Theory of Electroporation - Advances in Chemistry (ACS Publications)

21 Theory of Electroporation Downloaded by UNIV OF PITTSBURGH on January 27, 2016 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0235.ch021

James C . Weaver Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, M A 02139

Electroporation is a dramatic cell membrane phenomenon that is of growing importance to biology, biotechnology, and medicine. Electroporation caused by short pulses occurs at a transmembrane voltage of about 1000 mV for many different types of cell and artificial membranes. Here we describe the development of a theoretical model that predicts measurable quantities. An underlying theme of our general approach is the hypothesis that electroconformational or structural changes of (1) the membrane itself, (2) membrane macromolecules, and (3) membrane-macromolecule complexes can provide a general basis for electric field interactions with cells. Electroporation theory is presently based on the membrane itself and can be expected to also involve membrane-macromolecule complexes. In related nonelectroporation work, we hypothesized that membrane macromolecules are relevant to understanding possible weak electric field interactions, and we have estimated the threshold field, E associated with the thermal noise limit for the response of living cells to weak electric fields. e,min

Electroconformational Changes as a General Membrane Phenomenon A t l o w frequencies the quasistatic approximation can be used to separately describe b o t h the electric field, E(f), a n d the magnetic field, B(t). V e r y generally, cells are heterogeneous w i t h respect to their electrical properties ( I ) b u t u n i f o r m w i t h respect to their magnetic properties (excluding special i z e d magnetic material such as magnetite a n d ferratin) ( 2 ) . Specifically, the permittivity a n d electrical conductivity o f the m e m b r a n e is significantly different f r o m the permittivity a n d conductivity o f the intra- a n d extracellular 0065-2393/94/0235-0447 $09.08/0 © 1994 American Chemical Society

In Biomembrane Electrochemistry; Blank, Martin, et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

BIOMEMBRANE ELECTROCHEMISTRY

448

media. These differences i n electrical properties result i n interfacial polariza tion, w h i c h , t h r o u g h the c e l l geometry, leads to a n amplification i n w h i c h a change i n the external electric field, Δ Ε , causes a m u c h larger change i n the transmembrane field, AE = AU/d(3, 4). T w o relatively simple c e l l shapes are often considered a n d illustrate the basic concept: (1) a spherical cell, f o r w h i c h Δ17 = 1 . 5 E R cos θ i f θ is the angle between the direction o f E a n d the point o f interest o n the c e l l m e m b r a n e , a n d (2) a n elongated cell, f o r which A l / « E L/2 i f L is the characteristic l o n g d i m e n s i o n o f the c e l l a n d the c e l l is parallel to E . β

m

e

m a x

c e l l

e

e

Downloaded by UNIV OF PITTSBURGH on January 27, 2016 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0235.ch021

e

W i t h this i n m i n d , w e first focus o n the hypothesis that because o f a n increased U, electroconformational changes occur i n the membrane itself (Table I). O f specific interest to electroporation theory are those m e m b r a n e eleetroeonformation changes that lead to pore formation. T h e earliest studies k n o w n to us considered the overall compression o f a m e m b r a n e i n response to forces n o r m a l to the m e m b r a n e ( 5 ) . I n such models, a n increase i n U causes compression that leads to a mechanical instability a n d an associated critical transmembrane voltage, U , f o r m e m b r a n e rupture. T h i s early p r e d i c tion agreed w i t h experiments o n solvent-containing planar membranes, b u t not w i t h experiments o n the less compressible biological membranes. M o r e sophisticated treatments o f instabilities have b e e n developed (18, 19), b u t d o not describe the subsequent behavior o f the transmembrane voltage, [/(f), the membrane conductance, G(t), o r the n u m b e r o f transported molecules, N (t), that are c r u c i a l to a n understanding o f electroporation. M o r e o v e r , b u l k electrocompression models cannot distinguish between rupture a n d re versible electrical b r e a k d o w n , b o t h o f w h i c h have b e e n observed experimen tally i n oxidized cholesterol planar membranes ( 2 0 ) . T h e s e two very different outcomes are n o w b e l i e v e d to d e p e n d o n the relative dynamics o f pore c

s

population changes a n d m e m b r a n e discharge t h r o u g h the pore p o p u l a t i o n .

