## 2 Materials and Methods
### Kinetic Modelling  
To understand the kinetics of changes in lactic acid yield (La\({}_{\rm e}\)) in this study, a mathematical model was adopted that follows the kinetics of the subsequent reaction. The use of this model has been widely adopted to describe bio-processes, especially processes describing the formation of lactic acid [14; 50].  
In the first stage of conducting stem fermentation in the presence of LAB and molasses (which is a source of 6-carbon sugars), the sugars are bio-transformed to lactic acid, after which the LA formed is then bio-transformed to the corresponding metabolites. In a simplified way, the course of lactic acid formation and disappearance is described in Equation (2) below:  
\[Sug\stackrel{{ k_{1}}}{{\rightarrow}}LA\stackrel{{ k_{2}}}{{\rightarrow}}Met \tag{2}\]  
where:  
\(Sug\)--sugar,  
\(LA\)--lactic acid,  
\(Met\)--metabolite.  
The following Equation (3) describes changes in lactic acid efficiency (\(Y_{LA}\)) during the fermentation process (3):  
\[Y_{LA}=\frac{100\%\cdot k_{1}}{k_{2}-k_{1}}.(exp(-k_{1}\cdot t)-exp(-k_{2} \cdot t)) \tag{3}\]  
where:  
\(k_{1}\)--lactic acid (LA) formation rate constant,  
\(k_{2}\)--lactic acid (LA) disappearance rate constant [day\({}^{-1}\)],  
\(t\)--fermentation time [day].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters of the** & & & \\
**Fermentation Process** & **Unit** & **Ranges of Change** \\ \hline  & & _L. reuteri_ ML\_0168 \\  & & _L. salivarius_ LY\_0652 \\  & & _L. brevis_ LY\_1120 \\ Type of microorganism & - & _L. acidophilus_ MI-0078 \\  & & _L. rhamns_ MI-0272 \\  & & _L. plantarum_ MI-0102 \\ Initial sugar content & \% & 1.60–4.60 \\ Plant raw material content & \% & 0.032–12.80 \\ Fermentation time & days & 1–10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Technological parameters affecting the fermentation process of stems and the ranges of their changes.  
The lactic acid efficiency variation (\(Y_{LA}\)) function reaches its maximum (\(t_{max}\)) at the point defined in Equation (4):  
\[t_{max}=\frac{ln\frac{k_{2}}{k_{1}}}{k_{2}-k_{1}} \tag{4}\]  
where:  
\(k_{1}\)--lactic acid (LA) formation rate constant,  
\(k_{2}\)--lactic acid (LA) disappearance rate constant [day\({}^{-1}\)],  
\(t\)--fermentation time [day].  
The time (\(t_{max}\)) at which the lactic acid yield function reaches its maximum value was determined by calculating the reaction rate constants (k\({}_{1}\) and k\({}_{2}\)), which were determined through nonlinear estimation using the computer programme STATISTICA 13.3.  
In this study, we adopted a mathematical model for the kinetics of changes in antioxidant activity (AA) and total polyphenol content (TPC). This model mimics the kinetics of a first-order reversible reaction (5), for which the integral equation (\(Y_{i}\)) takes the form shown in Equation (6).  
\[RS\overset{k}{\leftrightarrow}R+S \tag{5}\]  
\[Y_{i}=Y_{ie}\cdot(1-exp(-k_{i}\cdot t)) \tag{6}\]  
where:  
\(Y\)--the corresponding parameter (AA or TPC),  
\(k\)--reaction rate constant for an increase in the relevant parameter (AA or TPC),  
The subscript e indicates the final (equilibrium) value.  
The use of this model (based on mass exchange) has been widely adopted to describe the evaluation of the antioxidant activity of antioxidants [51; 52].  
Earnings (X\({}_{i}\)) were defined as (7):  
\[X_{i}=\frac{(Y_{i}-Y_{i0})\cdot V}{m} \tag{7}\]  
where:  
\(m\)--mass of plant raw material [g],  
\(V\)--volume of FSE [L],  
\(Y_{i0}\)--parameter values for time 0.  
We assumed that the value X\({}_{i}\) = X\({}_{i0}\) corresponds to a raw material content of 0.032%. We determined the contents of the components responsible for AA or TPC in the obtained FSEs after the fermentation process.  
\[X_{i}=X_{i0}-\frac{(Y_{i}-Y_{i0})\cdot V}{m} \tag{8}\]  
The shape of the curves corresponds to the Langmuir adsorption equilibrium model (8):  
\[X=X_{0}\frac{b\cdot Y}{1+b\cdot Y} \tag{9}\]