Soft Matter PAPER. Mechano-capacitive properties of polarized membranes. Introduction. Lars D. Mosgaard, Karis A. Zecchi and Thomas Heimburg*

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1 Soft Matter PAPER Cite this: Soft Matter, 2015, 11, 7899 Mechano-capacitive properties of polarized ebranes Lars D. Mosgaard, Karis A. Zecchi and Thoas Heiburg* Received 19th June 2015, Accepted 20th August 2015 DOI: /c5s01519g Biological ebranes are capacitors that can be charged by applying a field across the ebrane. The charges on the capacitor exert a force on the ebrane that leads to electrostriction, i.e. a thinning of the ebrane. Since the force is quadratic in voltage, negative and positive voltage have an identical influence on the physics of syetric ebranes. However, this is not the case for a ebrane with an asyetry leading to a peranent electric polarization. Positive and negative voltages of identical agnitude lead to different properties. Such an asyetry can originate fro a lipid coposition that is different on the two onolayers of the ebrane, or fro ebrane curvature. The latter effect is called flexoelectricity. As a consequence of peranent polarization, the ebrane capacitor is discharged at a voltage different fro zero. This leads to interesting electrical phenoena such as outward or inward rectification of ebrane pereability. Here, we introduce a generalized theoretical fraework, that treats capacitance, polarization, flexoelectricity, piezoelectricity and theroelectricity in the sae language. We show applications to electrostriction, ebrane pereability and piezoelectricity and theroelectricity close to elting transitions, where such effects are especially pronounced. Introduction Many signaling processes in biology involve electrical phenoena. These processes are related to the oveent of ions and the orientation of polar olecules. Biological olecules typically contain charged groups that are at the origin of electrical fields and dipole oents. Furtherore, ebranes and acroolecules are surrounded by electrolytes containing charged ions. At physiological ionic strength, the Debye length of electrostatic interactions in the aqueous ediu is about 1 n. It is caused by the shielding of charges by ions. However, in the hydrophobic cores of ebranes and proteins, the dielectric constant is sall, and no ions that could shield electrostatic interactions are present. Thus, the length scale of electrostatic interactions is significantly larger. Generally, under physiological conditions the range of the electric fields is siilar to the size of biological acroolecules. In this publication we will focus on the electrostatics of ebranes that deterines capacitance, polarization, piezoelectricity, flexoelectricity and theroelectricity. There exist large concentration differences of ions across the ebranes of biological cells. For instance, the concentration of potassiu is about 400 M inside and only 20 M outside of a squid axon. If the ebrane is selective for potassiu, this results in a Nernst potential across the biological ebrane. Niels Bohr Institute, University of Copenhagen, Blegdasvej 17, 2100 Copenhagen Ø, Denark. E-ail: theibu@nbi.dk Lars D. Mosgaard and Karis A. Zecchi contributed equally to this work. The cobination of the Nernst potentials of different ions yields a resting potential, which for biological cells is in the range of 100 V. The central core of a ebrane is ostly ade of hydrophobic non-conductive aterial. Thus, the bioebrane is considered a capacitor, e.g., in the Hodgkin Huxley odel for the nervous ipulse. 1 During the nerve pulse, currents are thought to flow across ion channel proteins that transiently charge or discharge the ebrane capacitor. Within this odel, the ebrane is assued to be a hoogeneous planar capacitor with constant diensions. The capacitance can be calculated fro the relation C ¼ e A (1) d where e is the dielectric constant, A is the ebrane area and d is the ebrane thickness. Let us assue that the ebrane is surrounded by a conducting electrolyte solution. In the presence of an applied voltage, the charged capacitor consists of one plate with positive charges and one plate with negative charges at distance d. The field inside the capacitor can be deterined using the superposition of the fields of the two plates (illustrated in Fig. 1). The field inside a charged capacitor is different fro zero, while it is zero outside of the capacitor. If no field is applied, the capacitor is not charged. The charges on a capacitor generate echanical forces on the two ebrane layers. 2 These forces can change the diensions of the capacitor such that both the area and the thickness This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,

2 Paper Soft Matter Fig. 1 Illustration of capacitive effects. The field inside a charged capacitor can be obtained by the superposition of the fields of a positively and a negatively charged plate at distance d. The charged capacitor displays an internal field different fro zero, while the field is zero outside of the capacitor. Fig. 3 Illustration of polarization by cheical asyetry. Left: If the ebrane contains peranent electrical dipoles, it is charged even if the applied potential is zero. Both, the fields inside and outside of the capacitor are zero. Right: In order to discharge the capacitor, a potential of C = C 0 has to be applied. Fig. 2 Illustration of the electrostriction effect upon charging the ebrane capacitor. The potential difference, C, results in a force on the ebrane that leads to a copression of the ebrane to a state with larger area, A, and lower thickness, d. of the ebrane change (see Fig. 2). As a result, the capacitance is not generally a constant. 2 The capacitance increases upon charging the ebrane by an applied field because the ebrane thickness decreases and the area increases. This effect is known as electrostriction. Close to phase transitions in the ebrane (in which the copressibility of the ebrane is large 3 ), the ebrane should be considered as a nonlinear capacitor. A sall change in voltage can result in large changes in thickness and capacitance. The coupling between the ebrane voltage and its diensions renders the ebrane piezoelectric, i.e., echanical changes in the ebrane can create a ebrane potential and vice versa. Siilarly, a sall change in teperature can change the capacitance leading to theroelectricity. On average, about 80% of the lipids of biological ebranes are zwitterionic. Zwitterionic lipids possess peranent electrical dipole oents. Exaples of such lipids are phosphatidylcholines and phosphatidylethanolaines. About 10% of biological lipids carry a net negative charge, including phosphatidylinositol and phosphatidylserine. It is known that bioebranes often display asyetric distributions of lipids such that charged lipids are ostly found in the inner leaflet of the bilayer. 