Table I. Membrane-Related Electroconformational Change Candidates Type of Electroconformational Change Overall membrane compression Lipid-domain interface fluctuations Free volume fluctuations Local depressions and distortions Transient hydrophobic pores Transient hydrophilic pores Foot-in-the-door hydrophilic pores Composite hydrophilic pores Membrane enzyme changes Membrane macromolecule protrusion changes

Significant Feature Rupture and R E B not actually described (5) Suggested alternative to transient pores (6) Transport of nonpolar species (7) Possible precursors to hydrophilic pores (8, 9) Possible precursor to hydrophilic pores (JO) Key to quantitative descriptions (10-16) Candidate metastable pores" Candidate metastable pores Coupling to membrane macromolecules (17) Candidate signaling change mechanism

SOURCE: Data are taken from reference 2 3 A .


21.

WEAVER

Theory of

Electroporation

449

T h e important reversible nature o f electroporation that is often observed i n cell membranes is also not treated.


I n contrast, pore models consider the pore-expanding forces that are parallel to the m e m b r a n e a n d that arise f r o m n o r m a l forces acting o n pores as U is increased. P o r e models can quantitatively account for b o t h rupture a n d reversible electrical b r e a k d o w n ( R E B ) i n planar membranes a n d reversible behavior i n cells. F o r this reason, o u r discussion w i l l emphasize localized electroconformational changes, particularly h y d r o p h i l i c pores. T h e m o d e l i n g o f electroporation using such pores was first presented i n an impressive series of seven back-to-back papers; only the first paper is c i t e d here (10). T h e last entry i n T a b l e I, " m e m b r a n e macromolecule protrusion changes," ought to be a general p h e n o m e n o n . Recently it has b e e n consid e r e d theoretically (21 A, 2IB), although it was earlier identified as a possibil ity i n an experimental study (22). V i e w e d f r o m a macroscopic, c o n t i n u u m perspective, the m e a n molecular protrusion should reflect a force balance that involves electrostatic a n d h y d r o p h o b i c interactions that changes as the transmembrane voltage is varied. F o r example, as U changed, the electro static free energy o f the m e m b r a n e - i m m e r s e d p o r t i o n changes. T h e m e a n protrusion should, therefore, vary w i t h U, w h i c h may alter the accessibility o f b i n d i n g sites o n the p r o t r u d i n g molecule (e.g., a receptor) or at nearby sites o n the m e m b r a n e . T h i s protrusion provides a candidate mechanism for c o u p l i n g electric fields to b i o c h e m i c a l pathways o f the c e l l and, therefore, c o u l d alter c e l l f u n c t i o n because o f an external electric field.

Strong Electrical Fields: Electrical Behavior Due to Electroporation M o s t biological studies a n d applications o f electroporation involve short pulses (10 ~ < i < 1 0 " s) because longer pulses often l e a d to nonther m a l c e l l k i l l i n g a n d to intolerable heating i n physiological electrolytes. A striking aspect o f electroporation is that time scales that differ b y many orders o f magnitude are involved (see T a b l e II). B o t h planar membranes a n d closed membranes (cells a n d vesicles) have b e e n studied. Because the geometry is simpler, we first consider electroporation i n artificial planar bilayer m e m branes, some o f w h i c h (e.g., oxidized cholesterol) exhibit four different outcomes b y variation o f the properties o f the electrical pulse only (Table III). 6

p u l s e

3

Reversible electrical b r e a k d o w n ( R E B ) is particularly striking. ( R E B is actually a r a p i d discharge due to the h i g h ionic conductivity caused by the gentle structural m e m b r a n e rearrangement o f m u l t i p l e pore formation.) I n our models, subcritical pores (i.e., nonrupture-causing pores) are responsible for this h i g h conductance state ( I J). O u r first quantitative description o f R E B (33, 34) correctly p r e d i c t e d m a n y key features o f U(t) a n d G(t) but h a d


450


Table II. Time Scales Involved in Electroporation Time Scale (s)

