounds, with a retention time of 17.6 and 15.3 min for glycerol and glyceraldehyde, respectively. Commercial glycerol, DHA, and glyceric acid were employed for calibration purposes (supplied by Panreac AppliChem, Barcelona).

Mathematics and Calculations
Productivity as a specific rate of DHA bioproduction (P X ) was evaluated for each experiment employing Equation (8): where C max DHA (mol·L −1 ) is the maximum DHA concentration for each experiment, and t max (h) is the time in which this concentration is reached.
To calculate k L a in the runs in which the mass transfer rate exerts control over the global process rate, as well as in the applicable zones in which this control is not maintained throughout the whole reaction time, OriginLab software was employed, using a linear fitting algorithm. This software was also used for the correlation studies between k L a and stirring speed.
To fit the kinetic models to experimental data, the Aspen Custom Modeler © software was employed. For the purpose of validation of the different models and in order to quantify the goodness of the fit, several statistical parameters were employed, such as the following referring to DHA concentration: Fischer's F criterion (F), sum of residuals (SQR), standard error of estimates (SEE), residual mean square error (RMSE), and percentage of explained variance (VE). All of these statistical parameters are defined in Appendix A.

Conclusions
In summary, there is an important effect of OTR on DHA production rate employing G. oxydans as the biocatalyst. For low or very low mass transfer rates, the DHA production rate is controlled by the mass transfer rate, limiting the productivity of DHA in each experiment. However, once the maximum mass transfer rate is achieved, an increase in this rate by increasing agitation does not affect the DHA production rate, which is affected by other different factors related to the chemical reaction rate, such as biomass concentration and product/substrate inhibition, as previously reported in the literature [7,14,18,22,23]. These achieved results are in line with the previously published results regarding this system [16].

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Fischer's F criterion refers to the adequacy of the model to the observed values of the chosen variable, and it is based on the null hypothesis, and expressed by Equation (A1).
where y i , calc is the observed variable value predicted by the model, N is the number of data, P is the parameter number, and SQR is defined by Equation (A2).
where y i ,exp is the experimental value for the observed variable and y i ,cal is the value predicted by the model. SEE is calculated by Equation (A3).