9844
J. Phys. Chem. 1993,97, 9844-9851
Fluid-Gel Interphase Line Tension and Density Fluctuations in Dipalmitoylphosphatidylcholine Multilamellar Vesicles. An Ultrasonic Study D. P. Kbarakoz,? A. Colotto,'**K. Loher,# and P. Laggneri Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, 142292 Pushchino, Moscow Region, Russia, and Institute of Biophysics and X-ray Structure Research, Austrian Academy of Sciences, Steyrergasse 17, A-8010 Graz, Austria Received: April 13, 1993; In Final Form: June 25, I993@
Temperature dependence of sound velocity and absorption in dilute suspensions of dipalmitoylphosphatidylcholine multilamellar vesicles have been measured at 7.2 MHz in the temperature range 20-55 OC. Relaxational characteristics of heterophase fluctuations of density in the vicinity of the main phase transition have been determined, i.e. relaxational sound velocity, absorption, and effective relaxation time. Fluid-gel interphase line tension in the fluid bilayer has been evaluated for the first time from the acoustic data by means of a simple model based on Frenkel's theory of heterophase fluctuations in the vicinity of first-order phase transitions. The theory focuses on the formation of clusters of a new phase inside the parent phase. The line tension was found to be 5.6 X lo-' erg/cm. The rate constants of cluster creation and growth have also been evaluated. Ultrasonic absorption spectra calculated by means of the model are in agreement with known wide-frequency-range acoustic spectroscopy data.
1. Introduction
phatidylcholine (DPPC) multilamellar vesicles have been measured at 7.2 MHz. The extent and the spectrum of relaxation times of heterophase density fluctuations near the transition temperature are evaluated for the liquid crystalline state. Interphase line tension and kinetics of creation and growth of gel-like nuclei are evaluated from the single-frequency measurements by means of a simple kinetic model of nucleation.
Phase transitions in biological and model membranes are of interest from both biological and physical points of view. The biological interest is determined by the fact that phase transitions modulate important biological properties like permeability,' and there are numerous reports on observations related to the temperature adaptation of organisms which indicates the important physiological role of membrane phase transitions.z4 The 2. Materials and Methods physical mechanism underlying these biological observations is Sample Preparation. Synthetic 1,2-dipalmitoyl-sn-glycero-3still under question. phosphatidylcholine (DPPC), purissima (high degree of purity: Ultrasonicvelocity and absorption are among the innumerable >99%), was purchased from Avanti (Alabama) and used without physicochemical characteristics which serve to describe memfurther purification. branes (see, e.g. refs 5 and 6) and are very informative for the Multilamellar vesicles were prepared by dissolving the lipids study of intermolecular interactions and kinetics of biomolecular in CHCl,/CHsOH ( 2 1 by volume) and letting the solutions dry processes (refs 7 and 8 and references therein). Acoustic under a stream of NZ and finally in vacuum overnight, so that investigations of compressibility and fluctuation of density near a thin film of lipids was formed at the walls of the sample vials. the transition temperatures of lipid vesicles were started long Subsequently, the dry samples were dispersed in deionized water ago.7.e14 Intrinsic compressibilityof membranes evaluated from acoustic data is found to be (40-60) X 1od b a r 1 ,on a~erage,~>*S and incubated at 45 OC for 4 h. To obtain homogeneoussamples, the suspensions were vortexed periodically during this time. The twice less than the compressibilityof hydrocarbon liquids. In the final water concentration was 97.7 wt %. temperature range of the gel-fluid phase transition a remarkable Ultrasonic Measurements. Ultrasonic velocity and absorption increase of compressibilityis observed.I1 This observation is due were measured with a differential fixed-path interferometer using tovolume fluctuations in the time scale of nanoseconds and below. theacousticresonatorsdescribed in refs 19and 20. Deterpination Mitaku and coauthors considered the behavior of volume and of the resonance parameters was based on measuring the midpoints compressibility around the main transition as a pseudocritical and widths of the phasefrequency curves of the resonance peaks. phenomenon,"' in the frameworkof Landau theory. Alternative The measurements were performed by means of a home-made interpretations were discussed by van Osdol et al.5 electronic circuit based on a phasefrequency feedback. The Interphase surface tension (or line tension in case of tworesonators contained a sample volume of 0.8 mL; fundamental dimensional systems) is one of the most essential physical frequency of piezotransducers was 10 MHz; diameter of transproperties governing phase transitions: transition order, limiting ducers was 0.7 cm; and ultrasonic path length was 0.8 cm. temperature of supercooling,fluctuationsof density near transition Frequencies of the used resonance peaks ranged between 7.0 and temperature, and transition rate.16J7 Nevertheless-apart from 7.4 MHz. The results were interpolated to 7.2 MHz. Frequency, a recent work's where the solid-liquid interphase line tension in f, and half-power width, 6, of a resonance peak of the acoustic a fatty acid Langmuir monolayerhas been studied-this important resonator are related, respectively, to sound velocity, u, and parameter has never been estimated for phase transitions in lipid absorption per wavelength, d,as follows:21 bilayers. In the present work, the temperature dependence of sound Au/u = ( A ! / N + 4 (1) velocity and absorption in a dilute suspension of dipalmitoylphosand Author to whom correspondence should be sent. Russian Academy of Sciences. t Austrian Academy of Sciences. Abstract published in Advance ACS Abstracrs, September 1, 1993. t
AaX = Q(As/f) (2) where Au and Af are changes in sound velocity and resonance
0022-3654/93/2097-9844%04.00/0 0 1993 American Chemical Society
Ultrasonic Study of DPPC Multilamellar Vesicles
The Journal of Physical Chemistry, Vol. 97, NO. 38, 1993 9&45 and
[.A]
[ I:,
+Mnso4 0
znAc2
MMSURED RESONANCE PEAK WIDTH; LHZ
Figure 1. Calibration curve for absorption. Q vs measured peak width of the acoustic resonator (see eq 2).
