## 2 Materials and Methods
### Kinetic Model Development and Determination of Kinetic Parameters
#### 2.5.3 Sugar Consumption Kinetics  
The substrate consumption rate was represented by modified a Luedeking-Piret (MLP) model (Equation (14)) which takes account of substrate conversion to the cell mass and product and substrate consumption for maintenance [37]. MLP model was utilized to fit the actual data of substrate consumption in PCS-biofilm reactors including Media A-E under optimized conditions.  
\[-\frac{dS}{dt}\ =\ \frac{1}{Y^{0}{}_{X/S}}\frac{dX}{dt}+\frac{1}{Y^{0}{}_{P/S}} \frac{dP}{dt}+mX \tag{14}\]  
where \(m\) represents the substrate used to support cell maintenance. If Equation (9) is written in Equation (14), Equation (15) is obtained.  
\[-\frac{dS}{dt}\ =\ \Bigg{[}\frac{1}{Y^{0}{}_{\frac{S}{S}}}+\frac{a}{Y^{0}{}_{ \frac{S}{S}}}\Bigg{]}\frac{dX}{dt}+\Bigg{[}\frac{\beta}{Y^{0}{}_{\frac{S}{S}} }+m\Bigg{]}X \tag{15}\]  
At stationary phase, \(dX/dt=0\) and \(X=X_{max}\). From Equation (15),  
\[\frac{\beta}{Y^{0}{}_{P/S}}+m\ =\ Z\ =\ \frac{-(dS/dt)_{stationary}}{X_{max}} \tag{16}\]  
The substrate (\(S\)) is obtained as a function of time from Equation (15) as follows (Equation (17)).  
\[-dS\ =\ \Bigg{[}\frac{1}{Y^{0}{}_{X/S}}+\frac{a}{Y^{0}{}_{P/S}}\Bigg{]}dX+Z \int X(t)d(t) \tag{17}\]  
If Equation (7) is written in Equation (17) and is integrated at \(t=0\) for \(S=S_{0}\), the following equation is obtained (Equation (18)).  
\[S\ =\ S_{0}-\left[\frac{1}{Y^{0}\frac{1}{\xi}}+\frac{a}{Y^{0}\frac{b}{\xi}}\right] X_{0}\left(\frac{e^{\mu_{max}t}}{1-\frac{X_{0}}{X_{max}}(1-e^{\mu_{max}t})}-1 \right)-Z\frac{X_{max}}{\mu_{max}}ln\left(1-\frac{X_{0}}{X_{max}}\left(1-e^{\mu_ {max}t}\right)\right). \tag{18}\]  
To find \(\left[\frac{1}{Y^{0}X/S}+\frac{a}{Y^{0}\frac{b}{Y^{0}\frac{b}{Y^{0}}}}\right]\ =\ \gamma\), Equation (18) is written as follows (Equation (19)).  
\[S\ =\ S_{0}-\gamma A(t)-ZB(t). \tag{19}\]  
When \([S_{0}-S-ZB(t)]\) is plotted versus \(A(t)\), \(\gamma\) is found from the slope of the plot (Figure 1C,F,I,L,O).  
Besides, the following kinetics [2,8,9] were also calculated using the predicted data and compared with the experimental kinetics (Equations (20)-(31)):  
\[SUY\ =\ (\Delta S/S_{max})\times 100 \tag{30}\]  
\[\eta\ =\ (Y_{P/S}/51.1)\times 100 \tag{31}\]  
where, \(\Delta X\) is the biomass production (g/L), \(\Delta P\) is the ethanol production (g/L), \(\Delta S\) is the sugar consumption (g/L), \(X_{min}\), \(P_{min}\), and \(S_{min}\) are minimum biomass, ethanol, and sugar concentrations (g/L), respectively, \(X_{max}\), \(P_{max}\), and \(S_{max}\) are maximum biomass, ethanol, and sugar concentrations (g/L), respectively, \(Y_{X/S}\) is the biomass yield (%),\(Y_{P/S}\) is the ethanol yield (%),\(Y_{P/X}\) is the ethanol production per biomass (g ethanol/g biomass), \(Y_{S/X}\) is the sugar consumption per biomass (g sugar/g biomass), \(Q_{X}\) is the maximum growth rate (g/L/h), \(Q_{P}\) is the maximum production rate (g/L/h), \(Q_{S}\) is the maximum consumption rate (g/L/h), \(\mu_{max}\) is the maximum specific growth rate (h\({}^{-1}\)), \(SUY\) is the sugar utilization yield (%), and \(\eta\) is the theoretical ethanol yield (%). Additionally, the lag time (\(\lambda\), h) was also calculated. Based on Figure 2, the linear equation of the steepest part of the biomass was obtained in the form of \(y=ax+b\). Here, \(y\) is the lower asymptote for biomass and \(x\) is the lag time. Therefore, as the \(y\), \(a\), and \(b\) values are known, the \(x\) value (\(\lambda\)) is easily computed [2,38].