Fundamentals of electroporative delivery of drugs and genes

Bioelectrochemistry and Bioenergetics 48 Ž1999. 3–16

Mini-review

Fundamentals of electroporative delivery of drugs and genes Eberhard Neumann ) , Sergej Kakorin, Katja Tœnsing Physical and Biophysical Chemistry, Faculty of Chemistry, UniÕersity of Bielefeld, P.O. Box 100 131, D-33501 Bielefeld, Germany Dedicated to:Professor Hermann Berg on the occasion of his 75th birthday. Received 11 November 1998; revised 27 December 1998; accepted 4 January 1999

Abstract Electrooptical and conductometrical relaxation methods have given a new insight in the molecular mechanisms of the electroporative delivery of drug-like dyes and genes ŽDNA. to cells and tissues. Key findings are: Ž1. Membrane electroporation ŽME. and hence the electroporative transmembrane transport of macromolecules are facilitated by a higher curvature of the membrane as well as by a gradient of the ionic strength across charged membranes, affecting the spontaneous curvature. Ž2. The degree of pore formation as the primary field response increases continuously without a threshold field strength, whereas secondary phenomena, such as a dramatic increase in the membrane permeability to drug-like dyes and DNA Žalso called electropermeabilization., indicate threshold field strength ranges. Ž3. The transfer of DNA by ME requires surface adsorption and surface insertion of the permeant molecule or part of it. The diffusion coefficient for the translocation of DNA Ž Mr f 3.5 = 10 6 . through the electroporated membrane is Dm s 6.7 = 10y13 cm2 sy1 and Dm for the drug-like dye Serva Blue G Ž Mr f 854. is Dm s 2.0 = 10y12 cm2 sy1. The slow electroporative transport of both DNA and drugs across the electroporated membrane reflects highly interactive Želectro-. diffusion, involving many small pores coalesced into large, but transiently occluded pores ŽDNA.. The data on mouse B-cells and yeast cells provide directly the flow and permeability coefficients of Serva blue G and plasmid DNA at different electroporation protocols. The physico-chemical theory of ME and electroporative transport in terms of time-dependent flow coefficients has been developed to such a degree that analytical expressions are available to handle curvature and ionic strength effects on ME and transport. The theory presents further useful tools for the optimization of the ME techniques in biotechnology and medicine, in particular in the new field of electroporative delivery of drugs Želectrochemotherapy. and of DNA transfer and gene therapy. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Electroporation; Electrotransformation; Electrochemotherapy; Membrane permeability; Yeast cell; Membrane adsorption

1. Introduction The membrane electroporation ŽME. is an electric technique to render lipid and lipid–protein membranes porous and permeable, transiently and reversibly, by electric voltage pulses. The concept of ME has been derived from the electrically induced permeability changes which were indirectly judged from the partial release of intracellular components w1x or from the uptake of macromolecules such as DNA as indicated by electrotransformation data w2–4x. The electrically facilitated uptake of foreign genes is called direct electroporative gene transfer or electrotransformation of cells. Similarly, electrofusion of single cells to Abbreviations: SBG, Serva Blue G; C, closed lipid state; P, porous lipid state ) Corresponding author. Tel.: q49-521-106-20-53; fax: q49-52-10629-81.

large syncytia w5x and electroinsertion of foreign proteins w6x into electroporated membranes are also based on ME, i.e., electrically induced structural changes in the membrane phase. For the time being, the method of ME is widely used to manipulate all kinds of cells, organelles and even intact tissue. ME is also applied to enhance iontophoretic drug transport through skin, see, e.g., Pliquett et al. w7x, or to introduce chemotherapeuticals into cancer tissue, an approach pioneered by Mir w8x. Despite of common use of ME in biotechnology and medicine, the molecular mechanisms of ME and electrodelivery of macromolecules are not well understood. However, progress in the physico-chemical analysis of data on model systems, such as lipid bilayer membranes or unilamellar lipid vesicles, begins to establish a number of theory—based reliable directives for the various cell— manipulative applications w9,10x. When the primary processes are physico-chemically understood, the specific

0302-4598r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 2 - 4 5 9 8 Ž 9 9 . 0 0 0 0 8 - 2

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E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

electroporative properties of cell membranes and living tissue may also be quantitatively rationalized. Electrooptical and conductometrical data of unilamellar liposomes showed that the electric field causes not only membrane pores but also shape deformation of liposomes w11,12x. It appears that ME and shape deformation are strongly coupled, mutually effecting each other w4x. The primary field effect of ME and cell deformation triggers a cascade of numerous secondary phenomena, such as pore enlargement and transport of small and large molecules across the electroporated membrane. In this paper the discussion is focused on the chemical–structural aspects of ME and cell deformation, as well as the fundamentals of transport through electroporated membrane patches. The aim is to characterize the electroporative transport with specific parameters such as membrane permeation coefficient, flow coefficient, rate coefficients of the formation and resealing of transporting pores, membrane surface fraction as well as mean number and radius of these pores.

2. Fundamentals of membrane electroporation 2.1. Concept of pore It appears that electroporative uptake of DNA or drugs or release of cytosolic components is preceded by field-induced structural changes in the membrane phase, comprising transient, yet long-lived permeation sites, pathways, channels or pores w3,13–17x. On the same line, the massive ion transport through planar membranes, as observed in the dramatic conductivity increase when a voltage Ž100–500 mV. is applied, can hardly be rationalized solely by a permeation across the densely packed lipids of an electrically modified membrane without field-induced open passages or pores w17x. Actually, a small monovalent ion, such as NaqŽ aq. of radius r i s 0.22 nm and of charge q Na s e0 s 1.6 = 10y1 9 C, passing through a lipid membrane encounters the Born energy barrier of 68 kT. Hence, the required transmembrane voltage to overcome this barrier amounts to 1.75 V at T s 298 K Ž258C.. An even larger voltage of 3.5 V would be needed for divalent ions such as Ca2q Ž qCa s 3.2 = 10y1 9 C, r i s 0.22 nm.. Nevertheless, the transmembrane potential required to electroporate the cell membrane such that the ion transport gets measurable, usually does not exceed 0.5 V w16,17x. The reduction of the energy barrier can be readily achieved by a transient aqueous pore. Certainly, the stationary pores kept open by the transmembrane field can only be small ŽF 1 nm diameter. in order to prevent discharging of the membrane interface by ion conduction w4,9,18x. However, additional lateral tension due to long-lasting electrical Maxwell stress on the vesicle or cell may lead to further pore enlargements. The special structural order of a long-lived, potential permeation site may be modeled by the so-called inverted or hydrophilic ŽHI. pore ŽFig. 1., w17–19x.

Fig. 1. Model for the chemical state transitions C | ŽP. and C | HO | HI; molecular rearrangements of the lipids in the pore edges of the lipid vesicle membrane. C denotes the closed bilayer state. The external electric field causes ionic interfacial polarization of the membrane dielectrics which is analogous to condenser plates Žq,y.. Em s Eind is the membrane field induced by the external field E, leading to water entrance in the membrane to produce pores ŽP.; cylindrical hydrophobic ŽHO. pores or inverted hydrophilic ŽHI. pores. In the pore edge of the HI pore states the lipid molecules are turned to minimize the hydrophobic contact with water. Note that in the open ionic condenser the ion density adjacent to the aqueous pore Ž ´ W . is larger than in the interface lipid Ž ´ L .rmedium due to ´ W 4 ´ L .