4 Bioebranes also contain integral and peripheral proteins with asyetric distribution (or orientation) between inside and outside, which carry both positive and negative charges. Due to such copositional asyetries, a spontaneous electrical dipole oent of the ebrane can be generated in the absence of an externally applied potential. A redistribution or reorientation of polar olecules in an external field resebles the charging of a capacitor. If the ebrane possesses a spontaneous polarization, the ebrane capacitor in equilibriu can be charged even in the absence of an external field (illustrated in Fig. 3). In order to discharge this capacitor, a potential of C = C 0 has to applied. We call C 0 the spontaneous ebrane potential, or the offset potential. In a theoretical treatent one has to be very careful to correctly account for both capacitive and polarization effects. The polarization effects described above rely on an asyetric distribution of charges or dipoles on the two sides of a ebrane. Interestingly, even a cheically syetric lipid ebrane ade of zwitterionic (uncharged) lipid ay be polarized. The individual onolayers of zwitterionic lipids display trans-layer voltages on the order of 300 V. 5,6 Any geoetric deforation that breaks the syetry between the two onolayers of a ebrane results in a net polarization if these distortions alter the relative dipole orientation on the two layers. In particular, curvature induces different lateral pressure on the two sides of a ebrane. Thereby, curvature can induce polarization in the absence of an applied field. This consideration was introduced by Meyer in for liquid crystals. It was applied to curved lipid bilayers by Petrov in (see also ref. 9 and 10). He called this effect flexoelectricity. Upon bending (or flexing) the ebrane, both area, A, and volue, V, of the opposing onolayers change in opposite directions. If the polarization is a function of area and volue, the polarization of the outer onolayer is given by P o P o (A o, V o ) and that of the inner onolayer is given by P i P i (A i, V i ), respectively. Therefore, curvature can induce a net polarization across the ebrane. This is illustrated in Fig. 4. This polarization is counteracted by opposing charges adsorbing to the ebranes (Fig. 4b and c). In order to discharge the ebrane, a potential C = C 0 has to be applied. As in the case of cheical asyetry, at zero applied field, the field inside the capacitor is zero. The cases of a cheically asyetric planar ebrane and a cheically syetric curved ebrane are conceptually siilar. Charged capacitors, polarization, flexoelectricity and piezoelectricity all involve the spatial separation of charges. Thus, they all represent aspects of the sae electrostatic phenoena. However, in the literature they are often treated as different things and they are described by a different language. In this counication we forulate a general therodynaical description of the electrostatics of lipid ebranes, which represents a generalization of a study on the capacitance of ebranes previously published by our group. 2 It will be used to generalize the effect of an externally applied electric field on 7900 Soft Matter, 2015, 11, This journal is The Royal Society of Cheistry 2015

3 Soft Matter Paper The electric displaceent is related to the total polarization, P tot by D = e 0 E + P tot. (3) where e 0 is the vacuu perittivity. Most aterials have zero polarization at zero electric field, and polarization is only induced by an external field. For a linear dielectric aterial the induced polarization is P ind = e 0 w el E, where w el is the electric susceptibility. We are interested in extending our considerations to a dielectric aterial which can display spontaneous polarization, P 0, in the absence of an applied field such that P tot = e 0 w el E + P 0. (4) Fig. 4 Illustration of polarization by curvature. (a) The two onolayers of the syetric ebrane display opposite polarization. (b) Upon bending (flexing) the ebrane, the polarization in the two layers changes. (c) Effective polarization of the ebrane. (d) In order to discharge the capacitor, a potential of C = C 0 has to be applied. the lipid elting transition. We will introduce the therodynaics of a polarized lipid ebrane in an electric field, which then results in a generalization of electrostriction effects on lipid ebranes. Theory When the olecules of dielectric aterials are placed in an external electric field, they orient theselves to iniize the free energy. In capacitors, net acroscopic dipoles are induced in the dielectric ediu and tend to counteract the applied field. As a response to an applied electric field, echanical changes can be observed, e.g., in piezoelectric crystals. To deal with these effects, authors like Frank treated the electrostatic effects within a therodynaical fraework. 11 He considered the electrical work perfored on a fluid during any infinitesial and reversible change, dw el = Ed(vD). This type of consideration leads to expressing the electric displaceent, D, in a volue, v,as an extensive variable with the electric field, E, as its conjugated intensive variable. Vector notation has been dropped assuing planar geoetry. When we consider a ebrane capacitor, its hydrophobic core separates the two capacitor plates and acts both as a copressible and dielectric aterial. Choosing hydrostatic pressure ( p), lateral pressure (p), teperature (T) and applied electric field (E) as intensive variables, we can write the differential of the Gibbs free energy as dg = SdT + vdp + Adp (vd)de + (2) where the conjugated extensive variables are S (entropy), v (volue), A (area) and vd (electric displaceent). The electrical contribution to the free energy due to an applied electric field coes fro the final ter, which we will refer to as the electrical free energy, G el. The spontaneous polarization, P 0, can originate fro asyetric lipid bilayers, e.g., fro