View Article Online View Journal
Soft Matter Accepted Manuscript
This article can be cited before page numbers have been issued, to do this please use: P. F. Salipante and P. Vlahovska, Soft Matter, 2014, DOI: 10.1039/C3SM52870G.
This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.
www.rsc.org/softmatter
Page 1 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
Vesicle deformation in DC electric pulses Paul F. Salipantea and Petia M. Vlahovska⇤a First published on the web Xth XXXXXXXXXX 200X DOI: 10.1039/b000000x
The transient deformation of giant vesicles in square DC electric pulses is investigated. We experimentally observe the theoretically predicted transition from an oblate to prolate ellipsoidal shape in the case of a quasi-spherical vesicle encapsulating solution less conducting than the suspending medium 1 . The transition is detected by utilizing a two-step pulse in order to avoid electroporation and vesicle collapse. We develop a theoretical model to describe both the deformation under electric field and relaxation after the field is turned off. Agreement between experiment and theory demonstrates that the time-dependent vesicle shape can be used to measure membrane properties such as viscosity and capacitance.
1
Introduction
The electro-deformation of giant vesicles made of artificial bilayer membranes provides fundamental insights into the electromechanics of biomembranes, namely the coupling of membrane shape and transmembrane potential 2–4 . The steady shapes of vesicles in AC fields have been extensively studied experimentally 5,6 and theoretically 7–10 . The behavior in DC pulses has beed considered theoretically 1,11 , but only limited experimental data is available due to problems with visualization and vesicle fragility 12–15 . In this work we perform a systematic investigation of vesicle dynamics in square DC pulses. A theoretical model is developed to describe both the deformation under electric field and relaxation after the field is turned off. We show that the data can yield information about membrane properties such as viscosity, bending rigidity, and capacitance. Moreover, we confirm the recently predicted oblate-prolate shape transition during the pulse application for a vesicle filled with solution with lower salt content than the suspending liquid 1 . 1.1
Physical picture
Let us consider a vesicle made of an ion-impermeable chargefree lipid bilayer membrane with dielectric constant, emm , viscosity µmm , and thickness, d. The solutions inside and outside the vesicle have different conductivities, lin and lex (e.g., due to different salt concentrations). This mismatch in the bulk fluid electrical properties is characterized by the conductivity ratio lin R= . (1) lex a School of Engineering, Brown University, Providence RI 02912, USA. Fax: ++1 401 8639028; Tel: ++1 401 8639774; E-mail: petia [email protected]
The solutions’ viscosities are nearly the same (basically equal to the viscosity of water µ). Moreover, their dielectric constants are also very close and for simplicity we assume that ein = eex = e of water. Application of an electric field leads to accumulation of ions at the membrane surfaces and the vesicle will act as a capacitor, see Figure 1. For a quasi–spherical vesicle made of an insulating (ion-impermeable) membrane the characteristic time scale for the charging process is 16–18 , ◆ ✓ R aCmm , (2) 1+ tmm = lin 2 where Cmm = emm /d is the capacitance of the membrane and a is the vesicle radius. For systems with typical solution conductivity l ⇠ 0.1µS/cm and a ⇠ 10µm the charging timescale is tmm ⇠ 1ms. Membrane capacitance sets up a transmembrane potential, which in an uniform DC electric field, E• = E0 zˆ , evolves toward steady state as 1 3 Vmm = aE0 (1 2
exp ( t/tmm )) cos(q ) ,
(3)
where q is the angle between the electric field direction and position at the interface. During the charging process, t < tmm , the ion densities on the inner and outer membrane surfaces may become temporarily unbalanced. This is due to the difference in the conduction rates in the inner and suspending solutions, which is characterized by R. The polarity of the apparent charge (and resulting vesicle deformation) is illustrated in Figure 1. At steady state however, t tmm , irrespective of R the capacitor is fully charged, the apparent charge is zero, and the electric field is expelled from the vesicle interior. The physical picture suggests that a vesicle with R < 1 may initially deform into an oblate spheroid but eventually adopts 1–9 | 1
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
Soft Matter
Page 2 of 11 View Article Online
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
a)
b)
c)
Fig. 1 Sketch of the electric field and induced charge distribution around a
vesicle immersed in an electrolyte solution, following the imposition of a uniform DC field (a) t < tmm and R > 1 (b) t < tmm and R < 1 (c) fully charged membrane capacitor t tmm , any R. The dashed lines indicate the vesicle deformation.
a prolate shape 1 (see Figure 2). This theoretical prediction has not been confirmed experimentally, although it has been observed in numerical simulations 19 . In this work we experimentally investigate the conditions for this transition to occur. The challenge is to ensure that the transient oblate deformation is visible yet the field is not strong enough to cause vesicle poration. Poration makes the membrane conducting, which suppresses the prolate deformation for R < 1 19 and destabilizes the membrane 20 . In the next section we discuss the time scales relevant for the design of the experiment.
a
a a
a
Fig. 2 ak and a? are the axes parallel and perpendicular to the symmetry
axis respectively. Deformation is shown for prolate (left) and oblate (right). The deformation parameter is defined as D = ak /a?
1.2
Timescales
The electric stresses act to deform the vesicle on a typical time scale µ . (4) tel = eE02 An important ratio compares the rate of deformation, characterized by tel , to the membrane charging time, dm = tel /tmm .
(5)
As discussed in the previous section, the vesicle exists in the oblate shape while t < tmm . However, significant deformation implies t > tel . Hence, in order to detect the oblate shape dm must be less than 1 (in addition to R < 1). 2|
1–9
The electrodeformation timescale varies with field strength. A weak field condition can be defined as tmm < tel (dm > 1). In this limit, deformation occurs after the membrane has fully charged resulting in only prolate shape (independent of R). In strong fields, tmm > tel , the vesicle is charging and simultaneously deforming; the shape evolution depends on the conductivity ratio, R. Under these conditions, however, membranes may porate and disintegrate. The critical poration voltage Vc for lipid membranes is around 1V 2,21,22 and Vc ⇠ 4 8V for polymer membranes 12,23,24 . The corresponding critical electric field can be estimated using Eq.[3], Ec = 2Vc /3a. For a typical vesicle size, a = 20µm, a lipid membrane may porate above field strengths of about 30 kV /m. Even if the applied file E0 is above Ec , it takes time for the membrane potential to reach the critical value. We can estimate this time from Eq.[3] by setting Vmm = Vc and q = 0 (the poles are the location of maximum transmembrane potential)
tcrit = tmm ln 1
2Vc . 3aE0
(6)
Hence, the ideal conditions to detect the oblate–prolate deformation are tcrit > tel and tel < tmm . The typical lipid membrane charging timescale is tmm ⇠ 1ms, independent of field strength. In the weak field limit, the electric field is below the critical field for poration (e.g., E0 = 10kV /m). The vesicle deforms on a timescale tel ⇠ 10ms tmm . Increasing the field to E0 = 100kV /m (above Ec ) reduces the electrodeformation timescale tel ⇠ 0.1ms < tmm . This indicates that deformation will occur while the membrane is charging and show dependence on the conductivity ratio. However, according to Eq.[6] the membrane may porate at approximately tcrit ⇠ 0.5ms < tel . This analysis shows that a single pulse is either too weak to observe the oblate-prolate transition, or is too strong and will porate the membrane. 1.3
The oblate-prolate transition: pulse design
The proposed solution for this problem is to use a two-step pulse. First, a strong field is applied with duration t pulse ⇠ tel < tmm in order to observe oblate deformation. To avoid membrane poration, the duration of the first pulse must be shorter than the critical timescale, T1 < tcrit . A step decrease in the field strength, below the critical field strength, will produce a transition to a prolate shape if T2 > tmm . The second field strength must be below the critical field strength E0 < Ec to avoid poration. An example of the oblate-prolate transition observed using this method is shown in Figure 7. To optimize the experimental design, in the next section we investigate this protocol theoretically.