2.2. Visualization of pores Because of the small size there is up to now no visible evidence for electropores such as electromicrographs. The large pore-like crater structures or volcano funnels of 50 nm to 0.1 mm diameter, observed in electroporated red blood cells, most probably result from the enlargement of smaller primary pores by osmotic or hydrostatic pressure due to Maxwell stress w14x. Voltage-sensitive fluorescence microscopy at the membrane level of sea urchin eggs has shown that the transmembrane potential in the pole caps opposite to electrodes goes to a saturation level or even decreases, both as a function of pulse duration and external field strength, respectively. If the membrane conductivity would remain very low, the transmembrane potential should linearly increase with the external field strength. Leveling off and decrease of the transmembrane potential at higher fields indicate that the ionic conductivity of the membrane has increased, providing evidence for ion-conductive electropores w15x. On the same line, in DC electric fields the fluorescence images of the contour of elongated and electroporated giant vesicle shows large openings in the pole caps w20x. Most probably, these openings are appearing after coalescence of small primary pores invisible in microscopy. Theoretical analysis of the membrane curvature in the vesicle pole caps suggests that vesicle elongation under Maxwell stress must facilitate both pore formation and enlargement of existing pores w21x. 2.3. Transmembrane field According to the classical definition of the electric field strength as the negative electric potential gradient, the actual membrane field strength is given by: Em s yD wm rd

Ž 1.

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

where D wm is the intrinsic cross membrane potential difference and d f 5 nm is the dielectric membrane thickness. The potential difference D wm may generally consist of several contributions, e.g.,: natural Ž D wnat ., surface Ž D ws . and induced Ž D w ind . membrane potential differences, respectively. Here the locally limited, but high Ž150–600 mV. dipole potentials in the boundary between lipid head groups and hydrocarbon chains of the lipids are neglected w22,23x. Typically, D wnat f y70 mV, where the potential of the outside surface is taken as zero w24x. In living cell membranes D wnat is metabolically maintained. A finite value of D ws usually results from an excess of negatively charged groups at the interfaces between membrane surfaces and aqueous media and different ionic strengths of the cell internal and external medium. Therefore, even in the absence of an external field there can be a finite membrane field w24x. In static fields and lowfrequency alternating fields, for spherical geometry with cell or vesicle radius a, the solution of the Laplace equation yields the induced potential difference in the direction ™ of the external electric field vector E: D w ind s y1.5aEf Ž lm .
Ž 2. ™

where u is the polar angle between E and radius vector ™ r Žsee Fig. 2a.. The conductivity factor f Ž lm . can be expressed in terms of a and d and the conductivities lm , li ,

5

l0 of the membrane, the cell Žvesicle. interior and the external solution, respectively. Provided that additivity holds the overall cross-membrane potential difference is given by w25,26x: D wm s y

3a 2

Ef Ž lm . q

D wnat q D ws cos u


Ž 3.

Normally, D wnat and D ws are independent of u . For the special case when D wnat and D ws have equal signs, there can be a major asymmetry. At the left pole cap the sum D wnat q D ws is in the same direction as D w ind whereas at the right pole cap yŽ D wnat q D ws . is opposite to D w ind ŽFig. 2b.. Therefore, membrane electroporation will start at the left hemisphere where the field Em s yŽ D w ind q D wnat q D ws .rd is larger than at the right hemisphere Ž Em s yŽ D w ind y D wnat y D ws .rd .. In the case of opposite signs for D wnat and D ws , the natural potential D wnat may be compensated by D ws , then the asymmetry in the two hemispheres of cells gets smaller. Substitution of Eq. Ž3. in Eq. Ž1. shows that the field effect results in a size-dependent amplification of the membrane field relative to the external field E. The field amplification factor Ž3ar2 d . is especially significant for large objects with thin dielectric membranes. For typical values such as a s 10 mm and d s 5 nm, we have a field amplification of Ž3ar2 d . s 3 = 10 3. For elongated cells like bacteria aligned by the field in the direction of E, the contribution of Eind at the pole caps, where cos u s 1, amounts to Eind s Ž Lr2 d . E, where the amplification factor Ž Lr2 d . is proportional to the bacterium length L w27x. 2.4. The membrane as a dynamic electric condenser A lipid membrane is a highly dynamic phase of usually mobile lipid molecules hydrophobically held together by the aqueous environment. In the electric field, the redistribution of ions on both sides of the membrane dielectrics is equivalent to electric condenser plates at constant potential ŽFig. 2a.. Such a charged condenser with both mobile interior and mobile environment favors the entrance of water molecules to produce pores Ž P ., which may be hydrophobic ŽHO. andror hydrophilic ŽHI. pores, with higher dielectric constant Ž ´ f 80. compared with the lower dielectric constant Ž ´ L f 2. of the replaced lipids Žstate C, Fig. 1..

Fig. 2. Interfacial membrane polarization of a cell of radius a. Ža. Cross-section of a spherical membrane in the external field E. Žb. The profile of the electrical potential f across the cell membrane of thickness d in E as a function of distance x; Note that D fm Ž u s1808. s D f ind y D fs y D fnat and D fm Ž u s 08. s D f ind q D fs q D fnat , where D f ind is the drop in the induced membrane potential in the direction of E; D fs the surface potential difference at zero external field and D fnat the natural Ždiffusion. potential difference at zero external field, also called resting potential.

2.5. Electrochemical kinetics of pore creation and resealing The formation of pores in the field and pore resealing after the electric field pulse can be viewed as a state transition from the intact closed lipid state ŽC. to the porous state ŽP. according to the reaction scheme w26x: C|P

Ž 4.

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

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The state transition involves a cooperative cluster Ž L n . of n lipids L forming an electropore w19x. The degree of membrane electroporation f p is defined by the concentration ratio: f p s w P x r Ž w P x q w C x . s Kr Ž 1 q K .

Ž 5.

where K s wPxrwCx s k 1rky1 is the equilibrium distribution constant, k 1 the rate coefficient for the step C ™ P and ky1 the rate coefficient for the resealing step ŽC § P.. In an external electric field, the distribution between C and P is shifted in the direction of increasing wPx. Note, the frequently encountered observation of very small pore densities means that K < 1. For this case f p f K. Hence the thermodynamic, field-dependent quantity K is directly obtained from the experimental degree of poration. In the electric field Ž0 F t F t E , t E is the pulse duration., the time course of pore formation is obtained by the integration of the rate equation for the Scheme Ž4.: d wPx

d wCx sy

dt

dt

s k 1 w C x y ky1 w P x

Ž 6.

Applying the practical assumption that f p Ž0. s 0 at E s 0 and t s 0 we obtain: f pC ™ P Ž t . s

K Kq1

 1 y exp w ytrt x 4 ,

Ž 7.

where t s Ž k 1 q ky1 .y1 s Ž k 1Ž1 q K ..y1 is the relaxation time. For the after-field time range t ) t E , where ky1 4 k 1 and f Ž t E . s K Ž K q 1.y1  1 y expwyt Ert x4 , integration of Eq. Ž6. yields: f pP ™ C Ž t . s f Ž t E . exp yky1 Ž t y t E .

Ž 8.

