Modeling electroporation in a single cell

Electroporation uses electric pulses to promote delivery of DNA and drugs into cells. This study presents a model of electroporation in a spherical cell exposed to an electric field. The model determines transmembrane potential, number of pores, and distribution of pore radii as functions of time and position on the cell surface. For a 1-ms, 40 kV/m pulse, electroporation consists of three stages: charging of the cell membrane (0-0.51 micros), creation of pores (0.51-1.43 micros), and evolution of pore radii (1.43 micros to 1 ms). This pulse creates approximately 341,000 pores, of which 97.8% are small ( approximately 1 nm radius) and 2.2% are large. The average radius of large pores is 22.8 +/- 18.7 nm, although some pores grow to 419 nm. The highest pore density occurs on the depolarized and hyperpolarized poles but the largest pores are on the border of the electroporated regions of the cell. Despite their much smaller number, large pores comprise 95.3% of the total pore area and contribute 66% to the increased cell conductance. For stronger pulses, pore area and cell conductance increase, but these increases are due to the creation of small pores; the number and size of large pores do not increase.     

The influence of geometry, surface character, and flexibility on the permeation of ions and water through biological pores

A hydrophobic constriction site can act as an efficient barrier to ion and water permeation if its diameter is less than the diameter of an ion's first hydration shell. This hydrophobic gating mechanism is thought to operate in a number of ion channels, e.g. the nicotinic receptor, bacterial mechanosensitive channels (MscL and MscS) and perhaps in some potassium channels (e.g. KcsA, MthK and KvAP). Simplified pore models allow one to investigate the primary characteristics of a conduction pathway, namely its geometry (shape, pore length, and radius), the chemical character of the pore wall surface, and its local flexibility and surface roughness. Our extended (about 0.1 micros) molecular dynamic simulations show that a short hydrophobic pore is closed to water for radii smaller than 0.45 nm. By increasing the polarity of the pore wall (and thus reducing its hydrophobicity) the transition radius can be decreased until for hydrophilic pores liquid water is stable down to a radius comparable to a water molecule's radius. Ions behave similarly but the transition from conducting to non-conducting pores is even steeper and occurs at a radius of 0.65 nm for hydrophobic pores. The presence of water vapour in a constriction zone indicates a barrier for ion permeation. A thermodynamic model can explain the behaviour of water in nanopores in terms of the surface tensions, which leads to a simple measure of 'hydrophobicity' in this context. Furthermore, increased local flexibility decreases the permeability of polar species. An increase in temperature has the same effect, and we hypothesize that both effects can be explained by a decrease in the effective solvent-surface attraction which in turn leads to an increase in the solvent-wall surface free energy.     

Kinetics of pore size during irreversible electrical breakdown of lipid bilayer membranes.

The kinetics of pore formation followed by mechanical rupture of lipid bilayer membranes were investigated in detail by using the charge-pulse method. Membranes of various compositions were charged to a sufficiently high voltage to induce mechanical breakdown. The subsequent decrease of membrane voltage was used to calculate the conductance. During mechanical breakdown, which was probably caused by the widening of one single pore, the membrane conductance was a linear and not exponential function of time after the initial starting process. In a large number of experiments using various lipids and electrolytes, the characteristic opening process of the pore turned out to be independent of the actual membrane potential and electrolyte concentration. Our theoretical analysis of the pore formation suggested that the voltage-induced irreversible breakdown is due to a decrease in edge energy when the pore had formed. After initiation of the pore, the electrical contribution to surface tension is negligible. The time course of the increase of pore size shows that our model of the irreversible breakdown is in good agreement with mechanical properties of membranes reported elsewhere.     

Reversible electrical breakdown of lipid bilayers: formation and evolution of pores

The mechanism of reversible electric breakdown of lipid membranes is studied. The following stages of the process of pore development are substantiated. Hydrophobic pores are formed in the lipid bilayer by spontaneous fluctuations. If these water-filled defects extend to a radius of 0.3 to 0.5 nm, a hydrophilic pore is formed by reorientation of the lipid molecules. This process is favoured by a potential difference across the membrane. The conductivity of the pores depends on membrane voltage, and the type of this dependence changes with the radius of the pore. Hydrophilic pores of an effective radius of 0.6 up to more than 1 nm are formed, which account for the membrane conductivity increase observed. The characteristic times of changes in average radius and number of pores during the voltage pulse and after it are investigated.     