Soft Matter Accepted Manuscript
DOI: 10.1039/C3SM52870G
Page 3 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
2.1
Theoretical Model Evolution equations for shape and transmembrane potential
We extend the solution by Schwalbe et al 1 to describe the deformation of a quasi-spherical vesicle submitted to two square pulses with different strength. Vesicle shape is described by rs = a(1 + s2 P2 (cos q )), where P2 (cos q ) is the second-order Legendre polynomial and s2 quantifies the magnitude of deformation. Deformation is quantified by the aspect ratio (defined as ratio of the axes parallel and perpendicular to the applied field, see Figure 2) D = (1 s2 /2)/(1 + s2 ). Time is scaled by the capacitor time t = ttmm . In this notation a square DC electric pulse with duration T gives rise to a transmembrane potential Vmm (t) = aE0V¯mm (t) cos q , (7) ⇢ 3 t t tend 2 (1 e ) (8) V¯mm (t) = t+t end ¯ t > tend , Vmm (tend )e where tend = T /tmm . The resulting shape evolution is dm
∂ s2 = C(t) ∂t
sh 24 s2 (t), aeex E02 55 + 16Ls
(9)
where Ls = µmm /aµ is the dimensionless membrane viscosity, and sh is the membrane tension. The forcing term due to the electric field is C(t) =
2Q(t tend )]2 55 + 16Ls
2 [Pex (t)
8Pin (t)2
,
(10)
where Pex (t) and Pin (t) are electric field coefficients defined as Pex (t) = Pin (t) =
(1
tend ) + RV¯mm (t) , R+2 tend ) 2V¯mm (t) . R+2
R)Q(t
3Q(t
(11)
Q(t tend ) = 1 if t tend and zero otherwise. The evolution equations for a sequence of pulses are obtained from the above equations in a straightforward manner. In Eq.[9], the tension sh needs to be determined selfconsistently with deformation 25 . Under stress, a quasispherical vesicle deforms by pulling excess area stored in fluctuations. The area constraint requires that the total excess area ¯ is (stored in fluctuations, D f , and systematic deformation, D) ¯ constant. Accordingly, D = D + D f is solved numerically at each time step to determine sh . The area stored in fluctuations is ◆ ✓ 2 jmax + jmax + s¯ h kB T Df = . (12) ln 2k 6 + s¯ h
where jmax = a/d is the ratio of the vesicle radius to the membrane thickness, s¯ h = sh a2 /k is the dimensionless membrane tension, and k is the membrane bending rigidity. The excess area corresponding to the systematic deformation is D¯ = 8p s¯22 /5 (13) where s¯2 corresponds to a quasi-steady shape that satisfies Eq.[9] with ∂ s2 /∂ t = 0. 2.2
Model predicitions
Figure 3 illustrates the vesicle deformation with different conductivity ratios in response to two pulses. Only R < 1 displays the oblate-prolate transition. Furthermore, the relaxation is also affected by R due to the membrane discharging. When the field is turned off, the forcing term C(t) in the shape evolution equation Eq.[9] is C(t) ⇠ (R2 16) exp( t/tmm ). An interesting case arises for R < 4. If the vesicle shape at the end of the pulse is a prolate ellipsoid and the membrane capacitor is fully charged, the forcing term will be negative and the vesicle will be pushed towards an oblate shape due to the electric field created by the discharging membrane. This can be understood physically by considering the charges on the exterior of the vesicle, which relax rapidly into the highly conductive exterior, leaving an excess of charges on the interior of the vesicle membrane. The interior charges on the opposite poles attract each other until they fully relax. In the opposite case, R > 4, the forcing term will be positive and the vesicle will be forced into prolate deformation. However, for a fully charged membrane at steady state, the vesicle is already a prolate. The discharging will only prolong the prolate deformation before relaxing back to the equilibrium spherical shape. Figures 4–6 explore vesicle response to a two-step pulse. Figure 4(a) illustrates the effect of membrane equilibrium tension on vesicle dynamics. The initial tension has little influence on the shape dynamics during the first pulse because the forcing term dominates the relaxation term in Eq.[9]. However, the vesicle transition to prolate shape and its maximum deformation are sensitive to the initial tension. Higher initial tension leads to less elongation, as a result of having less excess area that can be transferred into systematic deformation. Once the field is turned off the vesicle relaxes back to equilibrium in two “steps”: a very fast snap, due to membrane discharging (tmm ⇠ 0.1 1ms), followed by slow contraction back to the quasi-spherical shape ( ts = µ/aseq ⇠ 10 100ms). Figure 4(b) zooms into the relaxation curves. Figure 5 shows that increasing the membrane viscosity slows the dynamics of the vesicle at all stages. There is significantly less deformation during the strong pulse and the vesicle takes longer to relax back to the equilibrium shape. However, 1–9 | 3
Soft Matter Accepted Manuscript
2
Soft Matter
Page 4 of 11 View Article Online
DOI: 10.1039/C3SM52870G
=0.1x10−7 N/m
eq
=1x10−7 N/m
eq
1.3
−7
=10x10
1.2
D e f orm a tio n D
D e f orm a tio n D
1.1
1
0.9
0.8
End Pulse 1 −2
10
End Pulse 2 0
10
2
tim e t / t
10
1
Experimental methods Preparation of giant vesicles
Giant vesicles are prepared from lipids (DOPC) and polyethyleneoxide-b-polybutadiene (PBdn -b-PEOm ) diblock 1–9
End Pulse 1 −2
10
End Pulse 2
0
10
mm
the membrane viscosity has no influence on the maximum deformation. The retraction due to discharging is inhibited by greater membrane viscosity as well. The effect of membrane capacitance is shown in Figure 6. A decrease in membrane capacitance leads to faster membrane charging (and discharging). While the field is on, this leads to a quicker transition from oblate to prolate and less time for the vesicle to deform into an oblate. If the capacitance is small, the membrane can be almost fully charged at the end of the first pulse and it will have to discharge in order to adjust to the weaker field. This effect can be seen in the solid line in Figure 6. The membrane charges rapidly during the strong pulse, and when the field is decreased the vesicle initially continues to deform into an oblate shape because of the stress created from discharging. In comparison, vesicles with high capacitance values quickly turn to prolate deformation because the transmembrane potential matches the fully charged state at the weaker field. After the field is turned off, vesicles with higher capacitance have a longer timescale for discharging because the membrane holds more charge. This creates a more persistent field due to the discharging membrane and a greater retraction.
4|
1.1
0.8
4
R = 2 (c) R = 10. Note the fast retraction at the end of the pulse in cases (a) and (b). Conductivity values (a) lin = 2.5µS/cm, lex = 5µS/cm (b) lin = 5µS/cm, lex = 2.5µS/cm (c) lin = 25µS/cm, lex = 2.5µS/cm. Other parameters are Cm = 0.7µF/cm2 , a = 20µm, µmm = 10 8 N.s/m, and D = 0.34.
3.1
1.2
10
2
tim e t / t
4
10
10
mm
(b)
Fig. 3 Shape evolution for a vesicle with R < 1 and R > 1. (a) R = 0.5 (b)
3
N/m
0.9
1.3
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
eq
1.2
1.1
1 =0.1x10−7 N/m
eq
0.9
−7
=1x10
eq
N/m
−7
0.8
=10x10
eq
End Pulse 2 −2
10
0
10
2
tim e t / t
10
N/m
4
10
mm
Fig. 4 Effect of the membrane initial tension on (a) vesicle deformation and
(b) relaxation (t = 0 is set at the end of the second pulse). The first pulse step is 400kV /m for 20µs and then 40kV /m for 50ms. Excess area is D = 0.39 (lowest tension), D = 0.34, D = 0.28 (highest tension). The other vesicle parameters are constant: lin = 3µS/cm, lex = 10µS/cm , Cm = 0.7µF/cm2 , a = 20µm, µmm = 10 8 N.s/m.
copolymers using the electroformation method 26 . The copolymers were obtained from Polymer Source Inc. (Montreal, Canada) and the lipids were purchased from Avanti. The same procedure as previous work is followed 6 . Briefly, a small volume (10 µl) of the lipid solution is spread on the conductive surfaces of two glass plates coated with indium tin oxide (ITO). Organic solvent is removed using a vacuum, then the chamber is assembled across a 2 mm Teflon spacer, conductive sides facing inwards. The chamber is gently filled with 100 mM sucrose solution, then connected to an alternating current (1.8V-4.0V, 10 Hz frequency) for 1-2h at room temperature. The vesicle solution is removed from the electroswelling chamber and diluted into an isotonic glucose and
Soft Matter Accepted Manuscript
1.3
(a)
R=10 R=2 R=0.5
Page 5 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
s
D e f orm a tio n D
µs=10x10−8 Ns/m µs=50x10−8 Ns/m
1.1
1
0.9
0.8 End Pulse 1
0.7
−2
End Pulse 2
0
10
10
2
tim e t / t
10
4
10
mm
Fig. 5 Membrane viscosity slows the rate of deformation, but does not
change the equilibrium shape. The first pulse step is 400kV /m for 20µs and then 40kV /m for 50ms. The other vesicle parameters are constant: lin = 5µS/cm, lex = 10µS/cm , Cm = 0.7µF/cm2 , a = 20µm, seq = 10 7 N/m, D = 0.34.
C =0.1 µF/cm2 m
1.3
C =0.3 µF/cm2 m
2
C =0.7 µF/cm m
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
1.2
1.2
1.1
Electrodeformation set-up
The electrodeformation experiments were conducted in a chamber (Eppendorf, Germany), which consists of a Teflon frame confined from above and below by two glass slides. A pair of parallel cylindrical electrodes with a radius of 92 µm and a separation of 0.5 mm are fixed to the lower glass. The field is applied using a function generator (Agilent 2300) connected to high voltage amplifier (Matsusada AMT1BG0). The function generator is programmed to apply either a single or two-step pulse using the arbitrary function command. The strength and duration of the pulses are varied by adjusting the signal strength from the function generator. The high voltage amplifier increases voltage amplitude from the function generator by 100 times. The pulse initiated from the function generator triggers the high speed camera (Photron SA1.1). 3.3
Optical microscopy and imaging
Due to the differences in density and refractive index between the sucrose and glucose solutions, the vesicles are stabilized by gravity close to the bottom of the chamber and are observed with phase contrast microscopy. An inverted microscope Axio Observer.A1 (Zeiss, Germany) equipped with 20⇥ or 40⇥ Ph2 objectives is used to image the vesicles. Images are recorded with the camera at 30,000 frames per second. The evolution of the vesicle shape is recorded using the high speed camera and analyzed using in-house image analysis software 27 . Using contour detection, two semi axes ak and a? are determined using the Fourier series expansion, where ak is parallel and a? is normal to the field direction.