curvature (flexoelectricity) or fro different coposition of the two onolayers. The electric displaceent takes the for D = e(e + E 0 ), (5) where e is the dielectric constant, e = e 0 (1 + w el ) and E 0 P 0 /e is the electric field related to the spontaneous polarization, P 0, at E =0. Using eqn (5), we, can deterine the electrical free energy: ð E G el ¼ ðvdþde 0 ¼ ev E2 2 þ E 0E 0 ¼ e 2 v ð E þ E 0Þ 2 2 E 0 ; where we have assued the volue of the lipid ebrane to be constant. Assuing that the dielectric properties of the ediu are hoogeneous across a ebrane with thickness d, wecan define Ed = C where C represents the applied electric potential difference. This leads to G el ¼ e A C 0 ; (7) 2 d where C 0 is the offset potential related to E 0 (E 0 d = C 0 ). The prefactor contains the capacitance of a planar capacitor (C = ea/d). Thus, the electric free energy is given by (6) G el = 1 2 C ((C + C 0 ) 2 C 0 2 ). (8) At C = 0 the electrical contribution to the free energy is zero. Capacitive susceptibility, piezoelectricity, flexoelectricity and theroelectric effect The polarization of a ebrane can change by charging, copressing, stretching, bending or heating of the ebrane. The corresponding electrostatic phenoena are called electrostriction, piezoelectricity, flexoelectricity and theroelectricity. In the past, soe siple relations were derived by A.G. Petrov. 12 For instance, piezoelectricity was described as the area-dependence of polarization. Correspondingly, flexoelectricity was described as the curvature-dependence of the polarization assuing that polarization is zero in the planar state of the ebrane. However, upon changing the ebrane area, its capacitance also changes. Thus, in the presence of a field not only the This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,

4 Paper polarization but also the charge on the capacitor can change upon changing area. In the case of ebrane curvature, the polarization ay be different fro zero in the planar state. In the following, we derive general equations for electrostriction, piezoelectricity, flexoelectricity and theroelectricity. We will find that soe relations previously derived by Petrov are special cases of our ore general description. The charge on a capacitor. The charge on a capacitor is given by q ¼ A D ¼ AðeE þ P 0 Þ ¼ e A ð d C þ C 0Þ ¼ C ðc þ C 0 Þ: (9) The dependence of the charge on potential, C, surface area, A, curvature, c, and teperature T is given by dq dc dc dt : C;A;c (10) Here, we assue that C, A, c and T are variables that can be independently controlled in the experient (which is not generally the case in all experients). Thus, the change of the charge on a capacitor as a function of potential, area, curvature and teperature is given by: " # dq ¼ ðc þ C A;c;T C;c;T þ C þ A;c;T " þ ðc þ C 0 0 þ C;A;T C;c;T " þ ðc þ C 0 0 þ C;A;T dc " þ ðc þ C 0 0 þ C C;A;c (11) or in abbreviated for as dq ðc þ C 0 Þa A;c;T þ C þ C b A;c;T dc þ ðc þ C 0 Þa C;c;T þ C b C;c;T da þ ðc þ C 0 Þa C;A;T þ C b C;A;T dc þ ðc þ C 0 Þa C;A;c þ C b C;A;c dt (12) The first ter describes the change of charge on a capacitor allowing for the possibility that both capacitance and polarization can depend on voltage. The second ter describes piezoelectricity, i.e., the change of charge by changing area, taking into account the area dependence of both capacitance and polarization. One could write siilar equations, if the lateral pressure, p, were controlled instead of the area, A. The third ter describes flexoelectricity, which relates to the change of charge caused by changes in curvature. Here, both dependence of capacitance and polarization on curvature are considered. The last ter describes the theroelectric (Seebeck) effect, i.e., the dependence of the charge on a capacitor on teperature. We will discuss experiental situations below where two variables, for instance teperature and area, are coupled and not independent. In such situation one has to adapt the above equations and adjust the coefficients of the paraeters a and b. Capacitive susceptibility. The capacitive susceptibility is given by Ĉ =(qq/qc). It is a susceptibility describing the increental change ofthechargeonacapacitoruponasall change in voltage. It was discussed in detail in Heiburg (2012). 2 It is different fro the capacitance, which only depends on the geoetry of the capacitor. In contrast to the capacitance, the capacitive susceptibility includes changes in geoetry induced by voltage. For this reason, it displays a axiu in a elting transition of lipid ebranes. If curvature and teperature are kept constant, and area is a function depending on voltage (due to electrostriction), we find fro eqn (12) that ¼ ðc þ C 0 Þa c;t þ C þ C b c;t (13) c;t Here, we have oitted the index A because area is not kept constant. Further, if the spontaneous polarization is constant, this reduces to ^C ¼ C þ ðc þ C 0 c;t If the spontaneous polarization is zero at all voltages, this reduces to ^C ¼ C þ c;t which is the relation given by Heiburg (2012). 2 If both capacitance and polarization are constant, Ĉ = C. However, eqn (13) also iplies that the capacitive susceptibility generally depends on the voltage-dependence of the polarization. Piezoelectricity. Piezoelectricity is the area or thickness dependence of the charge on a capacitor, or ore generally, the influence of geoetry on the charge of a capacitor. Let us assue that in eqn (12) C, c and T are constant. We then obtain dq =[(C + C 0 )a C,c,T + C b C,c,T ]da. (16) At C = 0, we obtain for a sall change in area, DA, Soft Matter Dq E (C 0 a C,c,T + C b C,c,T )DA. (17) If C 0 (DA = 0) is zero, the capacitor is uncharged for C = 0. Then the charge on the capacitor after a change in area of DA is given by q(da) =C b C,c,T DA or C 0 (DA) =b C,c,T DA. (18) A siilar relation was given by Petrov and Usherwood. 13 As evident in eqn (16), the piezoelectric effect is a function of the externally applied potential if the capacitance is influenced by area changes (which usually is the case). Inverse piezoelectric effect. The elastic free energy density of ebrane copression is given by g = 1 2 KA T(DA/A 0 ) 2, where K A T is 7902 Soft Matter, 2015, 11, This journal is The Royal Society of Cheistry 2015

