Cholesterol-Induced Variations in the Domain Structure Fluctuations

Article pubs.acs.org/JPCB

Cholesterol-Induced Variations in the Domain Structure Fluctuations and Microdynamics of Lipid Membranes W. Schrader, R. Behrends, and U. Kaatze* Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany ABSTRACT: Broadband ultrasonic attenuation spectra (100 kHz to 2 GHz) of aqueous solutions of vesicles from 1,2-dimyristoylL-3-phosphatidylcholine, with different amounts of cholesterol admixed, have been measured at temperatures between 20 and 28 °C. The spectra have been evaluated in terms of suitable relaxation functions. They are discussed in view of the effect of cholesterol on the membrane behavior around the gel−fluid phase transition temperature Tm. In addition to a frequency-independent asymptotic high-frequency term, all spectra reveal a critical term and a Debye-type relaxation term with relaxation time around 0.5 ns. The former is evaluated in the light of the Bhattacharjee−Ferrell dynamic scaling theory. It is assigned to the critical domain structure fluctuations of the membranes. Critical slowing of fluctuations is demonstrated. Also shown are relations of the critical amplitudes to thermodynamic parameters. The Debye term reflects the rotational isomerization of the phospholipid alkyl chains. The relaxation time of isomerization reveals a significant steplike change at Tm. At moderate cholesterol content an additional Debye relaxation term exists. It is assigned to the axial diffusion of the membrane molecules. Because it likewise shows effects of slowing near Tm, the diffusion appears to be coupled to the domain structure fluctuations. A further relaxation term at small cholesterol concentration is assumed to be due to small-range shape fluctuations of vesicles near the phase transition temperature. 30−50 mol %,16−18 it is only 5 mol %8 in the membrane of the endoplasmic recticulum. At cholesterol content above 20 mol % a so-called liquid-ordered (lo) membrane phase is formed which indeed lacks translational order but exhibits a high extent of conformational order. Hence the presence of cholesterol leads to a decoupling of translational and conformational degrees of freedom.9 Often lo microdomains are named “rafts”.7,15,16,19−22 These assemblies appear to participate in significant membrane processes, such as tension regulation, signaling, as well as lateral protein alignment and virus adaptation.22−28 The effect of the sterol at high concentration is quite different from that at low content ( Tm, quite remarkably, γ = 1.92 ± 0.04 agrees with the exponent Z0ν̃ (=1.93) which is theoretically predicted38 for critically demixing binary liquids near their consolute point. At T < Tm the exponent is slightly smaller (γ = 1.73 ± 0.15). Here Z0 (=3.06939,40) denotes the dynamics universal exponent and ν̃ (=0.6341) is the exponent of the fluctuation correlation length, given by

ξ = ξ0ε−ν̃

(3)

with amplitude ξ0. Figure 1 also shows that the relaxation time τ2 of the isomerization of alkyl chain conformations passes through a 2447

