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Kinetics and likelihood of membrane rupture during electroporation
Volume 143, number 8
PHYSICS LETTERS A
29 January 1990
KINETICS AND LIKELIHOOD OF MEMBRANE RUPTURE DURING ELECTROPORATION Mehmet TONER and Ernest G. CRAVALHO Harvard—MIT Division ofHealth Sciences and Technology, and Massachusetts Institute of Technology, Department of MechanicalEngineering, Cambridge, MA 02139, USA Received 7 November 1989; accepted for publication 30 November 1989 Communicated by A.A. Maradudin
The rate of production ofcritical size pores resulting in membrane rupture in the presence of an applied transmembrane potential is modelled based on the theory of nucleation in condensed systems.
The behavior of lipid bilayer membranes in the presence of an externally applied electrical field has been described in the literature [1,21. The membrane typically ruptures if transmembrane potentials between 150 and 400 mY occur for ~—l0—~ s or longer. This phenomenon is termed irreversible mechanical breakdown (1MB) and it is caused by the formation of several large transient pores. When larger voltages between 500 and 1000 mY are applied to the membranes by current pulses of 100 to 400 ns duration, a dramatic drop in membrane resistance is observed in several hundreds of nanoseconds due to the formation of many transient pores and reversible electrical breakdown (REB) occurs. In this case membrane rupture can be avoided and the membrane remains intact. Hence, electroporation is a dynamic competition between 1MB and REB and it can be quantitatively described by the dynamics of the transient aqueous pore formation in artificial bilayers and biological membranes. The rate of addition of water molecules to transient pores can be described in a way similar to the nucleation phenomenon in condensed systems [31 by modelling directly the dynamics ofpore formation [4]. This will eliminate many of the oversimplifications used in the derivation of diffusion-like equations in the present theories [5,6]. The following is a simple analysis of the electroporation using the nucleation theory in condensed systems to establish the basis for a more elaborate direct modelling of transient pore dynamics.
In 1975, Litster [7] proposed that pores can form and collapse in bilayer membranes by thermal fluctuations. Following this proposal, the thermal pore fluctuation theory was extended to include the effects of the applied transmembrane potentials which decrease the stability of the membrane against thermal fluctuations [8,9]. Abidor et al. [91 derived a diffusion-like equation to describe pore formation using an empirical value of pore radius diffusivity. Powell and Weaver [2,6] estimated this parameter using statistical mechanics and extended the model to a unified treatment of 1MB and REB. Kashchiev and Exerowa [101 have used the theory of nucleation of voids from vacancies in solids for determination of the work for pore formation in artificial bilayers. However, none of these studies treated the electroporation using the theory of nucleation in condensed systems. The energy required to form a cylindrical pore of radius r is [2] z~E(r, U) = 2ityr— ~tr2(1’+ a U2)
,
(1)
where U is the transmembrane potential, y is the edge energy, F is the surface tension, and a = ~ ( ~/ ~ 1) /2d, in which e~,e~,and e~are, respectively, the dielectric constants for vacuum, water and lipids, and d is the thickness of the hydrocarbon interior of the bilayer. Maximizing AE( r, U) with respect to r for constant y and F gives —
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
409
Volume 143, number 8
2)
= y/ (F+a U
,
PHYSICS LETTERS A
AE* (r*, U) = ity2/ (F+ aU2)
,
(2) where the asterisk denotes the properties of critical pore. When an additional water molecule is incor-
29 January 1990
in contact with the critical pore may be estimated as 0*~2(r*2/r~),where r~is the radius of a water molecule. The quasi steady-state number of critical pores can be assumed to be given by the Boltzmann distribution for N 0>> N i.e. N* N0 exp ( iIE/ kT), where N0 is the total number of pores in a poresaturated membrane per unit areawhere [21. N0 can ap2 ) lAm, A is thebetotal proximated by N0 (Am/2 bilayer thickness [21. The Zeldovich factor, Z= (~EI~/37tkTn*2)lI2,has been introduced to correct the pore formation rate based on a model assuming the applicability of equilibrium thermodynamics to a kinetic situation [12]. Here, n* is the ~‘,
porated in the critical pore from the suspending solution, and r increases beyond r*, the total pore energy decreases given by eq.The (1) formation which favors growth. Thus, as 1MB begins. of pore this critical pore leading to 1MB may be written as a series of bimolecular reactions
—
A_, +A, ±A 1,
number of water molecules in 2A/4r~. the critical Thus,pore the and rate cancritical be estimated from n* can 3r* be determined from of pore formation
A~ 1+A1~A~.,
eqs. (4)—(6) as (3)
~
where A~.represents a pore of The n~molecules A1 is a single water molecule. backwardandrate, ~ —‘0, at the critical cluster may be neglected and once the steady-state is reached, the rate of 1MB can be expressed as the rate of critical pore formation I(U)=Zk~.N*=Z0*wN*.
