Lipid monolayers have been shown to represent a powerful tool in studying
mechanical and thermodynamic properties of lipid membranes as well as their
interaction with proteins. Using Einstein's theory of fluctuations we here
demonstrate, that an experimentally derived linear relationship both between
transition entropy S and area A as well as between transition entropy and
charge q implies a linear relationships between compressibility \kappa_T, heat
capacity c_\pi, thermal expansion coefficient \alpha_T and electric capacity
CT. We demonstrate that these couplings have strong predictive power as they
allow calculating electrical and thermal properties from mechanical
measurements. The precision of the prediction increases as the critical point
TC is approached

A. Wixforth

C. L. Leirer

D. Steppich

E. Sackmann

H. Möhwald

J. Griesbauer

L. D. Landau

M. F. Schneider

O. Albrecht

T. Frommelt

T. Shimojo

W. Appelt

English

arXiv

Steppich, Daniel

Griesbauer, Josef

Frommelt, Thomas

Appelt, Wilhelm H.

Wixforth, Achim

Schneider, M. F.

OPUS Augsburg

Thermomechanic-electrical coupling in phospholipid monolayers near the critical pointD. Steppich,1 J. Griesbauer,1,2,* T. Frommelt,1 W. Appelt,1 A. Wixforth,1 and M. F. Schneider2,†1Experimental Physics I, University of Augsburg, Universitätstsr. 1, D-86159 Augsburg, Germany2Mechanical Engineering, Boston University, 110 Cummington Street, Boston, Massachusetts 02215, USAReceived 22 January 2010; revised manuscript received 24 March 2010; published 15 June 2010Lipid monolayers have been shown to represent a powerful tool in studying mechanical and thermodynamicproperties of lipid membranes as well as their interaction with proteins. Using Einstein’s theory of fluctuationswe here demonstrate that an experimentally derived linear relationship both between transition entropy S andarea A as well as between transition entropy and charge q implies a linear relationships between compress-ibility T, heat capacity c, thermal expansion coefficient T, and electric capacity CT. We demonstrate thatthese couplings have strong predictive power as they allow calculating electrical and thermal properties frommechanical measurements. The precision of the prediction increases as the critical point TC is approached.DOI: 10.1103/PhysRevE.81.061123 PACS numbers: 05.70.Np, 68.18.g, 68.35.MdI. INTRODUCTIONLipid monolayers have been of major interest for a varietyof reasons. They contain many interesting aspects of physicsin two dimensions, allowing us to study intermolecular inter-actions between lipids and/or proteins and mimic one leafletof a biomembrane in which proteins can be incorporatedfairly easy. Moreover, in contrast to most thin film physicsexperiments, the majority of measurements can be performedwithout a vast amount of technical equipment and prepara-tion 1. The prepared monomolecular film can then be trans-ferred on a solid support for further investigation 2. Mostmacroscopic conditions pressure, temperature, concentra-tion, etc. are easily accessible and thermodynamic proper-ties can be measured straight forwardly. Furthermore, usingthe appropriate thermodynamic relationships, the measuredquantities can be related to the heat of transition and severalresponse functions like compressibility, thermal expansioncoefficient, electrical capacity, and, as will be shown in thispaper, even heat capacity.As was demonstrated earlier for lipid vesicles 3, thereexists a linear relationship between the excess isothermalcompressibility T and the heat capacity cp at constantpressure,T  cP.This relationship implies a pronounced maximum of thecompressibility in the phase transition regime, following thewell-known profile of cp. However, there is no physicalexplanation of the origin of this relationship to date.Here, we show that the experimentally established propor-tionality between entropy and area and entropy and charge incombination with Einstein’s theory of fluctuations conse-quently leads to linear relationships between compressibilityT, heat