J . Phys. Chem. 1989, 93, 426-429
426
the energy needed to attain the proper transition state. Second, the distribution of energy into the electronic, vibrational, and translational modes of the products needs to be considered. Normally, electronic excitation in the products occurs when molecular orbital symmetries in the transition state correlate to those in the excited-state product fragments. If the electrons are constrained to move through this unimolecular fragmentation process governed by the Woodward-Hoffman rules, the a-electron density about the H-C1 bond would correlate to an electronically excited H-Cl molecule (the first excited electronic state of H-Cl is of a symmetry). However, due to the nonbonded electrons about the C1 and the partially filled orbitals about the phosphorus, electron motion in this concerted unimolecular decomposition is facilitated. Therefore, the orbital overlap constraints would be relieved, and the product fragments would probably form without electronic excitation. N o simple argument can be advanced concerning vibrational excitation. Quantitative values for energy deposition into the vibrational modes remain to be determined. Finally, excess energy in the translational modes must be considered. Coupling of the electron kinetic energy into translation of the products is probably very weak. The large differences in mass cause the momentum transfer from the electron to the nuclear frame to be inefficient, thus of little effect on the thermodynamic calculations. Therefore, the thermodynamic calculations are based upon the reasonable assumptions that the electronic and translational modes are not excited, while the vibrational mode excitation remains unclear. In spite of the uncertainty in these measurements, several qualitative and quantitative comparisons can be made that indicate that the values derived from these data are accurate to within the stated ranges. An extensive a b initio calculation was done by Bruna et a].," and their calculated value was 10.30 eV, which agrees well with our experiments. A qualitative prediction for the C=P bond strength (summation of u and a bond contributions) can be derived by considering the C=C analogue. The ratio of the C-C single-bond energy to the C=C double-bond energy is about 0.58. As an initial approximation, we assume that the ratio of the C-P single-bond energy to the C=P double-bond energy is the same as for the carbon analogue. Since the C-P bond strength is 63 kcal/mol, the C=P bond strength is predicted by this approximation to be 108 kcal/mol. This value represents an upper limit for the bond strength, since the overlap of the carbon 2p orbitals and the phosphorus 3p orbitals is expected to be less effective than the 2p-2p orbital overlap in the C=C case. A more quantitative
approximation of the double-bond strength between C and P can be obtained by plotting the relationship of the bond energy versus bond length for the single and triple bonds as done previously for hydrocarbon bonds.'9 Assuming a linear relationship, the double-bond strength can be estimated by using the bond length from past experiments.2021 Using this process, the double-bond energy would be about 98 kcal/mol. A further estimate is derived from the computations by Schmidt et who calculated the a-bond strengths between a variety of second- and third-row elements. They utilized an extensive basis set that included d orbitals in a multiconfigurational S C F calculation with second-order CI corrections. Their value for the C=P bond strength was 43 f 5 kcal/mol, and, by inclusion of the u-bond energy, the total double-bond energy would be 106 kcal/mol. The value of the P=C bond strength obtained in these experiments falls near these predicted values. Furthermore, since estimates of the a-bond strength in P=O and P=N compounds range from 30 to 40 kcal/m01,~~ a a-bond strength of about 38 kcal/mol in a P=C double bond seems reasonable. This value should be considered a first measurement since uncertainty exists in the amount of vibrational excitation of the fragments. If any energy was found in the vibrational modes, the final bond energy would have to be adjusted to a smaller value. For instance if the HCI fragment was predominantly formed in the v = 1 state (2950 cm-'), the double-bond strength would decrease by 7 kcal/mol, making the new estimate of the C=P bond strength to 94 f 7 kcal/mol. Further experimentation would be necessary to determine the state of the products.