1
a a
0.9
0.8
End Pulse 1 −2
10
2
tim e t / t
0 ms
End Pulse 2
0
10
10
0.4 ms
20 ms
50 ms
4
10
mm
Fig. 6 Effect of membrane capacitance. A greater capacitance leads to
slower charging. This causes greater deformation into oblate and a greater effect of membrane discharging. Membrane capacitance is varied from Cm = 0.1, 0.3, 0.7µF/cm2 . The other vesicle parameters are constant: lin = 5µS/cm, lex = 20µS/cm, a = 20µm, seq = 10 7 N/m, D = 0.34, µmm = 1 ⇥ 10 8 N.s/m.
Fig. 7 Transition from an oblate to a prolate spheroidal shape for a DOPC
vesicle. The lipid membrane has capacitance Cm = 0.71µF/cm2 , the vesicle initial radius a=24.5 µm, and the solution conductivities are lin =3 µS/cm, lex =10 µS/cm (R = 0.3). The membrane charging time is tmm =0.95 ms. A two pulse field is applied, pulse 1 is E0 = 400 kV/m with duration 20 µs followed by E0 =20 kV/m strength with duration 50 ms.
4 salt solution. The conductivity of the interior sucrose solution is measured to be in the range of 2 5µS/cm. The conductivity of the exterior glucose solution is varied by adjusting the concentration of NaCl in the range 0-1mM.
Experimental results
In this section, the oblate-prolate transition is demonstrated using a two-step pulse. All experiments are compared to the theory developed in Section 2. The value for the capacitance is taken from Salipante et al. 6 . The initial tension seq is determined from the relaxation curve 1–9 | 5
Soft Matter Accepted Manuscript
3.2
µ =1x10−8 Ns/m
Soft Matter
Page 6 of 11 View Article Online
DOI: 10.1039/C3SM52870G
First, theory and experiment are compared for a single long pulse (Tpulse > tmm ) where the field is below the critical field for poration, E0 < Ec . In such a weak field, the oblate to prolate transition is not observable because tel is longer than tmm (dm > 1, see Section 1.2). The shape evolution of the same vesicle at different field strengths is shown in Figure 8. The experiment is repeated at the new field strength immediately after the vesicle relaxes back to a sphere. The experimental data is fitted with identical set of parameters (membrane capacitance, viscosity, initial tension, and excess area). The theory correctly estimates the faster response as field strength is increased. The fast retraction due to membrane discharging after the field is turned off is clearly noticeable in the higher field strength (the retraction is more clearly seen in Figure 8(b), which zooms on the deformation data after the field is turned off). The theory captures the trend that the magnitude of this retraction increases with field strength. After the membrane is discharged, the vesicle relaxes back to a sphere over approximately the same timescale, ts , which is independent of field strength. 4.2
Two-step pulse
1.15 1.1 1.05 1 0.95
Pulse End
0
10
4
10
10 kV/m 15 kV/m 20 kV/m
1.25 1.2 1.15 1.1 1.05 1 0
10
1–9
2
10 tim e t / t m m
(b) 1.3
0.95
Vesicle poration limits the strength and duration for the first pulse and in general there is only a narrow window of pulse parameters which produce the oblate-prolate transition without poration. In most of our experiments, the first pulse is kept constant at E0 =400 kV/m, for a duration of 20 µs. The field is then decreased to a field strength below the poration threshold and turned off at t=50ms. Examples of the oblate-prolate transition for a lipid (DOPC) and polymer membrane (PS1) are shown in Figure 9 (a). When t/tmm ⇠ 1, both experiment and theory show the vesicle shape changes from an oblate to prolate spheroid. The vesicle and the polymersome were subjected to the same pulse strength and duration and were chosen to have similar size, solution conductivities, and initial tension. The parameters for the fits are given in the caption. Polymersomes have a thicker membrane than DOPC membranes, therefore they have a lower membrane capacitance. This leads to faster membrane charging and a shorter duration of forcing towards oblate deformation. Moreover, the polymer membranes have higher viscosity which reduces the rate of shape change and therefore limits the magnitude of oblate deformation. The slower rate of deformation is observed in the oblate to prolate transition as well as a slower relaxation after the field is 6|
1.2
2
10 tim e t / t m m
4
10
Fig. 8 (a) Prolate deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for a DOPC vesicle subjected to 10,15, and 20 kV/m 50 µs pulse. The vesicle parameters are a = 18µm, Cm =0.71 µF/cm, lin =3 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m, seq =2.4x10 7 N/m, D=0.59. ts = µa/seq ⇠ 0.1s
turned off. The theory is in a good overall agreement with the experiment with the exception of the deformation immediately after the first pulse when the field strength is decreased. This discrepancy may be due to errors in matching the image with the starting time for the pulse or limitations of the theory (for example, the theory assumes that the membrane deformation occurs in the steady Stokes limit). The value of the membrane viscosity extracted from the fit of the experimental data agrees well with published measurements 28 . The comparison between the two relaxation curves shows a more dramatic retraction due to discharging for the lipid membrane, see Figure 9(b). The higher membrane capacitance allows for more charge to be stored and the duration of the retraction, governed by tmm , is longer. In addition, the lower membrane viscosity reduces the viscous resistance to shape
Soft Matter Accepted Manuscript
Weak fields: prolate deformation
1.25 D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
4.1
10 kV/m 15 kV/m 20 kV/m
(a) 1.3
D e f orm a tio n D
(after the membrane has discharged) using Eq.[9] with C(t) = 0. The membrane viscosity and the excess area are used to fit the overall shape evolution curve.
Page 7 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
(a)
−8 µmm=0.3×10 Ns/m
(a) 1.15
−8
µmm=30×10
−7
=0.4×10
eq
−7
=5×10
Ns/m
1.2
eq
N/m
N/m
D e f orm a tio n D
D e f orm a tio n D
1.05 1 0.95 0.9
1.1
1
0.9
0.8
End Pulse 1 −2
0.8
End Pulse 2 0
10
2
10 tim e t / t m m
10
4
10
0.7
End Pulse 1 −2
(b) 1.15
(b)
End Pulse 2 0
10
2
10
tim e t / t
10
mm
1.3
−7
=0.4×10
eq
−7
=5×10 eq
1.25
1.1
4
10
N/m
N/m
1.2
D e f orm a tio n D
1.05 1 0.95 0.9
−8 µmm=0.3×10 Ns/m
µmm=30×10−8 Ns/m 0
10
2
tim e t / t m m
10
change. The effect of initial membrane tension is compared for two DOPC vesicles in Figure 10, which shows that only the dynamics after the field is turned off is very sensitive to the membrane tension. Both vesicles transition from oblate to prolate at approximately the same time. This is a result of the similar membrane charging timescales. The low tension vesicle deforms slightly more than the high tension membrane, but the effect of membrane tension is more noticeable during the relaxation, see Figure 10(b). Both vesicles retract a similar amount during membrane discharging because the corresponding timescale tmm and conductivity conditions are similar in the two vesicles. The high tension vesicle relaxes back to a spherical shape in about 102 ⇥ tmm while the low tension membrane takes almost 104 ⇥ tmm to fully relax to a sphere.
1.1 1.05 1
4
10
Fig. 9 (a)Deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for a vesicle ( ) and a polymersome (⇤) . An initial pulse of strength 400 kV/m for 20 µs is applied, and then the field strength is decreased to 20 kV/m for 50 ms. The DOPC vesicle parameters are a = 20.2µm, Cm =0.71 µF/cm, lin =2 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m. seq =4x10 7 N/m, D=0.27. The polymer vesicle parameters are a = 27.2µm, Cm =0.27 µF/cm, lin =2 µS/cm, lex =10 µS/cm, µmm =25 x10 8 N.s/m. seq =2.0x10 7 N/m, D=0.22.
1.15
0.95 0.9 −2 10
−1
10
0
10
1
2
10
tim e t / t
10
3
10
4
10
mm
Fig. 10 (a) Oblate to prolate deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for two different DOPC vesicle exposed to the same two-step pulse with different initial tensions. The vesicle with a higher initial tension is denoted by squares (⇤) and the vesicle with a lower initial tension is denoted by circles ( ). An initial field of strength 400 kV/m for 20 µis applied, the field strength is decreased to 20 kV/m for 50 ms. The high tension DOPC vesicle parameters are radius a = 19.9 µm, Cm =0.71 µF/cm, lin =2.5 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m. seq =5x10 7 N/m, D=0.27. The low tension DOPC vesicle parameters, radius a = 20.9 µm, Cm =0.71 µF/cm, lin =2.5 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m, seq =0.4x10 7 N/m, D=0.30.