capacity c, thermal expansion coefficient T, andelectric capacity CT. This allows to calculate the correspond-ing heat capacity of lipid monolayers which is shown to be ingood agreement to calorimetric data found for lipid vesiclesuspensions 3.II. MATERIAL AND METHODSThe phospholipids DMPC 1,2-Dimyristoyl-sn-Glycero-3-Phosphocholine, DPPC 1,2-Dipalmitoyl-sn-Glycero-3-Phosphocholine, DMPG 1,2-Dimyristoyl-sn-Glycero-3-Phospho-rac-1-glycerol, and DOTAP 1,2-Dioleoyl-3-Trimethylammonium-Propane were purchasedfrom avanti polar lipids Al, USA. Lipids were dissolved inpure chloroform and used without further purification. Formeasuring pressure-area and potential-area isotherms weused film balances of Nima Technologies Coventry, En-gland, which were partly equipped with a Kelvin Probe Sur-face Potential of TREK New York, USA for measuring thesurface potential of the lipid films. 500 ml of double distilledwater is filled into a trough and the lipid-chloroform solutionis spread onto the water surface. After evaporation of thesolvent the film was compressed with a speed of approxi-mately 3 cm2 /min or when accounting for the through size0.410−16 /min per lipid. At the same time the lateral pres-sure of the surface is measured using a Wilhelmy plate andthe surface potential is recorded if desired. To calculate thecompressibility we subdivided the pressure-area curve inthree sections and fitted each section by a polynomial func-tion of seventh degree or higher to assure only minimal de-viations less than the experimental error bar from the ex-perimental curve. The fits were then merged together toreach continuously differentiable plots. For conserving thestronger fluctuations of the surface potential curves, numeri-cal derivations were made for calculation of the electricalcapacity.*Corresponding author for electrical measurements and experi-ments: University of Augsburg Experimentalphysik 1 Univer-sitätsstr. 1 86159 Augsburg Germany;Josef.Griesbauer@physik.uni-augsburg.de†Corresponding author: Boston University Mechanical Engineer-ing, 110 Cummington Street, Boston, MA, 02215 USA;mfs@bu.eduPHYSICAL REVIEW E 81, 061123 20101539-3755/2010/816/0611235 ©2010 The American Physical Society061123-1III. RESULTS AND DISCUSSIONA. Experimental resultsLinear relationship among area, entropy, and chargeIn Figs. 1a and 1b a set of pressure-area isotherms isshown along with the calculated isothermal compressibility4,T = −1A AT. 1Here, A is the area per molecule and  the lateral pressure,which might not be mistaken with the global pressure p asdenoted in cp. Additionally a set of surface potential-areaisotherms −A are plotted in Fig. 1c to demonstrate theirsimilarity with the typical pressure-area isotherms Fig. 1aespecially in the phase transition regime, whereas this is awell-known behavior 5–7.From Figs. 1a and 1b, the change in phase transitionpressure t as a function of temperature T can be calculatedand is plotted in Fig. 2a. A linear relationship tT as hasbeen reported before 8,9 can be resolved. Interestingly thesame behavior is resembled for the transition potentialstT shown in Fig. 2b indicating a more general characterof the observed linearity, with a coupling of the thermody-namic observables studied.When examining the pressure-area data over a wider tem-perature range, one finds that this linear relationship is morepronounced in the high-temperature regime as the criticalpoint is approached but departs clearly for lower temperatureFig. 2a. This has been found earlier, e.g., by Albrecht etal. 8 and Hato et al. 10. Qualitatively, the observed lin-earity seems to be independent of the nature of the lipid headgroup and could be reproduced for different lipids DMPC,DMPG, and DOTAP and lipid mixtures of three differentlipids Fig. 2a. The linear character is further supported byrecent results 11 where the same qualitative relationshipwas found for a whole set of synthesized glycolipids contain-ing different size sugar lactose headgroups and no phos-phoric acid.