dx.doi.org/10.1021/jp2106007 | J. Phys. Chem. B 2012, 116, 2446−2454

The Journal of Physical Chemistry B

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filled with reference liquids of well-known attenuation coefficient. The sound velocity cs of the samples has been obtained from the series of resonance frequencies of the cavity resonator cells,46 carefully taking into account the nonequidistant distribution of these frequencies. At higher frequencies (ν > 15 MHz) ripples, as resulting from reflected waveforms in the transfer function of the pulse-modulated-wave-transmission mode of operation, have been utilized for cs determination. In addition, sound velocities have been measured as a function of temperature employing a high-precision twin cell method operated at some megahertz.48 2.3. Experimental Errors. Two resonator cells, five variable-path-length cells, and altogether five different electronic setups have been used to cover the frequency range of the broadband sonic attenuation coefficient measurements. Due to the overlap in the frequency range of different apparatus and cells, systematic errors exceeding the uncertainties given below can be excluded. Multiple data recording, followed by averaging and regression analysis, resulted in negligibly small statistical errors in the α values. The frequency ν of measurement was always known and kept constant with likewise negligibly small error. Temperature gradients within the cells and deviations between the temperatures of different cells did not exceed 0.05 K, corresponding to a relative error Δα/α < 0.0015. With the resonator measurements, the main sources of possible experimental errors are small disturbances in the cell geometry and cell adjustment that might result when the sample is exchanged for the reference liquid. Attenuation coefficients from the pulse-modulated-wave-transmission technique may be subject to incomplete parallelism of the transmitter and receiver transducers and, at low frequencies, to insufficient corrections for diffraction losses. Strictly, the uncertainties of α data depend on the magnitude of the attenuation coefficients themselves. Approximately, the uncertainty values are as follows: Δα/α = 0.1, 0.1−1 MHz; Δα/α = 0.05, 1−15 MHz; Δα/α = 0.01, 15− 440 MHz; Δα/α = 0.02, 440−1000 MHz; Δα/α = 0.01, 1000− 2000 MHz. The uncertainties in the sound velocity data, obtained as a byproduct of the attenuation coefficient measurements, are Δcs/cs = 0.005, 0.1−440 MHz; Δcs/cs = 0.01, 440−2000 MHz. The twin-resonator-cell method provides a resolution as high as Δcs/cs = 10−5 in the sound velocity measurements at fixed frequency. The uncertainty in the absolute Δcs data, however, may be larger (Δcs/cs < 10−4).

Figure 2. Relative shift ∂cs/cs of sound velocity cs with respect to the water value for vesicle solutions from mixtures of DMPC and cholesterol displayed vs temperature T.49 The DMPC concentration is 2 mg/mL; the mole fractions of cholesterol are xchol = 0 (●), 0.05 (○), 0.1 (▲), 0.3 (△), and 0.5 (▼).

Tm = 24.47 °C, xchol = 0; Tm = 24.12 °C, 0.05 ≤ xchol < 0.1; Tm = 24.11 °C, xchol = 0.1; Tm = 30.0 °C, xchol = 0.15. In Figure 3 the transition temperatures from the sound velocity and heat capacity measurements are shown along with

Figure 3. Detail of the phase diagram for DMPC−cholesterol in water (full lines50), literature data for the phase transition temperature Tm (○),51 as well as Tm values from heat capacity (●)49 and sound velocity (▲, Figure 2) measurements.

3. RESULTS 3.1. Sound Velocity, Phase Transition Temperature. For some solutions of vesicles from DMPC and cholesterol, relative deviations ∂cs/cs0 = (cs − cs0)/cs0 of sound velocities cs from the water value cs0 are shown as a function of temperature T in Figure 2. At low cholesterol content (mole fraction xchol ≤ 0.01), the ∂cs/cs0 values display a relative minimum at around the phase transition temperature Tm of the DMPC membranes. However, the minima become broader at increasing cholesterol content. At xchol = 0.3 just a flat minimum remains which, in addition, is shifted to a somewhat higher temperature. At even higher cholesterol content, any indication of a phase transition is missing at all. Heat capacity traces, recorded in an upscan mode on a differential scanning calorimeter (MicroCal, Northhampton, MA; scan rate 5 K/h49), had yielded the following phase transition temperatures for vesicle solutions used in the broadband ultrasonic attenuation measurements:

a phase diagram that had been obtained from discontinuities in ESR experiments.50 At xchol ≤ 0.2 our Tm values are slightly larger than those of the previously published phase diagram. They are even substantially larger than transition temperatures obtained from measurements of the ripple repeat distance.51 Those differences in the transition temperatures are taken to reflect the subtle effects of cholesterol on the structure and dynamic properties of membranes appearing variably in different parameters. Different experimental techniques, therefore, probe different influences of cholesterol so that no universally valid phase diagram exists.52 3.2. Ultrasonic Attenuation Spectra. In a broad frequency band the frequency-normalized ultrasonic attenuation coefficient α/ν2 of a DMPC solution with 7.5 mol % 2448



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cholesterol added decreases monotonously to asymptotically reach a frequency-independent value B′ = lim (α /ν 2) ν→∞

(4)

at high frequencies (Figure 4). The monotonous decrease in the spectrum, indicating an underlying broad distribution of

Figure 5. Ultrasonic excess attenuation per wavelength vs frequency ν for solutions of extruded vesicles from DMPC without (○, clipid = 10 mg/mL) and with (●, xchol = 0.075) cholesterol added.

relaxation term. For that reason, we finally used relaxation function 2

R(ν) = RBF(ν) +

∑ R Di(ν) + Bν i=0

(8)

with Debye-type relaxation terms Figure 4. Ultrasonic attenuation spectrum in the frequencynormalized format for an aqueous solution of extruded vesicles from DMPC (clipid = 10 mg/mL) and cholesterol (xchol = 0.075) at 28 °C. The dotted line shows the Bhattacharjee−Ferrell spectral function (eq 5 with scaling function according to eq 6), and the dashed line indicates the asymptotic high-frequency contribution B′ to the spectrum.