It is readily seen that from the experimentally accessible dependencies f p Ž t . in the field and in the post field time range, both rate coefficients, k 1 and ky1 , and the equilibrium constant K can be determined. The symbol P may include several different rapidly equilibrated pore states, for instance, HO | HI in the sequence C | HO | HI ŽFig. 1.. In this case, ky1 in the expressions for f p Ž t . must be replaced by ky1 rŽ1 q K 2 ., where K 2 s wHIxrwHOx is the equilibrium constant of the second step HO | HI w19x. 2.6. u-dependence of the membrane field For the curved membranes of cells and organelles, the dependence of the induced potential difference D w ind and thus the transmembrane field Eind s yD w indrd on the positional u-angle leads to the shape-dependent u-distribution of the values of K and k 1; ky1 is assumed to be independent of E and thus independent of u . Therefore, all conventionally measured quantities are u-averages. The stationary value of the actually measured u-average fraction f p of porated area is given by the integral: fp s

1

p

H 2 0

k 1Ž u . ky1 q k 1 Ž u .

sin u d u

Ž 9.

The actual pore density f pu in the cell pole caps, where u f 08 and 1808, respectively, can be a factor of 4 larger than the u-average fraction f p . For the electropores f p is usually very small, e.g., f p F 0.003 or 0.3% w11,12x. Even the pole cap value f pu Ž08, 1808. s 4 f p s 0.012 certainly corresponds to a small pore density. However, at longer pulses the hydrostatic pressure resulting from the Maxwell stress can enlarge the electropores Žand thereby the fraction f p . such that they can be easily seen in fluorescence microscopy w28x.

3. Thermodynamical description of electroporation The lipid membrane in an external electric field is an open system with respect to H 2 O molecules and surplus ions, charging the membrane condenser w18x. Therefore, to ensure the minimization of the adequate Gibbs energy with respect to the field Em , the normal Gibbs energy G with dG A Em d M, where M is the global electric dipole moment, must be transformed to the form Gˆ s G y Em M with dGˆ A yMd Em w29x. Now, Em in d Em is the explicit variable and membrane electroporation can be adequately described in terms of Em and the induced electric dipole moment M of the pore region. The global equilibrium constant K of the poration-resealing process is directly related to the standard value of the transformed reaction Gibbs energy D r Gˆ] by K s expŽyD r Gˆ]rRT. w30x, where RT is the molar thermal energy. The molar work potential difference D r Gˆ]s Gˆ] ŽP. y Gˆ] ŽC., between the two states C and P in the presence of an electric field generally comprises chemical and physical terms w18x: D r Gˆ]s Ý a

Ý Ž n j m]j .

a

q

j H

H0

q

Dr bd Hy

L

S

H0 D g d L qH0 D G d S r

Em

H0

D r Md Em

r

Ž 10 .

Note that D r s drd j , where d j s d n jrn j is the differential molar advancement of a state transition, n j is the amount of substance and n j is the stoichiometric coefficient of component j, respectively. The single terms are separately considered as follows. The first term of the right hand side of Eq. Ž10. is the so-called chemical contribution. The pure concentration changes of the lipid Ž j s L. and water Ž j s W. molecules involved in the formation of an aqueous pore with edges are described by y ja and the standard chemical potential mua of the participating molecule j, respectively, consj tituting the phase a , either state C or state P w29x; ].P . a s Ž n W m] here, Ý a Ý j Ž n j m] y Ž n W m] j W q n L mL Wq ]. C n L mL . For cylindrical pores ŽHO-pore, Fig. 1. of mean pore radius rp , the pore edge energy term can be specified as: H0LD r g d L s NA H0L Žg P y g C .d L s 2p NA g rp , where g is

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

the line tension or pore edge energy density, g P s g Žbecause g C s 0, no edge., L s 2p rp is the edge length and NA the Avogadro constant. At small field strengths Ž E F 2 MV my1 . and small cell radii Ž a F 50 nm. the surface tension term for spherical bilayers in water HD r G d S s NA H0s Ž G P y GC .d S is usually negligibly small because the difference in the surface energy density G between the states P and C is in the order of F 1.2 mN my1 for phosphatidylcholine in the fluid bilayer state w31x; S s p rp2 is the surface area of the average electropore in the surface plane of the membrane. At larger field strengths Ž E G 2 MV my1 . and larger cell radii Ž a G 50 nm., the Maxwell stress increases GC and thereby facilitates membrane electroporation and shape deformation. The explicit expression for the curvature energy term of lipid vesicles is given by w18,21x: HD r b d H s NA HŽ b P y bC .d H f y64 NAp 2ak rp2 z Ž1ra q H0elr2pa .rd, where differently to Ref. w21x here the total surface area difference refers to the middle of the two monolayers w32x. Note that the aqueous pore part has no curvature, hence the curvature term is reduced to bp y bC s ybC . H s H0 q 1ra is the membrane curvature inclusively the spontaneous curvature H0 s H0chem q H0el , where H0chem is the mean spontaneous curvature for different chemical compositions of the two membrane leaflets and H0el is the electrical part of the spontaneous curvature, e.g., at different electrolyte surroundings at the two membrane sides. If H0 s 0, then, in the case of spherical vesicles, we have H s 1ra. Furtheron, k is the elastic module, a Žf 1. is a material constant w32x, z is a geometric factor characterizing the pore conicity w18x. It appears that the larger the curvature and the larger the H0el term, the larger is the energetically favorable release of the Žtransformed. Gibbs energy during the pore formation. The curvature term HD r b d H can be as large as a few kT per one pore w21x, where k s RrNA is the Boltzmann constant and T the absolute temperature. For small vesicles or small organelles and cells the curvature term is particularly important for the energetics of ME. The effect of membrane curvature on ME has been studied with dye-doped vesicles of different size, i.e., for different curvatures. At constant transmembrane potential drop Že.g., D wm s y0.3 V., an increased curvature greatly increases the amplitude of the absorbance dichroism, characterizing the extent of pore formation ŽFig. 3a., w19,21x. This observation was quantified in terms of the curvaturedependent area difference elasticity energy ŽADE. resulting from the different packing density of the lipid molecules in the two membrane leaflets of curved membranes ŽFig. 3b., w18,33x. Strongly curved membranes appear to be electroporated easier than planar membrane parts w4x. Additionally, different electrolyte concentrations on the internal and external membrane sides cause different charge screening. The effect of this difference on ME is theoretically described in terms of the surface potential drop D ws

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Fig. 3. The effect of vesicle size on the extent and rate of electroporation. The amplitudes of Ža. the absorbance dichroism D Ayr A 0 as a function of the vesicle curvature H s1r a at the constant nominal transmembrane voltage drop D fmN sy1.5 aE. Here D fmN sy0.3 V. The unilamellar vesicles are composed of L-a-phosphatidyl-L-serine ŽPS. and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine ŽPOPC. in the molar ratio PS:POPC of 1:2 doped with 2-Ž3-Ždiphenylhexatrienyl.propanoyl.-1hexadecanoyl-sn-glycero-3-phosphocholine Žb-DPH pPC, Mr s 782.; total lipid concentration w L T x s1.0 mM; wb-DPH pPC T x s 5 mM; 0.66 mM HEPES ŽpH s 7.4., 130 mM CaCl 2 , T s 293 K Ž208C.; vesicle density r v s 2.1=10 15 dmy. Application of one rectangular electric pulse of the field strength E and pulse duration t E s10 ms at T s 293 K Ž208C., w9x. Žb. The membrane curvature is associated with a lipid packing difference between the two membrane leaflets and a lateral pressure gradient across the membrane. Membrane electroporation, causing conical hydrophobic ŽHO. pores, reduces the lipid packing density difference between the two monolayers and, consequently, decreases the gradient of lateral pressure across the membrane.

and the electrical part of the spontaneous curvature H0el . Extending previous approaches w34,35x, we obtain for a thin membrane Ž d < a., 1: 1 electrolyte and for small values of the dimensionless parameter sinŽout. s e 0 s inŽout.r Ž ´ 0 ´ inŽout. x inŽout. kT . < 1, that H0el is given by: H0el s 2 . Ž 2 2 Žy2r3.Ž sin2 y sout r sinrx in q sout rxout ., where ´ 0 is the permittivity of vacuum, e 0 s 1.6 = 10y1 9 C is the elementary charge, in the SI notation x inŽout. s w2 e 02 IcinŽout. NA r Ž ´ 0 ´ inŽout. kT .x1r2 is the inverse Debye length, ´ inŽout. the dielectric constant, s inŽout. and IcinŽout. the membrane surface charge density and the bulk ionic strength covering all ions of the inner Žin. and outer Žout. medium, respectively.