frequency, respectively, caused by a change of state in the measured liquid; d is a correction term which is negligible, d < 0.003 as shown theoreticallyz2 and experimentallyz3for the resonator used in this work; and Q is the calibration coefficient for sound absorption. For an ideal resonator Q = T if the peak width is measured at the half power level of the peak. For real systems the value of fl has to be obtained by calibration. The calibration was performed by using solutions of known ultrasonic absorption, MnS0, and Zn(CH3COO)z (the data were taken from the compilation presented in the review by Stuehr and Yeager24). The calibration curve is presented in Figure 1. The scatter of 52 plotted against the measured width is about 7% around the curve of an approximating calibration function. This value can be used as an estimate of the error of the absorption measurements. Precision of sound velocity depends on the measurement conditions. When sound velocity is measured in a liquid with low ultrasonic absorption (like water) and at a single temperature, the relative precision is about 2 X lo”%. In the cases where the temperature is varied, the reproducibilityof the base line (when both cells are filled with water) is within 1 X lO-3%. The latter value can serve as an estimateof the precision of our measurements. Reproducibility of the base line for absorption is the same as that at a single temperature, about 1%. Temperaturedependence of sound velocity and absorptionwere measured changing the temperature step-by-step: temperature setting, waiting for 3 min for the temperature equilibration (controlledby sound velocity), taking the reading, and setting up a new temperature, etc. Each of the steps took about 5 min overall (the average scan rate being about 0.2 K/min). Base lines of the device were determined in a preliminary experiment with both cells filled with pure water. The base lines were approximated by empiric polynomial functions used for further calculations of [u(T)] and [aX(T)]. Both sound velocity and absorptionwere measured at the same time with the same sample. The samples were degassed under a water pump vacuum during a short time (about 1 min) immediately prior to use to prevent bubble formation in the cell at high temperatures. To avoid sedimentationof the MLV’s due to their relatively higher density, samples were permanently homogenized by means of a rotating magnet rod placed in the bottom of the measurement cells. The intensity of ultrasound in the device was very small: less than 0.01 bar of the magnitude of alternative pressure in the ultrasonicwave. This is negligible to induce a possible ultrasonic influence on the phase transition. TheoreticalBackgroMd. Evaluationof theEffectiveRelaxation Time from SigleFrequency Ultrasonic Measurements. The theory of acoustic relaxation in aqueous solutions is described in a number of comprehensive reviews (e.g. ref 24). Below are listed the basic equations quoted in this work in terms of the so-called velocity number, [u], and absorption number, [ax], defined as
= (ax - aXo)/c
(4) where c is the concentration of the solution expressed in g/mL and the index “on refers to pure water. The sound velocity number consistsof two parts: instantaneous, [u] -,which is mainly defined by the stiffness of the intermolecular interaction potentials, and relaxational, [uIr,caused by pressureinduced changes in the structure of the system (e.g., in size distribution of heterophase fluctuations):
[ ~ =l [ul- + [ ~ l r (5) Similarly, the sound absorption number consists of classic absorption, [.A]-, whichisdue toshearviscosity,andrelaxation absorption, [ax],, caused by energy dissipation in relaxation processes taking place in the system: [ah1 = [ a X l , h i c + [.XI, (6) The classic part is negligible for dilute lipid suspensions in the MHz region of frequency, and the measured excess absorption can be attributed to the relaxation part only.11 In case of small distortions of relaxation processes taking place in a system, one can write24
and
where w is the circular frequency of sound, 71 is the relaxation time, and Ri is the relaxation strength which depends on the stoichiometryand adiabatic volume effect of the ith process. For the general case
(9) where V is the partial molar volume; p , pressure; S,entropy (because the process is adiabatic); M,the molecular weight; 6, the adiabatic compressibility coefficient of the system (which is close to the compressibility of water for the case of a small fraction of lipids); and K,J, the relaxational partial compressibility. The derivative of the partial molar volume over pressure stands here for the fraction of volume change which is caused by the pressureinduced shift of equilibrium in the ith process. Using eqs 7 and 8, one can write for a process with a single relaxation time the following relation between [.XI, and [u], (see, for example, ref 25):
In the case of more than one relaxation process occurring in the system, the calculated relaxation time is an effective value, T ~ E , defined as follows:
This is an average characteristic of that portion of the whole spectrum of relaxation processes which contributes to the velocity and absorption measured at the used single frequency w. The effective relaxation time defined this way can significantly differ from the apparent relaxation time which is usually determined from the frequency of the maximum absorption in the case of a complex spectrum.
