# 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
#### 3.6.2 HF-BP Pulses
As mentioned previously, the Peclet number (\(Pe=\frac{\mu Et_{p}}{\sqrt{2 Dt_{bp}}}\)) during a single phase (pulse + post pulse) of the 200 ns pulse was \(Pe=0.03\) (\(<1\)), indicating that diffusion dominated the process during the phase and that the initial narrow-band Gaussian distribution drifts very little due to electrophoresis and becomes nearly flat by spreading due to diffusion by the end of the phase. As a result, ns pulses can be studied in isolation. However, for a sequence of 32 bipolar pulses, pulse duration \(t_{p}=2\) us and time between pulses \(t_{bp}=2\) us is not large. For an electric field intensity of 1.25 kV/cm, the Peclet number for HF-BP pulses is \(Pe=4.69\) (\(>1\)), indicating the electrophoresis during the pulse dominates the diffusion in between pulses, and there is very little time for diffusion to spread the distribution. As a result, there could be a compounding effect of subsequent pulses due to which the entire sequence of 32 pulses was considered. However, if we consider pulse duration \(t_{p}=2\) us and the time between bursts \(t_{bp}=1\) s, the Peclet number for a single burst is \(Pe=0.01\) (\(<1\)), indicating that diffusion dominates over electrophoresis and spreads the distribution, making it almost flat by the end of the burst. As a result, the individual bursts can be studied in isolation. The assumption of \(t_{p}=2\) us as opposed to \(t_{p}=32\)\(\times\)\(2\) us is valid, as there is net zero electrophoresis/drift during the 32 pulses. For HF-BP pulses, the time between bursts (\(t=1\) s) is large enough for \(P(X,t)\) to become almost flat (by diffusion) before the onset of the second burst. For the case without the free energy barrier, \(PSC(t)\) for \((B_{L},B_{R})\) ranging from 77 nm to 433 nm reaches 1 before the onset of the second burst (Figure 15A). This should imply that without the free energy barrier, GET should be independent of concentration. This is unlike the 200 ns pulse and 500 ns pulse protocols, where, although the time between pulses was large (\(t=100\) ms), it was not large enough to ensure \(PSC(t)\) reaches 1 for \((B_{L},B_{R})\) ranging from 77 nm to 433 nm (Figure 13A,C), thus implying GET dependance on concentration even without the free energy barrier for 200 and 500 ns pulses. Furthermore, since \(PSC(t)\) reaches 1 before the onset of the second burst, the model does not account for the second burst since \(P(X,t)\sim 0\) in \(X\ \in\ [-B_{L},\ B_{R}]\). In this case, additional bursts can be thought to add more pDNA molecules to the cell membrane. For the case with free energy barrier, \(PSC(t)\) does not reach a value of 1, and the final value depends on \((B_{L},B_{R})\) ranging from 77 nm to 433 nm (Figure 15B). This implies that  
Figure 16: Evolution of \(PSC(t)\) for a single 200 ns pulse, a single 500 ns pulse and a single burst (32 bipolar pulses) for \((B_{L},B_{R})=134\) nm, corresponding to 500 \(\mu\)g/mL: (**A**) without free energy barrier and (**B**) with free energy barrier. Dashed lines represent the time at which the ns pulses end or the sequence of 32 pulses (i.e., 1 burst) ends. The solid vertical line represents the onset of the next 200 ns or 500 ns pulse (pulse repetition rate for ns pulses is 10 Hz). The next burst of HF-BP pulses starts at 1 s (burst repetition frequency is 1 Hz). GET should depend on concentration when the free energy barrier is present. However, since the time between bursts is large (\(t=1\) s), \(P(X,t)\) becomes almost flat (but non-zero) before the onset of the second burst (Figure 14I at \(t=1\) s). Therefore, as with 200 ns pulses, a uniform distribution of \(P(X,t)\) in \(X\ \in\ [-B_{L},\ B_{R}]\) can be considered as an initial condition for the second burst. 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 neglibly small at \(X=-B_{L},+B_{R}\).