10-

Molecular collision times within lipid membrane Cell and planar membrane charging times

io- -io~


7

10

7

-10

6

10

6

-10

5

10 ~ - 1 0 4

How Known

Significance

+ 4

Membrane discharging times (reversible electrical breakdown)

Membrane discharging times (planar membrane rupture) Phenomena that occur after the membrane has discharged (e.g., metastable pore phenomena, membrane recovery)

Statistical mechanics ( I I ,

23B)

Optical, microelectrode measurements consistent with theory (20, 24-26) Optical, microelectrode measurements; consistent with theory (16, 20, 24, 25) Electrical measurements; consistent with theory (16, 24) Optical, chemical, scanning electron microscopy ( S E M ) , and electrical measurements (27-32)

Table III. Four Observed (20) Outcomes for an Oxidized Cholesterol Planar Bilayer Membrane Characteristic Electrical Behavior R E B ; membrane discharge to Ό = 0 Incomplete R E B ; discharge halts at U > 0 Rupture (mechanical); slow, sigmoidal electrical discharge Membrane charging without dramatic behavior of U

Pulse Magnitude Largest Smaller Still smaller Smallest

several unsatisfactory assumptions. T h e most recent description ( L i s t 1 a n d T a b l e I V ) overcomes these difficulties a n d provides a quantitative description o f U(t) for all four outcomes for charge injection conditions (Figures 1 a n d 2) (16). A description o f R E B alone, w h i c h occurs i n closed membranes [e.g., vesicles (35)], is actually less challenging. A general approach to m o d e l i n g , i n w h i c h general hypotheses are interre lated i n a m o d u l a r f o r m ( L i s t 1 a n d T a b l e I V ) , is used. T h i s approach allows modules (e.g., a physical m o d e l o f the pore free energy) to be separately modified, whereas all o f the other features o f the simulation are retained. T h e elements given i n L i s t 1 are presently f o r m a l i z e d by using the physical ingredients identified i n T a b l e I V . Table V lists the simulation parameters a n d their values. H y d r o p h i l i e pores are assumed to be created b y transitions f r o m hy drophobic pores ( F i g u r e 3) (10). T h e details o f this process are not k n o w n .


21.

451

Theory of Electroporation

WEAVER


List 1. Elements of the General Electroporation Model 1. A membrane contains an equilibrium population (small at U = 0) of hydrophilic pores 2. The membrane conductance, G(t), is due to the pores; a cell membrane also has channels 3. Changes in the transmembrane voltage, Vit), cause changes in the pore population 4. Transient behavior is determined by calculating how the pore population changes because of changes in U(t) 5. Feedback between U(t) and G(t) involves both the pore population and external conduction pathways of the experimental system (membrane environment) 6. Reversible electrical breakdown ( R E B ) is caused by large ionic conduction through the transient pore population 7. Rupture of a planar membrane is caused by the appearance of one or more large, unstable pores; the closed membranes do not allow cell rupture (35), except by portions of the membrane that are bounded by other cellular structures (e.g., cytoskeleton) because these portions may behave like small planar membranes and therefore can rupture

Table IV. Physical Ingredients in a Recent Version of the Theory (16, 36) Ingredient

κ

Significance Rate of pore creation (pores of radius r

m i n

)

Rate of pore destruction (pores of radius r ) Pore density function that describes a wide range of pore sizes Dynamic behavior of the heterogeneous pore population Born energy-modified conductivity within pores Hindered transport through pores [Renkin equation (37)] Local transmembrane voltage reduced by the spreading resistance Membrane charging through external resistances Coupling to the pulse generator, electrodes, and electrolyte (Figure 4) Transport of charged molecules through pores by electrical drift* m i n

n(r, t) n(r, t) * (r) H(r, η) U r , w h e r e the critical radius f o r rupture is r = y/T. c

c

T h e electrical c o n t r i b u t i o n Δ Ε treats the pore as an electrical capacitor that explicitly includes conduction through the pore a n d external region near Ε

I

1—

ι

1

.8

r ι " ι r i ι ι 2 5 ncoulombs r\ 2 0 ncoulombs

j

ι

ι

.

,

ι

.