(4)
Here, Z is a factor called Zeldovich factor, k~.is the rate of addition of water molecules to a pore An., N * is the equilibrium number ofpores of critical size per unit area, w is the molecular jump rate and 0* is the number of monomers in contact with the critical pore. In a manner similar to the theory of nucleation in condensed systems as proposed by Turnbull and Fisher [31, cv can be written as w=(kT/h) exp(—z~.Gd/kT) ,
(5)
where k is Boltzmann’s constant, T is the absolute temperature, h is Planck’s constant and L~Gdis the activation energy for a diffusive jump across the liquid—critical pore interface. E,~Gdis taken equal to the activation energy for viscous flow using Eyring’s rate theory [11] z~Gd=kTln(i~v/NAh),
(6)
where i~is the viscosity of the suspending solution, v is the molar volume ofwater, and NA is Avogadro’s number. To obtain I( U), the number of monomers 410
4kTNArW
2exp(_~/kT), (7) (~E*/kT)hI where z~.E*is given by eq. (2). As by definition, the time constant for 1MB, t, can be then estimated from the rate of critical pore formation as ~=
t(U)= llI(U)Am,
(8)
where Am is the membrane surface area. Using the values r~ = 0.957 X 10’° m, A=42.4x 10’° m, i~=0.9xl0~ P, z’=18x 10—6 m3/mol, y=2x lO~ JIm, F= lx l0~ J/m2, a=0.l21,Am=2x106m2 and T=25~C;ris 0.2 ms at U= 233 mY. The present numerical values are based on a cylindrical pore and on the properties of water as suggested by Powell and Weaver [2]. This estimate of ~ is in reasonable agreement with experiments [1,2, 7]. Although excellent agreement can be obtained by adjusting y and F for a specific set of experimental conditions, this is beyond the scope of this communication. As can be seen from eqs. (2), (7) and (8), t is a very strong function of U since U2 appears in the exponential term. Fig. 1 illustrates the calculated t( U) curve which has been plotted on semi-logarithmic coordinates. 1MB time drops from 25000 ms at 200 mY to 0.024 ms at 240 mY. An increase of less than 25% in the transmembrane p0tential of a bilayer lipid membrane decreases the life span of the membrane more than six orders of magnitude. Eqs. (7) and (8) can also be used to predict
Volume 143, number 8
PHYSICS LETTERS A
io~5
The time constant for transient response is then given by rt=n*2,/40*w. The calculated values of ; are shown in fig. 1. For transmembrane potentials higher than -‘250 mY, the transients become important and cannot be ignored. At 240 mY, the transient time to reach 63% of the steady-state nucleation rate is about three orders of magnitude less than 1MB time. However, at a slightly higher transmembrane potential of 270 thethan transient time and is one orderbeofignored. magnitude mV, longer 1MB time cannot
\~ 8
\
/
___-._2t
u -~
/0
0.!
02
Q3
04
~ (~)
as
.
