1 Abstract
Reversible electroporation is a widely applied technique of creating transient pores in cell
membrane for the purpose of large molecules delivery. However, researchers are readily faced
with the failure of it in experiments due to a lack of vital knowledge of the electroporation, and
sometimes, the incorrect operations. And there is no doubt that it causes an increase in time and
economic cost. And the theory guidance to experiment seems nearly impossible due to complexity
of its principles and mathematical model construction. Therefore, for the purpose of successful
mega-sized plasmid (175,798 base pairs) electroporation in our project, an optimal model is
established to overcome these obstacles to explore the effective experimental conditions.
In our model, with the physical principles of pore creation and evolution and relevant
energy formulas[1], we optimize previous models and progressively simplify the algorithm
complexity. With this model, we can simulate and predict the applied voltage, hydrophobic and
hydrophilic pore formation on cell membrane, pore radius change in electric field and etc. By
this, we make it possible to guide our experiment with high DNA delivery efficiency and serve
it as a gift for future teams.
2 Background
The best-studied bacterium, Escherichia coli, is an ovoid about 2 µm long and a little
less than 1µm in diameter[5]
.E.coli is gram-negative and has a double membrane[5]. Its outer
membrane is studded with protein channels that allow small molecules, but not large molecules
to diffuse. Therefore, the rearrangements of lipids in electric field leading to sufficient radius
of pores formation in the intact membrane is the key substance to make high efficiency of
large plasmid delivery. Besides, in order to make cells sustain in electric field, irreversible
electroporation should be avoided by controlling voltage range reasonably. We can do it by
well-controlled electroporation. The electroporation process can be divided into three distinct
stages: charging, pore creation, and pore evolution[2]. On the basis of it, we explore the uptake
process of DNA plasmid to bring the model one step closer to experiments.
2.1 Pores part
Under correct conditions, hydrophilic pores of radius greater than a specific radius corresponding to energy
barriers for pore expansion occurs. It makes DNA delivery possible with
the reversible electroporation.
2.2 DNA plasmid delivery part
The uptake process of the large plasmid is revealed by the Nernst-Planck equation, which
accounts for both the diffusive and electrophoretic transport[3].
3 Model ideals and designs
Based on parts above, we constructed our models and combined them.
3.1 Pores creation
The membrane of E.coli has selective permeability that do not allow most ions to diffuse.
Therefore, it has very high resistance and we can see it as an open circuit. In contrast, there
are large quantities of ions in the cytoplasm environment, which can be seen a close circuit as
well. According to previous model[2], the cell shape is assumed to be spherical in suspensions.
The cytoplasm and outer solution environment are seen as pure conductive medium. And the
cell membrane is approximately considered as an aggregation of infinitesimal capacitors (AIC).
Furthermore, the culture medium and inner cytoplasm can be seen as a series of infinitesimal
resistances (SIR).
3.1.1 Voltage of the transmembrane
On the basis of these prerequisites, we divide the whole cell into two different parts (the
cell membrane and cell cytoplasm) to analyze. The cell transmembrane voltage is expressed as Schwan equation[2]:
And when electroporation proceeds, the electric field charges the AIC immediately. Therefore,
the charge time can be expressed as:
Then, we combine the equation (1) and (2) shown above to calculate the average transmembrane
voltage of the cell:
3.1.2 Pores density
After electroporation proceeds, the cell membrane spontaneously forms numerous hydrophobic pores, which can
be seen as capacitors and resistances[3]. If hydrophobic pores
of radius r > r∗ are created, they convert spontaneously to long lived hydrophilic
pores with
the minimum-energy radius of rm.
And with the transmembrane voltage increases, the radiuses of pores gradually become
stable. Relevant Schematic diagrams are shown above.
Here we assume all expandable pores’ radiuses is rm. The radius change process
follow
the equations below:
in equation(6) the first term accounts for the steric repulsion of lipid heads; the second, for the
edge energy of the pore perimeter; the third, for the effect of pores on the membrane tension;
and the fourth, for the contribution of the transmembrane voltage.