9846 The Journal of Physical Chemistry, Vol. 97, No. 38, 1993
'
+
MitaLuet a1 1%
-0.151
25
-o'%O
30
35
40
45
50
55
60
TEMpERATLiRE deg c
0.35 S
1
I
i
TEMPERATURE, dcg C
Figure 3. Temperature dependenceof sound velocity (A, top) and sound absorption (B, bottom) numbers. The lipid concentrationis 23 mg/mL. Dotted lines show the extrapolated curve for the instantaneous parts of both velocity and absorption. The vertical line indicates the maintransition temperature. Crosses and circles refer to two different runs on the same sample: (+) first run; ( 0 )second run after 12-h incubation at 4 OC.
3. Experimental Results
Concentration dependence of sound velocity and absorption numbers measured at 25 OC are shown in Figure 2. Both parameters decrease almost linearly with increase of concentration. This is similar to the behavior of aqueous solutions of low molecular weight compounds and proteins. For the lower concentrations our results coincide with available literature data (shown in Figure 2 by plus signs). Temperature dependence of sound velocity and absorption were measured as described in Section 2. Results of two independent runs are presented in Figure 3. In the pretransition region, around 34 OC,262* the temperature dependence of [u] has a simple
Kharakoz et al. sigmoidal shape. It means that the curve is determined just by the relative fraction of two states and the equilibrium between these states is almost not affected by alternating pressure at the ultrasonicfrequencies. This indicateatheabsenceof a relaxational contribution of the transition itself at the ultrasonic time scale.1° However, in the rippled phase, between pretransitions and main transitions, there is a relaxation process contributing to ultrasonic properties. This is reflected by an increase of absorption in the transition region. Probably the corresponding decrease of sound velocity is alsopartially determined by this relaxation in the rippled phase. The curve at the main transition has a more complex shape. Contrary to the pretransition curve, additional to the usual jump of the instantaneous part of sound velocity, a wedge-shaped decrease is seen in the high-temperature vicinity of the main transition. This feature is due to a relaxational contribution to the sound velocity reflecting highly intensive density fluctuations closely above the transition temperature. The magnitude of the jump of instantaneous sound velocity upon the main transition was evaluated by means of linear extrapolation of the low- and high-temperature branches to the midpoint of the transition (dotted lines in Figure 3). The so obtained change of [u] is -0.06 cm/g. ComparlsonwitbLiterature. There are only a few experimental works in the literature where temperature dependenceof ultrasonic absorption and velocity in suspensionsof pure DPPC multilamellar vesicles in pure water are presented.gJ1.29 Our data were compared to those in the literature. The temperature curves are of the same shape. The magnitudes of the velocity and absorption at 25 OC are the same if one takes into account the concentration dependence (see Figure 2). A discrepancy is found for the peak value of]A.[ at the main transition which is in our work about 65% of that reported by Mitaku.ll Note that our definition of "absorption number" differs from that adopted by Mitaku by a factor of 27r. The definitive cause for this discrepancy is not straightforward; a possible explanation would be differences in the liposomes size, whose effect on relaxation properties has been shown to be remarkable.11J3 Moreover, in the work of Mitaku, 5 mM sodium phosphate buffer (pH 7.3) has been used instead of pure water as used here. It is w e l l - k n ~ w nthat ~ ~ . salts ~ ~ have notable effects on the phase transitions of phosphatidylcholine bilayers. Another possible explanation would be the fact that our measurements have been done at temperatures different from those in ref 11. Noting, however, that the [.A] values also differ significantlyin the s l o p of the transition peak,it becomes unlikely that the discrepancy is due to interpolation errors. Differences in sample purities can also play a certain role, as our and Mitaku's DPPC, respectively, were obtained from different commercial sources. 4. Relaxation Sound Velocity and Absorption Caused by Heterophase Fluctuatio~ The relaxation parts of the sound velocity and absorption for the high-temperature branch of the main transition ( T - To > 0.5, to be away from the transition midpoint and thus to insure the applicability of eq 12, see below) are presented in Figure 4. They were determined as the differences between experimental curves and the instantaneous parts approximated by the straight lines drawn through the experimental points at 50 and 55 OC (shown in Figure 3A and B). For the absorption, since the instantaneous part is negligible, it is assumed to be independent of temperature and to have a magnitude determined by the absorption values at those temperatures at which no relaxational process takes place. Effectiverelaxation times evaluatedby means ofeq 11 are(1-2) X lWs(Figure5),whichiswithinthespectrum of relaxation times obtained directly by acoustic spe~troscopy.2~ The relaxation time has a tendency to increase when the temperature approaches the transition midpoint (Figure 5 ) . This
Ultrasonic Study of DPPC Multilamellar Vesicles
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9841 A
B Y
0.31
1
I
T .To
Figure 4. Relaxation parts of [u] (A) and [ah](B) as a function of the
differencebetweenthemeasurementtemperature ( T ) andmain-transition temperature (To). The relaxation parts were evaluated by taking the differencebetweenthemeasuredvalues(at 7.2 MHz) andtheextrapolated instantaneous values (see Figure 3). The lines show the results of the model calculations made under two different sets of assumptions: the best fit given by set no. 3 in Table I (solid line); and set no. 2 in Table I (dashed line). 2
-1
..