Lipid flow through fusion pores connecting membranes of different tensions.

When two membranes fuse, their components mix; this is usually described as a purely diffusional process. However, if the membranes are under different tensions, the material will spread predominantly by convection. We use standard fluid mechanics to rigorously calculate the steady-state convective flux of lipids. A fusion pore is modeled as a toroid shape, connecting two planar membranes. Each of the membrane monolayers is considered separately as incompressible viscous media with the same shear viscosity, etas. The two monolayers interact by sliding past each other, described by an intermonolayer viscosity, etar. Combining a continuity equation with an equation that balances the work provided by the tension difference, Deltasigma, against the energy dissipated by flow in the viscous membrane, yields expressions for lipid velocity, upsilon, and area of lipid flux, Phi. These expressions for upsilon and Phi depend on Deltasigma, etas, etar, and geometrical aspects of a toroidal pore, but the general features of the theory hold for any fusion pore that has a roughly hourglass shape. These expressions are readily applicable to data from any experiments that monitor movement of lipid dye between fused membranes under different tensions. Lipid velocity increases nonlinearly from a small value for small pore radii, rp, to a saturating value at large rp. As a result of velocity saturation, the flux increases linearly with pore radius for large pores. The calculated lipid flux is in agreement with available experimental data for both large and transient fusion pores.     

Reversible electrical breakdown of lipid bilayers: formation and evolution of pores

The mechanism of reversible electric breakdown of lipid membranes is studied. The following stages of the process of pore development are substantiated. Hydrophobic pores are formed in the lipid bilayer by spontaneous fluctuations. If these water-filled defects extend to a radius of 0.3 to 0.5 nm, a hydrophilic pore is formed by reorientation of the lipid molecules. This process is favoured by a potential difference across the membrane. The conductivity of the pores depends on membrane voltage, and the type of this dependence changes with the radius of the pore. Hydrophilic pores of an effective radius of 0.6 up to more than 1 nm are formed, which account for the membrane conductivity increase observed. The characteristic times of changes in average radius and number of pores during the voltage pulse and after it are investigated.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

Heteropore populations of bullfrog alveolar epithelium

Diffusional fluxes of a large number of hydrophilic solutes and water across bullfrog (Rana catesbeiana) alveolar epithelium were measured in the Ussing-type flux chamber. Lungs were isolated from double-pithed animals and studied as flat sheets. Radioactive solutes and water were added to the upstream reservoir, and the rate of change of downstream reservoir radioactivity was monitored. A permeability coefficient was estimated for each substance from a linear relationship between radiotracer concentration in the downstream reservoir and time. These permeability data were used to analyze the equivalent water-filled pore characteristics of the alveolar epithelial barrier. The data reveal that the alveolar epithelium is best characterized by two distinct pore populations rather than by a single homogeneous pore population. The large-pore population consists of pores with a radius of about 5 nm and occupies 4% of the available pore area. The small-pore population consists of pores with a radius of about 0.5 nm and occupies 96% of the available pore area. The number of small pores to large pores is 2.68 X 10(3). After the alveolar surface was damaged by acid, a large-pore population with a radius of about 27 nm was seen, allowing nearly free diffusion of solutes. A major implication of the presence of two populations of pores in the alveolar epithelium is that hydrostatically driven bulk water flow occurs predominantly through the large pores, while osmotically driven bulk water flow takes place predominantly through the small pores. As a result, in general, hydrostatic and osmotic gradients may not be equally effective driving forces for water flow across this tissue.     

Transients of perforin pore formation observed by fluorescence microscopic single channel recording.

