Frequency spectrum of enthalpy fluctuations associated with ...

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OBDULIo L. MAYORGA*, WILLIAM W. VAN OSDOL, J. L. LACOMBA, AND ERNESTO FREIREt. Department of Biology, The Johns Hopkins University, Baltimore, ...
Proc. Natl. Acad. Sci. USA Vol. 85, pp. 9514-9518, December 1988 Biophysics

Frequency spectrum of enthalpy fluctuations associated with macromolecular transitions (multifrequency calorimetry/heat capacity/thermodynamics/phospholipid bilayers)

OBDULIo L. MAYORGA*, WILLIAM W. VAN OSDOL, J. L. LACOMBA, AND ERNESTO FREIREt Department of Biology, The Johns Hopkins University, Baltimore, MD 21218

Communicated by Saul Roseman, August 31, 1988

processes associated with phospholipid phase transitions (5), thus providing the basic information to estimate the crosscorrelation function of volume and enthalpy fluctuations. By using temperature both in the perturbation and in the response signals, the instrument developed in this laboratory provides experimental access to the autocorrelation function of the enthalpy fluctuations, thus making the time dimension experimentally accessible to thermodynamic analysis.

A multifrequency calorimeter has been deABSTRACT signed to measure the amplitude and time regime of the enthalpic fluctuations associated with structural or conformational transitions in biological macromolecular systems. The heat capacity function at constant pressure is directly proportional to the magnitude of the enthalpic fluctuations in a system. Biological macromolecules undergo thermally induced transitions of different kinds. Within the transition region, these systems exhibit relatively large enthalpy fluctuations that give rise to the characteristic peaks observed by conventional differential scanning calorimetry. The multifrequency calorimeter developed in this laboratory has been designed to measure the frequency spectrum of the enthalpy fluctuations, thus allowing us to estimate thermodynamic parameters as well as relaxation times. This information is obtained from the attenuation in the amplitude or phase-angle shift of the response of the system to a periodic temperature oscillation. This instrument has been used to study the gel-liquid crystalline transition of phosphatidylcholine bilayers. The frequency-temperature response surface for large dimyristoyl phosphatidylcholine vesicles has been measured in the frequency range 0.04-1 Hz. The data are consistent with two enthalpic relaxation processes with time constants on the order of 3.8 s and 80 ms at the midpoint of the main gel-liquid crystalline transition.

THEORY General Formalism. The heat capacity at constant pressure, Cp, is directly proportional to the magnitude of the enthalpy fluctuations in a system (see for example ref. 6, for a complete discussion):

(H)2 [1] RT2 For proteins, membranes, and other macromolecular systems, enthalpy fluctuations arise from different sources ranging from vibrational and rotational motions at the atomic level, as well as backbone and other structural fluctuations within specific macromolecular conformations, to fluctuations between different conformational states of the macromolecule. For globular proteins, for example, the heat capacity of the native state is of the order of 0.32 ± 0.02 calK-1g-1 (1 cal = 4.184 J) (7). For a protein of 15,000 = (H2) -

Biological macromolecules are highly dynamic systems, characterized by different types of intramolecular motions covering a wide range of length and time scales (1). These motions can be classified as (i) those associated with spontaneous structural fluctuations within specific equilibrium states and (ii) those arising from transitions between different macromolecular conformational states (2). These fluctuations in macromolecular structure are coupled to enthalpy fluctuations whose overall magnitude is reflected in the heat capacity function. During the last decade, significant advances have been made in the characterization of the spatial distribution and time regime of these fluctuations (1); however, very little has been known regarding the dynamics of enthalpy fluctuations in biological macromolecular systems. The purpose of this paper is to present the basic theory and implementation of a multifrequency calorimeter designed to measure the frequency spectrum of enthalpy fluctuations. The multifrequency calorimeter developed in this laboratory measures the dynamic response of a system to a periodic temperature perturbation. Previously, temperature perturbations of this type have been used as a basis for the development of adiabatic ac calorimeters designed to measure heat capacities (3, 4); those instruments, however, have been restricted to operate at a single or very narrow frequency range. More recently, a periodic pressure perturbation has been used to measure the dynamics of temperature relaxation

daltons, this is indicative of enthalpy fluctuations of =30 kcal/mol within the native conformation. Macromolecular transitions, such as the folding-unfolding transitions in proteins, the double-stranded-single-stranded transitions in DNA, or the gel-liquid crystalline transitions in phospholipid bilayer membranes, are accompanied by characteristic peaks in the heat capacity function. These peaks reflect the existence of enhanced dynamic fluctuations between the states accessible to the macromolecule within the transition region. Accordingly, the heat capacity function, Cp(T), of a macromolecular system can be generally written as

Cp(T)

Cp(within states) + Cp(between states) = E Pi(T)Cpsi(T) + Cpex(T), =

[2]

where Pj(T) is the population of molecules in the ith macromolecular state, Cpj(T) is the heat capacity of that state, and Cpex(T) is the excess heat capacity arising from fluctuations between states (2). In general, the excess heat capacity is maximal within the transition region and tends to zero at temperatures away from this region. The exact functional form of the excess heat capacity function depends on the nature of the transition mechanism for a particular system (8).