2
3
4
5
6
7
I 8
T .To Figure 5. Temperature dependence of the effective relaxation time determined through eq 11. The solid line shows the calculated curve (set no. 1 in Table I).
effect known for density fluctuations around the main-transition temperature” is due to a steep temperature dependence of the size distribution of clusters of the new phase. Unfortunately it was not possible for the low-temperature branch to be studied by the same type of analysis because of the difficulty in approximating the instantaneous sound velocity in this region. 5. Interphase Line Tension and the Rate of Nucleation The phase-transition order is closely related to the interphase line tension. Until now, to our knowledge, thisvalue has not been determined for lipid bilayers. This is done here by means of a fitting procedure of the nucleation model described below to the experimental data presented in Section 4. In this calculation, the line tension, 7 , and the backward rate constant, bo, have been
used as adjustable parameters, as follows from the proposed physical model. 5.1. Model of Nucleation: Basic Framework. Density fluctuations occurring in the vicinity of a transition temperature can bedescribed as heterophase fluctuations, i.e. spontaneouscreation of nuclei of a new phase within the parent phase.16J7 The size distribution of the nuclei can be determined with the Frenkel semiempirictheory where the standard Gibbs chemical potential of the cluster of the new phase is given by the sum of a “volume“ term (b,proportional to the number of molecules contained in a cluster of the new phase) and an “interfacial” term (lS, proportionalto the perimeter of thecluster). Themainstatements and assumptions of the model used in this work are listed below: 1. If the mole fraction of lipids contained in all the clusters is assumed to be small, one can write for the concentration of the clusters16
where Nn is the number of the clusters containing i molecules, N is the total number of molecules in the system, N, is Avogadro’s number, and k is the Boltzmann constant. 2. A cooperative coupling between the monolayers is assumed. This means the cluster growth is considered as a sequential adjusting of pairs of lipids placed in two adjacent monolayers. 3. The minimum cluster of heterophase fluctuations is assumed to consist of seven pairs of molecules in the membrane plane. This is the size at which at least one molecule is surrounded by molecules of the same sort in a hexagonal lattice, which is the case for the rippled gel phase, Ps’, of DPPC below the maintransition temperat~re.3*-3~Fluctuations of smaller radius of correlation should rather be considered as homophase fluctuations which take place in the whole range of existence of the stable parent phase. Therefore, i = 7,8,9,...,N. An indirect argument for this assumption on the minimum size is that in condensed matter the distortion of structure caused by an interphase boundary takes place mainly in the nearest monomolecular layer at the boundary. For example, hydration effect of solute on the water structure is known to take place mainly within the first coordination sphere around the solute (cf. ref 35). 4. The “volume” terms of chemical potential, clv, and of other extensive characteristics of clusters, are calculated under the assumption that the partial molar values contributing to the “volume” terms are the same as those for the whole phase transition in an infinitesimal system. This assumption seems to be reasonable for the internal molecules of a cluster, which are surrounded by molecules of the same sort, but not for the external ones (subscript “ext”), which are surrounded by molecules of the parent phase. This is why the computations in our work were performed under three arbitrary assumptions about the contribution of the external molecules: the contribution is ( i )the same as that of internal ones, (ii) halfthat of the internal molecules, and (iii) zero. Therefore, where AH and A S are partial molar transition enthalpy and entropy, respectively,in an infinitesimal system; Toisthe transition temperature; and btfis the effective number of “internal” pairs of molecules, which is determined by the formula
ncff= n - (1 - e)nex, (14) where n is the number of pairs of lipid molecules in a cluster; next is the number of external lipid pairs (see below); and 8 is equal to 1, l/2, or 0, respectively, depending on the assumption of the contribution of external molecules adopted. 5. The interphase line tension is assumed to be constant all around the cluster which is considered to be of circular shape. The “interfacial” term of the chemical potential is, therefore,
9848 The Journal of Physical Chemistry, Vol. 97, No. 38, 1993
Pressure derivatives bxi@,t)/bp used as a measure of the pressure-induced distortion can be determined by
calculated as follows: clS = y 2 7 ~ N , ( m ~ / 7 ~ ) ~ / ~ (15) where y is the line tension; sa, the surface area of a lipid molecule in the plane of the bilayer; and n, the total number of pairs of molecules in the cluster. Also, for next
next= 2 ? r [ ( n / ~ )-' / ~ (16) 6. Kinetics of nucleation and growth of clusters of the new phase can be modeled by a chain of consecutive reversible firstorder reactions which correspond to the attachment of a pair of lipids from the parent phase to a developing cluster fl
fl
f2
fn-I
c, e c, e c, e ... e ,c, bl
bz
b3
6-1
f#
c,
F!
(17)
bn
This is a closed system of reactions which represents a thermodynamic equilibrium state. The scheme is similar to that used in ref 36 for the descriptionof phase transition, the only difference being the definition of the rate constants. The backward rate constantsare assumed to be proportional to the number of external molecules in the cluster, because the larger the number of external molecules, the larger the probability of one of them to leave the cluster
bi = boncxt,r (18) where 6, serves as an adjusting parameter (the physical meaning of bo is the probability of a single molecule pair to leave the cluster during a unit of time). The forward rate constants are determined from the equilibrium condition,ft = bixi/xi-,, where the concentrations of the clusters, xi, are calculated by eqs 1216. The kinetics of the system (eq 17) is described by the following system of differential equations:
x,
=o
. ;i
= ffli-1
Kharakoz et al.