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E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

Large salt concentration gradients across strongly curved charged membranes permit electroporative efflux of electrolyte ions at surprisingly low transmembrane potential differences, for instance < D wm < s 37.5 mV in the pole caps of a vesicle of radius of a s 50 nm at a pulse duration of t E s 100 ms compared with < D wm < f 500 mV for planar Žnon-curved. membranes w11,36x. In the electric polarization term HD r Md Em , the electric reaction moment D r M s Mm ŽP. y MmŽC. refers to the difference in the molar dipole moments Mm of state C and P, respectively. The field-induced reaction moment in the electrochemical model is given by w26x: D r M s NAVp D r P, where Vp s p rp2 d is the average Žinduced. pore volume of the assumed cylindrical pore. Inspired by the physical analysis by Abidor et al. w36x, we define the chemical reaction polarization as w19x: D r P s ´ 0 Ž ´ W y ´ L . Em . The difference ´ W y ´ L in the dielectric constants of water and of lipids, respectively, refers to the replacement of lipids by water at constant Em . Note that the possible difference in the values of Em ŽC. and EmŽP. is not too essential for the calculation of D r P, because usually ´ W 4 ´ L and EmŽC. f Em ŽP., thus we may approximate ´ 0 Ž ´ W EmŽP. y ´ L Em ŽC.. f ´ 0 Ž ´ W y ´ L . EmŽP.. In general, this approximation is valid only for small pores of radius - 1 nm, which are not yet too conductive. Since ´ W 4 ´ L , the formation of aqueous pores is strongly favored in the presence of a cross-membrane potential difference D wm , in particular at high E, when the contribution of D w ind in D wm is large Žsee Eq. Ž3... Explicitly, at the angle u , and in the boundaries Em s 0 and Em ,HD r Md Em s Ž9r8. p´ 0 a2 Ž ´ W y ´ L . rp2 NA f 2 Ž l m .cos 2 u E 2rd w18,19x. If the relation between K and E can be formulated as K s K 0 expw HD r Md Em rŽRT.x, where K 0 refers to E s 0, the mean pore radius rp can be calculated in a simple way from the field dependence of K or of f p Žthe degree of poration.. Typically, at D w ind s y0.42 V and pulse duration t E s 10 ms, the mean pore radius amounts to rp s 0.35 " 0.05 nm w19x.

4. Electroporative shape deformation of vesicles and cells 4.1. ElectroporatiÕe elongation at constant Õolume The initial very rapid Žms time range. electroporative elongation of salt filled lipid vesicles from the spherical shape to an ellipsoid in the direction of the field vector E was previously called phase 0, w4x. In this phase there is no measurable release of salt ions ŽFig. 4.. Hence the internal volume of the vesicle remains constant. Vesicle elongation is only possible if the membrane surface can be increased by ME and by membrane stretching and smoothing of membrane undulations under Maxwell stress. The formation of aqueous pores means entrance of water and thus increase in the overall membrane volume and surface.

Fig. 4. Electroporative deformation of unilamellar lipid vesicles Žor biological cells.. Phase 0: fast Žms. membrane electroporation rapidly coupled to Maxwell deformation at constant internal volume and slight Ž0.01–0.3%. increase in membrane surface area. Phase I: slow Žms–min. electromechanical deformation at constant membrane surface area and decreasing volume due to efflux of the internal solution through the electropores. Maxwell stress and electrolyte flow change the pore dimension from initially rp s 0.35"0.05 nm to rp s 0.9"0.1 nm w11x.

Thus, the vesicle elongation is rapidly coupled to ME according to the scheme: C | P m Ž elongation.

Ž 11 .

It is important to note that the characteristic time constant tdef of vesicle deformation is usually smaller than the u-average time constant of ME Žt f 0.5 to 1 ms.. Actually, for vesicles of radius a s 50 nm, a typical membrane bending rigidity of k s 2.5 = 10y2 0 J w11x and the viscosity h s 8.9 = 10y4 kg my1 sy1 of water at 208C, the upper limit of the shape deformation time constant at zero field is: tdef Ž0. s 0.38h a3rk s 0.9 ms w37x. It can be shown that in electric fields of typically 1 F ErMV my1 F 8, the shape relaxation time constant is 100-fold smaller than tdef Ž0., say tdef Ž E . s 10 ns ŽKakorin et al., unpublished.. The characteristic time constant of membrane stretching is even smaller: ts f h arK c s 0.32 ns, where, for instance, K c s 0.14 Nmy1 is the compression modulus w38x. Therefore, because t 4 tdef Ž E ., it is the structural change of pore formation, inherent in ME, that controls the major part of the extent and the rate of the vesicle deformation in the phase 0. The contributions of vesicle and cell deformations by stretching, smoothing and by ME can be easily measured by electrooptic dichroism, either turbidity dichroism or absorbance dichroism, if they occur on different time scales or if one of the contributions is dominant. Proper analysis of the respective electrooptic data provides the electroporative deformation parameter p s crb, where c and b are the major and minor ellipsoid axis, respectively, of the deformed vesicle or cell. Specifically, from the contribution of ME to p we obtain the u-average degree f p of ME w4x. 4.2. Elongation at constant surface area The initial phase 0 is followed by the slower phase I ŽFig. 4.. During the phase I there is an efflux of salt ions under Maxwell stress through the electropores created in phase 0, leading to a decrease in the vesicle volume under practically constant membrane surface Žincluding the surfaces of the aqueous pores.. The increase in the suspension

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

conductivity, D l Irl 0 , in the phase I reflects the efflux of salt ions under the electrical Maxwell stress through the electropores. Theoretically, the volume reduction during the phase I is described with a Lambert W function w12x. The kinetic analysis in terms of the volume decrease yields the membrane bending rigidity k s 3.0 " 0.3 = 10y2 0 J. At the field strength E s 1.0 MV my1 and in the range of pulse durations of 5 F t Erms F 60, the number of waterpermeable electropores is found to be Np s 35 " 5 per vesicle of radius a s 50 nm, with mean pore radius rp s 0.9 " 0.1 nm w11x. This pore size refers to the presence of Maxwell stress causing pore enlargement from an originally small value Ž rp s 0.35 " 0.05 nm. under the flow of electrolyte through the pores. The kinetic analysis developed for vesicles may be readily applied to tissue cells. The external electric field in tissue produces membrane pores as in isolated single cells and the electric Maxwell stress squeezes the cells w12x. The electromechanical cell squeezing can enlarge preexisting, or create new, pathways in the intercellular interstitial spaces, facilitating the migration of drugs and genes from the periphery to the more internal tissue cells.

