# 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  
Equation (4) is solved using \(B_{L}\), \(B_{R}\)= 134 nm (representing 500 \(\mu\)g/mL pDNA solution concentration) for 32 bipolar pulses (constituting 1 burst out of a 100 bursts) as described by the protocol given in Table 1 for C2C12 cells. Figure 14A-C show the evolution of \(P(X,t)\) during the 32 pulses, whereas Figure 14D shows shows the evolution of \(P(X,t)\) after termination of 32 pulses, i.e., 1 burst. The domain length considered was \(B_{L},B_{R}=134\) nm, corresponding to 500 \(\mu\)g/mL. Figure 14A shows the evolution of \(P(X,t)\) during the first bipolar pulse. During the first 2 \(\mu\)s of the pulse, when the electric field is applied in the negative \(X\) direction, \(P(X,t)\) moves to the right due to electrophoresis and only spreads slightly due to diffusion. During the next 2 \(\mu\)s, when the electric field is absent, the peak of \(P(X,t)\) remains at the same \(X\) location, and \(P(X,t)\) spreads slightly by diffusion. During the next 2 \(\mu\)s, the electric field is applied in the positive \(X\) direction and the peak of \(P(X,t)\) moves to the left due to electrophoresis and spreads slightly by diffusion. During the last 2 \(\mu\)s of the pulse, the electric field is absent again, and the peak of \(P(X,t)\) remains at the same location, spreading slightly by diffusion. The same pattern is repeated for 32 such bipolar pulses. Figure 14B shows the evolution of \(P(X,t)\) during pulse number 15 and Figure 14C shows the evolution of \(P(X,t)\) during pulse number 32. It can be seen from Figure 14B,C that the peak \(P(X,t)\) remains close to \(X=0\) nm during the 32 pulses, as there is a net zero drift in \(P(X,t)\) due to the bipolar nature of the pulses. \(P(X,t)\), however, spreads by diffusion during the 32 pulses. Finally, after the cessation of 32 bipolar pulses (i.e., after 1 burst), or after \(t=256\)\(\mu\)s, \(P(X,t)\) only spreads by diffusion and becomes almost flat until the onset of the next burst (at \(t=1\) s). The Probability of Successful Contact, \(PSC(t)\), is calculated as \(PSC(t)=1-\int_{-B_{L}}^{+B_{R}}P(X,t)dX\) and is shown in Figure 14E. \(PSC(t)\) remains negligibly small during the burst of 32 bipolar pulses, i.e., until \(t=256\)\(\mu\)s, as indicated by the dotted vertical line. Only from around \(t\sim 1\) ms does \(PSC(t)\) start to rise (due to diffusion), attaining a non-negligible probability. \(PSC(t)\) reaches a final probability of 1 before the onset of the next burst of bipolar pulses (at \(t=1\) s). To see the influence of the free energy barrier, Equation (5) is solved using \(B_{L},B_{R}\)= 134 nm (representing 500 \(\mu\)g/mL pDNA solution concentration) for 32 bipolar pulses, and the the evolution of \(P(X,t)\) is shown in Figure 14F-H during the 32 pulses and in Figure 14I after the termination of the 32 pulses. Similar to the case with no free energy barrier, the domain length considered was \(B_{L},B_{R}=134\) nm, corresponding to 500 \(\mu\)g/mL. The evolution of \(P(X,t)\) during 32 pulses for the case with the free energy barrier (Figure 14F-H) was similar to the the evolution of \(P(X,t)\) during the 32 pulses without the free energy barrier (Figure 14A-C). This could be due to the fact that \(P(X,t)\) is not able to spread enough by diffusion during the burst (and net electrophoresis is also negligible due to the bipolar nature of the pulses) to have a signficant portion other than the thin tail of \(P(X,t)\) close to the cell membrane where the free energy barrier is present. Therefore, the evolution of \(P(X,t)\) with the free energy barrier would be similar to the case without the free energy barrier for the large enough domain considered here (\(B_{L},B_{R}=134\) nm, corresponding to 500 \(\mu\)g/mL). For smaller domain lengths (or concentrations > 500 \(\mu\)g/mL), this might not be the case.