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

*Present address: Department of Chemical Physics, University of

Granada, Granada, Spain. tTo whom reprint requests should be addressed.

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Within-states and between-states macromolecular enthalpy fluctuations usually occur in well-separated time regimes. For example, protein backbone fluctuations occur in the picosecond to microsecond time scale, whereas betweenstates fluctuations determined by transition relaxation times are usually much slower, occurring in the millisecond and sometimes longer time scales. The same is true for nucleic acid and membrane structural transitions. For ergodic systems in thermal equilibrium, ensemble averages and time averages are identical, and accordingly, the heat capacity at any temperature can be written as

CP(T) =

[(H -

RT2 9)

[3]

where H(t) denotes the instantaneous value of the enthalpy function and the overbar denotes the time average. For small temperature perturbations, the autocorrelation function of the fluctuations in the enthalpy of the system can be expressed as a sum of exponentials, each one corresponding to a particular relaxation process (9, 10):

H(t)H(t + At) - 1 RT22 RT'2

Aiexp(-A/r,).

[4]

According to the fluctuation-dissipation theorem, the same equation describes the response of the system to a small external perturbation. In the case of a small periodic temperature perturbation, it is possible to observe the relaxation of the heat capacity and define a frequency-dependent "heat capacity." After the Fourier transform of the autocorrelation function is obtained, the frequency-dependent heat capacity can be written as C'(T V) =

CTP) -

1__ E ( '2- (H,)2) +J22Vr,) (1 1 __Hi) R T2 i 1 + (2iwf r,)2 ( +J2T7i) (2(1i) (1 + j~R V+TO, C,,,1T)

+

i1 +

[5]

[5]

(2irTVT,)2

where the integration constants are set so that the thermodynamic equilibrium heat capacity is equal to the limit of Eq. 5 as v -O 0, and j V_-. A similar conclusion has been obtained by Birge (11). Application to Multiple Independent Transitions. For macromolecular systems, Eq. 5 can be written in a convenient way as

C'(T,v) = C6(T,v) + Z i

CpexI(T)

1 + (2

[6]

where C6(T,v) contains all the contributions arising from within-states enthalpy fluctuations as well as those due to the solvent. These fluctuations occur in very fast time scales, and as such, they contribute the nonrelaxing components of C'(Tv) within the experimental frequency ranges used to study the conformational equilibrium. The second term on the right-hand side contains the contributions arising from between-states fluctuations associated with the structural or conformational transition. The quantity C'(T,v) defines a surface over the (temperature, frequency) plane containing sufficient information to develop a thermodynamic and dynamic characterization of the conformational equilibrium. The folding-unfolding equilibrium of globular proteins and multidomain and multisubunit proteins, as well as other types of structural transitions in different macromolecular systems, has been approximated as a sum of multiple independent

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transitions (7, 8). The excess heat capacity function associated with these transitions can be written in a general form as Cpex(T) = E Cp,ex1(T) Ki

-z 1

,

(1

+

+ K )2

AH AHVHi RT2

[7]

where Ki is the equilibrium constant and AHI and AHVHi are the calorimetric and van't Hoff enthalpies, respectively, for the ith transition. Within this approach, cooperative transitions are treated phenomenologically by allowing the van't Hoff enthalpy to be different than the calorimetric enthalpy. In addition, the van't Hoff enthalpy can be allowed to vary as a function of the fractional degree of completion of the transition, thus providing a very general phenomenological formalism.

RESULTS Multifrequency Calorimetry. The multifrequency calorimeter developed in this laboratory has been designed specifically to measure the frequency spectrum of the enthalpy fluctuations associated with thermally induced transitions in macromolecular systems in dilute solution. In brief, the multifrequency calorimeter is built around two very thin (1 mm) identical disc-shaped gold cells symmetrically placed on both sides of a specially designed ultrathin thermofoil heater of diameter equal to that of the cells (5.5 cm). This circuit is connected to a programmable modulated power supply driven by a function generator (Hewlett-Packard 3314A) and provides the input temperature oscillation at a preprogrammed frequency. Typically, the amplitude of the input temperature oscillation is less than 0.05°C. The response of the system is measured on the external face of the cells by a battery of miniature thermopiles (360 junctions on each side). This arrangement allows operation in single as well as differential configurations (i.e., either the input oscillation or the response of one of the cells can be used as reference). The response signal is amplified and processed by a HewlettPackard 3582A spectrum analyzer. The spectrum analyzer performs a Fourier analysis of the input and response functions in real time and provides amplitudes, phase angles, and their respective transfer functions as a function of frequency. The entire calorimeter cell assembly is thermostatted by a thermoelectric feedback control mechanism also developed in the laboratory. The frequency range of the current version of the calorimeter is 0.04-2 Hz and the temperature range 0-100°C. A block diagram of the instrument is shown in Fig. 1. The temperature oscillations on the measuring side of the calorimeter cells are attenuated in amplitude and retarded with respect to the oscillations on the excitation side. Mathematically, the calorimeter cells can be approximated with the equations derived for a semi-infinite cylindrical slab. If the temperature on the excitation face T(xot) is varied sinusoidally in time, then the temperature oscillations on the measuring face can be described by the following particular solution to the heat conduction equation derived by Eckert and Drake (12):