9

Fig. 5. The average fraction f p of the electroporated membrane area, ŽB. at a large NaCl concentration difference Žin the vesicle interior wNaClx in s 0.2 M, in the medium wNaClx out s 0.2 mM, osmotically balanced with 0.284 M sucrose., Ž'. at equal concentrations ŽwNaClx in s wNaClx out s 0.2 mM, smoothly increases with the field strength E, whereas the massive conductivity increase D l Ir l0 , Žv . of the suspension of the salt filled vesicles of radius as160"30 nm Ž l0 s 7.5 mS cmy1 , T s 293 K Ž208C., w19x. indicates an apparent threshold value Ethr s 7 MV my1 . The ratio f p s SŽ t E .rSm was calculated from the electrooptic relaxations, yielding characteristic rate parameters of the electroporation—resealing cycle in its coupling to ion transport.

5. Transport of small ions and macromolecules by membrane electroporation 5.1. ElectroporatiÕe transport of small ions It is recalled that the ion efflux from the salt-filled vesicles in an electric field is caused by membrane electroporation and by the hydrostatic pressure under Maxwell stress and that the electrooptic signals reflect electroporative vesicle deformations coupled to ME. The analysis of electrooptic dichroisms yields characteristic parameters of ME such as electrical pore densities for ion transport across the electroporated membrane patches. The fraction f p of the electroporated membrane surface Žderived from electrooptics. smoothly increases with the field strength ŽFig. 5.. In terms of the chemical model there is no threshold of the field strength w4,18x. Experimentally there is always a trivial threshold when the actual data points emerge out of the margin of measuring error. The conductivity increase Ž D l Irl0 . in the suspension of the salt filled vesicles however appears to have a ‘threshold value’ of the field strength ŽFig. 5.. The large pore dimensions refer to the pores maintained by medium efflow under Maxwell stress or reflect fragmentation of a small Ž- 1%. fraction of vesicles ŽU. Brinkmann et al., unpublished data.. 5.2. ElectroporatiÕe transport of ionic macromolecules and drugs The transport kinetics of larger macromolecules such as drugs and DNA is more complicated than the transport of

small salt ions and occurs in several temporally distinct stages. Obviously, the transport is greatly facilitated if there is at first adsorption of the macromolecules to the membrane surface w10,27x. For charged macromolecules, adsorption is followed by electrophoretic penetration into the surface of electroporated membrane patches. Further steps are the after-field diffusion, dissociation from the internal membrane surface and, finally, binding with cell components in the cell interior ŽFig. 6. w9,10x. The transient adsorption of potential permeants on the membrane surface may change both the local surface structure and the local membrane composition Žphase separation. in the outer membrane leaflet. The alterations of the molecular structure and redistributions of membrane components can lead to local changes in the membrane’s spontaneous curvature, bending rigidity and surface tension, respectively w32,33x. It is recalled that the spontaneous curvature is strongly affecting ME w21x. Increased spontaneous curvature can either hinder or facilitate ME. For instance, the Ca2q mediated adsorption of the protein annexin V to anionic lipids increases the lipid packing density by insertion of the tryptophan side chain into the membrane surface. This in turn, reduces the electroporatability of the remaining membrane parts w21x. Alternatively, the adsorption of plasmid DNA on the membrane surface, mediated by calcium or sphingosine, obviously facilitates ME and thus the transport of small ions Žleak. and DNA itself across the membrane w10,39,40x. The degree of transformation f T of yeast cells by plasmid DNA as a function of pulse duration is character-

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

10

ized by a long ‘delay phase’ ŽFig. 7a. w10x. The delay phase gets shorter with increasing field strength. The degree fC of coloring of B-cells by dye SERVA blue G ŽSBG. exhibits a similar functional dependence as f T of yeast cells ŽFig. 7b. w9x. These similarities of cell transformation and cell coloring suggest that the mechanism for the electroporative transport of both genes and drug-like dyes into the cell interior has essential features in common. Therefore, we developed a general formalism for the electroporative transport of drugs and genes. 5.3. Kinetics of drug and DNA uptake In line with Fick’s first law, the radial inflow Žvector. of macromolecules is given by: d n inc dt

s yDm Sm

d cm dx

Ž 12 .

where ninc is the molar amount of the transported molecule in the compartment volume Vc , c m and Dm are the concentration and the diffusion coefficient of the permeant in the membrane phase, respectively, Sm is the membrane surface through which the diffusional translocation occurs. The practical application of Eq. Ž12. includes several simplifying assumptions. Ž1. The diffusion coefficient as a phase-specific constant is independent of the position within the membrane phase of thickness D x s d. Ž2. The concentration gradient within the membrane is usually approximated by: d c m rd x s Ž c mout y c min . rd c mout

Ž 13 .

c min

where and are the concentrations of the permeant in the outer and inner membranermedium interfaces, respectively ŽFig. 8.. Ž3. The distribution of the molecules between the bulk solutions and the membrane interfaces is rapidly equilibrated with respect of the slower, rate-limiting crossmembrane translocation step. For this case, the partition may be

Fig. 7. Kinetics of the electroporative uptake of DNA and dye. Ža. The degree of transformation f T sT r Tmax of yeast cells by plasmid DNA Ž Mr s 3.5=10 6 ., where Tmax is the maximum number of transformants, and Žb. The degree of coloring fC sCr Cmax , where Cmax is the maximum number of colored mouse B cells by drug-like dye SERVA blue G Ž Mr s854. as a function of pulse duration t E 0 Ža. and t E Žb., respectively, at different field strengths: Ža. E0 rkV cmy1 s 2.5 Žl.; 3.0 Ž`.; 3.25 ŽI.; 3.5 Žv .; 4.0 ŽB., for cell transformation, and Žb. ErkV cmy1 s Ž`. 0.64; Žv . 0.85; ŽI. 1.06; ŽB. 1.28; Ž^. 1.49; Ž'. 1.7; Ž\. 1.91; Ž%. 2.13, for cell coloring, respectively. E0 is the amplitude and t E 0 is the characteristic time constant of an exponential pulse used for the transformation of yeast cells by plasmid DNA Ž Mr s 3.5=10 6 .. E is the amplitude and t E is the duration of the rectangular pulse used for the coloring of mouse B cells by the Ždrug-like. dye SERVA blue G Ž Mr s854..

quantified by a single distribution coefficient Žor distribution constant. according to:

gs

c mout c out

s

c min

Ž 14 .

c in

where c out and c in s n inc rVc are the bulk concentrations inside and outside the cell Žor vesicle., respectively. We define a flow coefficient k f for the crossmembrane transport by: Fig. 6. Scheme for the coupling of the binding of a macromolecule ŽD., either a dye-like drug or DNA, described by the equilibrium constant K D of overall binding, electrodiffusive penetration Žrate coefficient k pen . into the outer surface of the membrane and translocation across the membrane, in terms of the transport coefficient k f sg Dm Sm rŽ dVc .; and the binding of the internalized DNA or dye molecule ŽD in . to a cell component b Žrate coefficient k b . to yield the interaction complex D b as the starting point for the actual genetic cell transformation or cell coloring, respectively.

kf s

g Dm Sm d

s

Pm Sm

Vc

Vc

Ž 15 .

where the permeability coefficient Pm for the porated membrane patches is given by: Pm s

g Dm d

s kf

Vc Sm

Ž 16 .