- (bi
+ fi+l)*i
(23) A K = W t- (Cu,/PC,)~ where AV, is the isothermal molar volume effect of the phase transition in an infinitesimal system, per pair of lipid molecules; a, and C, are the thermal expansion coefficient and specificheat capacity at constant pressure, respectively; and p is the density of the suspension. Thevalues 7-and illare determined by means of diagonalization of the matrix A with use of the values 6xJ6p (as given by eq 22) as initial conditions, and using the parameters listed below (taken from refs 43, 37, and 38). All the molar extensive parameters are given in terms of pairs of lipid molecules to account for the cooperative coupling between the monolayers. transition temperature T = 41.3 O
C
molar transition enthalpy AH = 2 X 8.6 kcal/mol volume of a pair of molecules = 2(1200 x
cm3
area of a lipid molecule of a cluster in plane s, = 40
X
cmz
relative partial volume effect of transition = 0.037 (We used standard options of the PC-MATLAB computer program for the matrix diagonalization.) It should be noted that sincexfdepends exponentiallyon y (see eqs 12and 1S), thevalues of aredi directly related to this parameter (see eqs 21 and 22) and that, apart from the backward rate constant parameter (eq 18), all the other quantities appearing in this calculation are defined in the asumptions made for the nucleation model. This means that the line tension and the backward rate constant are theonly free parameters, and therefore the adjustable ones in the fitting procedure. For each relaxation time T ] one can find the relaxation partial compressibility measured at a given circular frequency w:
+
KrJ = 2/3R1(1 (24) where the relaxation strength Rj can be calculated from eq 9 if one takes into account that each particular component of the maximum relaxational compressibility is determined as
J
+ bi+lxi+l
where x, = 1. The system can be written in a matrix form
k=Axj;.
6xi(p,t)/ 6 p = -xincff,iAV,/N,k T (22) where AV, is the adiabatic molar volume effect which is given by
(20)
where A is the matrix of coefficients of the equations. 5.2. Model of Nucleation: Acoustical Consequences. In the case of a pressure-induced displacement of the equilibrium, the solution of the system (eq 19) can be given in the followinggeneral form: N
-(6VDP)iJ = -(AY,neff,i)(~x/6Pi)i]= - ( A m f f , i ) i i J where AV,n,, stands for the volume effect of formation of the cluster of the size nl. Then, N
R]= ( - A K / 2 B w ~ n c f f , i t , J
Finally, the total relaxational sound velocity and absorption numbers are calculated as additive functions of the particular contributions: N
where tIJare the amplitudes of the exponential components of the relaxation of X I ; t is time; and N is the number of steps of the cluster growth taken into account. It has been found that Ncan be practically limited by 30 because, due to their negligible concentration, the contribution of further steps to acoustic properties is negligible.
(25)
i= I
N
N
N
and the effective relaxation time, T,R, is calculated by means of eq 11.
Ultrasonic Study of DPPC Multilamellar Vesicles
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9849
TABLE I: Values of Interphase Line Tension y Obtained from the Fitting of the Experimental Data Using the Proposed Model, and the Rate Constants for Shrinking of the Cluster assumptions results minimum cooperative contribution of Y 60 least squares, set no. cluster sizeb coupling external molecules ( 8 ) c (IO-' erdcm) S-1 arbitrarv units ~
~ _ _ _ _ _ _ _
~~~
1 2 3 4
I 7 7
4
on on on on
1
'I2 0 '12
5.55 f 0.07 5.2f 0.1 4.5 f 0.3 6.4f 0.2
37f3
0.6
56 f 9
3.7
130 f 30
15 3.5
45 f 5 48 f 10
5 7 off 'I2 7.7 f 0.3 6.3 4 The errors of the adjusting parameters were estimated by calculation of the possible values in the 7-bo space when the least mean squares are doubled. Expressed in number of pairs of lipid molecules. Partial molar contribution of external molecules of a cluster was chosen to be equal to that of the internal molecules (e = l), half that of the internal molecules (e = l / z ) , or zero (e = 0). See eq 14.