A new type of single channel recording is described. Large pores were generated in the membranes of resealed human erythrocyte ghosts by incubation with perforin (cytolysin). The flux of the polar fluorescent probe Lucifer Yellow was measured in single ghosts by the fluorescence microphotolysis (photobleaching) technique. The distribution of flux rates for ghosts treated with a limiting perforin concentration showed equidistantly spaced peaks suggesting that subpopulations of ghosts with 0, 1 and 2 pores were resolved. Furthermore, distributions obtained for very different perforin concentrations could be well simulated by using one common value for the flux rate of the single pore (k = 4.65 x 10(-3) s) and assuming a Poisson distribution of pores among ghosts. The flux rate of the single pore corresponds to a pore radius of approximately 50 A, a value which is much smaller than that obtained previously by electron microscopic studies but which agrees well with recent electrical single channel recordings. Mature perforin pores were observed to be very stable. No closing events were detected at a time resolution of 0.2 s for a wide range of temperatures and Ca2+ concentrations. However, the formation of new pores was an unexpectedly slow process. Fluorescence microscopic single channel recording as introduced by this study is applicable to a variety of cellular systems and fluorescent probes and thus may complement the information obtainable by electrical single channel recording of anorganic ion fluxes.     

A thermodynamic approach to alamethicin pore formation

The structure and energetics of alamethicin Rf30 monomer to nonamer in cylindrical pores of 5 to 11 Å radius are investigated using molecular dynamics simulations in an implicit membrane model that includes the free energy cost of acyl chain hydrophobic area exposure. Stable, low energy pores are obtained for certain combinations of radius and oligomeric number. The trimer and the tetramer formed 6 Å pores that appear closed while the larger oligomers formed open pores at their optimal radius. The hexamer in an 8 Å pore and the octamer in an 11 Å pore give the lowest effective energy per monomer. However, all oligomers beyond the pentamer have comparable energies, consistent with the observation of multiple conductance levels. The results are consistent with the widely accepted “barrel-stave” model. The N terminal portion of the molecule exhibits smaller tilt with respect to the membrane normal than the C terminal portion, resulting in a pore shape that is a hybrid between a funnel and an hourglass. Transmembrane voltage has little effect on the structure of the oligomers but enhances or decreases their stability depending on its orientation. Antiparallel bundles are lower in energy than the commonly accepted parallel ones and could be present under certain experimental conditions. Dry aggregates (without an aqueous pore) have lower average effective energy than the corresponding aggregates in a pore, suggesting that alamethicin pores may be excited states that are stabilized in part by voltage and in part by the ion flow itself.     

Fluid mechanical and electrostatic study on the osmotic flow through cylindrical pores

When two solutions differing in solute concentration are separated by a porous membrane, the osmotic pressure will generate a net volume flux of the suspending fluid across the membrane; this is termed osmotic flow. We consider the osmotic flow across a membrane with circular cylindrical pores when the solute and the pore walls are electrically charged, and the suspending fluid is an electrolytic solution containing small cations and anions. Under the condition in which the radius of the pores and that of the solute molecules greatly exceed those of the solvent as well as the ions, a fluid mechanical and electrostatic theory is introduced to describe the osmotic flow in the presence of electric charge. The interaction energy, including the electrostatic interaction between the solute and the pore wall, plays a key role in determining the osmotic flow. We examine the electrostatic effect on the osmotic flow and discuss the difference in the interaction energy determined from the nonlinear Poisson-Boltzmann equation and from its linearized equation (the Debye-Hückel equation).     

The current-voltage relation of an aqueous pore in a lipid bilayer membrane

In a recent paper, Glaser et al. [1988) Biochim. Biophys. Acta 940, 275-287) use an inappropriate formula for the current-voltage relation of a pore to analyze the data from an electroporation experiment. They proceed to use their data analysis to support their contention that the voltage dependence of the membrane conductivity is caused entirely by the nonlinearity of the current-voltage relation of the pores, changes in pore size and number being unimportant. This contention is no longer justified. The formula that Glaser et al. use is valid for a one-dimensional problem, but they apply it to a pore that has a three-dimensional structure. Although the corresponding one-dimensional problem can be reduced to quadratures, only upper and lower bounds on the conductivity can be found for the three-dimensional problem without solving a partial differential equation.     

General continuum analysis of transport through pores. II. Nonuniform pores.