Fig. 1. Transmembrane potential dependence of the membrane life time, i; and of the transient pore formation time constant,;. r is obtained from eq. (8) and it is a measure of the membrane stability at a given transmembrane potential. ; is given in eq. (10) and it is a measure of the goodness ofthe steady state pore formation assumption,
the critical transmembrane potential causing 1MB for a given pulse length of an applied transmembrane potential. In most ofthe instances, a large population of cells is exposed to electroporation. The probability of 1MB in a population of cells as a result of random fluctuations of pores can be written by assuming that each cell is identical [13],
7
P(U, t)=l—expi
\
—
1I J(U)A,,~dtI.
j
(9)
0
Here, it is assumed that 1MB occurs by a stochastic process. For the steady-state case, this equation can be integrated to give P( U, M~)= 1 —exp( IAmL~T). For a given pulse duration of At, the demarcation zone for P( U, Ar) to increase from 0% to 100% can be calculated. For a 400 ns pulse, P=0 at 220 mY and rapidly increases to 1 at 240 mY. It is, therefore, possible for a membrane to contain a large population ofpores without 1MB, but can also rapidly cause 1MB at a slightly higher transmembrane potential. These results are in agreement with the existing theones of electroporation [2,91. It is also important to investigate the transient response of the creation of pores during electroporation. The approximate transient rate of critical pore formation is [121 2140*wt)
1
.
29 January 1990
(10)
.
The transient pore formation is especially important in describing REB [6]. REB consists of a very rapid decay (—~100 ns) of U to a low value following the transmembrane potential pulse. The analysis from this work can now serve the basis for a more elaborate transient modelling of the aqueous pores in the presence of an applied external electrical field directly simulating the dynamics of pore formation in eq. (3) to account for both 1MB and REB. Further understanding of the underlying mechanism by which electrical fields can affect cells is essential to the development of effective therapeutic strategies for electrical trauma patients [141. The authors are grateful to Dr. James C. Weaver of Massachusetts Institute of Technology and Dr. Raphael C. Lee of the University of Chicago for helpful discussions and review of the manuscript. This work has been supported by: The Boston Edison Company, Electric Power Research Institute, Empire State Electric Energy Research Corporation, EUA Service Corporation, Northeast Utilities Service Corporation, Pacific Gas and Electric Company, Pennsylvania Power and Light Company, The Public Service Company of Oklahoma and The Public Service Electric and Gas Company.
References .
.
[1] U. Zimmermann, Biochim. Biophys. Acta 694 (1982) 227. [2] K.T. Powell andJ.C. Weaver, Bioelectrochem. Bioenerg. 15 (1986) 2111. [3] D. Turnbull and J.C. Fisher, J. Chem. Phys. 17 (1949) 71. [41K.F. Kelton, A.L. Greer and C.V. Thompson, J. Chem. Phys. 79(1983) 6261. [5] V.F. Pastushenko, Yu.A. Chizmadznev and V.B. Arakelyan, Bioelectrochem. Bioenerg. 6 (1979) 53.
1(U, t) =I( U)exp( — n*
411
Volume 143, number 8
PHYSICS LETTERS A
[6] J.C. Weaver and K.T. Powell, in: Electroporation and electrofusion in cell biology, eds. E. Neumann, A. Sowers and C. Jordan (Plenum, New York, 1989). [7] J.D. Litster, Phys. LetI. A 53 (1975) 193. [8] J.C. Weaver and R.A. Mintzer, Phys. LetI. A 86 (1981) 57. [9] I.G. Abidor, LV. Chernomordik, Yu.A. Chizmadznev, V.F. Pastushenko and M.R. Tarasevich, Bioelectrochem. Bioenerg. 6 (1979) 37. [10] D. Kashchiev and D. Exerowa, Biochim. Biophys. Acta 732 (1983) 133.
412
29 January 1990
[11] S. Glasstone, K.J. Laidler and H. Eyring, The theory of rate processes (McGraw-Hill, New York, 1941). [12] D. Turnbull, in: Solid state physics, Vol. 3, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1956) p. 225. [13] A.E. Carte, Proc. Phys. Soc. 73 (1959) 324. [14] D.C. Gaylor, Physical mechanisms of cellular injury in electrical trauma, Ph.D. Thesis, Department of Electrical Engineering, MIT, Cambridge, MA.