ι A

-

-

---^^15^ncoulombs "

Φ

CD (0

? >

.6

10 ncoulombs ~

CD C

-Q

g

.4 5 ncoulombs

.2

1

0

1

.2

1

1

I

1

.4

i"*-—i—

t ^""H——1_

Time (μβ)

.6

.8

Figure 1A. Short time scale (0-1 με) behavior of the transmembrane voltage [U(t)] predicted by a recent version of the theoretical model for a planar bilayer membrane exposed to a single very short (0.4-με) pulse; that is, "charge injection" conditions (16). The key features of reversible electrical breakdown (REB) are predicted by the model, as is the occurrence of incomplete reversible electrical breakdown. In the case of incomplete reversible electrical breakdown, the membrane discharge is incomplete because U(t) does not reach zero after the pulse. Each curve is labeled by the corresponding value of the injected charge Q . The curves for Q = 25 and 20 nC show REB, whereas the other curves do not.


1

21.

10

10

=I s-

453


WEAVER

1 1 1 1

,

1

1

1

•|—r

1

τ ι

ι

ι

1

Γ Τ

Ί"

J

I

ι

ι

ι

9

10 10 10

Γ

7

"S

V ^ e .

8

i

^ 0 . 4 ^ : \ 0 . 5

Γ"M

-

B^

.1

-i

5

6


10

Γ \ \

5

10 10 10 10

4

3

\\

rf \\

Ν,

\0.3

ν

Ξ

\ \ "Ξ

N \

2

r f f \ 1

1

,

1

1 3

i X l

1

2

1

I

I

1

!

1

4

1

ι

ι

I

^

ι

ι

X I

5

6

Radius (nm)

Figure IB. Corresponding computed pore population distribution (probability density), n(r, t) (16). Each curve is labeled by the corresponding value of the injected charge Q . For Q = 25 and 20 nC (cases for which REB occurs), Ν increases to about 10 in less than 0.5 με and then decays exponentially with a time constant of » 4.5 με. For Q = 15 nC, Ν increases rapidly to about 10 and remains almost constant for about 4 με before the exponential decrease. For Q = 10 nC, Ν increases to about 2 X J O in about 5 με and remains almost constant for about 30 με before the decay phase. The membrane in thi8 case rupture8. For Q = 5 n C , Ν increase8 to about 40 in 80 με. Ν will return to it8 initial value as the membrane discharge8 with a time constant of about 2 s. 8

5

3

the pore m o u t h ( 1 3 , 14). T h i s gives Tr(8 w

E l

)t/

2

a rdr 2

where

1+

a(r)

^ r K p ( r )

2/ικ,

(2)

T h e function a ( r ) contains the "voltage divider ' effect associated w i t h the spreading resistance external to a pore a n d the internal pore resistance. W e assume that pores have a m i n i m u m radius, r = 1.0 n m , because head-group packing constraints require r to b e somewhat greater than the size o f the m i n

m i n


454


hydrophilic head groups (~ 0.7 n m ) that make u p the i n n e r surface o f the pore, a n d the pore w a l l must contain at least several p h o s p h o l i p i d molecules (35). W e also assume that the n u m b e r o f pores changes because o f the creation a n d destruction o f pores w i t h radius r (JO), w h i c h yields a boundary c o n d i t i o n at r = r for the flux o f pores i n radius space: m i n

m i n

J

p

Here N and N c

d

at r = r

= N -N c

d

(3)

m i n

are the pore creation a n d destruction rates, a n d the pore


flux is

I dn

Iι ι » ι I ι ι ι ι

η dAE\

1111

ιι ι ι ι ι ι ι ι

Α

25 ncoulombs 20 ncoulombs .8

.6 Φ C

2

εΦ

.4 5 ncoulombs .2

J 0 ncoulombs

10

20

30

15 ncoulombs _L _1_J. 40 50

Time (/is)

60

70

80

Figure 2A. Longer time scale (0-80 με) electrical behavior, predicted by the same model that shows rupture and simple charging of an artificial planar bilayer membrane (16). The characteristic sigmoidal behavior of U(t) is predicted by the model (16), but the time scale is somewhat shorter than found in experiments (61). Each curve is labeled by the corresponding value of the injected charge Q . The curves for Q = 25 and 20 nC are the spikes at t = 0. The curve for Q = 15 nC shows that the membrane underwent REB at t = 2 με, but the membrane recovered before it had time to discharge completely. The curve for Q = 10 nC shows rupture, whereas the curve for Q = 5 nC shows that the membrane conductance did not increase enough to d%8charge the membrane.


