2.5.2. Determination of Biomass Concentration

The measurement of biomass was followed by estimation of cell dry weight, expressed in g/L. one mL of yeast culture was centrifuged at 5000 rpm for 5 min. The supernatant obtained was washed twice with water and dried by incubation at 105 ◦C until at a constant weight [27].

## *2.6. Modeling*

Unstructured kinetic models using Monod, Verhulst, and Tessier [28] (Table 4) have been implemented to fit the experimental data. Kinetic parameters (*µmax*, *Ks* and *Xm*), were determined using the curve fitting method of each model. The fitness evaluation of experimental data on cell growth by models was performed using Excel software (Microsoft, Redmond, WA, USA).

#### Profile Prediction of Biomass and Substrate Concentration

To predict the experimental profile of biomass of *S. cerevisiae* during time fermentation, the integration of the Verhulst model was used to give a sigmoidal variation of *X* as a function of *t*, which may represent both an exponential and a stationary phase (Equation (3)):

$$\mathbf{X} = \frac{\mathbf{X}\_0 \mathbf{e}^{\mu\_m t}}{\left\{1 - \left(\mathbf{X}\_0 / \ \mathbf{X}\_m\right) \left(1 - \mathbf{e}^{\mu\_m t}\right)\right\}}\tag{3}$$

In addition, the substrate model (Leudeking Piret) as described below (Equation (4)) was also applied to predict an experimental profile for total reducing sugars consumption by *S. cerevisiae* during time.

$$-\frac{d\mathbf{S}}{dt} = p\frac{dX}{dt} + qX \tag{4}$$

where *p* = 1/*YX/S* and *q* is a maintenance coefficient.


**Table 4.**Unstructured kinetic models to determinate the kinetic parameters.

Equation (4) is rearranged as follows:

$$-dS = p\,dX + q\int X(t)dt\tag{5}$$

Substituting Equation (3) in Equation (5) and integrating with initial conditions (*S* = S0; *t* = 0) give the following Equation:

$$S = S\_0 - pX\_0 \left\{ \frac{e^{\mu\_m t}}{\left\{1 - \left(\frac{X\_0}{X\_m}\right) (1 - e^{\mu\_m t})\right\}} - 1 \right\} - q \frac{X\_m}{\mu\_m} \ln\left\{1 - \frac{X\_0}{X\_m} \left(1 - e^{\mu\_m t}\right)\right\} \tag{6}$$