R Di(ν) =

(5)

N

×

∑

w(νn)|αλ(νn) − R(νn, p1 , ..., pM )|

n=1

(10)

In this equation νn, n = 1, ..., N, denote the frequencies of measurement, and pm, m = 1, ..., M, the unknown parameters of R. Quantities w(νn) are weighing factors that are taken inversely proportional to the experimental uncertainties. The parameter values obtained by the fitting procedure are collected in Table 1.

4. DISCUSSION 4.1. Volume and Enthalpy Fluctuations. Due to the Newton−Laplace equation58

(6)

of the scaling function has been used, which fits to the critical ultrasonic attenuation near the isotropic−nematic transition of liquid crystals.56 Evidently the BF function provides a reasonable approximation of the experimental broad attenuation spectrum. It does, however, not perfectly fit. In the frequency range around 10 MHz, the experimental data clearly exceed the predictions by eqs 5 and 6. The deviations between RBF(ν) and experiment become even more obvious in the alternative representation of data shown in Figure 5. In that figure the excess attenuation per wavelength (αλ)exc = αλ − Bν

(9)

χ 2(p1 , ..., pM ) = (N − M − 1)−1

In this function, SBF denotes an amplitude, FBF a scaling function, δ (=0.058) a universal critical exponent, Ω = ω/Γ a scaled frequency, and ω = 2πν. Here Γ = 1/τξ is the relaxation rate of order parameter fluctuations, viz., of the fluctuations of the membrane domain structure. Originally, the BF function had been derived to represent the sonic attenuation near the demixing point of three-dimensional critical liquids.53−55 It has been shown, however, that the BF theory likewise applies to quasi-two-dimensional membrane systems. The reason for the applicability is the fact that fluctuations in the lateral area per membrane molecule are related to fluctuations in the thickness of the membrane.44,55 Indicated by the dotted line in Figure 4 is the BF contribution to the spectrum. For the numerical calculations the empirical form ̃ (Ω) = (2/Ω)1/2 [(1 + Ω2)1/4 cos(0.5 arctan Ω) − 1] FBF

1 + ω2τi 2

to analytically represent the measured spectra. Parameters Ai and τi, i = 0, 1, 2, are relaxation amplitudes and relaxation times, respectively. We mention that the same relaxation function applies well to the ultrasonic spectra for solutions of singlecomponent vesicles from DMPC29 and of vesicles from mixtures of phospholipids.30 Equation 8 has been fitted to the measured spectra using a regression analysis57 that minimizes the reduced variance

relaxation times, points at a Bhattacharjee−Ferrell (BF) relaxation spectral function R′BF (ν) = RBF(ν)/(νcs) = SBFν−(1 +δ)cs−1FBF(Ω)

A i ωτi

cs = (ρκS)−1/2

(11)

the sound velocity of a liquid is related to the density ρ and the isentropic compressibility κS = −1/V (∂V /∂p)S

(12)

According to ref 58 κS = κT CV /Cp

(13)

the isentropic compressibility is related to the isothermal compressibility κT = −1/V (∂V/∂p)T as well as the heat capacities CV and Cp at constant volume and constant pressure, respectively. Because of enhanced volume and enthalpy fluctuations at the gel−fluid phase transition of the membranes, we expect both the isothermal compressibility and the heat

(7)

with B = B′c, and λ = cs/ν, is plotted versus frequency ν. The measured spectra display definitely at least two further relaxation regions. A closer inspection of the experimental data reveals some spectra which require even a third additional 2449