A general continuum derivation of the nonelectrolyte (Js) and volume (Jv) flux through a pore whose cross section is a function of axial position (nonuniform) is given. In general, the flux equations cannot be reduced to the same form as for a uniform pore and it is not possible to characterize the pore kinetics by three constants as in the uniform pore case. However, it is shown that under certain conditions, the nonuniform pore equations can be approximated by the uniform pore form and can be characterized by three constants (omega, sigma, Lp). The only condition needed to reduce the Jv equation to the uniform form is that the solution be dilute. The deviation of the Js equation from the uniform form is characterized by an asymmetrical function of Jv whose maximum value is estimated. It is shown that the maximum posible fractional deviation of the Js equation from the uniform form is given by the parameter: 0:5sigmaJv/omegaRT. Since this parameter is less then 0.15 for most membrane studies, the nonuniform Js equation can usually be approximated by the uniform pore form. The general results are illustrated by explicit calculations on several models of nonuniform pores. It is shown, for example, that the "equivalent pore radius" defined in the usual way is a function of the experimental parameter that is measured and is not unique.     

Simulations of skin barrier function: free energies of hydrophobic and hydrophilic transmembrane pores in ceramide bilayers

Transmembrane pore formation is central to many biological processes such as ion transport, cell fusion, and viral infection. Furthermore, pore formation in the ceramide bilayers of the stratum corneum may be an important mechanism by which penetration enhancers such as dimethylsulfoxide (DMSO) weaken the barrier function of the skin. We have used the potential of mean constraint force (PMCF) method to calculate the free energy of pore formation in ceramide bilayers in both the innate gel phase and in the DMSO-induced fluidized state. Our simulations show that the fluid phase bilayers form archetypal water-filled hydrophilic pores similar to those observed in phospholipid bilayers. In contrast, the rigid gel-phase bilayers develop hydrophobic pores. At the relatively small pore diameters studied here, the hydrophobic pores are empty rather than filled with bulk water, suggesting that they do not compromise the barrier function of ceramide membranes. A phenomenological analysis suggests that these vapor pores are stable, below a critical radius, because the penalty of creating water-vapor and tail-vapor interfaces is lower than that of directly exposing the strongly hydrophobic tails to water. The PMCF free energy profile of the vapor pore supports this analysis. The simulations indicate that high DMSO concentrations drastically impair the barrier function of the skin by strongly reducing the free energy required for pore opening.     

A New Proposal for the Action of Vasopressin, Based on Studies of a Complex Synthetic Membrane

The total osmotic flow of water across cell membranes generally exceeds diffusional flow measured with labeled water. The ratio of osmotic to diffusional flow has been widely used as a basis for the calculation of the radius of pores in the membrane, assuming Poiseuille flow of water through the pores. An important assumption underlying this calculation is that both osmotic and diffusional flow are rate-limited by the same barrier in the membrane. Studies employing a complex synthetic membrane show, however, that osmotic flow can be limited by one barrier (thin, dense barrier), and the rate of diffusion of isotopic water by a second (thick, porous) barrier in series with the first. Calculation of a pore radius is meaningless under these conditions, greatly overestimating the size of the pores determining osmotic flow. On the basis of these results, the estimation of pore radius in biological membranes is reassessed. It is proposed that vasopressin acts by greatly increasing the rate of diffusion of water across an outer barrier of the membrane, with little or no accompanying increase in pore size.     

Smooth DNA Transport through a Narrowed Pore Geometry

Voltage-driven transport of double-stranded DNA through nanoscale pores holds much potential for applications in quantitative molecular biology and biotechnology, yet the microscopic details of translocation have proven to be challenging to decipher. Earlier experiments showed strong dependence of transport kinetics on pore size: fast regular transport in large pores (> 5 nm diameter), and slower yet heterogeneous transport time distributions in sub-5 nm pores, which imply a large positional uncertainty of the DNA in the pore as a function of the translocation time. In this work, we show that this anomalous transport is a result of DNA self-interaction, a phenomenon that is strictly pore-diameter dependent. We identify a regime in which DNA transport is regular, producing narrow and well-behaved dwell-time distributions that fit a simple drift-diffusion theory. Furthermore, a systematic study of the dependence of dwell time on DNA length reveals a single power-law scaling of 1.37 in the range of 35–20,000 bp. We highlight the resolution of our nanopore device by discriminating via single pulses 100 and 500 bp fragments in a mixture with >98% accuracy. When coupled to an appropriate sequence labeling method, our observation of smooth DNA translocation can pave the way for high-resolution DNA mapping and sizing applications in genomics.     