PHYSICS LETTERS A
29 January 1990
KINETICS AND LIKELIHOOD OF MEMBRANE RUPTURE DURING ELECTROPORATION Mehmet TONER and Ernest G. CRAVALHO Harvard—MIT Division ofHealth Sciences and Technology, and Massachusetts Institute of Technology, Department of MechanicalEngineering, Cambridge, MA 02139, USA Received 7 November 1989; accepted for publication 30 November 1989 Communicated by A.A. Maradudin
The rate of production ofcritical size pores resulting in membrane rupture in the presence of an applied transmembrane potential is modelled based on the theory of nucleation in condensed systems.
The behavior of lipid bilayer membranes in the presence of an externally applied electrical field has been described in the literature [1,21. The membrane typically ruptures if transmembrane potentials between 150 and 400 mY occur for ~—l0—~ s or longer. This phenomenon is termed irreversible mechanical breakdown (1MB) and it is caused by the formation of several large transient pores. When larger voltages between 500 and 1000 mY are applied to the membranes by current pulses of 100 to 400 ns duration, a dramatic drop in membrane resistance is observed in several hundreds of nanoseconds due to the formation of many transient pores and reversible electrical breakdown (REB) occurs. In this case membrane rupture can be avoided and the membrane remains intact. Hence, electroporation is a dynamic competition between 1MB and REB and it can be quantitatively described by the dynamics of the transient aqueous pore formation in artificial bilayers and biological membranes. The rate of addition of water molecules to transient pores can be described in a way similar to the nucleation phenomenon in condensed systems [31 by modelling directly the dynamics ofpore formation [4]. This will eliminate many of the oversimplifications used in the derivation of diffusion-like equations in the present theories [5,6]. The following is a simple analysis of the electroporation using the nucleation theory in condensed systems to establish the basis for a more elaborate direct modelling of transient pore dynamics.
In 1975, Litster [7] proposed that pores can form and collapse in bilayer membranes by thermal fluctuations. Following this proposal, the thermal pore fluctuation theory was extended to include the effects of the applied transmembrane potentials which decrease the stability of the membrane against thermal fluctuations [8,9]. Abidor et al. [91 derived a diffusion-like equation to describe pore formation using an empirical value of pore radius diffusivity. Powell and Weaver [2,6] estimated this parameter using statistical mechanics and extended the model to a unified treatment of 1MB and REB. Kashchiev and Exerowa [101 have used the theory of nucleation of voids from vacancies in solids for determination of the work for pore formation in artificial bilayers. However, none of these studies treated the electroporation using the theory of nucleation in condensed systems. The energy required to form a cylindrical pore of radius r is [2] z~E(r, U) = 2ityr— ~tr2(1’+ a U2)
,
(1)
where U is the transmembrane potential, y is the edge energy, F is the surface tension, and a = ~ ( ~/ ~ 1) /2d, in which e~,e~,and e~are, respectively, the dielectric constants for vacuum, water and lipids, and d is the thickness of the hydrocarbon interior of the bilayer. Maximizing AE( r, U) with respect to r for constant y and F gives —
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
409
Volume 143, number 8
2)
= y/ (F+a U
,
PHYSICS LETTERS A
AE* (r*, U) = ity2/ (F+ aU2)
,
(2) where the asterisk denotes the properties of critical pore. When an additional water molecule is incor-
29 January 1990
in contact with the critical pore may be estimated as 0*~2(r*2/r~),where r~is the radius of a water molecule. The quasi steady-state number of critical pores can be assumed to be given by the Boltzmann distribution for N 0>> N i.e. N* N0 exp ( iIE/ kT), where N0 is the total number of pores in a poresaturated membrane per unit areawhere [21. N0 can ap2 ) lAm, A is thebetotal proximated by N0 (Am/2 bilayer thickness [21. The Zeldovich factor, Z= (~EI~/37tkTn*2)lI2,has been introduced to correct the pore formation rate based on a model assuming the applicability of equilibrium thermodynamics to a kinetic situation [12]. Here, n* is the ~‘,
porated in the critical pore from the suspending solution, and r increases beyond r*, the total pore energy decreases given by eq.The (1) formation which favors growth. Thus, as 1MB begins. of pore this critical pore leading to 1MB may be written as a series of bimolecular reactions
—
A_, +A, ±A 1,
number of water molecules in 2A/4r~. the critical Thus,pore the and rate cancritical be estimated from n* can 3r* be determined from of pore formation
A~ 1+A1~A~.,
eqs. (4)—(6) as (3)
~
where A~.represents a pore of The n~molecules A1 is a single water molecule. backwardandrate, ~ —‘0, at the critical cluster may be neglected and once the steady-state is reached, the rate of 1MB can be expressed as the rate of critical pore formation I(U)=Zk~.N*=Z0*wN*.
