# Efficient Gene Transfection by Electroporation--In Vitro and In Silico Study of Pulse Parameters
## 3 Results
### Modeling the Probability of pDNA and Cell Membrane Contact during GET
#### 3.6.1 200 ns and 500 ns Pulses
Due to the nature of the free energy barrier considered in this study (50\(e^{2(X-B_{R})}\)\(k_{B}T\)), the free energy is negligibly small at distance greater than \(\sim\)2 nm away from the cell membrane, and it rises sharply as we approach the cell membrane. Therefore, for the electric field (and electrophoresis) to overcome this barrier, the pDNA molecule must already be present very close to the cell membrane, or the electric field should be applied for a longer duration such that electrophoresis first drags the pDNA molecule close to the cell membrane and then subsequently helps the pDNA molecule in overcoming this barrier. Thus, the pDNA molecule which relies on diffusion to reach the cell membrane is also repulsed by the free energy barrier, resulting in lower \(PSC(t)\) compared to the case when no free energy barrier was present (comparing Figure 12B,D vs. Figure 12F,H). Since the probability of pDNA being absorbed by the cell membrane is low in the presence of a free energy barrier, the probability of finding the pDNA molecule in the domain \(X\ \in\ [-B_{L},\ B_{R}]\) at \(t\)=100 ms is higher (comparing Figure 12A,C vs. Figure 12E,G at \(t\)=100 ms). For 200 ns pulses, the time between pulses (\(t\) = 100 ms) is large enough for \(P(X,t)\) to become almost flat (by diffusion) before the onset of the second pulse. This is the case for both - without the free energy barrier (Figure 12A,C) and the with free energy barrier (Figure 12E,G). This implies that it is almost equally probable to find the pDNA molecule at any \(X\) for \(X\ \in\ [-B_{L},\ B_{R}]\). Thus, for both cases, a uniform distribution of \(P(X,t)\) in \(X\ \in\ [-B_{L},\ B_{R}]\) can be considered as an initial condition for the second pulse. For a uniform distribution, there is a higher probabilty of pDNA being close to the cell membrane compared to the narrow-band Gaussian distribution with a standard deviation of 10 nm centered around \(X=0\), the tail of which is negligibly small at \(X=-B_{L},+B_{R}\). Therefore, we expect that electrophoresis provided by the second pulse would contribute more to the rise in \(PSC(t)\) during the second pulse compared to the first pulse for which the rise in \(PSC(t)\) was negligible during the pulse (Figure 12B,D,F,H). However, since the pulse duration is small, we further expect that the rise in \(PSC(t)\) will still not be significant to completely ensure the absorption of pDNA at the cell membrane. \(PSC(t)\) will still have to rely on the diffusion post-second pulse. We expect this process to continue in the subsequent pulses until diffusion increases \(PSC(t)\) to a value of 1. The role of any additional pulses after \(PSC(t)\) reaches 1 is not accounted for by the model since the pDNA is absorbed by the cell membrane and \(P(X,t)=0\) in \(X\ \in\ [-B_{L},\ B_{R}]\). In such cases, additional pulses can be thought to add more pDNA molecules to the cell membrane. To illustrate the point that ns pulses can be analysed in isolation, we can look at the Peclet number defined as the ratio of distance covered during electrophoresis to the distance covered during diffusion post pulse (\(Pe=\frac{\mu Et_{p}}{\sqrt{2 Dt_{bp}}}\), where \(t_{p}\) is the pulse duration and \(t_{bp}\) is the duration between pulses). For the 200 ns pulse, \(E=15.8\) kV/cm, \(t_{p}\) = 200 ns, and \(t_{bp}\) = 100 ms, which gives a \(Pe=0.03\ (<1)\), indicating that \(P(X,t)\) would spread more by diffusion and reach the cell membrane while becoming flat than it would be drifted towards the cell membrane by electrophoresis over a single phase (during pulse and post-pulse). As a result, the role of electric field pulses can be studied in isolation without the confounding/compounding effects of multiple pulses. Furthermore, for pedagogical purposes, and to study the effect of concentration on pDNA being absorbed by the cell membrane (a pre-requisite for GET) in isolation as a function different pulse durations and electric field intensities, we have decided to simulate only one pulse.