4.3
Phase diagram
The vesicle response to a two step pulse is summarized in a phase diagram shown in Figure 11. The first pulse is varied in strength and duration while the second pulse is the same for all experiments. The second pulse is below the critical field (E0 < Ec ) with a duration much longer than tmm . The results show that the pulse duration leading to crossover from prolate-only to oblate-prolate behavior correlates with the tel timescale. If the entire membrane became perfectly con1–9 | 7
Soft Matter Accepted Manuscript
0.85
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
1.1
Soft Matter
Page 8 of 11 View Article Online
DOI: 10.1039/C3SM52870G
0
t p uls e / t m m
10
6
−1
10
Acknowledgement
This work was partially supported by NSF grants CBET1117099 and CMMI-1232477. −2
10
0
2
4
6
E (k V / c m)
8
10
0
Fig. 11 Vesicle behavior as a function of the duration and field strength of
the first pulse, the duration of the first pulse is normalized by the capacitor charging time tmm . The second pulse is set at E0 = 20kV /m for 50µs. Three outcomes are observed: only prolate deformation, oblate-to-prolate transition, and only oblate deformation. The oblate only deformation occurs when the vesicle has porated the membrane capacitor is short circuited. The dashed line corresponds to tel and the solid line denotes tcrit evaluated at Ec = 1kV /cm. tmm = 0.5ms.
ducting due to poration (but poration does not destabilize the membrane), there should only be oblate deformation because the membrane capacitor is short-circuited. The tcrit timescale dictates the crossover between oblate-prolate and oblate-only behavior. A critical field strength of Ec = 1kV/cm, which corresponds to a Vc = 3V for a 20 micron vesicle, is used to estimate tcrit . Note that tcrit is determined from the transmembrane potential at the pole of the vesicle where the critical transmembrane potential is reached first. A stronger field is required to produce transmembrane conduction over enough membrane area such that the vesicle will not return to prolate.
5
Conclusions
This paper provides a quantitative comparison between theory and experiment for the deformation of giant quasi–spherical vesicles in DC fields. Upon application of a single square pulse, a vesicle either deforms into a prolate spheroid or undergoes poration. In a two-pulse sequence, the vesicle can first adopt oblate shape if the interior solution is less conducting than the exterior. We show that the first pulse should have duration shorter than the membrane charging time and strength above the poration threshold, while the second pulse must be longer and weaker. After the field is turned off, the vesicle undergoes a fast retraction associated with membrane discharge followed by much slower relaxation to equilibrium spherical shape driven by tension. Our results provide useful physical insights into the interac8|
1–9
References 1 J. Schwalbe, P. M. Vlahovska, and M. Miksis. Vesicle electrohydrodynamics. Phys. Rev E, 83:046309, 2011. 2 R. Dimova, N. Bezlyepkina, M. D. Jordo, R. L. Knorr, K. A. Riske, M. Staykova, P. M. Vlahovska, T. Yamamoto, P. Yang, and R. Lipowsky. Vesicles in electric fields: Some novel aspects of membrane behavior. Soft Matter, 5:3201 – 3212, 2009. 3 P. M. Vlahovska. Non-equilibrium dynamics of lipid membranes: deformation and stability in electric fields. In A. Iglic, editor, Advances in Planar Lipid Bilayers and Liposomes, vol. 12, pages 103–146. Elsevier, 2010. 4 F. Ziebert and D. Lacoste. A planar lipid bilayer in an electric field: membrane instability, flow field and electrical impedance. In A. Iglic, editor, Advances in Planar Lipid Bilayers and Liposomes, vol. 14, pages 63–95. Elsevier, 2011. 5 S. Aranda, K. A. Riske, R. Lipowsky, and R. Dimova. Morphological transitions of vesicles induced by ac electric fields. Biophys. J., 95:L19–L21, 2008. 6 P. F. Salipante, R. Knorr, R. Dimova, and P. M. Vlahovska. Electrodeformation method for measuring the capacitance of bilayer membranes. Soft Matter, 8:3810– 3816, 2012. 7 P. M. Vlahovska, R. S. Gracia, S. Aranda-Espinoza, and R. Dimova. Electrohydrodynamic model of vesicle deformation in alternating electric fields. Biophys. J., 96:4789– 4803, 2009. 8 T. Yamamoto, S. Aranda-Espinoza, R. Dimova, and R. Lipowsky. Stability of spherical vesicles in electric fields. Langmuir, 26:12390–12407, 2010. 9 P. Peterlin. Frequency-dependent electrodeformation of giant phospholipid vesicles in ac electric field. J. Biol. Phys., 36:339–354, 2010. 10 H. Nganguia and Y. N. Young. Ellipsoidal shapes of vesicles in ac electric fields. submitted, 2013. 11 J. Zhang, J. D. Zahn, W. Tan, and H. Lin. A transient solution for vesicle electrodeformation and relaxation. Phys. Fluids, 25:071903, 2013.
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
tion of biomembranes and pulsed electric fields used in electroporation. Moreover, the sensitivity of the dynamics to the membrane viscosity suggests that the two-pulse experiments can serve as a new method to measure this membrane property.
Oblate only Oblate−Prolate Prolate only
Page 9 of 11
Soft Matter View Article Online
12 K. A. Riske and R. Dimova. Electro-deformation and poration of giant vesicles viewed with high temporal resolution. Biophys. J., 88:1143–1155, 2005. 13 K. A. Riske and R. Dimova. Electric pulses induce cylindrical deformations on giant vesicles in salt solutions. Biophys. J., 91:1778–1786, 2006. 14 K.A. Riske, R.L. Knorr, and R. Dimova. Bursting of charged multicomponent vesicles subjected to electric pulses. Soft Matter, 5:1983–1986, 2009. 15 M. M. Sadik, J. B. Li, J. W. Shan, D. I. Shreiber, and H. Lin. Vesicle deformation and poration under strong dc electric fields. Phys. Rev. E., 83:066316, 2011. 16 C. Grosse and H. P. Schwan. Cellular membrane potentials induced by alternating fields. Biophys. J., 63:1632–1642, 1992. 17 H. P. Schwan. Dielectrophoresis and rotation of cells. In E. Neumann, A. E. Sowers, and C. A. Jordan, editors, Electroporation and electrofusion in cell biology, pages 3– 21. Plenum Press, 1989. 18 K. Kinosita Jr., I. Ashikawa, N. Saita, H. Yoshimura, H. Itoh, K. Nagayama, and A. Ikegami. Electroporation of cell membrane visualized under a pulsed laser fluorescence microscope. Biophys. J., 53:1015–1019, 1988. 19 L. C. McConnell, M. J. Miksis, and P. M. Vlahovska. Vesicle electrohydrodynamics in dc electric fields. IMA J. Appl. Math., 78:797–817, 2013. 20 J. Seiwert, M. J. Miksis, and P. M. Vlahovska. Stability of biomimetic membranes in dc electric fields. J. Fluid Mech., 706:58–70, 2012. 21 D. Needham and R. M. Hochmuth. Electromechanical permeabilization of lipid vesicles. role of membrane tension and compressibility. Biophys. J., 55:1001–1009, 1989. 22 T. Portet and R. Dimova. A new method for measuring edge tensions and stability of lipid bilayers: Effect of membrane composition. Biophys. J., 99:3264–3273, 2010. 23 H. Bermudez, H. Aranda-Espinoza, D. A. Hammer, and D. E. Discher. Pore stability and dynamics in polymer membranes. Europhys. Lett., 64:550–556, 2003. 24 H. Aranda-Espinoza, D. A. Hammer, and D. E. Discher. Electromechanical limits of polymersomes. Phys. Rev. Lett., 87:208301, 2001. 25 U. Seifert. Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles. Eur. Phys. J. B, 8:405–415, 1999. 26 M. I. Angelova and D. S. Dimitrov. Liposome electroformation. Faraday Discuss. Chem. Soc., 81:303– 311, 1986. 27 R. S. Gracia, N. Bezlyepkina, R. L. Knorr, R. L. Lipowsky, and R. Dimova. Effect of cholesterol on the rigidity of saturated and unsaturated membranes: fluctuation and electrodeformation analysis of giant vesicles. Soft Matter, 6:1472 – 1482, 2010.
28 R. Dimova, C. Dietrich, A. Hadjiisky, K. Danov, and B. Pouligny. Falling ball viscosimetry of giant vesicle membranes: finite-size effects. Eur. Phys. J. B, 12(4):589– 598, 1999.
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
DOI: 10.1039/C3SM52870G
1–9 | 9
Soft Matter
Page 10 of 11 View Article Online
DOI: 10.1039/C3SM52870G
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
A two-step-pulse reveals the transition from oblate to prolate ellipsoidal shape for giant vesicles in DC electric fields.
Page 11 of 11
Soft Matter
a
Soft Matter Accepted Manuscript
a
Ε
0 ms
0.4 ms
20 ms
50 ms
Soft Matter Accepted Manuscript
This article can be cited before page numbers have been issued, to do this please use: P. F. Salipante and P. Vlahovska, Soft Matter, 2014, DOI: 10.1039/C3SM52870G.