3.2 Pores evolution
The radiuses of these hydrophilic pores are further expanded in the process of current
conduction on the membrane. The radius changing process itself will change the transmembrane
voltage, which indirectly influence the pores density. This process follows equations shown
below:
3.3 DNA plasmid delivery
The mathematical models above can accurately predict the number of pores and their sizes
in the whole process. Although important, these predictions are not directly comparable to
DNA uptake experiments, which typically measure the fraction of transformed cells[3]. Therefore,
we construct the DNA plasmid delivery model (DPDM) next. The uptake of plasmid
concentration by E.coli is shown as Nernst-Planck equation below:
Zeff is an effective valence of the phosphate group obtained from [3].
D0 is estimated according to previously published experimental data[4] and
further curve-fitting.
The relevant estimation graph is shown below:
According to the curve-fitting graph below, the value of D0 can be calculated from
equations along with different initial conditions:
In the process of plasmid enter the cell through pores created and expanded by steps above, a
sevenfold decrease of plasmid speed inside the pore occurs[1]. Therefore, the
D0 value here is
divided by seven.
4 Model establishment
4.1 Initial conditions
(1)The plasmid size we use here is 175,798 base pairs and voltage range between 1000-3000
volts is available.
(2)An experimental equipment with square wave voltage generator is used in our project
and distance between electrodes applied here is 1mm and 2mm.
(3)The time scale of square wave voltage is 20-40 µs. Temperature applied in our model
is 293.15 kelvin degrees (20°C).
4.2 Simulation
4.2.1 Model and analysis
Based on the formula and the algorithm above, we construct a model with original differential equations.
According to our simulation result, we plot three characteristic curves with
three different initial applied voltages (1000 V, 2000 V, 3000 V).
As the graph shown above, these curves have a relatively big variant, as the applied
voltages differ from each other. AIC is immediately charged within 10 µs, which is reflected as
an increase of transmembrane voltage and pores radius. And then, the curve become stable as
a reflection of full charge of AIC.
And in order to avoid the irreversible electroporation and high efficiency of delivery, we
use initial voltage around 2000 V to create DNA plasmid uptake model by the Nernst-Planck
equation. And in most experiments, only one DNA plasmid delivery is not sufficient enough
to ensure the success of electroporation due to plasmids exclusion and other unknown reasons.
So, in order to make it successful, we need to deliver more plasmids into cells. According to
already known E.coli volume, We can calculate the [DNA]i concentration required for
at least
one plasmid delivery, which is 1.77×10-7mol/L. However, here we can’t simulate the process
of plasmid exclusion of cells. Therefore, we create our mathematical model by using an upper
limit of 100 DNA plasmids delivery in our range. On the basis of it, we can calculate the
voltages scope corresponding to the different concentration of plasmids delivery.
The graph shown above is the different voltages scope corresponding to the different concentration of
plasmids delivery. Here we use plasmids with different sizes, where 175.798kbp is
the plasmid size used in our experiment. The result show that the concentration of plasmids
delivery is proportional to the values of voltages. And we also simulate different plasmid with
different sizes in our model. The result shows that the size of plasmids is inversely proportional
to the corresponding values of voltages.
4.2.2 Conclusion
On the basis of our mathematical model and discussions above, 2136-2768V ,a reasonable
range of voltage for well-controlled electroporation is obtained.
5 Symbol & Explanations
6 Assumptions
(1) Concentration of DNA plasmids outside cells is assumed constant.
(2) All calculations assume a spherical cell with a 1.5 µm radius.
7 References
[1]Kyle, C., Smith, and, John, & C., et al. (2004). Model of creation and evolution of
stable electropores for dna delivery. Biophysical Journal.
[2]Ivorra, A. (2010). Tissue electroporation as a bioelectric phenomenon: Basic concepts.
In Irreversible electroporation (pp. 23-61). Springer, Berlin, Heidelberg.
[3]Neumann, E., Kakorin, S., Tsoneva, I., Nikolova, B., & Tomov, T. (1996). Calcium-mediated DNA adsorption
to yeast cells and kinetics of cell transformation by electroporation.
Biophysical journal, 71(2), 868-877.
[4]Prazeres, D. M. F. (2008). Prediction of diffusion coefficients of plasmids. Biotechnology
and bioengineering, 99(4), 1040-1044.
[5]White, A. (1954). Principles of biochemistry. McGraw-Hill Book, 6.
[6]Newman, J. (1966). Resistance for flow of current to a disk. J. electrochem. Soc, 113(5),
501-502.
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