(4)
Here, Z is a factor called Zeldovich factor, k~.is the rate of addition of water molecules to a pore An., N * is the equilibrium number ofpores of critical size per unit area, w is the molecular jump rate and 0* is the number of monomers in contact with the critical pore. In a manner similar to the theory of nucleation in condensed systems as proposed by Turnbull and Fisher [31, cv can be written as w=(kT/h) exp(—z~.Gd/kT) ,
(5)
where k is Boltzmann’s constant, T is the absolute temperature, h is Planck’s constant and L~Gdis the activation energy for a diffusive jump across the liquid—critical pore interface. E,~Gdis taken equal to the activation energy for viscous flow using Eyring’s rate theory [11] z~Gd=kTln(i~v/NAh),
(6)
where i~is the viscosity of the suspending solution, v is the molar volume ofwater, and NA is Avogadro’s number. To obtain I( U), the number of monomers 410
4kTNArW
2exp(_~/kT), (7) (~E*/kT)hI where z~.E*is given by eq. (2). As by definition, the time constant for 1MB, t, can be then estimated from the rate of critical pore formation as ~=
t(U)= llI(U)Am,
(8)
where Am is the membrane surface area. Using the values r~ = 0.957 X 10’° m, A=42.4x 10’° m, i~=0.9xl0~ P, z’=18x 10—6 m3/mol, y=2x lO~ JIm, F= lx l0~ J/m2, a=0.l21,Am=2x106m2 and T=25~C;ris 0.2 ms at U= 233 mY. The present numerical values are based on a cylindrical pore and on the properties of water as suggested by Powell and Weaver [2]. This estimate of ~ is in reasonable agreement with experiments [1,2, 7]. Although excellent agreement can be obtained by adjusting y and F for a specific set of experimental conditions, this is beyond the scope of this communication. As can be seen from eqs. (2), (7) and (8), t is a very strong function of U since U2 appears in the exponential term. Fig. 1 illustrates the calculated t( U) curve which has been plotted on semi-logarithmic coordinates. 1MB time drops from 25000 ms at 200 mY to 0.024 ms at 240 mY. An increase of less than 25% in the transmembrane p0tential of a bilayer lipid membrane decreases the life span of the membrane more than six orders of magnitude. Eqs. (7) and (8) can also be used to predict
Volume 143, number 8
PHYSICS LETTERS A
io~5
The time constant for transient response is then given by rt=n*2,/40*w. The calculated values of ; are shown in fig. 1. For transmembrane potentials higher than -‘250 mY, the transients become important and cannot be ignored. At 240 mY, the transient time to reach 63% of the steady-state nucleation rate is about three orders of magnitude less than 1MB time. However, at a slightly higher transmembrane potential of 270 thethan transient time and is one orderbeofignored. magnitude mV, longer 1MB time cannot
\~ 8
\
/
___-._2t
u -~
/0
0.!
02
Q3
04
~ (~)
as
.