This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.
www.rsc.org/softmatter
Page 1 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
Vesicle deformation in DC electric pulses Paul F. Salipantea and Petia M. Vlahovska⇤a First published on the web Xth XXXXXXXXXX 200X DOI: 10.1039/b000000x
The transient deformation of giant vesicles in square DC electric pulses is investigated. We experimentally observe the theoretically predicted transition from an oblate to prolate ellipsoidal shape in the case of a quasi-spherical vesicle encapsulating solution less conducting than the suspending medium 1 . The transition is detected by utilizing a two-step pulse in order to avoid electroporation and vesicle collapse. We develop a theoretical model to describe both the deformation under electric field and relaxation after the field is turned off. Agreement between experiment and theory demonstrates that the time-dependent vesicle shape can be used to measure membrane properties such as viscosity and capacitance.
1
Introduction
The electro-deformation of giant vesicles made of artificial bilayer membranes provides fundamental insights into the electromechanics of biomembranes, namely the coupling of membrane shape and transmembrane potential 2–4 . The steady shapes of vesicles in AC fields have been extensively studied experimentally 5,6 and theoretically 7–10 . The behavior in DC pulses has beed considered theoretically 1,11 , but only limited experimental data is available due to problems with visualization and vesicle fragility 12–15 . In this work we perform a systematic investigation of vesicle dynamics in square DC pulses. A theoretical model is developed to describe both the deformation under electric field and relaxation after the field is turned off. We show that the data can yield information about membrane properties such as viscosity, bending rigidity, and capacitance. Moreover, we confirm the recently predicted oblate-prolate shape transition during the pulse application for a vesicle filled with solution with lower salt content than the suspending liquid 1 . 1.1
Physical picture
Let us consider a vesicle made of an ion-impermeable chargefree lipid bilayer membrane with dielectric constant, emm , viscosity µmm , and thickness, d. The solutions inside and outside the vesicle have different conductivities, lin and lex (e.g., due to different salt concentrations). This mismatch in the bulk fluid electrical properties is characterized by the conductivity ratio lin R= . (1) lex a School of Engineering, Brown University, Providence RI 02912, USA. Fax: ++1 401 8639028; Tel: ++1 401 8639774; E-mail: petia [email protected]
The solutions’ viscosities are nearly the same (basically equal to the viscosity of water µ). Moreover, their dielectric constants are also very close and for simplicity we assume that ein = eex = e of water. Application of an electric field leads to accumulation of ions at the membrane surfaces and the vesicle will act as a capacitor, see Figure 1. For a quasi–spherical vesicle made of an insulating (ion-impermeable) membrane the characteristic time scale for the charging process is 16–18 , ◆ ✓ R aCmm , (2) 1+ tmm = lin 2 where Cmm = emm /d is the capacitance of the membrane and a is the vesicle radius. For systems with typical solution conductivity l ⇠ 0.1µS/cm and a ⇠ 10µm the charging timescale is tmm ⇠ 1ms. Membrane capacitance sets up a transmembrane potential, which in an uniform DC electric field, E• = E0 zˆ , evolves toward steady state as 1 3 Vmm = aE0 (1 2
exp ( t/tmm )) cos(q ) ,
(3)
where q is the angle between the electric field direction and position at the interface. During the charging process, t < tmm , the ion densities on the inner and outer membrane surfaces may become temporarily unbalanced. This is due to the difference in the conduction rates in the inner and suspending solutions, which is characterized by R. The polarity of the apparent charge (and resulting vesicle deformation) is illustrated in Figure 1. At steady state however, t tmm , irrespective of R the capacitor is fully charged, the apparent charge is zero, and the electric field is expelled from the vesicle interior. The physical picture suggests that a vesicle with R < 1 may initially deform into an oblate spheroid but eventually adopts 1–9 | 1
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
Soft Matter
Page 2 of 11 View Article Online
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
a)
b)
c)
Fig. 1 Sketch of the electric field and induced charge distribution around a
vesicle immersed in an electrolyte solution, following the imposition of a uniform DC field (a) t < tmm and R > 1 (b) t < tmm and R < 1 (c) fully charged membrane capacitor t tmm , any R. The dashed lines indicate the vesicle deformation.
a prolate shape 1 (see Figure 2). This theoretical prediction has not been confirmed experimentally, although it has been observed in numerical simulations 19 . In this work we experimentally investigate the conditions for this transition to occur. The challenge is to ensure that the transient oblate deformation is visible yet the field is not strong enough to cause vesicle poration. Poration makes the membrane conducting, which suppresses the prolate deformation for R < 1 19 and destabilizes the membrane 20 . In the next section we discuss the time scales relevant for the design of the experiment.
a
a a
a
Fig. 2 ak and a? are the axes parallel and perpendicular to the symmetry
axis respectively. Deformation is shown for prolate (left) and oblate (right). The deformation parameter is defined as D = ak /a?
1.2
Timescales
The electric stresses act to deform the vesicle on a typical time scale µ . (4) tel = eE02 An important ratio compares the rate of deformation, characterized by tel , to the membrane charging time, dm = tel /tmm .
(5)
As discussed in the previous section, the vesicle exists in the oblate shape while t < tmm . However, significant deformation implies t > tel . Hence, in order to detect the oblate shape dm must be less than 1 (in addition to R < 1). 2|
1–9
The electrodeformation timescale varies with field strength. A weak field condition can be defined as tmm < tel (dm > 1). In this limit, deformation occurs after the membrane has fully charged resulting in only prolate shape (independent of R). In strong fields, tmm > tel , the vesicle is charging and simultaneously deforming; the shape evolution depends on the conductivity ratio, R. Under these conditions, however, membranes may porate and disintegrate. The critical poration voltage Vc for lipid membranes is around 1V 2,21,22 and Vc ⇠ 4 8V for polymer membranes 12,23,24 . The corresponding critical electric field can be estimated using Eq.[3], Ec = 2Vc /3a. For a typical vesicle size, a = 20µm, a lipid membrane may porate above field strengths of about 30 kV /m. Even if the applied file E0 is above Ec , it takes time for the membrane potential to reach the critical value. We can estimate this time from Eq.[3] by setting Vmm = Vc and q = 0 (the poles are the location of maximum transmembrane potential)
tcrit = tmm ln 1
2Vc . 3aE0
(6)
Hence, the ideal conditions to detect the oblate–prolate deformation are tcrit > tel and tel < tmm . The typical lipid membrane charging timescale is tmm ⇠ 1ms, independent of field strength. In the weak field limit, the electric field is below the critical field for poration (e.g., E0 = 10kV /m). The vesicle deforms on a timescale tel ⇠ 10ms tmm . Increasing the field to E0 = 100kV /m (above Ec ) reduces the electrodeformation timescale tel ⇠ 0.1ms < tmm . This indicates that deformation will occur while the membrane is charging and show dependence on the conductivity ratio. However, according to Eq.[6] the membrane may porate at approximately tcrit ⇠ 0.5ms < tel . This analysis shows that a single pulse is either too weak to observe the oblate-prolate transition, or is too strong and will porate the membrane. 1.3
The oblate-prolate transition: pulse design
The proposed solution for this problem is to use a two-step pulse. First, a strong field is applied with duration t pulse ⇠ tel < tmm in order to observe oblate deformation. To avoid membrane poration, the duration of the first pulse must be shorter than the critical timescale, T1 < tcrit . A step decrease in the field strength, below the critical field strength, will produce a transition to a prolate shape if T2 > tmm . The second field strength must be below the critical field strength E0 < Ec to avoid poration. An example of the oblate-prolate transition observed using this method is shown in Figure 7. To optimize the experimental design, in the next section we investigate this protocol theoretically.
