## 3 Results and Discussion
### Plasmid Release Kinetics  
Figure 5 shows the experimental data of the release of pDNA(t) at various times of bead milling, using mill frequencies of 15 Hz and 30 Hz. The continuous line in the figures represents the best fit of the model given in Equation (5). The kinetic parameters \(R_{m}\) and \(k\) were determined using the experimental kinetic data and Equation (5) using a MATLAB code with a _nlinfit_ function. The corresponding values were an \(R_{\rm mt}\)-value of 14.16 mg pDNA(t)/g dcw and a \(k\)-value varied from 0.14 to 1.07 min\({}^{-1}\) (Figure 5). Figure 4: Response surface plots. (**A**) Effect of mill frequency and bead size in the release of pDNA(t). (A\({}_{1}\)) Low cell concentration (10 g wcw/L) and (A\({}_{2}\)) high cell concentration (20 g wcw/L). (**B**) Effect of mill frequency and bead size in the release of pDNA(sc). (B\({}_{1}\)) Low cell concentration (10 g wcw/L) and (B\({}_{2}\)) high cell concentration (20 g wcw/L). It can be observed in Figure 5A that the model provided a good fit to the release of pDNA(t) data obtained in the bead mill in Experiments 1, 3, 5, and 7 with R\({}^{2}\)-values of 0.82, 0.87, 0.99, and 0.98, respectively. A faster kinetics of the release of pDNA(t) was obtained with high cell concentration and smaller beads. This could be due to the high probability of the collision between the beads against the cells. The opposite effect occurs when using a low cell concentration and large beads. This is probably because a greater bead creates greater void spaces, resulting in a reduced number of collisions against the cells. Figure 5b shows that the model given in Equation (5) provided a good fit to the release of pDNA(t) data obtained in the bead mill in Experiments 4, 6, and 8 (R\({}^{2}\)-values of 0.99, 0.90, and 0.94, respectively) except for Experiment 2 (R\({}^{2}\) = 0.54). Considering the results of the analysis of Experiments 1 and 2, a poorer model fit is obtained with low cell concentration and smaller beads, suggesting that, under these experimental conditions, the probability of disruption decreases more rapidly than a first-order kinetics as the process takes place. A faster kinetic of the release of pDNA(t) at 30 Hz (Figure 5B) can be achieved using a high cell concentration and small beads analogous to the results at 15 Hz (Figure 5A). On the contrary, slower kinetics is obtained using a high cell concentration and large beads. This behavior can be explained as discussed by [PERSON] et al. [18]: large beads create larger void spaces in the milling chamber, resulting in a reduced number of collisions against the cells. Moreover, given a high cell concentration, the release of intracellular components increases the viscosity of the cell solution, so the impaction between beads is reduced, which results in an incomplete lysis. Figure 6 shows the experimental data of the release of pDNA(sc) at various times of bead milling, using a mill frequency of 15 Hz. The continuous line in the figure represents the best fit of the model given in Equation (5). The corresponding fitted values were an \(R_{\text{msc1}}\)-value of 8.98 mg pDNA(sc)/g dcw and a \(k\)-value varied from 0.14 to 1.24 min\({}^{-1}\) (Figure 6). It can be observed that the model provided a good fit of the release of pDNA(sc) data obtained in the bead mill in Experiments 1, 3, 5, and 7, with R\({}^{2}\) values of 0.99, 0.78, 0.88, and 0.99, respectively. One atypical behavior was obtained in Experiment 3 and requires deeper research. Faster kinetics of the release of pDNA(sc) at 15 Hz can be achieved using a low cell concentration and small beads. [PERSON] et al. [37] reported that smaller beads are effective at lower viscosities, but these become fluidized when run at higher cell loads, at higher viscosities, or at a high agitation frequency, which in turn results in a lower momentum of impaction and incomplete cell lysis, and negatively affects the process productivity.