The values of [u],, [ak],, and 7,fl calculated from the model this way were fitted to the experimental data on relaxation sound velocity and absorptionby means of the least mean square criterion (see Table I and Figures 4 and 5). As said before, since in this fitting procedure the interface line tension and the backward rate are the adjustable parameters, these quantities are so evaluated. From the satisfactory fit obtained as shown in Figures 4 and 5, this procedure appears to be adequate with respect to the validity of our above model assumptions and also to yield good precision in the resultingvalues of y and bo. The uniqueness of this solution, however, will have to be further tested. One possibility is atrialand-error search for other possible solutions in the 7-6, space; the other, perhaps more convincing one, would involve additional boundary conditions from experiments with variable acoustic frequencies. As discussed below, however, our model allows prediction of the frequency dispersion of the absorption, which is indeed in very good agreement with data from the literature." 5.3. Assessment of the Model. To test the sensitivity of the model to the assumptionsmade, different parameters (contribution of the external molecules, minimum size, cooperative coupling) were varied. The best fit has been obtained assuming the full contribution of external molecules to the "volume" term (0 = l), a minimum size of 7, and the existence of a cooperative coupling (see Table I and Figures 4 and 5 ) . All the other variations made the slope of the curves too shallow and therefore not adjustable to the experimental data. This is indirectly seen from the values of least mean squares presented in Table I. To illustrate this result, one example of the variations is presented in the figures by the dashed line corresponding to a change of 0 from 1 to 1/2. All the other variations gave a similar or even more pronounced deviation of the calculated curve's shape from the experimental one. The followingconclusionscan be made: First, a cooperative coupling between monolayers takes place. Whether this is due to a specific interaction of the polar heads and hydration water between two adjacent monolayers in the multilamellar system or between opposite monolayers in a bilayer cannot be established with the present data. It should be noted that an application of our model to data on large unilamellar vesicles (large to preclude effects of curvature strain in the transition characteristics) could give more insight in this respect. Unfortunately, the complete set of data necessary for the fitting procedure is not available at present. Second, the minimum size of 7 has been properly chosen. And third, external molecules of a cluster contribute fully to the "volume" terms. The last conclusion has to be commented upon. Certainly, the interface between a cluster and the parent phase is not infinitesimally narrow, and the properties of the external molecules of the clusters are more fluid-like. But this deviation of the external molecules from the real gel state is compensated by the same effect occurring in the surrounding molecules of the parent phase which should be slightly more gel-like compared to the fluid phase. In other words,effectively,the external molecules do not have a full volume contribution. However, since the surrounding molecules belonging to the parent phase also contribute to the total cluster volume, the two contributions sum
up. The result is that, apparently, the external molecules have a full volume contribution. The resulting best value of 0 depends on the accuracy of the evaluation of the relaxational part of sound velocity from the experimental data, made by means of extrapolation of the instantaneous sound velocity from a short-range temperature above 50 'C. The shape of the temperature-dependent curve of [ u ] , is sensitive to the slope of the straight line approximating the instantaneous sound velocity (Figure 3A). Nevertheless the estimation of the line tension is not very sensitive to 0. The cooperativecoupling and the minimum size assumptions are more relevant to the estimated value of the line tension. The errors of the adjusting parameters were estimated by calculating the possible values in the y-bo space when the least mean squares are doubled. If one compares the results from the different sets of assumptions, one can see that the actual divergences are larger than the calculated errors. Considering the data in Table I, it can be seen that the most probable values for the evaluated parameters are y = 5.6 X le7erg/cm and bo = 40 ps-l (Table I). Through this analysis it has been possible to determine the fluid-gel interphase line tension in the melted lipid bilayer. Recently a work has appeared where the fluid-gel transition in Langmuir monolayers of fluorescence-labeled stearic acid has been studied'* by fluorescence microscopy. The interphase line tension was calculated from the rate of nucleation and found to be (4 f 1.5) X le7 erg/cm. The result is in good agreement with ours if one takes into account considerable differences in the systems under study (phospholipid multibilayers and fatty acid monolayers) and in the methods used for the measurements. The line tension calculated per a single pair of external lipid molecules is about 0.7-0.8 kT. This value is useful to give an estimation of the energetic price of creation of an interphase line. The second fitting parameter bo, with the value 40 ps-', has the physical meaning of the probability that a given external molecule leaves the cluster in a unit of time and therefore may lend itself for further analysis of diffusional properties and phase propagation. The kinetic behavior of the model adjusted to the singlefrequencyacoustic data can be compared to acoustic spectroscopy data taken from the literature." Figure 6A shows the calculated components of the spectra of acoustic relaxation caused by heterophase fluctuations at different temperatures. The total spectrum of sound absorption is a result of superposition of the particular components. The result is presented in Figure 6B by dashed lines. The shape of the absorption band caused by the heterophase nucleation is close to that of a single relaxation time process (dotted line), but the band width is about 30% broader. Therefore single relaxation time approximation used in the literature for the description of this process is not fully adequate (ref 29). The "asterisk-made" line in Figure 6B corresponds to the contribution of trans-gauche transformations calculated as a singlerelaxation process.& The solid lines are the resulting spectra (the sum of the partial contributions of the heterophase fluctu-
9850
Kharakoz et al.
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 A
0.09
,
,
, , , , , , ,
T - To (dcg) =
,
' , ' , ' , ,
,
,
'
I
"
.
'
.