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Table 1. Parameters of the Relaxation Spectral Function Defined by Eq 8 and Sound Velocity cs for Solutions of Vesicles from DMPC (10 mg/mL) with and without Cholesterol (Mole Fraction xchol) Added xchol, Tm °C 0 24.6

0.05 24.4 0.075 24.4 0.1 24.11 0.15 24.4

0.33 −

T °C

cs (ms−1) ±0.5%

SBF (10−6 s0.94 m−1) ±10%

τξ−1 (106 s−1) ±15%

A0 10−3 ±25%

τ0 (ns) ±25%

A1 10−3 ±25%

τ1 (ns) ±25%

A2 10−3 ±25%

τ2 (ns) ±25%

B (ps) ±2%

20.0 23.5 24.0 25.0 28.0 20.0 24.0 28.0 20.0 24.0 28.0 20.0 24.0 28.0 20.0 23.0 24.0 25.0 28.0 20.0 23.0 24.0 25.0 28.0

1482.6 1492.0 1453.7 1497.0 1503.1 1483.2 1494.1 1502.7 1483.0 1494.0 1504.4 1482.6 1494.4 1504.4 1482.0 1491.2 1494.0 1496.7 1504.4 1482.4 1491.2 1494.0 1496.7 1504.4

2.0 3.5 3.7 2.5 1.0 3.1 4.1 1.8 3.1 4.5 2.2 2.6 3.7 2.6 3.0 3.0 4.4 2.8 3.1 0.65 0.66 0.68 0.73 0.79

0.4 0.023 0.01 0.009 0.4 0.36 0.001 0.26 0.32 0.001 0.27 0.21 0.001 0.21 0.12 0.03 0.02 0.04 0.05 0.14 0.004 0.006 0.011 0.007

− 1.3 2.4 2.5 0.5 − 2.2 1.7 − 1.1 1.7 − 1.1 1.7 − − − − − − − − − −

− 150 172 147 108 − 151 15 − 130 14.5 − 101 13 − − − − − − − − − −

0.15 1.5 3.5 4.0 1.0 0.44 3.2 1.9 0.7 2.6 1.1 0.7 1.9 0.8 0.95 1.2 1.0 1.6 2.2 − − − − −

11 13 21 21 9.3 9.9 20 2.5 11.2 19 3.7 23 18 2.5 14 18 18 11.6 8.8 − − − − −

0.9 1.5 3.5 5.0 4.1 0.78 3.2 4.0 1.6 2.5 3.9 1.7 1.9 3.5 5.0 2.5 2.0 4.5 3.0 2.0 1.7 1.8 1.8 1.7

0.7 0.3 0.12 0.10 0.09 0.89 0.12 0.09 0.23 0.10 0.10 0.35 0.12 0.12 0.19 0.15 0.12 0.10 0.09 0.77 0.82 0.79 0.61 0.67

38.6 34.5 32.5 31.9 30.3 39.7 32.6 30.2 37.7 33.0 29.7 38.9 33.3 29.5 37.0 33.9 33.2 31.9 29.6 37.3 34.1 33.1 32.1 29.4

a very flat minimum still exists in the ∂cs/cs vs T relation and the transition temperature is shifted to about 35 °C (Figure 2). As mentioned before, no relative minimum is found in the cs data at even higher cholesterol content. Hence, at large mole fraction, the compressibility of the membrane passes through a continuous transition from a more rigid to a more compressible state. For mole fractions of cholesterol up to 0.2, the width ΔT of the gel−fluid phase transition for DMPC−cholesterol mixtures is shown and compared to some data for mixtures of DMPC with DPPC in Figure 6. The ΔT values of both systems

capacity at constant pressure to increase at Tm. Combining eqs 11 and 13 yields cs = (ρκT CV /Cp)−1/2

(14)

This relation reveals the relative minima in the cs data at low cholesterol content (xchol ≤ 0.1, Figure 2) to reflect contrarian impacts of κT and Cp on the sound velocity when approaching Tm. Experiments have shown that, close to the melting transition of phospholipid alkyl chains, the changes in the volume and excess enthalpy of membranes are proportional to one another59−61 ΔVlipid(T ) = γΔHlipid(T )