Cascades of transient pores in giant vesicles: line tension and transport

Under ordinary circumstances, the membrane tension of a giant unilamellar vesicle is essentially nil. Using visible light, we stretch the vesicles, increasing the membrane tension until the membrane responds by the sudden opening of a large pore (several micrometers in size). Only a single pore is observed at a time in a given vesicle. However, a cascade of transient pores appear, up to 30-40 in succession, in the same vesicle. These pores are transient: they reseal within a few seconds as the inner liquid leaks out. The membrane tension, which is the driving force for pore opening, is relaxed with the opening of a pore and the leakage of the inner liquid; the line tension of the pore's edge is then able to drive the closure of a pore. We use fluorescent membrane probes and real-time videomicroscopy to study the dynamics of the pores. These can be visualized only if the vesicles are prepared in a viscous solution to slow down the leakout of the internal liquid. From measurements of the closure velocity of the pores, we are able to infer the line tension,. We have studied the effect of the shape of inclusion molecules on. Cholesterol, which can be modeled as an inverted cone-shaped molecule, increases the line tension when incorporated into the bilayers. Conversely, addition of cone-shaped detergents reduces. The effect of some detergents can be dramatic, reducing by two orders of magnitude, and increasing pore lifetimes up to several minutes. We give some examples of transport through transient pores and present a rough measurement of the leakout velocity of the inner liquid through a pore. We discuss how our results can be extended to less viscous aqueous solutions which are more relevant for biological systems and biotechnological applications.     

Filtration,diffusion, and molecular sieving through porous cellulose membranes

1. A study has been made of the diffusion and filtration of a graded series of molecules (including tritium-labelled water, urea, glucose, antipyrine, sucrose, raffinose, and hemoglobin) in aqueous solution through porous cellulose membranes of three degrees of porosity. 2. Experimental results were in close agreement with predictions based on the membrane pore theory of Pappenheimer et al. (1,2). Restriction to molecular diffusion is a function of pore radius and molecular radius described by equation (11) in the text. Molecular sieving during ultrafiltration is a function of total pore area per unit path length, pore radius, molecular radius, and filtration rate given by equations (16) and (19). 3. Estimates of average pore radius made by means of this theory were considerably larger than estimates made by the method of Elford and Ferry (3) (Table II). Sources of error in the latter method are discussed and a new method of membrane calibration is proposed in which the total cross-sectional area of the pores is measured by direct diffusion of isotope-labelled water. 4. Steady-state osmotic pressures of solutions of sucrose and raffinose measured during molecular sieving through cellulose membranes were found to be close to the "ideal" osmotic pressures calculated by van't Hoff's law. Thus the present experimental data support the methods used by Pappenheimer et al. in their studies on living capillary walls as well as their theory of membrane pore permeability.     

A model for sieving of macromolecules by the glomerular membrane of the kidney.

The transport of water and of macromolecules across the glomerular membrane of the kidney depends on the membrane parameters (radius, length and number of pores) as well as on the hydrostatic and oncotic pressures on either side of the membrane. The filtration pressure decreases along the capillary loops from afferent to efferent end. Water and solute flows are thus given by a system of two differential equations. The sieving coefficient of the macromolecules is the ratio of solute to water flow. In the program described the differential equations are solved by the Runge-Kutta method (fourth order). Rosenbrock's method of minimization is used to adjust the theoretical to the experimental sieving coefficients. The pore radius, total pore area per unit of path length and conductance of the membrane, as well as the intracapillary hydrostatic pressure and its gradient can thus be determined.     

A Brownian ratchet for protein translocation including dissociation of ratcheting sites

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate γ. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901–925, 2010) we allow the ratcheting molecules to dissociate (at rate δ) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length.