# Efficient Gene Transfection by Electroporation--In Vitro and In Silico Study of Pulse Parameters
## Materials and Methods
### Modeling the Probability of pDNA and Cell Membrane Contact during GET
The simplicity of such a 1D assumption enhances the pedagogical aspect of inferring the role of diffusion and electrophoresis in bringing the pDNA molecule to the permeabilized membrane (a prerequisite for GET) for a wide variety of experimental conditions used (e.g., pulse durations ranging from nanosecond to millisecond and concentrations ranging from 40 \(\upmu\)g/mL to 500 \(\upmu\)g/mL). The purpose of the model is not an estimation and/or prediction of experimental outcomes such as GET efficiency or MFI. The probability density (\(P(X,t)\)) of finding the pDNA molecule at particular location along the \(X\)-axis at time \(t\) can be described using the following drift-diffusion equation of probability density [57]:  
\[\frac{\partial P(X,t)}{\partial t}=-v\frac{\partial P(X,t)}{\partial X}+D \frac{\partial^{2}P(X,t)}{\partial X^{2}} \tag{4}\]  
Equation (4) does not assume any interaction between the pDNA molecule and the cell membrane. However, a pDNA and cell membrane interaction free energy \(F(X)\) can be assumed between the pDNA molecule and the cell membrane. In this case, the probability density of finding a particle at a particular location along the \(X\)-axis at time \(t\) can be described using the one-dimensional Fokker-Planck equation [57]:  
\[\frac{\partial P(X,t)}{\partial t}=\frac{D}{k_{B}T}\frac{\partial}{\partial X }\left[\frac{\partial F(X)}{\partial X}P(X,t)\right]-v\frac{\partial P(X,t)}{ \partial X}+D\frac{\partial^{2}P(X,t)}{\partial X^{2}} \tag{5}\]  
Figure 2: Schematic of the biased random walk of pDNA molecule along the \(X\)-axis due to drift (electrophoresis), diffusion, and a free energy barrier. Electric field acts along the negative \(X\)-axis, which creates a drift in the pDNA molecule along the positive \(X\)-axis. The pDNA molecule is initially centered at \(X=0\) with the initial condition defined by a narrow-band Gaussian distribution. The boundary conditions at the right (\(X=+B_{R}\)) and left (\(X=-B_{L}\)) of the domain are the absorbing and reflecting boundary conditions, respectively. A free energy barrier between the pDNA molecule and the cell membrane is present at the right boundary (\(X=+B_{R}\)). The free energy barrier \(F(X)\) is assumed to be \(F(X)=50e^{2(X-B_{R})}\ k_{B}T\), i.e., a barrier with a height of 50 \(\ k_{B}T\) that steeply decays exponentially away from the cell membrane at \(X=+B_{R}\) (Figure 2). Such a barrier is inspired from [63]. The drift term in Equations (4) and (5) is due to the electrophoresis of the pDNA molecule and is given by \(v=\mu E\), where \(\mu\) is the electrophoretic mobility of the pDNA molecule (\(\mu=-3.75\times 10^{-8}\) m\({}^{2}\)V\({}^{-1}\)s\({}^{-1}\)) [59; 64; 65] and \(E\) is the electric field intensity. The diffusion coefficient of pDNA is taken as \(D=10^{-12}\) m\({}^{2}\)s\({}^{-1}\)[65]. Equation (4) describes the process of finding pDNA at a particular location along the \(X\)-axis as a biased random walk, where the bias is due to electrophoresis of pDNA molecule. Equation (5) describes the same process of finding pDNA at a particular location along the \(X\)-axis based on a bias (electrophoresis), diffusion, and free energy interaction (barrier) with the cell membrane. The drift term in Equations (4) and (5) is non-zero only when the electric field pulse is present and is zero otherwise. Equations (4) and (5) are solved for the probability density function \(P(X,t)\) in the domain \(X\in[B_{L},B_{R}]\) with appropriate boundary and initial conditions using the pdepe(0) function in MATLAB(r) (R2021b). The initial condition for the pDNA molecule is considered as a narrow-band Gaussian/normal distribution with zero mean (centered at \(X=0\), Figure 2) and a standard deviation of 10 nm. The boundary at the right (positive end of the domain), i.e., the cell membrane, is considered an absorbing boundary condition, implying that once pDNA reaches the cell membrane, it is absorbed into the cell membrane, i.e., it is internalized and expressed by the cell.