Acknowledgment. We gratefully acknowledge support of the Army Research Office under Grant DAAL03-86-K-0172 and the USC Loker Hydrocarbon Institute for partial funding of this project. We also thank Dr. Sidney W. Benson and Dr. Michael T. Bowers for the many insightful and invaluable discussions. Registry NO.CHZPH, 61 183-53-7; C , 7440-44-0; P, 7723-14-0; ClCH2PH2, 7237-08-3. (19) Pauling, L . The Nature of the Chemicul Bond, 3rd ed.;Cornell University Press: Ithaca, NY, 1960; p 221. (20) Kroto, H.; Nixon, J. F.; Ohno, K.; Simmons, N. P. C. Chem. Commun. 1980, 709. (21) Brown, R. D.; Gcdfrey, P. D.; McNaughton, D. Ausf.J. Chem. 1981, 34, 465. (22) Schmidt, M. W.; Truong, P. N.; Gordon, M. S.J . Am. Chem. SOC. 1987, 109, 5217-5227. (23) Goldwhite, H. Introduction to Phosphorus Chemistry; Cambridge University Press: New York, 1981; pp 30-31.
Effect of Temperature on the Compresslblllty of Native Globular Proteins Kunihiko Gekko* and Yasunobu Hasegawa Department of Food Science and Technology, Faculty of Agriculture, Nagoya University, Nagoya 464, Japan (Received: March 16, 1988; In Final Form: June 6, 1988)
a,,
The adiabatic compressibility, of two native globular proteins (lysozyme and bovine serum albumin) was determined by measuring the sound velocity in aqueous solutions at various temperatures (10, 15,25, and 40 " C ) . 8, was positive at room temperature, but it decreased to a negative value with decreasing temperature. The isothermal compressibility, &, of the two proteins was estimated from the 8, value by using the thermal expansion coefficient and heat capacity data. With the pT values obtained, the partial specific volumes of the two proteins were simulated as a function of temperature and pressure. The results were discussed in terms of the interdependent effects of temperature and pressure on the cavity and hydration of the protein molecules. Introduction As demonstrated by recent X-ray studies,'**the thermal fluetuation of a protein structure is a function of temperature and pressure. Since the fluctuation in volume is directly related to the compre~sibility,~ at present, the effect of temperature on the *To whom correspondence should be addressed. 0022-3654/89/2093-0426$01.50/0
compressibility of globular proteins is a matter of interest. During the past 10 years, a considerable amount of data has accumulated (1) Kundrot, C. E.; Richards, F. M. J . Mol. Biol. 1987, 193, 157. (2) Frauenfelder, H.; Hartmann, H.; Karplus, M.; Kuntz, I. D., Jr.; Kuriyan, J.; Parak, F.; Petsko, G. A.; Ringe, D.; Tilton, R. F., Jr.; Connolly, M. L.; Max, N. Biochemistry 1987, 26, 254.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 427
Compressibility of Proteins
on the adiabatic compressibility of proteins, since the accurate measurement of sound velocity became possible in dilute solutions.‘” An important finding in those studies as well as earlier ones1w12was that globular proteins have positive compressibility, indicating the great contribution of the internal cavity in overcoming the hydration effect. The contributions of both the cavity and hydration to the compressibility have been quantitatively analyzed for some globular proteins with appropriate assumptions for the hydration term.6 Recently, we found that the compressibility or the volume fluctuation of proteins rather sensitively depends on their structural characteristics, such as hydrophobicity, secondary structure, or amino acid composition? However, most compressibility studies have been performed under a fixed condition of temperature and solvent composition. For a full understanding of the volume-structure relationship of globular proteins, further detailed investigations are required on the effect of temperature on the compressibility, since such data could be useful for deriving the partial specific volume of a protein as a function of temperature and pressure, that is, the “equation of state” of protein molecules, which has remained unknown. From this point of view, we have measured the adiabatic compressibility of two typical globular proteins, lysozyme and bovine serum albumin, at various temperatures, at which the two proteins are in their native states. The results will be discussed in terms of the interdependent effects of temperature and pressure on the cavity and hydration of the protein molecules.