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

11

The time dependent flow coefficient can now be expressed as: k f Ž t . s k f0 f p Ž t .

Ž 19 .

where the characteristic flow coefficient for the radial inflow is defined by k f0 s

Pm Sc

s

Vc

3 Pm

Ž 20 .

a

Note that k f0 and thus Pm are independent of the electrical pulse parameters E and t E . Hence these transport quantities are suited to compare vesicles and cells of different size and different lipid composition. Substitution of Eq. Ž19. into Eq. Ž18. and integration yields the practical equation for the time course of internal permeant concentration: Fig. 8. Profile of concentration of a lipid-soluble or surface adsorbed permeant across the lipid plasma membrane of the thickness d, between the outer Žout. and inner Žin. cell compartments, respectively, in the direction x. Because of adsorption of permeant on the cell surface, the bulk concentrations c out and c in of the permeant are smaller than c mout and c min , respectively; cm refers to the very small volume of a shell with thickness Ø, where Ø is given by the diameter of the Žflatly adsorbed parts of. DNA. For the data in Fig. 7 Ža., the distribution constant is g s cmout r c out s1.3=10 3 as an upper limit.

Pm can be calculated from the experimental value of k f , provided Sm is known. Substitution of Eqs. Ž13. – Ž15. into Eq. Ž12. yields the linear flow equation: d c in dt

s yk f Ž c out y c in .

Ž 17 .

Frequently, the external volume V0 is much larger than the intracellular or intravesicular volume, i.e., Nc Vc < V0 , where Nc is the number of cells or vesicles in suspension. Mass conservation dictates that the amount n out of permeant in the outside volume is given by nout s n 0 y n inc Nc . Hence the inequality Nc Vc < V0 yields: c out s n outrV0 s c 0 y c in Nc VcrV0 f c 0 , where n 0 and c 0 s n 0rV0 are the initial amount and the initial total concentration of the permeant in the outside volume, respectively. Substitution of the approximation c out s c 0 into Eq. Ž17. yields the simple transport equation: d c in dt

s yk f Ž c 0 y c in .

Ž 18 .

If the effective diffusion area Sm changes with time, for instance, due to electroporation-resealing processes, the flow coefficient k f Ž t . is time dependent. In this case we may specify Sm Ž t . with the degree of electroporation f p according to Sm Ž t . s f p Ž t . Sc , where Sc s 4p a 2 is the total area of the outer membrane surface. The explicit form of the pore fraction f p Ž t . is dependent on the model applied. See, for instance, Eqs. Ž7. and Ž8..

½

c in s c 0 1 y exp yk f0 tobs P ™ C

Ht

q

fp

tE C ™ P

žH /5 t0

fp

Ž t.dt

E

Ž t.dt

Ž 21 .

If the transported molecules are added before the pulse, we have t 0 s 0. For the post-field addition the first integral for f pC ™ P in Eq. Ž21. cancels and we set t E s t 0 s tadd , where tadd is the time point of adding the molecules after pulse termination Ž t E .. Usually, the appearance of the transported molecules becomes noticeable at observation times t obs which are much larger Žmin. than the characteristic time of pore resealing Ž ky1 .y1 which is in the ms–s time range. For these cases the approximation t obs ™ ` holds w9,10x. Note that the integrals in Eq. Ž21. contain implicitly the pulse duration t E and the field strength E in the degree of poration f p Ž t,t E , E .. In the case of charged macromolecules like DNA or the dye SBG, the presence of an electric field across the membrane causes electrodiffusion. The enhancement of the transport of a macroion only refers to that side of the cell or vesicle where the electric potential drop D wm is in the favorable direction. The electrodiffusive efflux of the macromolecules from the cell cytoplasm is usually negligibly small compared with the influx and may be neglected. Formally, for the boundaries t 0 and t E , Dm in Eq. Ž21. must be replaced by the electrodiffusional coefficient w10x:

ž

Dm Ž E . s Dm 1 q

< z eff < e 0 D wm kT

/

Ž 22 .

where D wm is the u-average transmembrane potential drop D wm s yŽ3r8. aEf Ž lm ., lm is the angular and time average of the membrane conductivity, z eff is the effective charge number Žwith sign. of the transported macromolecule.

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

12

On the same line, the permeability coefficient with respect to electrodiffusion is given by: Pm Ž E . s

g Dm Ž E . d

Ž 23 .

It is instructive to compare the present analysis of Želectro. diffusion through porous membrane patches characterized by the quantities k f0 , Pm and f p with the conventional approach with the permeability coefficient P in the context of formally f p s 1. The conventional coefficient P is related to Pm of the present analysis by: P s f p Ž t E . Pm

Ž 24 .

The analysis of the kinetic data of cell transformation and cell coloring by dyes ŽFig. 7. suggests that the ratelimiting step is the binding of the permeants to intracellular components. The simplest binding scheme is given by Žsee Fig. 6.: kb

D in q b ™ D b

Ž 25 .

where D in symbolizes the macromolecules in the cell interior Ž c in ., b the yet unoccupied binding sites in the cell and k b is the overall rate coefficient of binding. The degree of binding of molecule D is defined by: f b s wD b x r wb0 x

Ž 26 .

where wD b x is the concentration of bound maromolecules and wb 0 x is the total concentration of binding sites in the cell interior. The integration of the binding rate equation dwD b xrd t s k b c in wbx for the Scheme Ž25., and substitution of Eq. Ž21. yields w10x:

f b Ž t E ,t obs . s

c in  1 y exp A4

w b 0 x y c in exp A

Ž 27 .

where the dependence on t E and t obs is explicitly in c in Ž t E ,t obs . and AŽ t E ,t obs . s k b tobsŽ c in Ž t obs ,t E . y wb 0 x.. For the cell transformation the time of observation is t obs f 2 h. Note that c in Ž t E ,t obs . refers to the total amount of the transported molecule which enters the cell interior in the time interval t 0 F t F t obs when a pulse of duration t E is applied. In a previous study the equation for f contains a misprint w10x. As previously suggested w27x, the degree of transformation f T s TrTmax , where Tmax is the maximum number of transformants, may be equated with the degree of bound molecules f b . Hence the data analysis uses f TrC s f b and Eq. Ž27.. Obviously, at least one binding site b has to be occupied with DNA to permit transformation. In the fol-

lowing we present the reevaluation of previous data in terms of the transport parameters k f0 , Pm and f p .