Fig. 1. Transmembrane potential dependence of the membrane life time, i; and of the transient pore formation time constant,;. r is obtained from eq. (8) and it is a measure of the membrane stability at a given transmembrane potential. ; is given in eq. (10) and it is a measure of the goodness ofthe steady state pore formation assumption,
the critical transmembrane potential causing 1MB for a given pulse length of an applied transmembrane potential. In most ofthe instances, a large population of cells is exposed to electroporation. The probability of 1MB in a population of cells as a result of random fluctuations of pores can be written by assuming that each cell is identical [13],
7
P(U, t)=l—expi
\
—
1I J(U)A,,~dtI.
j
(9)
0
Here, it is assumed that 1MB occurs by a stochastic process. For the steady-state case, this equation can be integrated to give P( U, M~)= 1 —exp( IAmL~T). For a given pulse duration of At, the demarcation zone for P( U, Ar) to increase from 0% to 100% can be calculated. For a 400 ns pulse, P=0 at 220 mY and rapidly increases to 1 at 240 mY. It is, therefore, possible for a membrane to contain a large population ofpores without 1MB, but can also rapidly cause 1MB at a slightly higher transmembrane potential. These results are in agreement with the existing theones of electroporation [2,91. It is also important to investigate the transient response of the creation of pores during electroporation. The approximate transient rate of critical pore formation is [121 2140*wt)
1
.
29 January 1990
(10)
.
The transient pore formation is especially important in describing REB [6]. REB consists of a very rapid decay (—~100 ns) of U to a low value following the transmembrane potential pulse. The analysis from this work can now serve the basis for a more elaborate transient modelling of the aqueous pores in the presence of an applied external electrical field directly simulating the dynamics of pore formation in eq. (3) to account for both 1MB and REB. Further understanding of the underlying mechanism by which electrical fields can affect cells is essential to the development of effective therapeutic strategies for electrical trauma patients [141. The authors are grateful to Dr. James C. Weaver of Massachusetts Institute of Technology and Dr. Raphael C. Lee of the University of Chicago for helpful discussions and review of the manuscript. This work has been supported by: The Boston Edison Company, Electric Power Research Institute, Empire State Electric Energy Research Corporation, EUA Service Corporation, Northeast Utilities Service Corporation, Pacific Gas and Electric Company, Pennsylvania Power and Light Company, The Public Service Company of Oklahoma and The Public Service Electric and Gas Company.
References .
.
[1] U. Zimmermann, Biochim. Biophys. Acta 694 (1982) 227. [2] K.T. Powell andJ.C. Weaver, Bioelectrochem. Bioenerg. 15 (1986) 2111. [3] D. Turnbull and J.C. Fisher, J. Chem. Phys. 17 (1949) 71. [41K.F. Kelton, A.L. Greer and C.V. Thompson, J. Chem. Phys. 79(1983) 6261. [5] V.F. Pastushenko, Yu.A. Chizmadznev and V.B. Arakelyan, Bioelectrochem. Bioenerg. 6 (1979) 53.
1(U, t) =I( U)exp( — n*
411
Volume 143, number 8
PHYSICS LETTERS A
[6] J.C. Weaver and K.T. Powell, in: Electroporation and electrofusion in cell biology, eds. E. Neumann, A. Sowers and C. Jordan (Plenum, New York, 1989). [7] J.D. Litster, Phys. LetI. A 53 (1975) 193. [8] J.C. Weaver and R.A. Mintzer, Phys. LetI. A 86 (1981) 57. [9] I.G. Abidor, LV. Chernomordik, Yu.A. Chizmadznev, V.F. Pastushenko and M.R. Tarasevich, Bioelectrochem. Bioenerg. 6 (1979) 37. [10] D. Kashchiev and D. Exerowa, Biochim. Biophys. Acta 732 (1983) 133.
412
29 January 1990
[11] S. Glasstone, K.J. Laidler and H. Eyring, The theory of rate processes (McGraw-Hill, New York, 1941). [12] D. Turnbull, in: Solid state physics, Vol. 3, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1956) p. 225. [13] A.E. Carte, Proc. Phys. Soc. 73 (1959) 324. [14] D.C. Gaylor, Physical mechanisms of cellular injury in electrical trauma, Ph.D. Thesis, Department of Electrical Engineering, MIT, Cambridge, MA.