(15) −4

The proportionality factor resulted as γ = 7.8 × 10 cm3/J3 for various lipids, lipid mixtures, and also biological membranes.60,61 Based on this finding, a relation between the sound velocity and the excess heat capacity of a membrane systems has been derived which allows the heat capacity changes at Tm to be calculated from sound velocities and vice versa. This relation has been verified for monocomponent lipid membranes, for mixtures of phospholipids with different chain length, and for mixtures of phospholipids with diacyl glycerides,62 as well as mixtures of DMPC with cholesterol.49 At low cholesterol content, the sound velocity minimum at Tm broadens with increasing xchol (Figure 2). Along with the finding of an almost constant phase transition temperature, this broadening of the sound velocity (and thus heat capacity) profiles supports the idea that, at low cholesterol content, the sterol does neither reduce the order of phospholipid alkyl chains in the gel phase nor enhance the conformational order in the fluid phase. It has been suggested that cholesterol is accumulated in the interfaces of gel and fluid domains, thereby reducing the cooperativeness in the transition.9,63 At xchol = 0.3

Figure 6. Width ΔT of the gel−fluid phase transition for solutions of extruded vesicles from DMPC−cholesterol mixtures displayed as a function of mole fraction xchol of the sterol. Data from two series of measurements are given: (▲) clipid = 2 mg/mL; (●) clipid = 4 or 5 mg/mL. The inset show some ΔT data for DMPC−DPPC mixtures62 (DPPC = 1,2-dipalmitoyl-L-3-phosphtidylcholine).

increase with mole fraction of the second constituent. At DPPC content larger than about 50 mol % the width of the transition 2450



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of DMPC−DPPC mixtures decreases to reach a small value at xDPPC = 1. The finding of ΔT(xDPPC=1) < ΔT(xDPPC=0) reflects the higher cooperativeness of the transition at longer alkyl chains of the lipid molecules. 4.2. Relaxation Rate of Domain Structure Fluctuations. We have investigated the critical contribution to the ultrasonic attenuation spectra of bilayer systems by subtracting the noncritical background part from the total attenuation per wavelength. In this way the critical part is obtained as 2

(αλ)c (ν) = αλ − ( ∑ R Di(ν) + Bν) i=0

(16)

Figure 8. Relaxation rate Γ (=τξ−1 = 2D/ξ2) of order parameter fluctuations vs temperature T for DMPC−cholesterol mixtures with different contents of cholesterol: (○) xchol = 0; (▲) xchol = 0.05; (●) xchol = 0.075; (▼) xchol = 0.1; (■) xchol = 0.15.

From these data the scaling function in the BF model can be determined as the ratio F = (αλ)c (ν, T )/(αλ)c (ν, Tc)

(17)

relaxation rate data using eq 1. A rather uniform behavior of ξ as a function of temperature near Tm has been found at mole fractions xchol = 0.05, 0.075, and 0.1. Similar to singlecomponent DMPC membrane systems at 20 °C (Figure 9), the ξ values hardly exceed the nearest-neighbor distance d =

In Figure 7, as an example, normalized attenuation-per-wavelength data for a DMPC system with substantial amount of cholesterol

Figure 7. Normalized ultrasonic attenuation coefficient excluding noncritical background contributions (eqs 16 and 17) displayed as a function of reduced frequency Ω for solutions of extruded vesicles from DMPC without (open symbols; clipid = 10 mg/mL) and with cholesterol (closed symbols; xchol = 0.15, Tm = 30 °C) added. All data refer to temperatures below the gel−fluid phase transition temperature (●, 23 °C; ■, 24 °C; ▼, 25 °C; ▲, 28 °C; Tm = 30 °C). The line is the graph of the analytical scaling function defined by eq 6.