5 Soft Matter the lateral copression odulus and A 0 is the equilibriu area prior to copression. In the presence of an applied potential, the free energy density is given by g ¼ 1 DA 2 2 KA T 1 C C 0 ; (19) A 0 2 A 0 which contains an elastic and an electrostatic ter. In order to obtain the free energy, G, this has to be integrated over the surface area of the lipid ebrane. At constant copression odulus, K A T, and constant potential C, the area change DA equilibrates ¼ DA KA T A A ¼ 0 Therefore, DAðCÞ ¼A 0 2A C b C;c;T K A T C;c;T C þ a C;c;T K A T C C;c;T C 0 (20) C 0 : (21) Here, the first linear ter is due to the area dependence of the ebrane polarization, while the second quadratic ter originates fro the area dependence of the capacitance. It is evident that the area change induced by an applied voltage will be uch larger with the transition regie of a lipid ebrane where the copression odulus is uch saller than in the pure phases. 3 Flexoelectricity. Flexoelectricity describes the dependence of the charge on the ebrane capacitor induced by ebrane curvature. Let us assue that in eqn (11) C, and T are constant. It is questionable whether one can define experiental conditions where the ebrane area is independent of curvature. Therefore, we consider area a variable depending on curvature, which is not independently controlled. Then we find dq =[(C + C 0 )a C,T + C b C,T ]dc. (22) This is the (direct) flexoelectric effect. If we ake the siplifying assuption that the capacitance C does not depend on curvature and that the coefficient b C,T is constant, we obtain q(c) =C (C + C 0 (0)) + C b C,T c, (23) where C (C + C 0 (0)) is the ebrane charge at c = 0. If further the applied potential, C, is zero and the polarization in the absence of curvature is also assued being zero, we obtain q(c) =C b C,T c or C 0 (c) =b C,T c. (24) Thus, the offset potential C 0 is proportional to the curvature. This relation is a special case of the flexoelectric effect described in eqn (22). It was previously discussed by Petrov. 12 He introduced a flexoelectric coefficient, f, which is given by f eb C,T. Petrov found experientally that f =10 18 [C], or b C,T = [] for e =4e 0, respectively. Inverse flexoelectric effect. In the absence of a spontaneous curvature, the elastic free energy density of bending is given by g = 1 2 K Bc 2, where K B is the bending odulus. In the presence of an applied potential and assuing that C does not depend on curvature, the free energy density is given by g ¼ 1 2 K Bc 2 1 C ð 2 A C þ C 0Þ 2 2 C 0 ; (25) which contains an elastic and an electrostatic ter as in the expression for piezoelectricity. In order to obtain the free energy, G, this has to be integrated over the surface area of the lipid ebrane. At constant potential C, the curvature c equilibrates ¼ K Bc C A cðcþ ¼ C A Paper C ¼ K B c C C;T A b C;TC ¼ 0 (26) b C;T C ¼ e b C;T C (27) K B d K B This effect is called the inverse flexoelectric effect. It describes how curvature is induced by an applied potential. It depends on the bending odulus and the dependence of polarization on curvature, b C,T. In elting transitions, the curvature-induction by voltage is enhanced because K B approaches a iniu. 3 This iplies that in the presence of an applied field, the curvature of a ebrane changes upon changing the teperature in particular close to transitions. Both, the investigation of flexoelectric and inverse flexoelectric effects have been pioneered by Petrov. 12 In Petrov s noenclature, eqn (27) assues the for c(c) =(f/dk B )C. Theroelectricity. Let us assue that in eqn (11) C, and c are constant, and A is a function depending on teperature. Then we find dq =[(C + C 0 )a C,c + C b C,c ]dt. (28) This is the Seebeck effect. It describes the charging of a capacitor by changing teperature. Since the changes in capacitance are especially strong in the elting transition of ebranes, the coefficient a C,c (which is positive) typically displays a axiu at T. 2 The coefficient b C,c that describes the change in spontaneous polarization with teperature will also display an extreu. In the absence of spontaneous polarization, eqn (28) reduces to dq dt ¼ : C;c Due to changes in area and thickness, the dependence of the capacitance on teperature is especially large in the elting transition of a ebrane. This effect was described in Heiburg (2012). 2 A related effect, the induction of teperature changes by charging a ebrane is the Peltier effect. It can occur in ebranes due to charge-induced transitions that lead to an absorption or release of latent heat fro the ebrane. This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,

6 Paper Applications Electrostriction The charges on a capacitor attract each other. These attractive forces can change the diensions of the ebrane and thereby change the capacitance. If C 0 = 0, the electric contribution to the free energy according to eqn (8) is G el = 1 2 C C 2.ForA E const. and C =const.,theforcef acting on the layers is ¼ C 2 ¼ 1 C C 2 2 d : (30) Soft Matter This is the force acting on a planar capacitor given in the literature (e.g., ref. 2). If there exists a constant offset potential C 0, we find instead (eqn (8)) F ¼ 1 2 C d C 0 : (31) Thus, one expects that the force on a ebrane is a quadratic function of voltage which displays an offset voltage when the ebrane is polarized. This force can reduce the ebrane thickness and thereby increase the capacitance of a ebrane. Note, however, that for (C + C 0 ) 2 C 0 2 o 0, the force F is negative. As a consequence, capacitance will be decreased. Let us assue a ebrane with constant area and sall thickness change, Dd { d. Then the change in capacitance, DC, caused by a change of thickness, Dd, is given by DC ¼ e A d2dd (32) Thus, the change in capacitance is proportional to the change in thickness. If the thickness is a linear function of the force (F p Dd), one finds that the capacitance is proportional to the force F. Therefore, it is a quadratic function of voltage with an offset of C 0, DC p ((C + C 0 ) 2 C 0 2 ). (33) The agnitude of the change in capacitance depends on the elastic constants of the ebrane. Relation (33) was studied by various authors. Using black lipid ebranes, Alvarez and Latorre 14 found a quadratic dependence of the capacitance on voltage (Fig. 5). In a syetric ebrane ade of the zwitterionic (uncharged) lipid phosphatidylethanolaine (PE), the offset potential C 0 in a 1 M KCl buffer was found to be zero. In an asyetric ebrane with one onolayer ade of PE and the other ade of the charged lipid phosphatidylserine (PS), a polarization is induced. In a 1 M KCl buffer, the offset potential was C 0 = 47 V, while it was C 0 = 116 V in a 0.1 M KCl buffer. It is obvious fro Fig. 5 that within experiental error the shape of the capacitance profile is unaffected by the nature of the ebrane. Only the offset potential is influenced by coposition and ionic strength. This suggests that the offset potential has an ionic strength dependence. In this publication, we do not explore the theoretical background of this experiental fact. In a range of 300 V around the iniu capacitance, the change in capacitance, DC, is of the order of o1.5 pf, Fig. 5 The change in capacitance as a function of the applied potential in a black lipid ebrane. Solid circles: syetric ebrane in 1 M KCl. Both onolayers are ade fro zwitterionic bacterial phosphatidyl ethanolaine (PE). Open circles: asyetric ebrane in 1 M KCl. One onolayer is consists of bacterial PE, while the other onolayer consist of the charged bovine brain phosphatidylserine (PS). Open squares: sae as open circle, but with saller salt concentration (0.1 M KCl). The absolute capacitance, C,0 at C = 0 V is approxiately 300 pf. Raw data adapted fro ref. 14. while the absolute capacitance, C,0,atC = 0 is approxiately 300 pf. 14 Thus, the change in capacitance caused by voltage is very sall copared to the absolute agnitude of the capacitance. Mebrane conductance and rectification A very interesting application of electrostriction is the influence of voltage on ebrane pereability. It has been found that ebranes can for pores that appear as quantized conduction events upon the application of potential difference across the ebrane (see right insert in Fig. 6). The likelihood to for a pore is thought to be proportional to the square of the applied electric potential. 20,21 This assuption is based on the hypothesis that an increase in voltage thins the ebrane (as described in the previous section) and eventually leads to an electric breakdown linked to pore foration. Laub et al. 22 found that the current voltage (I V) relation for a cheically syetric phosphatidylcholine ebrane patch fored on the tip of a glass pipette was a non-linear function of voltage which was not syetric around C = 0, but rather outward rectified (Fig. 6). Blicher et al. 23 proposed that an offset potential can explain the outward-rectification. The free energy difference between an open and a closed pore, DG p, can be expressed by DG p = DG p,0 + a((c + C 0 ) 2 C 0 2 ), (34) where DG p,0 and a are coefficients describing the difference in free energy between open and a closed pore in the absence and the presence of externally applied voltage. The equilibriu 7904 Soft Matter, 2015, 11, This journal is The Royal Society of Cheistry 2015

7 Soft Matter Fig. 6 Current through a lipid ebrane (10 : 1 DMPC : DLPC, T =301C, 150 M) on a patch pipette as a function of voltage. The current voltage relation can nicely be fitted by the siple odel described in the text. The offset potential is C 0 = 110 V. Left inset: Open probability of lipid ebrane channels. Right inset: Opening and closing of individual lipid channels as a function of tie at a voltage of 50 V. Data adapted fro ref. 24 and 23. constant between open and closed pores is given by K p = exp( DG p /kt), and the likelihood of finding an open pore is given by P open = K p /(1 + K p ), shown in the left insert of Fig. 6. The I V relation can be expressed as I = g p P open C. (35) This relation perfectly describes the experiental current voltage data if a offset potential of C 0 = 110 V was assued (solid line in Fig. 6). Thus, inward and outward rectified I V profiles can be found in pure lipid ebranes in the coplete absence of proteins. They find their origin in the polarization of the ebrane and the effect of electrostriction. The ost likely origin of the offset potential is polarization of the ebrane in the patch pipette due to curvature (flexoelectricity 23 ). The above analysis iplies that any asyetric phenoena of bioebranes could have their origin in a spontaneous polarization of the ebrane as a whole. Piezoelectricity and capacitive susceptibility: influence of the potential on the ebrane diensions and capacitance close to a elting transition As discussed above, the influence of voltage on the capacitance is sall in the gel and in the fluid phase because ebranes are not very copressible in their pure phases. However, close to the phase transition between gel and fluid, ebranes becoe very copressible. 3 In this transition, the thickness of the ebrane, d, decreases by about 16% and the area, A, increases by about 24% 3 for the lipid dipalitoyl phosphatidylcholine (DPPC), with a elting teperature T = K. Therefore, the capacitance of the fluid ebrane is about 1.5 tie higher than the capacitance of the gel phase. 2 In the following we wish to describe the influence of voltage on ebrane area, capacitance and charge. According to eqn (8), the Gibbs free energy difference caused by an external electric field can be written as DG el ¼ G el fluid Gel gel ¼ DC C 0 ; (36) 2 where DC is the difference between the capacitance of gel and fluid phase. Here, we assued that both the offset potential C 0 and the dielectric constant e do not change with the state. We have confired the latter in experients on the dielectric constant in the elting transition of oleic acid using a parallel plate capacitor (data not shown). We found that the changes of the dielectric constant caused by the elting of oleic acid (T E 17 1C) are very sall. It has been shown experientally that in the vicinity of the lipid elting transition changes of various extensive variables are proportionally related. 