TABLE I: Partial Specific Volume and Compressibility of Proteins in Water temp, Do, du/dc, lim, x loL2, BT x 10’2, OC mL/g (m.mL)/(g.s) (@/Bo - V o ) / c cm2/dyn cm*/dyn Bovine Serum Albumin 10 15 25 40
0.730 0.731 0.736 0.743
281.8 256.9 222.4 211.1
10
0.703 0.706 0.712 0.717
307.4 297.2 257.1 240.1
0.0571 0.0911 0.171 0.195
3.73 5.87 10.4 11.2
7.54 9.75 14.4
15.5
Lysozyme 15 25 40 12
-0,0251 0.0134 0.0743 0.0978
-1.71 0.884 4.67 5.87
1.16 3.81 7.73 9.11
-
C
0 \
0 -
Y
n
-2
4 -
x
Id 0 -
Experimental Section Materials. Two proteins, commercial products, were used without further purification: bovine serum albumin from Sigma (crystallized, Lot 41F-9300) and chicken egg white lysozyme from Seikagaku Kogyo (six times crystallized, Lot E83Y03). Both proteins were completely deionized by exhaustive dialysis against distilled water at 4 OC. Six sample solutions of different protein concentrations (0.2-1 .O%) were prepared by diluting the dialyzed stock solution with the dialyzate after purification through a glass filter. The solutions prepared were degassed under vacuum and kept as still as possible to avoid surface denaturation or bubbling. Sound Velocity Measurements. The sound velocity in a protein solution was measured, with an accuracy of 1 cm/s, by means of a “sing-around pulse” method a t 3 MHz. The apparatus and procedures were essentially the same as those used in the previous s t u d i e ~ . ~The , ~ partial specific adiabatic compressibility of the solute, p,, was calculated with the following equation p, = -(i/oo)(aao/aP) = (po/oo) lim @/Bo - V 0 ) / c (1) ro
where Do = lim [(l r-0
- Vo)/c]
P is the pressure; p, the adiabatic compressibility of the solution; Po, the adiabatic compressibility of the solvent; d, the density of the solution; do, the density of the solvent; c, the concentration of the solute in grams per milliliter of solution; Vo,the apparent volume fraction of the solvent in solution; and Do, the partial specific volume of the solute. The values of and bo can be calculated from the sound velocity, u, and the density, d, of the solution or solvent with the Laplace equation, = l/du2. (3) Cooper, A. Proc. Natl. Acad. Sci. U.S.A. 1976, 73, 2740. (4) Millero, F.J.; Ward, G. K.; Chetirkin, P. J. Biol. Chem. 1976, 252, 4001.
P.;Hemmes, P. Biopolymers 1979, 18, 3015. (6) Gekko, K.;Noguchi, H. J. Phys. Chem. 1979,83,2706. (7) Eden, D.;Matthew, J. B.; Rosa, J. J.; Richards, F. M. Proc. Narl. ( 5 ) Sarvazyan, A.
Acad. Sci. U.S.A. 1982, 79, 815. (8)Gavish, B.; Gratton, E.; Hardy, C. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 750. (9)Gekko, K.;Hasegawa, Y . Biochemistry 1986, 25, 6563. (10)Jacobson, B. Ark. Kemi 1950, 2, 177. (11) Miyahara, Y.Bull. Chem. SOC.Jpn. 1956, 29, 741. (12) Anderson, G. R. Ark. Kemi 1963, 20, 513.
10
20
30
40
Temperature (‘C)
Figure 1. Effect of temperature on
a,
of bovine serum albumin ( 0 )and lysozyme (0). The solid lines represent the least-squares fitting of the data according to the second-order function of temperature, t (“C): B, = -4.04 + 0.878f- 0.0124t2(bovine serum albumin); = -8.87 + 0.826~ - 0.01 15t2 (lysozyme).
Density Measurements. The densities of the solvent and solutions were measured with a precision density meter, DMA-02C (Anton Paar, Gratz). The partial specific volumes of the proteins were calculated from eq 2 and 3 according to the standard procedure. Protein Concentration Determination. The concentrations of the two proteins, bovine serum albumin and lysozyme, were determined by absorption measurement using extinction coefficients of 6.58 dL/(gcm) at 278 nm13 and 27.4 dL/(gcm) at 281 nm,I4 respectively. The absorption measurements were carried out with a Jasco UVIDEC-6 1OC spectrophotometer after sound velocity and density measurements.