,
,
, . . , '
0
0.08 I
RELAXATION FREQUENCY,HI
B
.--
Nucleation
*****
Trsnr-puck
.....
the sum of (- - -)and
(***I
Appmximntion by single
relaxationtimc
FREQUENCY,Hz
Figure 6. Spectralcomponentsof acoustic relaxation (A) and of ultrasonic absorption spectra (B) of heterophase fluctuations calculated by means of the developed model: dashed lines, heterophase fluctuations; "asterisksmade" line, trans-gauche transformation; solid lines, resulting spectrum ofheterophase fluctuations and trans-gauche transformations. Thedotted line shows a hypothetical single relaxation process for comparison with the real shape of the heterophase fluctuations spectrum. The spectrum at T = To corresponds to a hypothetical metastable situation in which the membranes are fluid.
the presented model. The values are higher than that of T ~ R . This fact should not cause confusion: this difference is due to the different ways of defining the two quantities which coincide only in the case of a single relaxation process. We stress again that both relaxationtimes given here result from the interplaybetween the many components of the spectrum of single relaxation processes. We wish to emphasize also that the relaxation times we refer to in this work are concerned with only one of the many molecular and supramolecular processes apparently involved in the main transition of DPPC. The local cluster volume fluctuations probed by the acoustical approach may justifiably be regarded as the essential factor in the transition process, as has been pointed out by Bilt~nen.'~The good agreement of our present data with those reported earlier by Mitaku et al.11 reinforces the confidence that this cluster fluctuation occurs characteristically in the time scale between 60 and 10 ns (Figure 7). Faster and slower processes have been probed by a host of other techniques (for reviews see refs 3, 5 , 6, and 40). Some of these processes are even found to be only very slowly reversible.41 A complete descriptionof the main transition of DPPC will therefore require an integrated presentation of all these processes, which is not yet available. Therefore, further experimentaland theoretical efforts should be made in order to extend this study to the whole main transition of DPPC and other transitions.
Acknowledgment. We thankDr. D. Tikhonov, Dr. G. Sarkisov, and Dr. A. Finkelstein for helpful discussions of some aspects of the model and Prof. A. Sarvazyan,who initiated this collaborating work. We also want to thank the Austrian Academy of Sciences for financial supportunder the scheme of the East-West Program. Support by the Osterreichischer Fonds zur Farderung der wissenschaftlichen Forschungunder Grant n. S4614 is gratefully acknowledged. References and Notes (1) Cullis, P. R.; Hope, M. J.; de Kruijff, B.; Verldeij, A. J.; Tilwk, C. P. S.In Phospholipids and Cellular Regulations; Kou, J. F., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. I, p 1. (2) Shnol, S. E. Physico-chemical Factors of Eiological Evolution; Hardwood Academic PublishersGmbH Amsterdam, TheNetherlands, 198 1; Chapter 11. (3) Bloom, M.; Evans, E.; Mouritsen, 0. G. Q. Rev. Eiophys. 1991,24, 293. (4) Kinnuen, P. K. J. Chem. Phys. Lipids 1991,57, 375.
( 5 ) van Osdol, W. W.; Biltonen, R. L.; Johnson, M. L. J. Eiochem. Eiophys. Methods 1989, 20, 1. (6) Laggner, P.; Kriechbaum, M. Chem. Phys. Lipids 1991, 57, 121. (7) Eggen, F.; & Funck, T.Natunvissenshaften 1976, 63, 200. (8) Sarvazyan, A. P. Annu. Rev. Eiophys. Eiophys. Chem. 1991,20,321. (9) Mitaku, S.; Ikegami, A.;Sakanishi, A. Eiophys. Chem. 1978,8,295. (10) Mitaku, S.;Date, T.Eiophys. Eiochim. Acta 1982, 688, 411. (11) Mitaku, S.;Jippo, T.;Kataoka, R. Eiophys. J. 1983, 42, 137. (12) Harkness, J. E.; White, H . D. Eiophys. Eiochim. Acta 1979, 552, T - TO
Flgure 7. Temperature dependence of relaxation times evaluated from the frequencies of the peaks of the theoretical absorption bands (shown in Figure 6B), solid line, compared to those observed by ultrasonic spectroscopy,ll circles. Crosses show the experimental data" at the lowtemperature side of the main transition, not being compared here.
ations and trans-gauche transformations) and have a shape very similar to that of the experimental spectrum obtained by Eggers and Funck in the frequency range 15-145 MHz. Acoustic relaxationtime is usually determined in experimental studies, assuming a single relaxation time, as the reciprocal of the circular frequency of the absorption maximum. Here this quantity is called apparent relaxation time, T ~ to ~distinguish , from the effective one, T ~ R defined , by eq 11. The apparent relaxation times obtainedfrom the spectra of the model of nucleation (Figure 6) are presented in Figure 7 as a function of temperature. The temperature dependence is well consistent with the direct experimental data taken from ref 11. The excellent agreement of the calculated spectrum with the experimentaldata reinforces
450. (13) Maynard, V. M.; Magin, R. L.; Dunn, F. Chem. Phys. Lipids 1985, 37, 1. (14) Sano, T.;Tomaka, J.; Yasumaga, T.; Toyoshima, Y.J . Phys. Chem. 1982, 86, 3013. (15) Buckin, V . A.; Sarvazyan, A. P.; Passechnik, V. I. EiofIrika 1979, 24, 61. (16) Frenkel, J. Kinetic Theory ofliquids; Dover: New York, 1946; p 366.
(17) Ubellohde, A. B. Melting and Crystal Structure; Clarendon Press: Oxford, England, 1965; Chapter 11. (18) Muller, R.; Gallet, F. Phys. Rev. Lett. 1991, 67, 1106. (19) Sarvazyan, A. P.;Kharakoz, D. P. Instrum. Exp. Tech. 1981, 24, 782.