Axsin2irvtAxI, [8] T(Ax,t) = Toexpa J J [ L a where To is the amplitude of the temperature oscillation on the excitation side (Ax = 0) and a is the thermal diffusivity, equal to the ratio of the thermal conductivity and heat capacity per unit volume. For a frequency-dependent heat

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FIG. 1. Block diagram of multifrequency calorimetry. C, calorimeter cells; S, measuring thermopiles; P, Peltier element for temperature control; E, heat exchangers; JI and JII, adiabatic jackets; TCA and TCB, thermocouples for temperature control; CHN, channel.

capacity, the solution to the heat conduction equation is of the same form as Eq. 8 except that a is complex-valued. However, since temperature is a real quantity, only the real part of the right-hand side of Eq. 8 must be considered. The validity of the semi-infinite approximation as well as the accuracy of Eq. 8 to calculate the absolute magnitude and the frequency dependence of the calorimeter response were evaluated by using pure solvents (water or ethanol). It was found that, within the experimentally accessible frequency range, Eq. 8 provided a satisfactory description with a correlation coefficient better than 0.99994. The absolute magnitude of the specific heat of ethanol could be obtained with an accuracy better than 99% by using Eq. 8. Since the thermal diffusivity is inversely proportional to the heat

capacity, from a practical standpoint Eq. 8 can be rewritten as

T(Ax,t) =

Toexp[- VBvC'(T,0)]sin[2Ivt - V/BvC'(Tvj ],

[9]

where the parameter B can be considered as an empirical response constant containing instrument and sample contributions, and C'(T,v) represents the contributions to the heat capacity at the specified temperature and frequency. As indicated in Eq. 9, the same information can be obtained from the response amplitude or the phase-angle shift. However, with our current experimental setup, amplitudes can be

Temperature, 0C FIG. 2. Temperature and frequency (v) dependence of the amplitude response function (A) associated with large unilamellar dimyristoyl phosphatidylcholine vesicles of 4000 A diameter (see text for details). For presentation purposes, the curves have been normalized by subtracting

the minimum value.

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Proc. Natl. Acad. Sci. USA 85 (1988)

1000

0.06

600

00

E

0.2

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0.4 0.6 Frequency, Hz

200 -

-200

0

0.2

0.4 0.6 Frequency, Hz

0.8

FIG. 3. Frequency dependence of amplitude response function (in arbitrary units) and relaxing contributions to heat capacity (Inset) at the phase-transition temperature of the large unilamellar dimyristoyl phosphatidylcholine vesicles. In both plots, the open circles are the experimental values and the solid lines the theoretical curves obtained by nonlinear least-squares analysis as described in the text. The broken lines in the Inset represent theoretical curves obtained by using short relaxation times of 0.10 and 0.06 s, respectively.

determined more accurately than phase angles at these low frequencies. Application to Phospholipid Phase Transitions. Aqueous dispersions of large unilamellar dimyristoyl phosphatidylcholine vesicles of 4000-A diameter were prepared by the extrusion method in 50 mM KCI (13) followed by centrifugation (15,000 x g for 5 min) to eliminate remaining multilamellar contaminants. The gel-liquid crystalline transition of this preparation was characterized by conventional highsensitivity differential scanning calorimetry (data not shown). Vesicles prepared in this way remain in suspension and do not settle with time as do conventional multilamellar preparations. This is an important requirement for this instrument. The calorimetric scan of this membrane preparation gave a single sharp peak characterized by a melting temperature of 24. 1C, a half width of 0.40C, and an enthalpy change (AR) of 6.2 kcal/mol. These values are in excellent agreement with previous calorimetric studies (14). For accurate comparison, the same samples were analyzed with the multifrequency calorimeter. The frequency-temperature surface defined by the amplitude of the response function measured by the multifrequency calorimeter is shown in Fig. 2. For these experiments, the phospholipid concentration was 25 mg/ml. The phospholipid sample was equilibrated at the desired temperatures inside the calorimeter cells, and then the response function was measured at various excitation frequencies ranging between 0.04 and 1 Hz. As seen in Fig. 2, at all frequencies the amplitude shows a pronounced minimum at the temperature of the maximum in the heat capacity function of these vesicles (24.10C). The half-height width of the transition is similar to that obtained by differential scanning calorimetry. Also, as predicted by Eqs. 8 and 9, at all temperatures the amplitude of the response function is maximal at the lowest frequency and decreases monotonically as the driving frequency increases. The data in Fig. 2 can be analyzed in a model-independent fashion by considering the demodulation function BC'(Tv) in Eq. 9. This term can be written as BCo(1 + CQ1/Co), where BC' represents the experimentally measured demodulation function