Results and Discussion The partial specific volume of a protein at infinite dilution, Do, was determined, with an accuracy of h0.002 mL/g, by extrapolating the apparent partial specific volume to zero protein concentration. The o0 values obtained at four temperatures are listed in Table I. Evidently, the Do value increases with increasing temperature: the temperature dependence, aa0/aT, was roughly estimated (with about 30% error) to be 4.6 X lo-” and 4.5 X 10-4 mL/(gK) for lysozyme and bovine serum albumin, respectively. These values are considerably larger than those determined with a dilatometer’* or by the floating method16 (3.03 X 10-4and 3.66 X lo-“ mL/(gK), respectively) but were within the range of the reported values.” The sound velocity of solutions of a protein increased in proportion to its concentration. The concentration dependence of the sound velocity, duldc, calculated by the (13) Noelken, M. E.; Timasheff, S. N. J. Biol. Chem. 1967, 242, 5080. (14)Roxby, R.; Tanford, C. Biochemistry 1971, 10, 3348. (15) Bull, H. B.; Breese, K. Biopolymers 1973, 12, 235 1. (16)Hunter, M. J. J. Phys. Chem. 1967, 72, 3717. (17) Durchschlag, H. In Thermodynamic Data for Biochemistry and Biotechnology; Hinz, H. J., Ed.; Springer-Verlag: West Berlin, 1986;Chapter 3.
428
The Journal of Physical Chemistry, Vol. 93, No. 1 , 1989
least-squares method is listed in the third column of Table I. The value of limd [(BIBo - V o ) / c ]was determined by linear extrapolation of @/Bo - V o ) / cto zero protein concentration. The p, values thus obtained at the four temperatures are listed in the fifth column of the same table and plotted against temperature in Figure 1. In most cases, the 6, values were determined with an experimental error of less than 10%. As shown in Figure 1, the 8, values of both proteins increased with temperature following the second-order function of temperature, t ("C). However, this relationship should be regarded as only holding a t temperatures below 40 "C, the highest temperature examined, since the p, of lysozyme continuously increases at higher temperatures as revealed by a Brillouin scattering study.'* An interesting finding is that there was a critical temperature for 6, = 0, which was estimated to be 5 and 13 "C for bovine serum albumin and lysozyme, respectively. A similar temperature dependence of adiabatic compressibility has been observed for small compounds with a hydrophobic moiety such as alcohol^'^-^^ and was tentatively interpreted in terms of the gradual loosening of the water of hydration and the consequent aggregation of the solute molecules. However, protein systems are more complicated on account of the cavities in the molecules. Effect of Cavity and Hydration. The experimentally determined adiabatic compressibility of a protein would mainly consist of two contributions of the cavity and hydration as follows6 p, = -( 1/D")(dV,,/dP dAV,i/dP) (4) where V,, is the volume of the cavity in a protein molecule generated by imperfect atomic packing and AVmlis the volume change due to solvation or hydration. Increased pressure may squeeze out cavity in the protein molecule and force water into the cavity.22 An explicit formulation of such processes seems difficult at present, but they should also be included in the two terms of eq 4. It is known that the first term on the right-hand side of this equation contributes positively and the second term negatively to p,, and both terms are canceled out at absolute (mL.cm2)/(gdyn).6 Thus, compression of the order of 10 X the positive p, values observed can be ascribed to the large cavity effect overcoming the hydration effect. At low temperature, however, the hydration effect would oppositely overcome the cavity effect due to the increased amount of hydration, resulting in a negative p, value. The temperature for p, = 0 can be regarded as a compensation temperature for both factors, the packing state in the protein molecule and the protein-solvent interaction. The temperature dependence of p, should be better understood if the temperature derivatives of each term on the right-hand side of eq 4 are determined separately. However, such quantitative analysis seems difficult at present. A successful approach may be to estimate the cavity term through X-ray diffraction analysis of a protein crystal at different pressures and temperatures. Such investigations have been recently started by two groups,'J although coupling analysis of the pressure and temperature effects has not yet been performed. From X-ray diffraction data obtained at 80 and 300 K, Frauenfelder et aL2 estimated the thermal expansion coefficient of metmyoglobin to be 1.15 X lo4 K-l, which is mainly due to expansion of the cavity. Although there are no such data for lysozyme or bovine serum albumin, a similar expansion coefficient might be expected for the latter protein since the p, value and cavity volume of myoglobin are close to those of bovine serum albumin? If this is the case, more than 75% of the thermal expansion of bovine serum albumin in water could be ascribed to the hydration effect, that is, a volume inrease due to dehydration of the protein molecule. Therefore, it is reasonable to expect that the temperature-induced increase in p, of proteins is also mainly
+
(18) Doster, W.; Simon, B.; Schmidt, G . ;Mayer, W. Biopolymers 1985, 24, 1543. (19) Nakajima, T.; Komatsu, T.; Nakagawa, T. Bull. Chem. SOC.Jpn. 1975, 48, 788. (20) Kaulgud, M. V.; Rao, K. S. M. J . Chem. SOC.,Faraday Trans. 1 1979, 75, 2231. (21) Cabani, S.;Conti, G.; Matteoli, E. J. Solution Chem. 1979, 8, 11. (22) Lumry, R.; Gregory, R. B. In The Fluctuating Enzyme; Welch, G . R., Ed.; Wiley: New York, 1986; Chapter 1.