21.

455


WEAVER

ι ιι Β

10 ' 1


S

10

9

10

8

10

7

10 10

5

10

4

10

3

10

2

e

15

20 Radius (nm)

40

25

Figure 2B. Corresponding pore population distributions as described by n(r, t) (16). Each curve is labeled by the corresponding value of the injected charge Q. For Q = 25 and 20 nC (cases for which RE Β occurs), Ν increases to about 10 in less than 0.5 μ$ and then decays exponentially with a time constant of ^4.5 μ-s. For Q = 15 nC, Ν increases rapidly to about 10 and remains almost constant for about 4 μ« before the exponential decrease. For Q = 10 nC, Ν increases to about 2 X 1 0 in about 5 μ,-s and remains almost constant for about 30 μβ before the decay phase. The membrane in this case ruptures. For Q = 5 nC, Ν increases to about 40 in 80 /xs. Ν will return to its initial value as the membrane discharges with a time constant of about 2 s. s

5

3

D u r i n g pore formation the m e m b r a n e achieves energetically unfavorable configurations; that is, an energy barrier, Λ , is overcome. A l t h o u g h the important details are u n k n o w n , we assume that Λ depends o n the t r a n s m e m brane voltage U. T h e contribution o f permanent dipoles associated w i t h pore structures (41) is presently neglected, so Λ is assumed to d e p e n d o n U (viz. Λ = h — all , w h e r e ô and a are constants). T h e corresponding absolute rate estimate is 2

c

2

c

ν exp

8

C

~

aU

2

(5)

kT

w h e r e ν is an attempt rate based o n an attempt rate density, v , a n d the v o l u m e of the m e m b r a n e ( J J ) . T o provide continuity w i t h i n our work, we have used the barrier contained i n e q 5 a n d have not yet attempted to include 0


456


T a b l e V . Simulation Parameters a n d Values Description

Value

Coefficient of U in Λ Membrane area Capacitance of membrane Diffusion constant in pore radius space Time step size (in units of D / ( A r ) ) Membrane thickness Boltzmann's constant times temperature Series resistance of electrolyte, electrodes, and wires Internal resistance of current source (pulse generator) Large pore initial size Minimum pore radius Radius of positive ions Radius of negative ions Time Pulse length Charge of positive ions (in units of proton charge) Charge of negative ions (in units of proton charge) Pore edge energy density Membrane surface energy Pore creation energy barrier Pore destruction energy barrier Numerical grid spacing of simulation Dielectric constant of l i p i d Dielectric constant of water Pore creation rate prefactor Conductivity of bulk solution Pore destruction rate prefactor

1.9 X 1 0 " F 1.45 X 1 0 " m 9.61 X 1 0 " F 5 Χ 10" m /s 0.5 2.8 nm 4 Χ ΙΟ J 30 i l

Parameter a

2

0

C

0

dt h kT


RE

B

N

Γ Γmax · mm +

r

r_ t épuise

z_ Ί Γ

2

p

42

0

0

0

0

h

ΔΓ

0

0

V

0

Χ a

6

2

9

1 4

2

- 2 1

50 Ω 40 nm 1 nm 0.2 nm 0.2 nm s 0.4 ILS + 1 -1 2 X 10" J / m 1 Χ 10" J / m " 2.04 X 1 0 " J 2.04 X 1 0 ~ J 0.0195 nm 1 1 3

2

1 9

19

2.180

E

80e 10 s " 0.98 i l / m 5 Χ 10 m / s b

0

28

1

16

Parameters that characterize the membrane (16). = 8.85 X H T F/m. 0

s

b

0

2 0

1 2

other estimates ( J 5 ) o f t h e barrier. A l t h o u g h pore destruction is also not understood i n detail, w e assume that the probability that a pore o f radius r is destroyed is independent o f U. T h i s assumption gives m i n

= X^(

r

m i n

)exp|--^J

(6)

where χ is a constant. A combination o f physical forces a n d diffusion governs pore evolution. A s a result, pores w i t h a w i d e range o f sizes appear i n the m e m b r a n e . T h i s distribution o f sizes is described b y a " p o r e population f u n c t i o n ; " that is, a probability density function, n ( r , t) A t any time t, there are n ( r , t) Δ Γ pores w i t h radii between r a n d r + Δ Γ .