3,25,26 For instance, changes in enthalpy are proportional to changes in area, in volue and we assue that a siilar relation holds for changes in thickness. Further, close to transitions the elastic constants are closely related to the heat capacity. For instance, the teperature-dependent change of the isotheral copressibility is proportional to heat capacity changes. Thus, ebranes are ore copressible close to transitions, and it is to be expected that the effect of potential changes on ebrane capacitance is enhanced. This will be calculated in the following. We assue that the lipid elting transition is described by a two-state transition governed by a van t Hoff law, so that the equilibriu constant between the gel and the fluid state of the ebrane can be written as 2,27 KðT; CÞ ¼exp n DG (37) RT where n is the cooperative unit size which describes the nuber of lipids that change state cooperatively (for LUVs of DPPC we used n = ). The free energy difference between gel and fluid ebranes is given by DG =(DH 0 TDS 0 )+DG el, (38) where DH 0 = 35 kj ol 1 and DS 0 = J ol 1 K 1 (for DPPC). Fro the equilibriu constant we can calculate the fluid fraction, the average fraction of the lipids that are in the fluid state, f f ðt; CÞ ¼ Paper KðT; CÞ 1 þ KðT; CÞ : (39) For DPPC LUV, the thickness in the gel and fluid state is given by d g = 4.79 n and d f = 3.92 n, respectively. The area per lipid is A g = n 2 and A f = n 2. 3 We assue a dielectric constant of e =3e 0 independent of the state of the ebrane, and that the offset potential C 0 = 70 V is also a constant. The area is described by A(T,C) =A g + f f DA, and the ebrane thickness by d(t,c) =d g f f Dd, respectively. This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,

8 Paper Soft Matter Fig. 7 Change of capacitance, capacitive susceptibility, area and ebrane charge as a function of applied voltage at five teperatures close to the elting transition of a ebrane. Paraeters are for LUV of DPPC, where DC E 656 J (ol V 2 ) 1, and the elting teperature T = K. C 0 was chosen to be 70 V. The inset in the botto left panel shows the noralized capacitance changes for coparison with Fig. 5. In contrast to the capacitance, the capacitive susceptibility displays axia at the voltage-induced elting transition. Both, teperature and voltage-dependent area, A(T,C), and capacitance, C = ea(t,c)/d(t,c) are shown in Fig. 7. For sall variations of the potential around C = C 0, the change in capacitance is well approxiated by a quadratic function (insert in Fig. 7). The gel phase capacitance is saller than the fluid phase capacitance. Due to the effect of electrostriction, an increase in voltage can induce a elting transition. In this transition, the capacitance increase until the value for the fluid phase is reached. By necessity, the charge on the ebrane also undergoes a stepwise change (Fig. 7, top right). For this reason, the capacitive susceptibility Ĉ =(qq/qc) given by eqn (14) displays axia at the transition voltage if the teperature is within or below the elting regie. Due to the presence of an offset potential, the transition profiles are not syetric around C = 0. One can recognize that the sensitivity of the capacitance to voltage changes close to the transition is uch larger than that of thepurephases(fig.5).itisalsoasensitivefunctionofthe teperature. Fig. 7 shows C (C) for five different teperatures close to the elting teperature of DPPC at C. At T = K, the change in capacitance at C C 0 = 300 V is approxiately 3% copared to the about 0.5% experientally easured in the absence of a transition (Fig. 5). Due to the presence of a elting transition, the curve profile in Fig. 7 is only a quadratic function of potential close to C = C 0. If the teperature is lower than T, one finds voltage-induced transitions with a teperature-dependent transition voltage. Above this critical voltage the change in capacitance can be as high as 50%. The coefficient a c,t in eqn (14) is given by 1 a c;t ¼ ðc þ C 0 Þ ^C C : (40) Close to the transition, it is a non-linear function of the therodynaic variables (Fig. 8). In the absence of a transition, a c,t is zero because (Ĉ C ) E 0. If the spontaneous polarization C 0 is independent of the applied potential, as assued here, the coefficient b c,t is zero. However, generally the coefficients a i,k and b i,k are nonlinear functions of the variables close to transitions. Under these conditions, the electrostatics of ebranes is especially interesting. The dependence of the elting teperature on the applied potential. The total free energy difference between gel and fluid phase, DG, given by eqn (38) consists of an enthalpic and an Fig. 8 The coefficient a c,t =(qc /qc) c,t as a function of applied voltage at five different teperatures close to the elting teperature T. Paraeters as in Fig Soft Matter, 2015, 11, This journal is The Royal Society of Cheistry 2015

9 Soft Matter Paper entropic contribution. At the elting teperature, T, the Gibbs free energy difference DG is zero, so that T ¼ T ;0 1 þ DGel DS 0 ¼ T ;0 1 1 DC (41) C 0 ; 2 DS 0 where T,0 = DH 0 /DS 0 is the elting teperature in the absence of an external field (for DPPC: DH 0 = 35 kj ol 1, T,0 = K and DS 0 = J ol 1 K 1 (ref. 3)). This result describes the effect of electrostriction on the lipid elting transition in the presence of spontaneous polarization. It is a generalization of the electrostriction effect described by Heiburg 2 who treated this phenoenon in the absence of polarization effects. Fig. 9 shows the dependence of T on an applied voltage for three different offset potentials, C 0. It can be seen that in the presence of an applied potential, the spontaneous polarization and its sign influences the elting teperature. Generalization for W 0 a const. The orientation of lipid dipoles can change upon lipid elting. It sees to be obvious fro lipid onolayer experients that the polarization of liquid expanded and solid condensed layers is different. 28 Weassuethesaetobetrue for bilayers. Let us assue that the net offset potentials originating fro ebrane polarization in the gel and the fluid phase are given by C g 0 and C f 0, respectively. The free energy is now given by DG el ¼ DC 2 C0;g 2 C þ C 0;g C f CC 0;f C 0;g 2 This can be inserted in eqn (38) and (41) to obtain the change in elting teperature due to an applied field. The theroelectric effect The charge on a capacitor is given by q = C (C + C 0 ), where the capacitance C can be a function of teperature. If the spontaneous polarization C 0 is independent of teperature, Fig. 9 The lipid elting teperature as a function of applied potential with three different offset potentials, C 0 =0.1V,C 0 = 0 V, and C 0 = 0.1 V. The paraeters are taken fro LUV of DPPC, where DC E 656 J (ol V 2 ) 1 for e =4e 0. Fig. 10 The theroelectric effect close to the elting transitions. Given is the charge, its derivative and the coefficient a C,c =(qc /qt) as a function of teperature for two different voltages, C = 100 V and C = 100 V with C 0 = 70 V. The paraeters for DPPC are the sae as used for Fig. 7. the dependence of the charge on the