Gekko and Hasegawa caused by the diminished amount of hydration. Isothermal Compressibility. The volume fluctuation and the pressure-dependent properties of a protein molecule are theoretically related to the isothermal compressibility, pT, rather than the adiabatic compressibility. However, the direct measurement of ,?ITis generally difficult for protein systems since high pressure often induces the denaturation or aggregation of protein molecules. Thus, it has been estimated from the 6, value via the following equation4g6 jjT = a2T/dC,
B, +
where T is the absolute temperature, C, is the isobaric specific heat, and a is the thermal expansion coefficient. The C, values of lysozyme and bovine serum albumin in their native states have been determined by calorimetry to be respectively 0.30 and 0.32 cal/(gK), which are only slightly influenced by ternperat~re.~~." The a values can be estimated from the temperature dependence of the partial specific volume of proteins. As shown in the present study, however, the 8" values determined by density measurement include too great experimental error for accurate estimation of the a values. Therefore, assuming the deo/dT values more precisely determined with a dilat~meter'~ or by the floating methodI6 (as mentioned above), and setting the standard state at 25 "C and 1 atm, the a values were estimated to be 4.26 X lo4 and 4.97 X lo4 K-I for lysozyme and bovine serum albumin, respectively. The pT values calculated from eq 5 with these Cpand a values are listed in the last column of Table I. It can be seen that pT is greater by (3-4) X cm2/dyn than p, at any temperature. This difference in the two types of compressibilities is comparable with that observed for amino acids in water.25 Very few, but noble, experiments have been carried out to directly measure the BT of these two proteins. AndersonI2 measured the partial specific volume of bovine serum albumin in water under a hydrostatic pressure of 2900 atm and at a temperature of 20.2 "C. A value of 13.7 X cm2/dyn was predicted, from the result, for pT of bovine serum albumin at this temperature, which is close to our estimation based on p,. Sharp et a1.% reported cm2/dyn, for bovine serum mera similar value, 13.4 X captalbumin, determined by means of ultracentrifugation analysis with a high concentration of cesium chloride. Recently, Kundrot and Richards' estimated the PT of lysozyme to be 4.7 X cm2/dyn from X-ray analysis of the protein crystal at 1000 and 1 atm (probably at room temperature). Although this value is not a partial specific quantity in solution, it is comparable with the pTvalues obtained in the present study. Clearly, pT of both proteins shows a similar temperature dependence to 8, while the critical temperature for zero compressibility shifts to a lower temperature by about 5 "C for each protein. Pressure-Volume-Temperature Relationship. From the temperature dependence of isothermal compressibility, we could derive the partial specific volume of a protein as an interdependent function of temperature and pressure. Such a pressure-volume-temperature relationship corresponds to the "equation of state" of a protein molecule, which remains unknown. As a first approximation, we assumed a linear relationship between the partial specific volume and pressure in the low-pressure region and took the pT value at 1 atm as the standard state. Thus, the partial specific volumes of the two proteins were calculated at each temperature ( e 4 0 "C) and pressure (C3000atm) and regressed for the following equation with temperature, t ("C), and pressure, P (atm) 0" =
A
+ Bt + Ct2 + DP + EtP
(6) where A , B, C, D,and E are constants. The results of regression are listed in Table I1 and are simulatively illustrated by means (23) Khechinashvili, N. N.; Privalov, P. L.; Tiktopulos, E. I. FEBS Lett. 1973, 30, 51. (24) Privalov, P. L.; Monaselidze, D. R. Biofizika 1963, 8, 420. (25) Cabani, S.; Conti, G . ; Matteoli, E.; Tine, M. R. J . Chem. SOC., Faraday Trans. 1 1981, 177, 2385. (26) Sharp, D.; Fujita, N.; Kinzie, K.; Ifft, J. B. Biopolymers 1978, 17, 817.