outside the transition region (i.e., containing all instrumental, solvent, and nonrelaxing sample terms). By analyzing the data within and outside the transition region, it is possible to calculate CQX/C6 as a function of frequency and temperature. This is shown in Fig. 3 for the frequency attenuation of the response amplitude at the midpoint of the transition. The experimentally calculated C'x/QC values are plotted in Fig. 3 Inset together with the best curve obtained by nonlinear least-squares fitting of the data. According to this analysis, the best fit corresponds to two processes with relaxation times of 3.8 ± 0.4 s and 0.08 ± 0.02 s, respectively. Approximately 80% of the enthalpic relaxation amplitude is associated with the fast relaxation process. Due to the limited frequency range of the current multifrequency calorimeter, the fast relaxation process should be considered as an upper bound since other, faster relaxation processes beyond the limits of detectability are theoretically possible and have been measured by others (15). These relaxation times compare very well with those obtained by other techniques. In particular, these values are similar to those obtained for both dipalmitoyl and dimyristoyl phosphatidylcholines by van Osdol (16) using a volume-perturbation calorimeter. It should be pointed out that these two instruments are highly complementary in the sense that, in combination, they should provide a measure of the magnitude of the enthalpies and volume changes associated with each relaxation process. Also, these relaxation times are consistent with the values of 2.5 s and 0.04 s found for dimyristoyl phosphatidylcholine by Tsong and Kanehisa (15) using temperature jumps and turbidity measurements. The ability to experimentally access the frequency spectrum of the enthalpy fluctuations opens the time dimension to the experimental thermodynamic analysis of macromolecular systems. For complex systems characterized by multiple relaxation effects, this technique will allow a simultaneous evaluation of the dynamics and energetics of these processes. We thank Dr. R. L. Biltonen for many helpful discussions and Dr. I. Hatta for important suggestions regarding the analysis of the data. The excellent technical assistance and machine work of Zygmunt

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Schroeder are greatly appreciated. This work was supported by National Institutes of Health Grants GM37911 and NS24520. 1. Petsko, G. A. & Ringe, D. (1984) Annu. Rev. Biophys. Bioeng. 13, 331-371. 2. Lumry, R. & Biltonen, R. (1969) in Structure and Stability of Biological Macromolecules, Biological Macromolecules, eds. Timasheff, S. & Fasman, G. (Dekker, New York), Vol. 2. 3. Imaizumi, S., Suzuki, K. & Hatta, I. (1983) Rev. Sci. Instrum. 54, 1180-1185. 4. Black, S. G. & Dixon, G. S. (1981) Biochemistry 20, 6740-

6744. 5. Johnson, M. L., van Osdol, W. W. & Biltonen, R. L. (1986) Methods Enzymol. 130, 534-551. 6. Hill, T. L. (1960) An Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, MA). 7. Privalov, P. L. (1979) Adv. Prot. Chem. 33, 167-241.

Proc. Nadl. Acad. Sci. USA 85 (1988) 8. Biltonen, R. L. & Freire, E. (1978) CRC Crit. Rev. Biochem. 5, 85-124. 9. Eigen, M. & de Maeyer, L. (1963) in Techniques of Organic Chemistry, ed. Weissberger, A. (Wiley, New York), Vol. 3, Part 2. 10. Feher, G. & Weissman, M. (1973) Proc. Natl. Acad. Sci. USA 70, 870-875. 11. Birge, N. 0. (1986) Phys. Rev. B 34, 1631-1642. 12. Eckert, E. R. G. & Drake, R. M. (1972) Analysis of Heat and Mass Transfer (McGraw-Hill, New York). 13. Mayer, L. D., Hope, M. J. & Cullis, P. R. (1986) Biochim. Biophys. Acta 858, 161-168. 14. Mabrey, S. & Sturtevant, J. M. (1976) Proc. Natl. Acad. Sci. USA 73, 3862-3866. 15. Tsong, T. Y. & Kanehisa, M. I. (1977) Biochemistry 16, 26742680. 16. van Osdol, W. W. (1988) Ph.D. Dissertation (Univ. of Virginia, Charlottesville).