teperature changes is given by (see eqn (28)) dq =(C + C 0 )a C,c dt. (42) This is shown in Fig. 10 for two different fixed voltages, C =100V and C = 100 V, and an offset potential of C 0 =70V.Itcanbe seen that close to a transition the charge changes in a stepwise anner. Heiburg (2012) 2 called this change an excess charge. This absorption or release of charge upon teperature change could be called a theroelectric effect. Close to transitions, the sudden change in teperature can lead to a capacitive current. One can also alter the ebrane charge in an adiabatically shielded syste. The induction of a phase transition by charging the ebrane leads to a release or an absorption of the latent heat which would alter the teperature in the aqueous reservoir. This is the Peltier effect. The dielectric susceptibility In ref. 2 we defined a capacitive susceptibility, Ĉ =(qq/qc) = C + C(qC /qc). This susceptibility has a axiu at the elting teperature, which is a consequence of the fact that the capacitances of gel and fluid lipid phases differ. By analogy, we now introduce a dielectric susceptibility, ^e =(qd/qe), which is given by: ee þ P ¼ e þ This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,

10 Paper Therodynaic susceptibilities are linked to fluctuation relations. For instance, in ref. 2 we showed that the capacitive susceptibility is given by Ĉ =(hq 2 i hqi 2 )/kt, i.e., it is proportional to the fluctuations in charge. This fluctuation relation is valid as long as the distribution of states is described by Boltzann statistics and the area and thickness are kept constant. Analogously, for constant volue, v, the dielectric susceptibility, ^e, isgivenby ^e ¼ v D2 hdi 2 : (44) kt Since this is a positive definite for, ^e is always larger than zero. The ean displaceent, hdi, always increases with an increase in the electric field, E. If either e or the peranent polarization P 0 are different in the gel and the fluid state of a ebrane, one can induce a transition. In this transition, the dielectric susceptibility displays an extreu. Suary and discussion In this publication, we provided a general therodynaic treatent of polarization effects on the properties of lipid ebranes. When applied to a ebrane in an electrolyte, these electric effects can all be related to the charging (or discharging) of capacitors by either potential, curvature, area (or lateral pressure) or teperature changes. Curvature and area changes can lead to an offset potential or a spontaneous polarization. This is iportant because biological ebranes are known to be polar and changes in voltage are generally considered to be central to the understanding of the functioning of cells. We show that a peranent or spontaneous polarization of a ebrane influences the properties of a ebrane capacitor such that it is discharged at a voltage different fro zero. We relate this voltage to an offset potential. The existence of this potential has the consequence that ebrane properties even of cheically syetric ebranes are controlled differently for positive and negative voltages. We derived equations for the piezoelectric and inverse piezoelectric effect. The first considers the change in the offset potential when changing the ebrane area. The second considers the change in ebrane area by an applied field, which depends on the elastic odulus of the ebrane. Finally, we derived general relations for the flexoelectric and the inverse flexoelectric effect. We showed that in soe siple liiting cases, our derivations lead to relations identical to those of Petrov 12 who pioneered the field of ebrane flexoelectricity (e.g., ref. 8, 12, 13, 29 33). An electric field applied across a lipid ebrane generates a force noral to the ebrane surface due to the charging of the ebrane capacitor. The resulting reduction in ebrane thickness is called electrostriction. 2 For fixed ebrane diensions, the electrostrictive force is a quadratic function of voltage. Due to ebrane thinning induced by the forces, one finds an increase in ebrane capacitance. This has been deonstrated for syetric black lipid ebranes ade fro phosphatidylethanolaines (Fig. 5, Alvarez et al., ). However, for an asyetric ebrane ade of charged lipids on one side and zwitterionic lipids on the other side (thus displaying polarity) the Soft Matter iniu capacitance is found at a voltage different fro zero (Fig. 5, ref. 14). This indicates that a spontaneous electric polarization of the ebrane influences the capacitive properties of a ebrane. This has also been found in biological preparations. Huan ebryonic kidney cells display an offset potential of 51 V. 34 This indicates that the capacitance in electrophysiological odels such as the Hodgkin Huxley odel 1 is incorrectly used because offset potentials are not considered. However, it is very likely that the offset potentials are closely related to the resting potentials of ebranes. It should also be noted that the capacitance is typically dependent on the voltage. This effect has also not been considered in classical electrophysiology odels. We treat that here in ters of a capacitive susceptibility (eqn (13), cf. ref. 2). Electrostrictive forces also influence elting transitions of lipid ebranes. Since the fluid state of the ebrane displays a saller thickness than the gel phase, an electrostrictive force will shift the state of the ebrane towards the fluid state. Heiburg 2 calculated a decrease of the elting teperature, T, which is a quadratic function of voltage. Since the ebrane was considered being syetric, the largest T is found at C = 0. Here, we showed that a ebrane which displays a spontaneous polarization in the absence of an applied electric field possesses an offset potential, C 0, in the free energy (eqn (8)). The respective equation contains the ter ((C + C 0 ) 2 C 0 2 )=C 2 +2CC 0, which is approxiately linear for C { C 0 (eqn (8)). In fact, Antonov and collaborators found a linear dependence of the elting teperature on voltage. 35 This indicates that the ebranes studied by Antonov and collaborators 35 were polar. Antonov s experient deterined the voltage-dependence of the elting teperature by easuring the pereability changes in the transition. The authors ade use of the fact that ebranes display axiu conductance in lipid phase transitions. 