Sarvazyan, A. P. Ultrasonics 1982, 20, 151. Eggers, F.; Funck, Th. Rev. Sci. Instrum. 1973,44, 969. Sarvazyan, A. P.; Chalikian, T.V . Ultrasonics 1991, 29, 119. Kharakoz, D. P. J. Phys. Chem. 1991,95,5634. (24) Stuehr, J.; Yeager, E. In Physical Acoustics; Mason, W. P., Ed.; Academic Ress: New York and London, 1965; Vol. 2, Part A, Chapter 6. (25) Kharakoz, D. P. J. Acousr. Soc. Am. 1992,287. (26) Salsbury, N. J.; Darke, A.; Chapman, D. Chem. Phys. Lipids 1971, 8. ~,142. -. (27) Janiak, M. J.; Small, D. M.; Shipley, G.; Graham. Biochemistry 1976, 15,4515. ~
Ultrasonic Study of
DPPC Multilamellar Vesicles
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9851
(28) Lentz, B. R.; Freire, E.; Biltonen, R. L. Biochemistry 1978,17,4475. (29) Eggers, F.; Funck, Th. Be?. Bunsen-Ges. Phys. Chem. 1978,82,927. (30) Chapman, D.; Peel, W. E.; Kingston, B.; Lilley, T. H. Biochim. Biophys. Acta 1977, 464, 260. (31) Akutsu, H.; Seelig, J. Biochemistry 1981, 20, 7366. (32) Tardieu, A.; Luzzati, V.; Reman, F. C. J. Mol. Biol. 1973, 75,711. (33) Ranck, J. L.; Mateu, L.;Sadler, D. M.; Tardieu,A.; Gulik-Krzywicki, T.; Luzzati, V. J. Mol. Biol. 1974, 85, 249. (34) Nagle, J. F.; Wiener, M.C. Biochim. Bfophys. Acta 1988, 942, 1 . (35) Buckin, V. A.; Sarva2yan.A. P.; Kharakoz, D. P. In Waterin Disperse Systems; Churaev, N. V., Ovcharenko, F. D., Eds.;Khimiya: Moscow, 1989; p 45. (36) Tsong, Y.; Kanehisa, I. Biochemistry 1977, 16, 2674. (37) Laggner,P.; Lohner, K.; Degovics, G.; Muller, K.; Shuster, A. Chem. Phys. Lipids 1987, 44, 31. (38) Wiener, M. C.; Tristram-Nagle, S.;Wilkinson, D. A,; Campbell, L. E.; Nagle, J. F. Biochim. Biophys. Acta 1988, 938, 135. (39) Biltonen, R. L. J. Chem. Thermodyn. 1990, 22, 1. (40) Caffrey, M. Annu. Reu. Biophys. Biophys. Chem. 1989, 18, 159. (41) Tenchov, B. Chem. Phys. Lipids 1990, 57, 165. (42) Kell, G. S.J . Chem. Eng. Data. 1975, 20, 97. (43) Posch, M.; Rakusch, U.; Mollay, Ch.; Laggner, P. J . Biol. Chem. 1983, 10, 1761. (44) Genz, A.; Holzwarth, J. F. Colloid Polym. Sci. 1985, 263, 484. (45) Cevc, G.; Marsh, D. Phospholipid Bilayers. Physical Principles and Models; Wiley: New York, 1987; Chapter 1. (46) Partial compressibility, K, of vesicles and its changes upon the transitions are calculated from the sound velocity numbers [u] and partial volumes, V,using the followingequations, which arevalid for dilutesolutions:8
K
+
B(2V- 2 [ ~ ] l / p )
(28)
AK = 2@(AV- A[u]) (29) where p is the density of water. The data of p were taken from ref 42, partial volumes were taken from ref 43, and the 6values were calculated from sound velocities by means of the Laplace equation, @ = l/pu2. It results that LW = 8 X 1od cm3 bar1 g*for the main transition. Upon a pressure perturbation like the one induced by the propagation of ultrasound, the compressibility of the system changes due to a large volume variation associatedwith a shift of the gel-liquid phase equilibrium. A contribution to this compressibility change from trans-gauche transformations taking place on a nanosecond time scaleU is expected, since these transformations are significant for the comprcssibility in the fluid state. The volume effect of formation of one kink in a hydrocarbon chain is about 25 A3,45 which compares to the experimental volume effect of the main transition, 22 A3 per chain.43 This indicatesan average fraction of x = 0.9 for the transformed hydrocarbon chains. Kink formation can be considered as a monomolecular two-state transition of hydrocarbon chains, and therefore the relaxation strength is expressed as follows:24
RI-
Av2
@NakT4(' - X)/W
(30)
where AYis the volume effect of the kink formation and M the molecular weight per hydrocarbon chain. The relaxation time of the trans-gauche transformation studied by laser temperature jump has been found to be 4 ns.32 However, according to acoustic s p c c t r ~ s c o p ythe ~ ~ acoustic relaxation time is lower than 0.3 ns. Therefore we used the latter value in our calculations. The value [u], calculated with eqs 7 and 30 for the frequency 7.2 MHz and T = 1 ns is equal to 4 . 0 2 5 cm3/g if the above value x = 0.9 is assumed.