5.3.1. Uptake of DNA by yeast cells For an efficient uptake, DNA should be present, preferably adsorbed already before pulse application. Both the adsorption of DNA, directly measured with 32 P-dC DNA, and the number of transformants are collinearly enhanced with increasing total concentrations wDt x and wCa t x of DNA and of Ca2q, respectively. At the total bulk concentration wDt x s 2.7 nM, the molar concentration of DNA bound to the membrane surface amounts to wD sb x s 2 nM w10x. At the cell density rc s 10 9 cmy3 , there are NDNA s NA wD sb xrrc s 1.2 = 10 3 DNA molecules per cell of radius a s 2.7 mm. Presumably all the adsorbed parts of DNA are located in the head group region of the outer leaflet of the membrane bilayer. Formally the actual concentration of DNA in the membrane surface refers to a thin layer of thickness Ø s 2.37 nm, where Ø is the diameter of the B-helix of DNA. We obtain c mout s wD sb xrŽ rc Sc Ø. s 9.2 mM ŽFig. 8.. Since the bulk concentration of DNA is c out s wDt x y wD sb x s 0.7 nM, the partition coefficient amounts to g s c moutrc out s 1.3 = 10 3. Note that this value is an upper limit because not all parts of the DNA macromolecules are adsorbed. Nevertheless, if only one part of the DNA is adsorbed, the whole DNA can be electrophoretically drawn into the membrane. Hence, for transport the concentration of the absorbed DNA is about 10 3-fold larger than the bulk concentration. This feature was not considered so far and requires a partial reevaluation of previous data w10x, Fig. 7 Ža., where it was found that the direct electroporative transfer of plasmid DNA ŽYEp 351, 5.6 kbp, supercoiled, Mr f 3.5 = 10 6 . in yeast cells Ž Saccharomyces cereÕisiae, strain AH 215. is basically due to Želectro.diffusive processes. At the field strength E0 s 4.0 kV cmy1 , the diffusion coefficient ratio is Dm Ž E .rDm f 10.3. Hence electrodiffusion of DNA is about 10 times more effective than simple diffusion. Addition of DNA after the field pulse only occasionally leads to transformants. The most decisive stage in the cell transformation is the electrodiffusive surface penetration of DNA followed either by further electrodiffusive, or by passive Žafter field. diffusive, translocation of the inserted DNA into the cell interior ŽFig. 6.. Actually, the rather long sigmoid phase of f T Ž t E . in Fig. 7 Ža. requires a description in terms of an at least two-step process: kp

kp

C ™ P1 ™ P2

Ž 28 .

where the state P1 denotes pore structures of negligible permeability for DNA; P2 is the porous membrane state of finite permeability for DNA. The electroporation rate coefficient k p is assumed to be the same for both steps,

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

associated with the same reaction volume D r Vp . This assumption is theoretically justified by the corresponding minima in the hydrophobic force profiles as a function of pore radius w41x. Pore resealing, i.e., the reverse reaction steps ŽP2 ™ P1 ™ C., may be neglected for the time range 0 F t F t E in the presence of the external field. We recall that k p explicitly occurs in the integral: tE C ™ P

H0

fp

Ž t . d t s f p0 ½ t E q ky1 Ž 2 q k p t E . eyk p t E y 2 5 , p Ž 29 .

where f pC ™ P s f p0 Ž1 y Ž1 q k p t . eyk p t . for the reaction Scheme Ž28. and f p0 is the amplitude value of f pC ™ P Ž t .. Applying Eq. Ž27. for the exponential pulse of the initial field strength E0 s 4.0 kV cmy1 and the decay time constant t E s 45 ms, we find with t E s t E that k p s 7.2 sy1 . The mean minimum radius of DNA-permeable pores has been calculated from the field dependence of k p Ž E0 .: rp ŽP2 . s 0.39 " 0.05 nm w10x. If we assume that deviations of the data points from the relationship lnŽ k prk p Ž E s 0.. s b ) cos 2u E 2 , where b ) s Ž9r8.p ´ 0 a 2 Ž ´ W y ´ L . rp2 NA f 2 Ž l m . E 2rŽ dkT . at higher field strengths is due to the increase in the average transmembrane conductivity by D lm s 2.5 = 10y7 S cmy1 from l mŽ E0 s 0. to l mŽ E0 s 4 kV cmy1 . s lm Ž E0 s 0. q D l m . This conductivity increase corresponds to a replacement of 0.0025% of the membrane area by pores filled with the intracellular medium of conductivity li s 1.0 = 10y2 S cmy1 under Maxwell stress. The fractional increase in the transport area for small ions ŽNaq, Cly . is given by f pi s D l m rli s 2.5 = 10y5 w15x. For these conditions the mean number of conductive pores per cell is Np s Sc f prp rp2 s 4.8 = 10 3 , corresponding to an average minimum distance between the pore centers l p s ScrNp s 138 nm. In order to estimate the permeability coefficient Pm of DNA, one may identify the fraction f p of DNA permeable membrane area Žpore state P2 . with that of small ions: f p s f pi. If the DNA permeable membrane area is smaller than the area of ion permeable pores: f p - f pi , we obtain only an upper limit of Pm for DNA. Apparently, the mean radius rp ŽP2 . s 0.39 nm of the pores in DNA-permeable pore patches is too small for free diffusion of large plasmid DNA. Such a small pore radius is not even sufficient for the entrance of a free end of a linear DNA molecule, because the diameter of the type B-DNA is Ø f 2.37 nm. Nevertheless, small parts of the adsorbed DNA may interact with many small pores, and the DNA-polymer may penetrate part by part into the membrane. The total length of a 6.5 kbp DNA is about l DN A s 6.5 = 10 3 = 0.34 nm s 2.2 = 10 3 nm and the corresponding surface area on the membrane is S DN A s l DN AØ s 5.2 = 10 3 nm2 . On average, one totally adsorbed DNA may cover only 4 Np S DNA rSc f 1 membrane electro-

(

13

pore in the cell pole caps Žsee Section 2.. Since the DNA is probably only partially inserted into porous patches, the regions can be considered as closed, but leaky. If the occlusions locally decrease the membrane conductivity, the transmembrane field gets larger such that the membrane somewhere in the vicinity of the inserted DNA part is electroporated. As a consequence, a neighboring part of DNA can penetrate into the newly porated membrane patch. In any case the interaction of the adsorbed DNA with the lipid membrane appears to largely facilitate ME, yielding larger transiently occluded pores. Leaky pore-like channel structures are indicated by ionic current events if DNA interacts with lipid bilayers. Furtheron, if DNA is present in the medium, there is a sharp increase in the membrane permeability of Cos-1 cells to fluorescent dextrin molecules in the electric field w42x. The reevaluation of the data ŽFig. 7. for E0 s 4.0 kV cmy1 and t E s t E s 45 ms yields k f s 2 = 10 2 sy1 . With f p Ž t E . f f pi s 2.5 = 10y5 the characteristic flow coefficient is k f0 s g Dm Ž E . ScrdVc s 8.0 = 10 6 sy1 at T s 293 K. From Eq. Ž16. we obtain the corresponding permeability coefficient Pm s k f0 ar3 s 7.2 = 10 2 cm sy1 . Because Dm Ž E . s 10.3 Dm , we see that at E s 0 formally Pm0 s Pm r10.3 s 70 cm sy1 . Note that the conventional membrane permeability coefficient P 0 refers to the total membrane surface area by P 0 s Pm0 f p Ž t E . s 1.8 = 10y3 cm sy1 . With g s 1.3 = 10 3 and d s 5 nm, the electrodiffusion coefficient DmŽ E . of DNA in the electroporated membrane patches at E s 4 kV cmy1 is DmŽ E . s Pm drg s 2.8 = 10y7 cm2 sy1 , and at E s 0 we have Dm s Dm Ž E .r10.3 s 2.7 = 10y8 cm2 sy1. If the diffusion of DNA is formally related to the total membrane surface Želectroporated patches and the larger nonelectroporated part., D s Dm f p Ž t E . s 6.7 = 10y1 3 cm2 sy1. Compared with the diffusion coefficient of free DNA in solution D free f 5 = 10y8 cm2 sy1 w43x, the bulk diffusion is about 7 = 10 4-fold faster than the interactive diffusion of DNA through the electroporated membrane, reflecting the occluding interaction of DNA with perhaps many small membrane electropores. For practical purposes of optimum transformation efficiency, 1 mM Ca2q is necessary for sufficient DNA binding and the relatively long pulse duration of 20–40 ms is required to achieve efficient electrodiffusive transport across the cell wall and into the outer surface of electroporated cell membrane patches.

