# Efficient Gene Transfection by Electroporation--In Vitro and In Silico Study of Pulse Parameters
## 4 Discussion
### Modeling the Probability of pDNA Cell Membrane Contact during GET
This high frequency of pulse repetition, or the corresponding short duration between successive pulses, could imply that the cumulative effect of 100 pulses at 1 MHz pulse repetition frequency is equivalent to a single pulse of duration 100 us. To test this, Equation (4) was solved for 100 pulses of duration 300 ns and an electric field intensity of 7 kV/cm at a pulse repetition frequency of 1 MHz and a pDNA cell membrane distance of 170 nm (corresponding to 250 ug/mL, Supplementary Table S1), similar to the conditions used in [106]. The evolution of \(P(X,t)\) is shown in Figure 23 for pulse numbers 1, 5, 10, 20, and 30. From Figure 23A, it can be seen that during the first pulse, the peak shifts towards the right due to electrophoresis (until 300 ns) and also spreads only slightly due to diffusion. After the cessation of first pulse, the peak stays at the same place, and the distribution spreads only slightly due to diffusion from 300 ns to 1 us. During the second pulse, this peak is then further driven to the right by electrophoresis, spreading only slightly by diffusion. After the end of the second pulse, the distribution spreads only slightly by diffusion, until the time the third pulse acts and the process repeats (see Figure 23B for pulse number 5). By the 10 th pulse, the peak of the distribution has already migrated a distance greater than 50 nm towards the right, spreading only slightly by diffusion (as seen by a broadening of \(P(X,t)\) and a decrease in the maximum value of \(P(X,t)\)) (Figure 23C). By pulse number 20 (starting at \(t=19\) us), the peak of the distribution has already drifted to the right by \(\sim\)150 nm by electrophoresis indicating a non negligible \(PSC(t)\) by \(t\sim\)19 us (Figure 23D,F). Figure 23E, corresponding to the 30 th pulse (or \(t=29\) us), indicates that the pDNA molecule has already been absorbed at the cell membrane as \(P(X,t)=0\) in \(X\ \in\ [B_{L},B_{R}]\) by \(t=29\) us. This is further evident from Figure 23F, which shows that \(PSC(t)\) has risen sharply and reached a final value of 1 before \(t\sim 30\) us. The results in Figure 23 corresponding to the high frequency of 1 MHz differ from the results of Figure 12 corresponding to the low frequency of 10 Hz. In both cases, electrophoresis drives the peak of \(P(X,t)\) towards the right (cell membrane) during the pulse while spreading it only slightly by diffusion. Once the pulse ends, the distribution of \(P(X,t)\) is allowed to spread by diffusion. For the low frequency of 10 Hz case, the time between the subsequent pulses or the time allowed for the distribution to relax by diffusion is significantly higher (\(t\ \sim 100\) ms). This large time allows the spread of \(P(X,t)\) by diffusion and flattens it out in a way that enough flux of \(P(X,t)\) can be collected at the cell membrane and \(PSC(t)\) starts to rise by diffusion. However, for the high frequency of 1 MHz, the time between subsequent pulses or the time allowed for the distribution to relax by diffusion is small, corresponding to \(t<1\) us. This time is too small for the distribution of \(P(X,t)\) to spread by diffusion and reach the cell membrane. Soon after, the next pulse acts on the pDNA molecule driving the still narrow \(P(X,t)\) distribution to the right. Cumulatively, this is how \(P(X,t)\) is driven to the right by electrophoresis; if enough pulses are delivered (in this case \(\sim\)20), the pDNA molecule reaches the cell membrane by electrophoresis, as seen by the early and sharp rise in \(PSC(t)\). The process of high-frequency ns pulses can also be analyzed from the point of view of Peclet number defined as \(Pe=\frac{\mu Et_{p}}{\sqrt{2D/f}}\). For experimental conditions in [106], \(t_{p}=300\) ns, \(E=7\) kV/cm, and \(f=1\) MHz, which gives \(Pe=5.57\), indicating that electrophoresis is dominating the transport of pDNA to the cell membrane, and multiple pulses at such a high pulse repetition frequency can be thought of as a single long pulse providing the electrophoresis.