added are displayed versus scaled frequency Ω. Also shown for comparison are data for single-component DMPC bilayer systems and likewise the graph of the BF scaling function as given by eq 6. The experimental data have been fitted to the analytical function using relaxation rate Γ as the only adjustable parameter. The consistency of the scaling function data for the DMPC−cholesterol system with those for the singlecomponent membranes and also with the analytical scaling function may be taken an indication of the appropriateness of the relaxation model. The elevated scatter in the data at Ω > 103 reflects the fact that, at high frequencies, the critical part (αλ)c is obtained as the small difference between the total attenuation per wavelength and the noncritical contributions. At 0 ≤ xchol ≤ 0.15 the relaxation rates Γ = 1/τξ of membrane structure fluctuations are displayed as a function of temperature in Figure 8. At around Tm all membrane systems display a minimum in the relaxation rate Γ, thus clearly disclosing slowing down of fluctuations near the transition temperature. 4.3. Fluctuation Correlation Length. Using diffusion coefficients D as obtained by fluorescent-labeling of phospholipids64,65 and cholesterol,66 the fluctuation correlation length ξ of DMPC−cholesterol mixtures has been calculated from the

Figure 9. Fluctuation correlation length ξ (full symbols), as following from eq 1, and diffusion coefficient data (open symbols) from the literature64−66 plotted vs temperature T for mixtures of DMPC and cholesterol at different cholesterol contents: (○,●) xchol = 0; (◊,⧫) xchol = 0.05, 0.075, and 0.1, respectively; (□,■) xchol = 0.15.

0.38 nm of the phospholipid molecules. Consistent with the diffusion coefficient, the mean lateral extent of fluctuations increases dramatically on approach of Tm (Figure 9) and reaches values on the order of the vesicle radius. For membrane systems with vanishing or moderate cholesterol content (xchol < 0.15) the correlation length decreases again when going to higher temperatures. At xchol = 0.15, however, the extent of fluctuations in the fluid phase persists at a large value (ξ ≈ 10 nm) at least at temperatures up to 28 °C. At this point, we mention two soft spots in the above estimation of fluctuation correlation length. First, the smallest relaxation rates derived from the ultrasonic spectra correspond with frequencies below our frequency range of measurement. They have thus been obtained by extrapolation from the high-frequency branch of the scaling function. In addition, diffusion coefficients for the lateral mass diffusion have been used, whereas the diffusion of 2451



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reflect the variation in the slope ∂V(T)/∂T of the temperature dependence of the membrane volume. Interestingly, at xchol = 0.33, where no phase transition occurs within the temperature range of measurement, there seems to still exist a smallamplitude low-frequency BF contribution in the spectra (Figure 10). However, the amplitude of that contribution does not reveal a relative maximum, in accordance to the missing steplike change in the membrane volume. The maximum in the SBF data at xchol ≤ 0.1 extends over a broader frequency range than the change in the specific volume of the membranes. This finding may be taken as an indication that the range of SBF data around Tm is compressed by a noticeable temperature dependence of Cpb (eq 18), counteracting the effect of the thermal expansion coefficient. The heat capacity increases around the phase transition temperature not just because of the growing domain structure fluctuations but also due to the melting of phospholipid alkyl chains. Though both processes are coupled, it is unclear presently whether a simple representation of the heat capacity of membrane systems in terms of eq 19 is adequate to the sonic attenuation spectra. Likely, a noticeable temperature variation in the background part of the heat capacity has to be taken into account. As the critical amplitude (eq 18) is inversely proportional to Cpb2, an increase in the background part of the heat capacity may substantially reduce the SBF enhancement due to the increment in the thermal expansion coefficient near Tm. 4.5. Axial Diffusion and Chain Isomerization. As already mentioned in the Introduction and in accordance with a previous assignment29 of relaxation times of DMPC membrane systems, τ1 (2.5 ns ≤ τ1 ≤ 23 ns, Figure 11) is assumed to

domains concerns rather the diffusion of the molecular structure. For these reasons, the absolute ξ values should not be overemphasized. 4.4. Critical Amplitude. As revealed by Figure 10, not just the relaxation time τξ but also the BF amplitude SBF displays a

Figure 10. Amplitude SBF of the critical term (eq 5) of the relaxation spectral function (eq 8) for mixtures of DMPC and cholesterol at different cholesterol contents (○, xchol = 0; ▲, xchol = 0.05; ●, xchol = 0.075; ▼, xchol = 0.1; ■, xchol = 0.15; ⧫, xchol = 0.33) displayed vs temperature T. The shaded area accentuates the trends in the data at xchol = 0.05, 0.075, and 0.1. The full (xchol = 0) and dashed (xchol = 0.15, 0.33) lines are given to guide the eyes.