Soft Matter Accepted Manuscript
DOI: 10.1039/C3SM52870G
Page 3 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
2.1
Theoretical Model Evolution equations for shape and transmembrane potential
We extend the solution by Schwalbe et al 1 to describe the deformation of a quasi-spherical vesicle submitted to two square pulses with different strength. Vesicle shape is described by rs = a(1 + s2 P2 (cos q )), where P2 (cos q ) is the second-order Legendre polynomial and s2 quantifies the magnitude of deformation. Deformation is quantified by the aspect ratio (defined as ratio of the axes parallel and perpendicular to the applied field, see Figure 2) D = (1 s2 /2)/(1 + s2 ). Time is scaled by the capacitor time t = ttmm . In this notation a square DC electric pulse with duration T gives rise to a transmembrane potential Vmm (t) = aE0V¯mm (t) cos q , (7) ⇢ 3 t t tend 2 (1 e ) (8) V¯mm (t) = t+t end ¯ t > tend , Vmm (tend )e where tend = T /tmm . The resulting shape evolution is dm
∂ s2 = C(t) ∂t
sh 24 s2 (t), aeex E02 55 + 16Ls
(9)
where Ls = µmm /aµ is the dimensionless membrane viscosity, and sh is the membrane tension. The forcing term due to the electric field is C(t) =
2Q(t tend )]2 55 + 16Ls
2 [Pex (t)
8Pin (t)2
,
(10)
where Pex (t) and Pin (t) are electric field coefficients defined as Pex (t) = Pin (t) =
(1
tend ) + RV¯mm (t) , R+2 tend ) 2V¯mm (t) . R+2
R)Q(t
3Q(t
(11)
Q(t tend ) = 1 if t tend and zero otherwise. The evolution equations for a sequence of pulses are obtained from the above equations in a straightforward manner. In Eq.[9], the tension sh needs to be determined selfconsistently with deformation 25 . Under stress, a quasispherical vesicle deforms by pulling excess area stored in fluctuations. The area constraint requires that the total excess area ¯ is (stored in fluctuations, D f , and systematic deformation, D) ¯ constant. Accordingly, D = D + D f is solved numerically at each time step to determine sh . The area stored in fluctuations is ◆ ✓ 2 jmax + jmax + s¯ h kB T Df = . (12) ln 2k 6 + s¯ h
where jmax = a/d is the ratio of the vesicle radius to the membrane thickness, s¯ h = sh a2 /k is the dimensionless membrane tension, and k is the membrane bending rigidity. The excess area corresponding to the systematic deformation is D¯ = 8p s¯22 /5 (13) where s¯2 corresponds to a quasi-steady shape that satisfies Eq.[9] with ∂ s2 /∂ t = 0. 2.2
Model predicitions
Figure 3 illustrates the vesicle deformation with different conductivity ratios in response to two pulses. Only R < 1 displays the oblate-prolate transition. Furthermore, the relaxation is also affected by R due to the membrane discharging. When the field is turned off, the forcing term C(t) in the shape evolution equation Eq.[9] is C(t) ⇠ (R2 16) exp( t/tmm ). An interesting case arises for R < 4. If the vesicle shape at the end of the pulse is a prolate ellipsoid and the membrane capacitor is fully charged, the forcing term will be negative and the vesicle will be pushed towards an oblate shape due to the electric field created by the discharging membrane. This can be understood physically by considering the charges on the exterior of the vesicle, which relax rapidly into the highly conductive exterior, leaving an excess of charges on the interior of the vesicle membrane. The interior charges on the opposite poles attract each other until they fully relax. In the opposite case, R > 4, the forcing term will be positive and the vesicle will be forced into prolate deformation. However, for a fully charged membrane at steady state, the vesicle is already a prolate. The discharging will only prolong the prolate deformation before relaxing back to the equilibrium spherical shape. Figures 4–6 explore vesicle response to a two-step pulse. Figure 4(a) illustrates the effect of membrane equilibrium tension on vesicle dynamics. The initial tension has little influence on the shape dynamics during the first pulse because the forcing term dominates the relaxation term in Eq.[9]. However, the vesicle transition to prolate shape and its maximum deformation are sensitive to the initial tension. Higher initial tension leads to less elongation, as a result of having less excess area that can be transferred into systematic deformation. Once the field is turned off the vesicle relaxes back to equilibrium in two “steps”: a very fast snap, due to membrane discharging (tmm ⇠ 0.1 1ms), followed by slow contraction back to the quasi-spherical shape ( ts = µ/aseq ⇠ 10 100ms). Figure 4(b) zooms into the relaxation curves. Figure 5 shows that increasing the membrane viscosity slows the dynamics of the vesicle at all stages. There is significantly less deformation during the strong pulse and the vesicle takes longer to relax back to the equilibrium shape. However, 1–9 | 3
Soft Matter Accepted Manuscript
2
Soft Matter
Page 4 of 11 View Article Online
DOI: 10.1039/C3SM52870G
=0.1x10−7 N/m
eq
=1x10−7 N/m
eq
1.3
−7
=10x10
1.2
D e f orm a tio n D
D e f orm a tio n D
1.1
1
0.9
0.8
End Pulse 1 −2
10
End Pulse 2 0
10
2
tim e t / t
10
1
Experimental methods Preparation of giant vesicles
Giant vesicles are prepared from lipids (DOPC) and polyethyleneoxide-b-polybutadiene (PBdn -b-PEOm ) diblock 1–9
End Pulse 1 −2
10
End Pulse 2
0
10
mm
the membrane viscosity has no influence on the maximum deformation. The retraction due to discharging is inhibited by greater membrane viscosity as well. The effect of membrane capacitance is shown in Figure 6. A decrease in membrane capacitance leads to faster membrane charging (and discharging). While the field is on, this leads to a quicker transition from oblate to prolate and less time for the vesicle to deform into an oblate. If the capacitance is small, the membrane can be almost fully charged at the end of the first pulse and it will have to discharge in order to adjust to the weaker field. This effect can be seen in the solid line in Figure 6. The membrane charges rapidly during the strong pulse, and when the field is decreased the vesicle initially continues to deform into an oblate shape because of the stress created from discharging. In comparison, vesicles with high capacitance values quickly turn to prolate deformation because the transmembrane potential matches the fully charged state at the weaker field. After the field is turned off, vesicles with higher capacitance have a longer timescale for discharging because the membrane holds more charge. This creates a more persistent field due to the discharging membrane and a greater retraction.
4|
1.1
0.8
4
R = 2 (c) R = 10. Note the fast retraction at the end of the pulse in cases (a) and (b). Conductivity values (a) lin = 2.5µS/cm, lex = 5µS/cm (b) lin = 5µS/cm, lex = 2.5µS/cm (c) lin = 25µS/cm, lex = 2.5µS/cm. Other parameters are Cm = 0.7µF/cm2 , a = 20µm, µmm = 10 8 N.s/m, and D = 0.34.
3.1
1.2
10
2
tim e t / t
4
10
10
mm
(b)
Fig. 3 Shape evolution for a vesicle with R < 1 and R > 1. (a) R = 0.5 (b)
3
N/m
0.9
1.3
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
eq
1.2
1.1
1 =0.1x10−7 N/m
eq
0.9
−7
=1x10
eq
N/m
−7
0.8
=10x10
eq
End Pulse 2 −2
10
0
10
2
tim e t / t
10
N/m
4
10
mm
Fig. 4 Effect of the membrane initial tension on (a) vesicle deformation and
(b) relaxation (t = 0 is set at the end of the second pulse). The first pulse step is 400kV /m for 20µs and then 40kV /m for 50ms. Excess area is D = 0.39 (lowest tension), D = 0.34, D = 0.28 (highest tension). The other vesicle parameters are constant: lin = 3µS/cm, lex = 10µS/cm , Cm = 0.7µF/cm2 , a = 20µm, µmm = 10 8 N.s/m.
copolymers using the electroformation method 26 . The copolymers were obtained from Polymer Source Inc. (Montreal, Canada) and the lipids were purchased from Avanti. The same procedure as previous work is followed 6 . Briefly, a small volume (10 µl) of the lipid solution is spread on the conductive surfaces of two glass plates coated with indium tin oxide (ITO). Organic solvent is removed using a vacuum, then the chamber is assembled across a 2 mm Teflon spacer, conductive sides facing inwards. The chamber is gently filled with 100 mM sucrose solution, then connected to an alternating current (1.8V-4.0V, 10 Hz frequency) for 1-2h at room temperature. The vesicle solution is removed from the electroswelling chamber and diluted into an isotonic glucose and
Soft Matter Accepted Manuscript
1.3
(a)
R=10 R=2 R=0.5
Page 5 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
s
D e f orm a tio n D
µs=10x10−8 Ns/m µs=50x10−8 Ns/m
1.1
1
0.9
0.8 End Pulse 1
0.7
−2
End Pulse 2
0
10
10
2
tim e t / t
10
4
10
mm
Fig. 5 Membrane viscosity slows the rate of deformation, but does not
change the equilibrium shape. The first pulse step is 400kV /m for 20µs and then 40kV /m for 50ms. The other vesicle parameters are constant: lin = 5µS/cm, lex = 10µS/cm , Cm = 0.7µF/cm2 , a = 20µm, seq = 10 7 N/m, D = 0.34.
C =0.1 µF/cm2 m
1.3
C =0.3 µF/cm2 m
2
C =0.7 µF/cm m
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
1.2
1.2
1.1
Electrodeformation set-up
The electrodeformation experiments were conducted in a chamber (Eppendorf, Germany), which consists of a Teflon frame confined from above and below by two glass slides. A pair of parallel cylindrical electrodes with a radius of 92 µm and a separation of 0.5 mm are fixed to the lower glass. The field is applied using a function generator (Agilent 2300) connected to high voltage amplifier (Matsusada AMT1BG0). The function generator is programmed to apply either a single or two-step pulse using the arbitrary function command. The strength and duration of the pulses are varied by adjusting the signal strength from the function generator. The high voltage amplifier increases voltage amplitude from the function generator by 100 times. The pulse initiated from the function generator triggers the high speed camera (Photron SA1.1). 3.3
Optical microscopy and imaging
Due to the differences in density and refractive index between the sucrose and glucose solutions, the vesicles are stabilized by gravity close to the bottom of the chamber and are observed with phase contrast microscopy. An inverted microscope Axio Observer.A1 (Zeiss, Germany) equipped with 20⇥ or 40⇥ Ph2 objectives is used to image the vesicles. Images are recorded with the camera at 30,000 frames per second. The evolution of the vesicle shape is recorded using the high speed camera and analyzed using in-house image analysis software 27 . Using contour detection, two semi axes ak and a? are determined using the Fourier series expansion, where ak is parallel and a? is normal to the field direction.