(b)(a)(c)FIG. 1. a and b DMPC pressure-area isotherms measuredfor different temperatures. For the sake of clarity only four iso-therms are shown. From the raw data of the upper panel, the iso-thermal compressibility is calculated, showing a pronounced maxi-mum during the phase transition regime lower panel. c DMPCpotential-area isotherms measured for different temperatures. Theregion around phase transition is plotted in a separate inset forevery temperature, demonstrating the disappearance of the plateauwith increasing temperature.(b)(a)FIG. 2. a Plot of the transition pressure t of the phospholipidsDOTAP, DMPC, DMPG, and a multicomponent system consistingof 40 mol% DMPC, 40 mol% DMPG, and 20 mol% DPPC as afunction of temperature T. A linear relationship tT is clearlyconserved for all systems. b Transition Potentials t of DMPC asa function of temperature. Clearly, the surface potential tT in-creases linearly with temperature in transition regime.STEPPICH et al. PHYSICAL REVIEW E 81, 061123 2010061123-2B. Theoretical resultsThe last section has experimentally shown that the changein phase transition pressure t and potential t with tem-perature T occurs linear in the regime of the critical point. Asa consequence, the change in area, charge, and entropy dur-ing the phase transition are linearly related according to theClausius-Clayperon equation 5: tTTrans=SA= const = B, 2a tTTrans=Sq= const = B 2band thereforeS = BA , 3aS = Bq . 3bFor the transition pressures this relationship seems to be wellconserved and becomes more accurate in the vicinity of thecritical point.When approaching the critical point the double minimumof the free energy gradually disappears and changes into asingle rather flat minimum. In the near vicinity the potentialcan therefore be approximated by S=S0+S /xixi+2S /xixjxixj. The corresponding fluctuations arexixj = − kB	 2Sxixj−1. 4Here, xi and xj are extensive thermodynamic variables area,charge, particle number, etc., kB is the Boltzmann constant,and Sxi ,xj is the proper thermodynamic entropy potential.We have recently shown 12,13 that S of the monolayer canindeed be considered as decoupled from the bulk, whichholds for adiabatic properties as well as for reversible fluc-tuations down to the nanoscale. Now using Eq. 4, one gets	A2 = AkBTT, 5a	q2 = kBT qT,A = kBTCT, 5b	S2 = kBc, 5cwhere CT is the electrical capacity. For the fluctuation corre-lations between area and entropy	S	A = AkBT, 6a	S	q = kBT qT,A. 6bIn other words, area fluctuations are proportional to the iso-thermal compressibility T, entropy fluctuations are propor-tional to the isobaric heat capacity c and charge fluctuationsare proportional to the electrical capacity CT. The correlatedfluctuations are proportional to the isobaric area expansioncoefficient  or the derivation of the chargeq /T,A. Welike to note here that while Eqs. 5 and 6 are acceptedtextbook representations, the physically interpretation of afluctuating entropy potential Eqs. 6a and 6b is physi-cally not clear to these authors.Under the assumption that the linear relationship in Eq.3 holds also for small fluctuations 	 one finds usingEqs. 3, 5, and 6:c = B2 ATT, 7c = BAT, 8c = B2 TCT, 9and thereforeT    c  CT. 10Partly, for thermal and mechanical properties, this was al-ready put forward by Heimburg Eq. 3 for lipid vesiclesuspensions. He demonstrated that one can assume A	A when the “excess” heat capacity c and compressibil-ity T are proportional for a wide T range, the latter ofwhich has been derived experimentally.Equation 10 states, that the excess response functionsexhibit a simple linear relationship, which predicts, accord-ing to Fig. 1, a pronounced maximum in area expansioncoefficient, the electrical capacity, and the heat capacity closeto the phase transition regime.In the following section we will compare our theoreticalpredictions with our experimental results.C. Discussion1. Coupling between thermal and elastic properties during phasetransitionIn Fig. 3a the area per molecule as a function of T isplotted for three different  values. When calculating thecorresponding  