distinct maximum at the phase transition temperature. The amplitude in eq 5 is given by the relation53−55 δ

πδCpccs(Tm) ⎛ Ω1/2 ⎞ 2 ⎜⎜ ⎟⎟ g (T ) SBF(T ) = 2TmCp b2 ⎝ 2πτξ0 ⎠

(18)

Here Cpc is the amplitude in the heat capacity contribution due to the domain structure fluctuations and Cpb is the background contribution to the heat capacity at constant pressure. For binary critical mixtures power law behavior Cp = Cpc ε−α 0 + Cp b

(19)

is assumed. Furthermore, in eq 18 Ω1/2 is a scaled halfattenuation frequency, defined by FBF(Ω1/2) = 0.5. Parameter g is the adiabatic coupling constant which, within the framework of the BF theory, is given by the slope ∂Tm/∂p in the pressure dependence of the critical temperature and the thermal expansion coefficient αp at constant pressure: g (T ) = ρ(Tm)Cp(T )∂Tm/∂p − T α p(T )

Figure 11. Relaxation times τ1 (open symbols) and τ2 (full symbols) of Debye-type relaxation terms in the spectral function (eq 8) shown as a function of temperature T. The data refer to DMPC−cholesterol mixtures at different cholesterol content: (○,●) xchol = 0; (△,▲) xchol = 0.05; (◊,⧫) xchol = 0.075; (□,■) xchol = 0.15.

(20)

With critically demixing binary liquids, no pronounced temperature dependence in SBF is normally found. An exception is systems for which the second term on the right-hand side of eq 20 adopts values on the order of the first term.67 Attempting to estimate the relative importance of both terms in the coupling constant, we used the following data for monolamellar DMPC vesicles: ρ(Tm) = 1.04 g·cm−3, Cp = Cp0 + Cpexc with Cp0 = 1.65 kJ/(mol·K)68 as well as Cpexc = 18.4 kJ/(mol·K),49,62 ∂Tm/ ∂p = 2.5 × 10−7 K/Pa,69 and αp = 9.4 × 10−2/K.70 With those data the thermal expansion coefficient term Tαp in the coupling constant exceeds the term ρCp∂Tm/∂p by a considerable amount. This is also true when somewhat smaller thermal expansion coefficient values are used as derived from our own density measurements of a DMPC sample in a range of temperatures around Tm. Hence the maximum in the BF amplitude at xchol ≤ 0.1 (Figure 10) seems to predominantly

reflect the axial diffusion of the phospholipid molecules. Outside the transition region, the τ1 values for pure DMPC membranes correspond with NMR correlation times for the axial diffusion71 and also with the dielectric relaxation times for the reorientational motions of the dipolar phospholipid head groups.29,72 Obviously, apart from Tm the headgroup reorientations and the reorientational motions of the complete molecule are coupled. Around Tm, however, the dielectric relaxation time displays a steplike change, indicating the ability of the dipolar head groups to reorientate faster in the fluid phase than in the gel phase. The rotation of the whole molecules slows down near the transition temperature. This result reflects the need 2452



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maximum in the amplitude of the critical term at Tm reflects the enhanced thermal expansion coefficients of the membranes. Seemingly, however, this effect interferes with a counteracting temperature variation in the background part of the heat capacity at constant pressure. The Debye relaxation terms are due the rotational isomerization of the DMPC alkyl chains, the axial diffusion of the membrane molecules, and, probably, short-scale deformations of the vesicle shape. The steplike change in the relaxation time τ2 of chain isomerization is less pronounced at cholesterol mole fraction 0.075 ≤ xchol ≤ 0.15, indicating a reduced free energy of activation of rotational isomerization below Tm. At higher cholesterol content the steplike change in τ2 is missing at all. Within the temperature range of measurement, the alkyl chain fluctuations correspond with those of pure DMPC membranes in the gel state. The axial diffusion term does not exist in the spectra for cholesterol mole fraction xchol ≥ 0.33. At lower cholesterol content the diffusive rotations appear to be coupled to the domain structure fluctuations as their relaxation times τ1 display a relative maximum at Tm. At temperatures below Tm the τ1 values vary only insignificantly with xchol. Above Tm, however, membranes with moderate content of cholesterol show a noticeably enhanced rotational mobility of the membrane molecules with respect to pure DMPC membranes or to such with larger cholesterol concentration. The third Debye term, tentatively assigned to fluctuations in the vesicle shape, has been found at xchol ≤ 0.1 only. This result may reveal a stiffening of the vesicles at higher content of cholesterol, attended by reduced small-range fluctuations of the vesicle shape.