17,36 In the Applications section we showed how voltage can influence ebrane pereability in the presence of spontaneous polarization. Surprisingly, the corresponding current voltage (I V) relations of ebrane conductance can be inward or outward rectified, and reseble the I V relations of any proteins. 19 Here, we investigated two possible echaniss that can give rise to spontaneous polarization in the absence of an applied field, which both break the syetry of the ebrane. The first (flexoelectricity) acts by allowing the ebrane to be curved (thus introducing a curvature, c) and a difference of the lateral tension within the two onolayers. The second echanis acts by assuing a cheically or physically asyetric lipid coposition on the two leaflets. An exaple for a physically asyetric ebrane is a situation where one onolayer is in a fluid state while the other onolayer is in a gel state. Cheical asyetry assues a different lipid coposition on the two sides of the ebrane. The agnitude of the resulting offset, C 0, is strongly influenced by experiental conditions such as the lipid coposition, salt concentration, ph, or the presence of divalent ions. Peranent polarization of the lipids can not only lead to an electrical offset but also to an enhanced dielectric constant. For biological ebranes, polarization asyetries 7908 Soft Matter, 2015, 11, This journal is The Royal Society of Cheistry 2015

11 Soft Matter Fig. 11 The polarization of bioebranes can have various reasons. For instance, it can originate fro curvature, fro an asyetry of zwitterionic and negatively charged lipids, fro polarized or charged integral and peripheral proteins (green objects), fro concentration differences of ions such as Na +,K +,Cl and Ca 2+, or fro ph difference. can originate fro any constituting eleent of the ebrane including integral ebrane proteins (see Fig. 11 for a scheatic description). We can also speculate that other ebrane adhesive olecules with large dipoles can be used to create an asyetric ebrane, e.g., soluble proteins or lipid-associated olecules such as long-chain sugars. Depending on the nature of the asyetry, the syste can display piezoelectric properties. The offset potential can have interesting consequences for capacitive currents. The charge on a capacitor is given by q = C (C + C 0 ). Therefore, for constant C 0 the capacitive current is given by I c ðtþ ¼ dq dt ¼ C dc dt þ ð C þ C 0Þ dc (45) dt For a positive change in potential, the first ter in eqn (45) is positive and leads to a positive current. If the change in voltage happens instantaneously, the corresponding current peak is very short. The second ter describes the teporal change in the capacitance induced by the voltage change. It depends on the relaxation tie of the capacitor diensions, which close to transitions can range fro illiseconds to seconds. Thus, it can be distinguished fro the first ter. Let us consider the situation shown in the insert of Fig. 7 (C 0 = 70 V, T = K) with a ebrane capacitance of E1 F c 2. Here, a jup fro C = 70 V to C = 10 V yields a positive change in capacitance of DC = 2.6 nf c 2. If the offset potential were C 0 = 70 V instead, the sae jup would change the capacitance by DC = 7.8 nf c 2. Therefore, the second ter in eqn (45) is positive in the first situation but negative in the second situation. For this reason, depending on the offset potential and holding potential, the capacitive current associated to the second ter in eqn (45) can go along the applied field or against the applied field. Siilarly, for a jup in potential of +60 V, the capacitive current would depend on the holding potential before the jup. For C 0 = 70 V, the change in capacitance is DC = 2.6 nf c 2 for a jup fro 130 V to 70 V. It is DC = +8.9 nf c 2 for a jup fro +70 V to +130 V. The typical tie-scale of processes in bioebranes is a few illiseconds to a few ten illiseconds. It can be different for different voltages. Thus, slow currents on this tie-scale against an applied field can originate fro voltage-induced changes in lipid ebrane capacitance. If the offset-potential also depends on voltage, this situation is ore coplicated. Flexoelectric and piezoelectric phenoena have also be considered to be at the origin of an electroechanical echanis for nerve pulse propagation. 37 In 2005, Heiburg and Jackson proposed that the action potential in nerves consists of an electroechanical soliton. The nerve pulse is considered as a propagating local copression of the ebrane with a larger area density. According to the piezoelectric effect treated here (eqn (16)), a change in ebrane area can lead to the charging of the ebrane capacitor. Alternatively, due to the inverse piezoelectric effect a change in the applied ebrane potential can induce area changes (eqn (21)) and thus induce a density pulse. The inverse piezoelectric effect is very dependent on the lateral copressibility of a ebrane. Thus, is largely enhanced in the elting transition where the copressibility is high. Further, these effects will largely depend on ebrane polarization. Finally, it should be entioned that soe of the polarization effects on artificial ebranes are not very pronounced because changes in polarization due to changes in area are not very large. For instance, a voltage change of 200 V changes the transition teperature by only 0.12 K. However, the absolute agnitude of the effect largely depends on offset polarizations. These could be influenced by lipid-ebrane-associated olecules (such as proteins) with large dipole oents. Acknowledgeents We thank to Prof. Andrew D. Jackson fro the Niels Bohr International Acadey for useful discussions and for a critical reading of the anuscript. This work was supported by the Villu foundation (VKR ). References Paper 1 A. L. Hodgkin and A. F. Huxley, J. Physiol., 1952, 117, T. Heiburg, Biophys. J., 2012, 103, T. Heiburg, Biochi. Biophys. Acta, 1998, 1415, J. E. Rothan and J. Lenard, Science, 1977, 195, V. Vogel and D. Möbius, J. Colloid Interface Sci., 1988, 126, H. Brockan, Che. Phys. Lipids, 1994, 73, R. B. Meyer, Phys. Rev. Lett., 1969, 22, A. G. Petrov, Physical and cheical basis of inforation transfer, Plenu Press, New York, 1975, pp K. Hristova, I. Bivas, A. G. Petrov and A. Derzhanski, Mol. Cryst. Liq. Cryst., 1991, 200, F. Ahadpoor, Q. Deng, L. P. Liu and P. Shara, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 88, (R). 11 H. S. Frank, J. Che. Phys., 1955, 23, This journal is The Royal Society of Cheistry 2015 Soft Matter, 2015, 11,