1
a a
0.9
0.8
End Pulse 1 −2
10
2
tim e t / t
0 ms
End Pulse 2
0
10
10
0.4 ms
20 ms
50 ms
4
10
mm
Fig. 6 Effect of membrane capacitance. A greater capacitance leads to
slower charging. This causes greater deformation into oblate and a greater effect of membrane discharging. Membrane capacitance is varied from Cm = 0.1, 0.3, 0.7µF/cm2 . The other vesicle parameters are constant: lin = 5µS/cm, lex = 20µS/cm, a = 20µm, seq = 10 7 N/m, D = 0.34, µmm = 1 ⇥ 10 8 N.s/m.
Fig. 7 Transition from an oblate to a prolate spheroidal shape for a DOPC
vesicle. The lipid membrane has capacitance Cm = 0.71µF/cm2 , the vesicle initial radius a=24.5 µm, and the solution conductivities are lin =3 µS/cm, lex =10 µS/cm (R = 0.3). The membrane charging time is tmm =0.95 ms. A two pulse field is applied, pulse 1 is E0 = 400 kV/m with duration 20 µs followed by E0 =20 kV/m strength with duration 50 ms.
4 salt solution. The conductivity of the interior sucrose solution is measured to be in the range of 2 5µS/cm. The conductivity of the exterior glucose solution is varied by adjusting the concentration of NaCl in the range 0-1mM.
Experimental results
In this section, the oblate-prolate transition is demonstrated using a two-step pulse. All experiments are compared to the theory developed in Section 2. The value for the capacitance is taken from Salipante et al. 6 . The initial tension seq is determined from the relaxation curve 1–9 | 5
Soft Matter Accepted Manuscript
3.2
µ =1x10−8 Ns/m
Soft Matter
Page 6 of 11 View Article Online
DOI: 10.1039/C3SM52870G
First, theory and experiment are compared for a single long pulse (Tpulse > tmm ) where the field is below the critical field for poration, E0 < Ec . In such a weak field, the oblate to prolate transition is not observable because tel is longer than tmm (dm > 1, see Section 1.2). The shape evolution of the same vesicle at different field strengths is shown in Figure 8. The experiment is repeated at the new field strength immediately after the vesicle relaxes back to a sphere. The experimental data is fitted with identical set of parameters (membrane capacitance, viscosity, initial tension, and excess area). The theory correctly estimates the faster response as field strength is increased. The fast retraction due to membrane discharging after the field is turned off is clearly noticeable in the higher field strength (the retraction is more clearly seen in Figure 8(b), which zooms on the deformation data after the field is turned off). The theory captures the trend that the magnitude of this retraction increases with field strength. After the membrane is discharged, the vesicle relaxes back to a sphere over approximately the same timescale, ts , which is independent of field strength. 4.2
Two-step pulse
1.15 1.1 1.05 1 0.95
Pulse End
0
10
4
10
10 kV/m 15 kV/m 20 kV/m
1.25 1.2 1.15 1.1 1.05 1 0
10
1–9
2
10 tim e t / t m m
(b) 1.3
0.95
Vesicle poration limits the strength and duration for the first pulse and in general there is only a narrow window of pulse parameters which produce the oblate-prolate transition without poration. In most of our experiments, the first pulse is kept constant at E0 =400 kV/m, for a duration of 20 µs. The field is then decreased to a field strength below the poration threshold and turned off at t=50ms. Examples of the oblate-prolate transition for a lipid (DOPC) and polymer membrane (PS1) are shown in Figure 9 (a). When t/tmm ⇠ 1, both experiment and theory show the vesicle shape changes from an oblate to prolate spheroid. The vesicle and the polymersome were subjected to the same pulse strength and duration and were chosen to have similar size, solution conductivities, and initial tension. The parameters for the fits are given in the caption. Polymersomes have a thicker membrane than DOPC membranes, therefore they have a lower membrane capacitance. This leads to faster membrane charging and a shorter duration of forcing towards oblate deformation. Moreover, the polymer membranes have higher viscosity which reduces the rate of shape change and therefore limits the magnitude of oblate deformation. The slower rate of deformation is observed in the oblate to prolate transition as well as a slower relaxation after the field is 6|
1.2
2
10 tim e t / t m m
4
10
Fig. 8 (a) Prolate deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for a DOPC vesicle subjected to 10,15, and 20 kV/m 50 µs pulse. The vesicle parameters are a = 18µm, Cm =0.71 µF/cm, lin =3 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m, seq =2.4x10 7 N/m, D=0.59. ts = µa/seq ⇠ 0.1s
turned off. The theory is in a good overall agreement with the experiment with the exception of the deformation immediately after the first pulse when the field strength is decreased. This discrepancy may be due to errors in matching the image with the starting time for the pulse or limitations of the theory (for example, the theory assumes that the membrane deformation occurs in the steady Stokes limit). The value of the membrane viscosity extracted from the fit of the experimental data agrees well with published measurements 28 . The comparison between the two relaxation curves shows a more dramatic retraction due to discharging for the lipid membrane, see Figure 9(b). The higher membrane capacitance allows for more charge to be stored and the duration of the retraction, governed by tmm , is longer. In addition, the lower membrane viscosity reduces the viscous resistance to shape
Soft Matter Accepted Manuscript
Weak fields: prolate deformation
1.25 D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
4.1
10 kV/m 15 kV/m 20 kV/m
(a) 1.3
D e f orm a tio n D
(after the membrane has discharged) using Eq.[9] with C(t) = 0. The membrane viscosity and the excess area are used to fit the overall shape evolution curve.
Page 7 of 11
Soft Matter View Article Online
DOI: 10.1039/C3SM52870G
(a)
−8 µmm=0.3×10 Ns/m
(a) 1.15
−8
µmm=30×10
−7
=0.4×10
eq
−7
=5×10
Ns/m
1.2
eq
N/m
N/m
D e f orm a tio n D
D e f orm a tio n D
1.05 1 0.95 0.9
1.1
1
0.9
0.8
End Pulse 1 −2
0.8
End Pulse 2 0
10
2
10 tim e t / t m m
10
4
10
0.7
End Pulse 1 −2
(b) 1.15
(b)
End Pulse 2 0
10
2
10
tim e t / t
10
mm
1.3
−7
=0.4×10
eq
−7
=5×10 eq
1.25
1.1
4
10
N/m
N/m
1.2
D e f orm a tio n D
1.05 1 0.95 0.9
−8 µmm=0.3×10 Ns/m
µmm=30×10−8 Ns/m 0
10
2
tim e t / t m m
10
change. The effect of initial membrane tension is compared for two DOPC vesicles in Figure 10, which shows that only the dynamics after the field is turned off is very sensitive to the membrane tension. Both vesicles transition from oblate to prolate at approximately the same time. This is a result of the similar membrane charging timescales. The low tension vesicle deforms slightly more than the high tension membrane, but the effect of membrane tension is more noticeable during the relaxation, see Figure 10(b). Both vesicles retract a similar amount during membrane discharging because the corresponding timescale tmm and conductivity conditions are similar in the two vesicles. The high tension vesicle relaxes back to a spherical shape in about 102 ⇥ tmm while the low tension membrane takes almost 104 ⇥ tmm to fully relax to a sphere.
1.1 1.05 1
4
10
Fig. 9 (a)Deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for a vesicle ( ) and a polymersome (⇤) . An initial pulse of strength 400 kV/m for 20 µs is applied, and then the field strength is decreased to 20 kV/m for 50 ms. The DOPC vesicle parameters are a = 20.2µm, Cm =0.71 µF/cm, lin =2 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m. seq =4x10 7 N/m, D=0.27. The polymer vesicle parameters are a = 27.2µm, Cm =0.27 µF/cm, lin =2 µS/cm, lex =10 µS/cm, µmm =25 x10 8 N.s/m. seq =2.0x10 7 N/m, D=0.22.
1.15
0.95 0.9 −2 10
−1
10
0
10
1
2
10
tim e t / t
10
3
10
4
10
mm
Fig. 10 (a) Oblate to prolate deformation and (b) relaxation (t = 0 is set at the end of the second pulse) for two different DOPC vesicle exposed to the same two-step pulse with different initial tensions. The vesicle with a higher initial tension is denoted by squares (⇤) and the vesicle with a lower initial tension is denoted by circles ( ). An initial field of strength 400 kV/m for 20 µis applied, the field strength is decreased to 20 kV/m for 50 ms. The high tension DOPC vesicle parameters are radius a = 19.9 µm, Cm =0.71 µF/cm, lin =2.5 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m. seq =5x10 7 N/m, D=0.27. The low tension DOPC vesicle parameters, radius a = 20.9 µm, Cm =0.71 µF/cm, lin =2.5 µS/cm, lex =10 µS/cm, µmm =0.3 x10 8 N.s/m, seq =0.4x10 7 N/m, D=0.30.