using =1 / ĀA /T Fig. 3b a pro-nounced maximum with a shoulder on the low-temperaturesite and a steep decrease in the liquid expanded phase isfound. This is very similar to the experimental results of TFig. 1b, where we also resolve a shoulder toward theliquid condensed phase and a steep change in T when leav-ing the liquid expanded phase. The enormous predictivepower of the Eqs. 7–10 can be demonstrated by calculat-ing the thermal expansion from the compressibility usingT=− /B. When taking for example Tmax at 13 °C285 K and 23 mN/m Fig. 1b, we find 0,4 K−1,which is close to max=0,35 K−1 at 13.6 °C extractedfrom the expansion measurement directly Eq. 3b. Thisquantitative and qualitative agreement indicates that the ex-trapolation from A to area fluctuations 	A used to deriveEq. 10 is well justified.2. Coupling between electrical and elastic properties duringphase transitionAs  was measured for DMPC, the electrical capacity CTcan be calculated according to CT=dq /d=THERMOMECHANIC-ELECTRICAL COUPLING IN… PHYSICAL REVIEW E 81, 061123 2010061123-3−A /q,T /A,T where we have used the Maxwell re-lation: −A /q,T=q /A,T. By using tT to determineB from a linear fit Eq. 2b, CT can directly be comparedto T. Although the correlation is more pronounced in Figs.3a and 3b, the coupling is clearly visible in Fig. 3c aswell.3. Heat capacity of lipid monolayersThe isobaric heat capacity c compare to cp as measuredby differential scanning calorimetry DSC for small unila-melar lipid vesicles SUVs of lipid monolayer is not di-rectly experimentally accessible 12. However using the re-lation shown in Eq. 10 we are now able to calculate cTfrom  14.The result plotted in Fig. 3b contains a pronouncedmaximum, which qualitatively resembles the heat-capacityprofile of a lipid vesicle suspension. Remarkably, consider-ing the calculated monolayer heat-capacity profile at 32mN/m both the half width at half height HWHH as well asthe absolute value of the maximum in cT agree within20% with the heat-capacity profile of lipid SUVs of the sametype 15. Surprisingly, even the shoulder in the liquid or-dered gel phase known from DSC measurements is foundin our monolayer experiments. We believe that further theo-retical investigations on this observation might give insighton topological and pure in plane effects on the heat-capacityprofile of lipid suspensions such as SUVs since lipid mono-layers simply cannot undergo morphological transitions asopposed to lipid vesicles 16–19.4. Role of the critical pointThe general character of the linear relationship between, T, c, and CT calls for a more profound understandingof the origin of this linear relationship. Obviously, the char-acter of the first-order transition e.g., plateau width be-comes increasingly weaker as the temperature rises Fig.1a, a typical fingerprint for the existence of a critical point.The same holds when looking at the area jump in Fig. 3a:as the pressure is increased toward the critical point theabrupt change in area decreases. Indeed, it has been stated,that the lipid monolayer phase transition is only of weak firstorder in the vicinity of a second-order transition 20,21.In summary one can say, that all experiments indicate thetrend toward a critical point as the temperature is increased.Since the symmetry between liquid expanded and liquid con-densed phase remains beyond the critical point, the first-order transition must converge into a second-order transitiontricritical point 8. At this point the Clausius-Clayperonequation for second-order phase transition states 5:T=cAT11orT=cTCT. 12The vicinity of a critical point therefore implies cCT Eqs. 11 and 12 near the phase transition regime,once transition pressure t and potential t increase nearlylinear with temperature T.