5.3.2. Uptake of drug-like dyes by mouse B-cells The color change of electroporated intact Fc gRy mouse B cells Žline IIA1.6, cell diameter 25 mm. after direct electroporative transfer of the drug-like dye Serva Blue G ŽSBG. Ž Mr s 854. into the cell interior is shown to be prevailingly due to diffusion of the dye after the electric field pulse w9x. The net influx of the dyes ceases, even if the pores stay open, when the concentration equality c in f

14

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

c 0 is attained. For this limiting case, the fraction fC s c in rc0 of the colored cells equal unity. The data ŽFig. 7. suggest that at least three different pore states ŽP. in the reaction cascade C | P1 | P2 | P3 are required to model the sigmoid kinetics of pore formation as well as the biphasic pore resealing. The rate coefficient for pore formation k p was taken equal for all the three steps: C | P1 , P1 | P2 and P2 | P3 . At E s 2.1 kV cmy1 and T s 293 K, we find from an expression similar to Eq. Ž29. that k p s 2.4 " 0.2 = 10 3 sy1 . The resealing rate coefficients are ky2 s 4.0 " 0.5 = 10y2 sy1 and ky3 s 4.5 " 0.5 = 10y3 sy1 , independent of E as expected for E s 0. Analysis of the field dependence of k p Ž E . yields the mean radius of the dye permeable pore state r ŽP3 . s 1.2 " 0.1 nm w9x. The SBG molecule may be modeled as an elongated parallelepiped of dimensions Žnm. 2.4 = 1.4 = 0.2. The diameter Ø s 2.4 nm of the pore state P3 equals the long axis Ž2.4 nm. of the SBG molecule. Therefore, the dye molecule most likely passes the membrane with the long axis oriented along the membrane normal. Note that during the translocation step larger pore is transiently occluded. The maximum value of the fractional surface area of the dye-conductive pores is approximated by the fraction of conductive pores: f p s D lm rli s 1.0 = 10y3 , where D lm s 1.3 = 10y5 S cmy1 is the increase in the transmembrane conductivity at E s 2.1 kV cmy1 and li s 1.3 = 10y2 S cmy1 . Hence the maximum number of dye permeable pores is Np s Sc f prp rp2 ŽP3 . s 4.4 = 10 5 per average cell, where Sc s 4p a 2 s 2.0 = 10y5 cm2 . Data reevaluation yields k f s 1 = 10y2 sy1. From Eq. Ž19. we obtain the characteristic flow coefficient k f0 s 10 " 1 sy1 . Since there is no evidence for adsorption of SBG on the membrane surface, the partition coefficient was assumed to be g f 1. The corresponding permeability coefficient of dye in the pores is: Pm s k f0 ar3 s 4.2 = 10y3 cm sy1 . If the permeability coefficient is related to the total membrane surface area we obtain P s Pm f p s 4.2 = 10y6 cm sy1 . The diffusion coefficient of SBG is Dm s Pm d s 2.1 = 10y9 cm2 sy1 and D s Dm f p s 2.1 = 10y1 2 cm2 sy1 , respectively. It is seen that Dm is by the factor D freerDm s 2.4 = 10y5 smaller than D free s 5 = 10y6 cm2 sy1 estimated for free dye diffusion. This difference apparently indicates transient interaction of the dye with the pore lipids during translocation and partial occlusion of the pores. 5.4. Field–time relationship for the electroporatiÕe transport Obviously the two pulse parameters E and t E are of primary importance to control extend and rate of the transmembrane transport. Within certain ranges of E and t E a relationship of the type E 2 t E s c holds, where c is a constant w9,10,25x. However, very large field strengths or

very long pulse durations may lead to secondary effects like bleb formation w9x or fragmentation of the vesicles and cells under Maxwell stress. Therefore in the range of massive cell deformation and fragmentation the constant c has a different value than in the range of short pulse durations. In any case, the empirical correlation E 2 t E s constant is theoretically rationalized in terms of the interfacial polarization mechanism of ME w25,27x.

6. Summary and conclusions Since the electroporative transport of permeants is caused by ME, the transport quantities f T Ž t . and fC Ž t . are closely connected to the degree f p Ž t . of ME, permitting to investigate the mechanism of formation and development of membrane pores by the electric field. The results of our theoretical approach, based on electrooptical data of vesicles, as well as on the kinetics of cell electrotransformation and cell coloring, can be used to specify conditions for the practical purposes of gene transfer and drug delivery into the cells. In electrochemotherapy, for instance, the optimization of the electroporative channeling of the cytotoxic drugs into the tissue cells may be refined by using the electroporative transport theory w4,44–46x. Future work may include optical probes like DPH in cell plasma membranes to elucidate the sequence of events of the electroporative DNA and protein transfers as well as to investigate molecular details of other electroporation phenomena such as electrofusion and electroinsertion. In conclusion, the theory of ME has been developed to such a degree that analytical expressions are available for the optimization of the ME techniques in biotechnology and medicine, in particular in the new fields of electroporative drug delivery and gene therapy. The electroporative gene vaccination is certainly a great challenge for modern medicine.

7. Table of definitions wb 0 x wCa t x wDt x s c o wD b x wP2 x wP3 x a c mout and c min c out and c in

initial amount of cell components binding the penetrated DNA or SBG total concentration of Ca total concentration of DNA concentration of bound DNA concentration of DNA-permeable pores concentration of SBG-permeable pores cellrvesicle radius molar concentrations of the permeant in the outer and inner membranermedium interfaces, respectively bulk concentrations of the permeant inside and outside the cell Žor vesicle., respectively

E. Neumann et al.r Bioelectrochemistry and Bioenergetics 48 (1999) 3–16

c0 Dm Dm Ž E . D wD b x wD sb x E Em eo ´o ´W ´L fT fC fb fp f Ž lm . g D fm k1 ky1 kB kb kp k f0 kf

lm lm l0 li Np n inc n out Pm P rp rc Sc Sm Sp tE tE Vc

initial total concentration of permeant in the outside medium diffusion coefficient in electroporated membrane patches electrodiffusion coefficient in electroporated membrane patches diffusion coefficient related to the total membrane surface area concentration of bound macromolecules to the intracellular sites concentration of bound macromolecules to the membrane surface field strength transmembrane field strength elementary charge vacuum permittivity dielectric constant of water dielectric constant of the lipid phase degree of cell transformation degree of cell coloring degree of binding of permeants to intracellular sites average fraction of porated membrane area conductivity factor partition coefficient mean electrical potential difference across the electroporated membrane patches rate coefficient for the step C ™ P rate coefficient for the step P ™ C Boltzmann constant rate coefficient of binding wMy1 sy1 x electroporation rate coefficient wsy1 x characteristic flow coefficient wsy1 x flow coefficient for cross-membrane transport wsy1 x transmembrane conductivity u-average transmembrane conductivity conductivity of bulk solution conductivity of cell interior number of pores per cell molar amount of DNA or SBG in one cell molar amount of DNA or SBG in the bulk solution permeability coefficient for the electroporated membrane patches conventional permeability coefficient Žrelated to the total membrane. mean pore radius cell density cell surface area electroporated area of cell surface surface area of the average pore electrical pulse duration decay time constant of an exponential field pulse volume of an average cell

V0 z eff

15

external volume effective charge number Žwith sign. of the DNA-phosphate group

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