# 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  
Equation (4) is first solved for a single pulse of 200 ns at an electric field intensity of 15.8 kV/cm. These were the conditions used in the experiments for C2C12 cells (Table 1). The results of the solution of Equation (4) are shown in Figure 12A below for \(B_{L}\), \(B_{R}\) = 134 nm, representing 500 \(\upmu\)g/mL pDNA solution concentration. The solid lines represent the evolution of \(P(X,t)\) during the pulse, and the dashed lines represent the evolution of \(P(X,t)\) after the pulse. Equation (4) is solved for \(t\) = 100 ms since the ns pulses are applied at a frequency of 10 Hz (implying 100 ms between pulses). During the pulse (\(t\) = 0 ns to \(t\) = 200 ns), the initial narrow band normal/Gaussian distribution drifts towards the right by electrophoresis and spreads only a little by diffusion. After the pulse, there is no drift, and the distribution spreads purely by diffusion. The time between two successive pulses is long enough for \(P(X,t)\) to become almost flat before the onset of the second pulse (\(t\) = 100 ms). To calculate the Probability of Successful Contact \(PSC(t)\), the flux \(P(X,t)\) is collected at the cell membrane, i.e., at \(X=B_{R}\). \(PSC(t)\) is calculated as \(PSC(t)=1-\int_{-B_{1}}^{+B_{R}}P(X,t)dX\). This represents the probability that a pDNA molecule will successfully establish contact with, and be absorbed by, the cell membrane, which is pre-requisite for a successful GET. As observed from Figure 12B, \(PSC(t)\) remains negligibly small during and after the 200 ns pulse, implying a negligible probability of pDNA establishing contact with the cell membrane. Only from around \(t\)\(\sim\)1 ms does \(PSC(t)\) start to rise (due to diffusion), attaining a non-negligible probability. To determine the influence of concentration (i.e., of the distance between the pDNA and the cell membrane (in Figure 1)), another simulation using \(B_{L}\), \(B_{R}\) = 433 nm, representing 15 \(\upmu\)g/mL pDNA solution concentration, is shown in Figure 12C,D. Due to the lower concentration (or a larger distance between the pDNA and the cell membrane), the domain has expanded (Figure 12C), and the pDNA molecule has to travel a larger distance before it can reach the cell membrane. Due to insufficient electrophoresis, the pDNA molecule relies on diffusion after the cessation of pulse to cover this large distance and reach the cell membrane. For \(B_{L}\), \(B_{R}\) = 433 nm, \(PSC(t)\) starts to rise at a much later time and attains a lower \(PSC(t)\) at \(t\) = 100 ms (Figure 12D), compared to \(B_{L}\), \(B_{R}\) = 134 nm (Figure 12B). Evolution of \(P(X,t)\), as calculated using Equation (5), i.e., in the presence of a free energy barrier between the pDNA and the cell membrane, is shown in Figure 12E for  
Figure 12: The 200 ns pulse: Evolution of \(P(X,t)\) and the corresponding Probability of Successful Contact \(PSC(t)=1-\int_{-B_{L}}^{+B_{R}}P(X,t)dX\) for \(B_{L}\), \(B_{R}\) = 134 nm in (**A**,**B**,**E**,**F**) and for \(B_{L}\), \(B_{R}\) = 433 nm in (**C**,**D**,**G**,**H**); (**A**–**D**) represent cases without free energy barrier and (**E**–**H**) represent cases with free energy barrier. Solid lines in (**A**,**C**,**E**,**G**) correspond to times during the pulse, and dashed lines in (**A**,**C**,**E**,**G**) correspond to times after the pulse. The dashed lines in (**B**,**D**,**F**,**H**) represent the time at which the 200 ns pulse ends. \(B_{L},B_{R}=134\) nm (500 \(\mu\)g/mL) and in Figure 12G for \(B_{L},B_{R}=433\) nm (15 \(\mu\)g/mL); the corresponding \(PSC(t)\) values are shown in Figure 12F,H, respectively. The presence of a high free energy barrier (barrier height of 50 \(k_{B}T\)) signficantly reduces the probability of pDNA reaching the cell membrane. For a 200 ns pulse, electrophoresis acts for an insufficient amount of time to bring the pDNA molecule close to the cell membrane and for it to influence the pDNA molecule in overcoming the free energy barrier.