(b)(a)(c)FIG. 3. a By keeping the lateral pressure constant and varyingthe temperature, the area per molecule can be measured as a func-tion of temperature. During the phase transition, the slope of thegraph significantly increases. b From a the isobaric expansioncoefficient  can be calculated. Similar to the heat-capacity pro-files, calculated using Eq. 11, it displays a maximum in the phasetransition regime. The graphs display qualitatively and quantita-tively heat of transition very similar features as the heat capacityobtained by DSC for giant unilamellar vesicles. c Our relationsEq. 10 allow to directly compare the electrical capacity CT withthe mechanical compressibility T at 8 °C. The compressibility ascalculated from the electrical capacity CT using Eqs. 7–9 is plot-ted in the same graph as the actual measured using pressure-areaisotherms compressibility. Both curves show the typical finger-prints of a phase transition.STEPPICH et al. PHYSICAL REVIEW E 81, 061123 2010061123-4IV. CONCLUSIONSOur results reveal a thermal-mechano-electrical couplingin lipid monolayers. Using Einstein’s theory of fluctuationsin line with the experimental results, a linear relationshipbetween electrical capacity, heat capacity, compressibility,and the thermal expansion coefficient has been found. Theseresults enable us to calculate the heat capacity of lipid mono-layers which has been shown to be within reasonable rangewith the heat capacity known from lipid vesicle suspensions.The relationship established in this work carries an enormouspredictive power allowing to calculate heat and electricalproperties from mechanical measurements. In future, it willbe of enormous interest whether the same couplings hold formore complex or biological systems.ACKNOWLEDGMENTSM.F.S. likes to thank Dr. K. Kaufmann for stimulatingdiscussions and lectures concerning the thermodynamicfoundation of soft interfaces. Furthermore, we would like toacknowledge very helpful remarks from Professor ErichSackmann. Financial support is acknowledged from the Ger-man government through the Cluster of Excellence “NIM”and the DFG SPP 1313. M.F.S. and D.S. like to thank theFond der Chemischen Industrie as well as the ElitenetzwerkBayern CompInt.1 H. Mohwald, Annu. Rev. Phys. Chem. 41, 441 1990.2 E. Sackmann, Science 271, 43 1996.3 T. Heimburg, Biochim. Biophys. Acta 1415, 147 1998.4 L. D. Landau and E. M. Lifschitz, Course of Theoretical Phys-ics Butterworth-Heinemann, Frankfurt, 1987, Vol. 6.5 T. Shimojo and T. Ohnishi, J. Biochem. Tokyo 61, 891967.6 H. Beitinger, V. Vogel, D. Möbius, and H. Rahmann, Biochim.Biophys. Acta 984, 293 1989.7 V. Vogel and D. Möbius, J. Colloid Interface Sci. 126, 4081988.8 O. Albrecht, H. Gruler, and E. Sackmann, J. Phys. I 39, 3011978.9 H. Möhwald, Structure and Dynamics of Membranes Elsevier,Amsterdam, 1995.10 M. Hato, H. Minamikawa, K. Tamada, T. Baba, and Y. Tanabe,Adv. Colloid Interface Sci. 80, 233 1999.11 M. F. Schneider, G. Mathe, C. Gege, R. R. Schmidt, and M.Tanaka, J. Phys. Chem. B 105, 5178 2001.12 B. Wunderlich, C. Leirer, A.-L. Idzko, U. F. Keyser, A. Wix-forth, V. M. Myles, T. Heimburg, and M. F. Schneider, Bio-phys. J. 96, 4592 2009.13 J. Griesbauer, A. Wixforth, and M. F. Schneider, Biophys. J.97, 2710 2009.14 V. V. Plotnikov, J. M. Brandts, L.-N. Lin, and J. F. Brandts,Anal. Biochem. 250, 237 1997.15 E. Sackmann, in From Vesicles to Cells, edited by E. Sack-mann and R. Lipowsky Elsevier, Amsterdam, 1996, Vol. A.16 M. F. Schneider, D. Marsh, W. Jahn, B. Kloesgen, and T. He-imburg, Proc. Natl. Acad. Sci. U.S.A. 96, 14312 1999.17 C. L. Leirer, B. Wunderlich, V. M. Myles, and M. F. Schneider,Biophys. Chem. 143, 106 2009.18 C. L. Leirer, B. Wunderlich, A. Wixforth, and M. F. Schneider,Phys. Biol. 96, 6 2009.19 T. Franke, C. L. Leirer, A. Wixforth, N. Dan, and M. F.Schneider, ChemPhysChem 10, 2852 2009.20 O. G. Mouritsen, Biochim. Biophys. Acta 731, 217 1983.21 O. G. Mouritsen, Chem. Phys. Lipids 57, 179 1991.THERMOMECHANIC-ELECTRICAL COUPLING IN… PHYSICAL REVIEW E 81, 061123 2010061123-5

Thermomechanic-electrical coupling in phospholipid monolayers near the critical point

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Thermo-mechanic-electrical coupling in phospholipid monolayers near the critical point

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