4.3
Phase diagram
The vesicle response to a two step pulse is summarized in a phase diagram shown in Figure 11. The first pulse is varied in strength and duration while the second pulse is the same for all experiments. The second pulse is below the critical field (E0 < Ec ) with a duration much longer than tmm . The results show that the pulse duration leading to crossover from prolate-only to oblate-prolate behavior correlates with the tel timescale. If the entire membrane became perfectly con1–9 | 7
Soft Matter Accepted Manuscript
0.85
D e f orm a tio n D
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
1.1
Soft Matter
Page 8 of 11 View Article Online
DOI: 10.1039/C3SM52870G
0
t p uls e / t m m
10
6
−1
10
Acknowledgement
This work was partially supported by NSF grants CBET1117099 and CMMI-1232477. −2
10
0
2
4
6
E (k V / c m)
8
10
0
Fig. 11 Vesicle behavior as a function of the duration and field strength of
the first pulse, the duration of the first pulse is normalized by the capacitor charging time tmm . The second pulse is set at E0 = 20kV /m for 50µs. Three outcomes are observed: only prolate deformation, oblate-to-prolate transition, and only oblate deformation. The oblate only deformation occurs when the vesicle has porated the membrane capacitor is short circuited. The dashed line corresponds to tel and the solid line denotes tcrit evaluated at Ec = 1kV /cm. tmm = 0.5ms.
ducting due to poration (but poration does not destabilize the membrane), there should only be oblate deformation because the membrane capacitor is short-circuited. The tcrit timescale dictates the crossover between oblate-prolate and oblate-only behavior. A critical field strength of Ec = 1kV/cm, which corresponds to a Vc = 3V for a 20 micron vesicle, is used to estimate tcrit . Note that tcrit is determined from the transmembrane potential at the pole of the vesicle where the critical transmembrane potential is reached first. A stronger field is required to produce transmembrane conduction over enough membrane area such that the vesicle will not return to prolate.
5
Conclusions
This paper provides a quantitative comparison between theory and experiment for the deformation of giant quasi–spherical vesicles in DC fields. Upon application of a single square pulse, a vesicle either deforms into a prolate spheroid or undergoes poration. In a two-pulse sequence, the vesicle can first adopt oblate shape if the interior solution is less conducting than the exterior. We show that the first pulse should have duration shorter than the membrane charging time and strength above the poration threshold, while the second pulse must be longer and weaker. After the field is turned off, the vesicle undergoes a fast retraction associated with membrane discharge followed by much slower relaxation to equilibrium spherical shape driven by tension. Our results provide useful physical insights into the interac8|
1–9
References 1 J. Schwalbe, P. M. Vlahovska, and M. Miksis. Vesicle electrohydrodynamics. Phys. Rev E, 83:046309, 2011. 2 R. Dimova, N. Bezlyepkina, M. D. Jordo, R. L. Knorr, K. A. Riske, M. Staykova, P. M. Vlahovska, T. Yamamoto, P. Yang, and R. Lipowsky. Vesicles in electric fields: Some novel aspects of membrane behavior. Soft Matter, 5:3201 – 3212, 2009. 3 P. M. Vlahovska. Non-equilibrium dynamics of lipid membranes: deformation and stability in electric fields. In A. Iglic, editor, Advances in Planar Lipid Bilayers and Liposomes, vol. 12, pages 103–146. Elsevier, 2010. 4 F. Ziebert and D. Lacoste. A planar lipid bilayer in an electric field: membrane instability, flow field and electrical impedance. In A. Iglic, editor, Advances in Planar Lipid Bilayers and Liposomes, vol. 14, pages 63–95. Elsevier, 2011. 5 S. Aranda, K. A. Riske, R. Lipowsky, and R. Dimova. Morphological transitions of vesicles induced by ac electric fields. Biophys. J., 95:L19–L21, 2008. 6 P. F. Salipante, R. Knorr, R. Dimova, and P. M. Vlahovska. Electrodeformation method for measuring the capacitance of bilayer membranes. Soft Matter, 8:3810– 3816, 2012. 7 P. M. Vlahovska, R. S. Gracia, S. Aranda-Espinoza, and R. Dimova. Electrohydrodynamic model of vesicle deformation in alternating electric fields. Biophys. J., 96:4789– 4803, 2009. 8 T. Yamamoto, S. Aranda-Espinoza, R. Dimova, and R. Lipowsky. Stability of spherical vesicles in electric fields. Langmuir, 26:12390–12407, 2010. 9 P. Peterlin. Frequency-dependent electrodeformation of giant phospholipid vesicles in ac electric field. J. Biol. Phys., 36:339–354, 2010. 10 H. Nganguia and Y. N. Young. Ellipsoidal shapes of vesicles in ac electric fields. submitted, 2013. 11 J. Zhang, J. D. Zahn, W. Tan, and H. Lin. A transient solution for vesicle electrodeformation and relaxation. Phys. Fluids, 25:071903, 2013.
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
tion of biomembranes and pulsed electric fields used in electroporation. Moreover, the sensitivity of the dynamics to the membrane viscosity suggests that the two-pulse experiments can serve as a new method to measure this membrane property.
Oblate only Oblate−Prolate Prolate only
Page 9 of 11
Soft Matter View Article Online
12 K. A. Riske and R. Dimova. Electro-deformation and poration of giant vesicles viewed with high temporal resolution. Biophys. J., 88:1143–1155, 2005. 13 K. A. Riske and R. Dimova. Electric pulses induce cylindrical deformations on giant vesicles in salt solutions. Biophys. J., 91:1778–1786, 2006. 14 K.A. Riske, R.L. Knorr, and R. Dimova. Bursting of charged multicomponent vesicles subjected to electric pulses. Soft Matter, 5:1983–1986, 2009. 15 M. M. Sadik, J. B. Li, J. W. Shan, D. I. Shreiber, and H. Lin. Vesicle deformation and poration under strong dc electric fields. Phys. Rev. E., 83:066316, 2011. 16 C. Grosse and H. P. Schwan. Cellular membrane potentials induced by alternating fields. Biophys. J., 63:1632–1642, 1992. 17 H. P. Schwan. Dielectrophoresis and rotation of cells. In E. Neumann, A. E. Sowers, and C. A. Jordan, editors, Electroporation and electrofusion in cell biology, pages 3– 21. Plenum Press, 1989. 18 K. Kinosita Jr., I. Ashikawa, N. Saita, H. Yoshimura, H. Itoh, K. Nagayama, and A. Ikegami. Electroporation of cell membrane visualized under a pulsed laser fluorescence microscope. Biophys. J., 53:1015–1019, 1988. 19 L. C. McConnell, M. J. Miksis, and P. M. Vlahovska. Vesicle electrohydrodynamics in dc electric fields. IMA J. Appl. Math., 78:797–817, 2013. 20 J. Seiwert, M. J. Miksis, and P. M. Vlahovska. Stability of biomimetic membranes in dc electric fields. J. Fluid Mech., 706:58–70, 2012. 21 D. Needham and R. M. Hochmuth. Electromechanical permeabilization of lipid vesicles. role of membrane tension and compressibility. Biophys. J., 55:1001–1009, 1989. 22 T. Portet and R. Dimova. A new method for measuring edge tensions and stability of lipid bilayers: Effect of membrane composition. Biophys. J., 99:3264–3273, 2010. 23 H. Bermudez, H. Aranda-Espinoza, D. A. Hammer, and D. E. Discher. Pore stability and dynamics in polymer membranes. Europhys. Lett., 64:550–556, 2003. 24 H. Aranda-Espinoza, D. A. Hammer, and D. E. Discher. Electromechanical limits of polymersomes. Phys. Rev. Lett., 87:208301, 2001. 25 U. Seifert. Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles. Eur. Phys. J. B, 8:405–415, 1999. 26 M. I. Angelova and D. S. Dimitrov. Liposome electroformation. Faraday Discuss. Chem. Soc., 81:303– 311, 1986. 27 R. S. Gracia, N. Bezlyepkina, R. L. Knorr, R. L. Lipowsky, and R. Dimova. Effect of cholesterol on the rigidity of saturated and unsaturated membranes: fluctuation and electrodeformation analysis of giant vesicles. Soft Matter, 6:1472 – 1482, 2010.
28 R. Dimova, C. Dietrich, A. Hadjiisky, K. Danov, and B. Pouligny. Falling ball viscosimetry of giant vesicle membranes: finite-size effects. Eur. Phys. J. B, 12(4):589– 598, 1999.
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
DOI: 10.1039/C3SM52870G
1–9 | 9
Soft Matter
Page 10 of 11 View Article Online
DOI: 10.1039/C3SM52870G
Soft Matter Accepted Manuscript
Published on 05 February 2014. Downloaded by St. Petersburg State University on 08/02/2014 16:52:35.
A two-step-pulse reveals the transition from oblate to prolate ellipsoidal shape for giant vesicles in DC electric fields.
Page 11 of 11
Soft Matter
a
Soft Matter Accepted Manuscript
a
Ε
0 ms
0.4 ms
20 ms
50 ms