21.

WEAVER

liUUUMUiiU

UUUUUilUUUU

ÏÏÏÏTÎÎ& ( / Downloaded by UNIV OF PITTSBURGH on January 27, 2016 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0235.ch021

457


MM

MM

|ÏÏÏÏTÎTÎ ÎTÎÎÎT iUUT

MU

D Figure 3. Hypothetical structures of transient and metastable membrane conformations that are believed relevant to electroporation (17). A, Free volume fluctuation believed to be involved in the transport of nonpolar molecules across membranes (7) and a possible early precursor to a hydrophilic pore. B, Aqueous protrusion or "dimple" that is envisioned as a more direct precursor to a hydrophilic pore (8). C, Hydrophobic pore, a high-energy transient structure that is believed to be a direct precursor to a hydrophilic pore (10). D, Hydrophilic pore that is believed to be the primary participant in short-term electrical behavior and probably involved in molecular transport (10). E, Composite pore with one or more proteins at the inner pore edge, a speculative possibility that might account for a metastable pore that persists after U has decayed because of reversible electrical breakdown (58). F, Composite pore due to a "foot-in-the-door" mechanism, which involves insertion of a linear charged macromolecule into a hydrophilic pore, such that screened coulombic repulsion prevents shrinkage of the pore. This is another candidate for a metastable pore, which can persist and transport small ions and molecules long after U has decayed to a small value by reversible electrical breakdown. The present view of electroporation assumes that transitions from A -> Β -> C -> D have a nonlinear increasedfrequency of occurrence as Ό is increased. Similar transitions may lead to Ε and F. Partially anchored cytoplasmic molecules, components of extracellular matrix, and "in transit" transported molecules are examples of pore-entering molecules that may lead to persistent pores of type F. Indeed, observation of enhanced transport of a small charged molecule due to electroporative uptake of DNA (a charged molecule "in transit" during uptake) has been reported (53).

T h e kinetics of the pore population are S m o l u c h o w s k f s equation (14, 42): dn

dn l

+

1? Here D

p

η dr \ kT

quantitatively described

3ΔΕ

p

(7)

dr

is the effective diffusion constant for the pore radius (43).

our previous models showed that D

by

One

of

should be independent of pore radius,


458


but the m o d e l used an estimate i n w h i c h water was treated as an ideal gas (14) a n d gave a value o f D that was 2 0 times larger than the present value (which assumes that molecular interactions w i l l decrease D ) . Δ Ε is i m p o r tant because — (d AE/dr) is the effective driving force that acts to change the pore size. E q u a t i o n 7 is valid for r < r < r (we used r = 2 r in o u r simulations). Relatively few pores are p r e d i c t e d for small U(t). H o w e v e r , for large U(t) a tremendous n u m b e r o f rapidly changing pores w i t h a w i d e range o f sizes is predicted, a n d this p h e n o m e n o n corresponds to "electro p o r a t i o n . " T o quantitatively describe metastable pores (not yet accomplished), additional interactions w i l l b e needed. p

p

v


m i n

m a x

m a x

c

W e assume that t h e transport o f ions across t h e m e m b r a n e occurs b y passage through pores large enough to accommodate small hydrated ions (e.g., N a a n d C l ~ ) . H o w e v e r , t h e presence o f ions w i t h i n small pores requires that the " B o r n energy" (44) a n d h i n d e r e d motion b o t h b e consid ered. T h e b u l k electrolyte conductivity σ is a function o f the concentrations, Cj, a n d o f the mobilities, η o f its ions: +

β

ί ?

Theory of Electroporation - Advances in Chemistry (ACS Publications)

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