## Materials and Methods
### Discussion about the Obtained Polynomial Models  
Although a black-box model does not allow one to ascertain why some operational variables are influential whereas others are not, it could be interesting to identify the most influential variables and their interaction terms. The influence (statistical significance) of each term in a polynomial equation can be assessed through statistic \(F\), which was used here to decide whether a term was to be included or excluded. By way of example, Table S3.14 in file \"S3.doc\" in Supplementary Materials reveals that the highest \(F\) values were those for \(E_{u1}\) and \(E_{u1}^{2}\). Therefore, the variable \((r_{A})_{global\;est}\) was especially sensitive to the ethanol concentration at the time the first bioreactor was unloaded--it was directly influenced by \(E_{u1}\) and by its quadratic term.  
As a rule, the operational variables associated to the first bioreactor were more markedly influential on most of the dependent variables than were those pertaining to the second. This is unsurprising if one considers that the first reactor not only contributed to the total acetic acid production but also supplied the second with the microorganisms which must work under more extreme conditions in the second bioreactor, since one of the main goals was to deplete ethanol in the medium. Therefore, the conditions prevailing in the first reactor should allow a high concentration of very active acetic acid bacteria to be maintained. As stated in the introduction, such conditions are obtained by keeping the ethanol and acetic acid concentration at not too high levels. It is thus unsurprising that the polynomials used to estimate \((r_{A})_{global\;est}\) and \(P_{m\;est}\) were so strongly dependent on \(E_{u1}\) and \(E_{l1}\) as the latter two variables are directly related to acidity in the reaction medium. \(V_{u1}\) is also highly influential; in fact, the greater the volume unloaded into the second bioreactor is, the more marked will be the potential changes in ethanol and acidity levels in the first as a result of the need for a greater volume of fresh medium for replenishment. As expected, the interaction term \(E_{u1}\)-\(E_{l1}\) in the polynomial for \(EtOH_{m1\;est}\) is especially important; in fact, changes in \(E_{u1}\) and \(E_{l1}\) must have a strong impact on the mean ethanol concentration in each transformation cycle in the first bioreactor.  
One other interesting inference is that the polynomials for \(EtOH_{m1\;est}\) and \(HAc_{m1\;est}\) are complementary; in fact, they only differ in the independent term and in the signs of the others. The sum of the independent term coincides with the overall content of the medium [% (\(v\)/\(v\)) ethanol + % (\(w\)/\(v\)) acetic acid]. Since the total concentration remains constant, in the absence of volatile losses by sweeping--which was the case with our experiments--this result is unsurprising and provides support for the correlation procedure used to develop the equations. Similar reasoning can be applied to \(EtOH_{m2\;est}\) and \(HAc_{m2\;est}\).  
The variable \(E_{d2\;est}\) is of special interest as it is a measure of ethanol depletion in each biotransformation cycle. One aim of the acidification process may be not to operate at the highest possible rate but rather to deplete or nearly deplete the substrate in each cycle--in which case \(E_{d2\;est}\) will be zero or near-zero. Again, the variables \(E_{u1}\), \(E_{l1}\) and \(V_{u1}\) will be especially influential--not directly, but through their interaction terms--, but so will \(T_{2}\) and \(E_{l2}\).  
Once the previously described models have been obtained, the operating conditions can only be optimized, for specific purposes, through a well-designed optimization process using several objective functions. Hence, additional work would be necessary in this regard.