## Materials and Methods
### Mathematical Methods
#### 2.6.1 Polynomial Models and Fitting Methods
\begin{table}
\begin{tabular}{l c} \hline \hline
**Variable** & **Value** \\ \hline Total cycle duration (h) & \(39.7\pm 0.8\) \\ Duration of the fast loading stage in the first bioreactor (h) & \(0.9\pm 0.1\) \\ Duration of the slow loading stage in the first bioreactor (h) & \(28.3\pm 0.6\) \\ Duration of the depletion stage in the first bioreactor (h) & \(10.5\pm 1.0\) \\ Duration of the fast loading stage in the second bioreactor (h) & \(0.2\pm 0.1\) \\ Duration of the slow loading stage in the second bioreactor (h) & \(39.4\pm 1.2\) \\ Time fraction of the fast loading stage in the first bioreactor over the cycle duration & \(0.023\pm 0.003\) \\ Time fraction of the slow loading stage in the first bioreactor over the cycle duration & \(0.713\pm 0.021\) \\ Time fraction of the depletion stage in the first bioreactor over the cycle duration & \(0.264\pm 0.026\) \\ Time fraction of the fast loading stage in the second bioreactor over the cycle duration & \(0.006\pm 0.001\) \\ Time fraction of the slow loading stage in the second bioreactor over the cycle duration & \(0.994\pm 0.043\) \\ Mean volume in the fast loading stage in the first bioreactor (L) & \(4.21\pm 0.05\) \\ Mean volume in the slow loading stage in the first bioreactor (L) & \(6.35\pm 0.05\) \\ Mean volume in the depletion stage in the first bioreactor (L) & \(8.02\pm 0.05\) \\ Mean volume in the first bioreactor (L) & \(6.74\pm 0.25\) \\ Mean volume in the fast loading stage in the second bioreactor (L) & \(4.38\pm 0.05\) \\ Mean volume in the slow loading stage in the second bioreactor (L) & \(5.86\pm 0.05\) \\ Mean volume in the second bioreactor (L) & \(5.85\pm 0.25\) \\ Mean overall volume during a cycle in the two bioreactors as a whole (L) & \(12.59\pm 0.35\) \\ Mean volume of fermentation medium unloaded from the second reactor (L) & \(7.2\pm 0.05\) \\ Final ethanol concentration at the time the second bioreactor was unloaded (\% v/v) & \(2.3\pm 0.2\) \\ Mean ethanol concentration in the first bioreactor (\% v/v) & \(3.7\pm 0.1\) \\ Mean ethanol concentration in the second bioreactor (\% v/v) & \(3.9\pm 0.1\) \\ Final acetic acid concentration at the time the first bioreactor was unloaded (\% v/v) & \(9.4\pm 0.2\) \\ Final acetic acid concentration at the time the second bioreactor was unloaded (\% v/v) & \(9.2\pm 0.2\) \\ Mean acetic acid concentration in the first bioreactor (\% w/v) & \(7.8\pm 0.1\) \\ Mean acetic acid concentration in the second bioreactor (\% w/v) & \(7.6\pm 0.1\) \\ Mean rate of acetic acid formation in the first bioreactor (\% w/v-h\({}^{-1}\)) & \(0.15\pm 0.01\) \\ Mean rate of acetic acid formation in second bioreactor (\% w/v-h\({}^{-1}\)) & \(0.11\pm 0.01\) \\ Mean overall rate of acetic acid formation in the two bioreactors as a whole (\% w/v-h\({}^{-1}\)) & \(0.13\pm 0.01\) \\ Total acetic acid production in the first bioreactor (g acetic acid-h\({}^{-1}\)) & \(10.1\pm 0.3\) \\ Total acetic acid production in the second bioreactor (g acetic acid-h\({}^{-1}\)) & \(6.6\pm 0.5\) \\ Total acetic acid production in the two bioreactors as a whole (g acetic acid-h\({}^{-1}\)) & \(16.7\pm 0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The results obtained in Experiment 1. A comparison of the \((r_{A})_{global}\) predictions with the experimental values (Figure 5, which is the same as Figure S3.1 in in file \"S3.docx\" in Supplementary Materials) reveals that the two coincided exactly in 10 of the 18 experiments and differed only very slightly in the other 8. Also, all predictions fell within the 95% prediction interval. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Exp. No. ** & \(\mathbf{(r_{A})_{global}}\)**, g Acetic** & \(\mathbf{P_{m}}\)**, g Acetic** & \(\mathbf{E_{u2}}\)**, \%(r/v)** & \(\mathbf{V_{u2}}\)**, L** & \(\mathbf{t_{cycle}}\)**, h** & \(\mathbf{V_{m}}\)**, L** \\  & **Acid (100 mL-h)-1** & **Acid-h-1** & **\(\mathbf{E_{u2}}\)**, \%(r/v)** & \(\mathbf{V_{u2}}\)**, L** & \(\mathbf{t_{cycle}}\)**, h** & \(\mathbf{V_{m}}\)**, L** \\ \hline
1 & \(0.13\pm 0.01\) & \(16.