for some space for reorientation and thus the coupling of the axial diffusion to the lateral diffusion. In fact, near Tm the τ1 values correspond with the correlation times of the lateral diffusion of lipid molecules.71 In the gel phase and at temperatures around Tm only minor differences in the τ1 values of membranes with different cholesterol content are found at xchol ≤ 0.15. In the fluid phase τ1 decreases by a factor of about 4 when going from the pure DMPC membrane to a cholesterol content xchol = 0.05. Addition of small amounts of cholesterol evidently results in an enhanced rotational mobility of the membrane molecules. However, τ1 increases at higher xchol to almost reach the original value again at xchol = 0.15 (Figure 11). Relaxation time τ2 (0.09 ns ≤ τ2 ≤ 0.89 ns, Figure 11) displays a quite different behavior. First of all, it shows a steplike decrease when temperature exceeds Tm. The τ2 values agree with the correlation times of the segmental motions of phospholipid alkyl chains and also with the rotational correlation times of the terminal methyl groups of the chains.70 Furthermore, they correspond to the ultrasonic relaxation times of liquid alkanes of similar length.73 The alkyl chain isomerization of DMPC membranes has thus been considered29 in terms of a torsional oscillator model74,75 to show that above Tm the free energy of activation ΔG⧧ of chain isomerization agrees with that of n-tetradecane at 25 °C. Below Tm, because of the high degree of order of the chains, ΔG⧧ is larger by the amount of 3.4 kJ/mol than above Tm. Up to xchol = 0.05 there is only a minor effect of cholesterol on the relaxation time of alkyl chain isomerization. At T < Tm relaxation time τ2 is considerably smaller at higher cholesterol content (0.075 ≤ xchol ≤ 0.15) than at xchol < 0.075, thus indicating an enhanced fluidity in the chain structural isomerization in the gel phase (Figure 11). However, at still higher cholesterol content τ2 increases again up to the value of about 0.7 ns for DMPC membranes below the phase transition temperature (Table 1). This value is maintained when the temperature is raised up to 28 °C. The results for τ2 show again that, in the temperature range under consideration, no gel−fluid phase transition occurs at xchol = 0.33. The source of the additional Debye-type relaxation term with relaxation time τ0 around 100 ns (Table 1) is still unclear. Referring to optical measurements,76 it has been tentatively assigned to short-scale deformations of the vesicle shape.29 This Debye term exists only at temperatures around Tm where the fluctuations in the domain structure are particularly strong. If our assignment is true, the ultrasonic spectra show also that fluctuations of the vesicle shape are below limit of detection at higher cholesterol content (xchol ≥ 0.15).

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +49 551 39 7715. Fax: +49 551 39 7720. E-mail: uka@ physik3.gwdg.de. Notes

The authors declare no competing financial interest.

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REFERENCES

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5. CONCLUSIONS Close to the gel−fluid phase transition temperature Tm, broadband ultrasonic spectra of aqueous solutions of vesicles from DMPC with different amounts of cholesterol added expose critical fluctuations and up to three Debye-type relaxation processes. The critical contributions can be well represented by the Bhattacharjee−Ferrell theory of concentration fluctuations in critically demixing binary systems. In the membrane systems it is not the local concentration that fluctuates but the domain structure of the membranes, that is, the local correlation with a state: gel or fluid. At cholesterol mole fraction between 0.05 and 0.1 the slowing in the domain structure fluctuations around Tm is significantly more pronounced than in pure DMPC membrane systems. At higher cholesterol content, however, the slowing effect in the relaxation rate of domain structure fluctuations appears to be reduced. At xchol ≤ 0.15 a relative 2453



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Cholesterol-Induced Variations in the Domain Structure Fluctuations

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