Compressibility of Proteins
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 429
TABLE II: Regression Analysis of the Function V" = A A,
B x 104, mL/(g.deg)
+ Bt + Ct* + DP + E t P c x 105, mL/(g.deg2) Bovine Serum Albumin
0.7278(*0.0008)
-1.200 (f0.689)
1.405 (f0.150)
0.6991 (*0.0008)
1.459 (f0.674)
0.983 (f0.147)
lo6,
mL/(g-atm)
E x 107, mL/ (gatmedeg)
-1.571 (f0.908)
-2.915 (*0.179)
2.657 (f0.888)
2.684 (f0.175)
D
X
Lysozyme 1,
temperature ("C); P,pressure (atm). The values in parentheses represent the standard deviation.
0.74 0
0.7 2 0
results might be expected if the water structure in the solvent and hydration phases is modified under high pressure. Assuming negligibly low expansion of the atomic volume, aoo/dT can be expressed as follows aao/aT = av,,/aT aAvmI/aT (7)
I"
+
v'(mLr) 0.720
00
Figure 2. P-Do-T diagrams of bovine serum albuminn (A) and lysozyme (B), simulated with the temperature dependence of the compressibility. The results of regression are listed in Table 11.
of three-dimensional graphics in Figure 2. Through such regression, the Do of bovine serum albumin was predicted to be 0.708 mL/g at 2900 atm and 20.2 OC, which is very close to the value of 0.703 mL/g directly measured under the same conditions. This coincidence indicates that some assumptions used in our simulation are essentially acceptable. As shown in Figure 2, the P-BO-T plane for both proteins is not flat but shows curvature due to the effective contribution of the cross factor of temperature and pressure (the last term on the right-hand side of eq 6). doo/aT is positive over all the temperature region at low pressure but changes to a negative value in the low-temperature and high-pressure regions, and the critical temperature for dao/aT = 0 shifts to a higher temperature with increasing pressure. Similar inversion effects of temperature and pressure have been observed for the critical micelle concentration of s ~ r f a c t a n t s ~and ~ - ~the * solubility of hydrocarbon^.^^ Such (27) Brun, T.S.;Hoiland, H.; Vikingstad, E. J . Colloid Interface Sci. 1978,63, 89. (28)Nishikido. N.: Tanaka. M. Hvomen 1979. 17. 215. (29)Bradley, R.S.;Dew, M. J.; Munro, D. C. High Temp.-High Pressures 1973,5 , 169.
The first term on the right-hand side should always be positive while it is a decreasing function of pressure, since the cavity is compressed with pressure. The second term is also positive at low temperature since the amount of hydration should decrease with increasing temperature. Thus, both factors, the cavity and hydration, would positively contribute to aoo/dT at low pressure. Under high pressure, however, a water molecule in the bulk solvent phase is compressed so that its molar volume decreases, the extent being greater at lower temperature due to the increased compressibility of water.30 For example, the molar volume of water is 17.0 mL/mol at 25 OC and 1500 atm and 16.2 mL/mol a t 0 "C and 3000 atm. These values are comparable with the molar volumes of water of hydration around the solute at atmospheric pressure.6 Water of hydration also might be compressed, but the extent of the compression would be lower than that of solvent water since the compressibilityof water of hydration is considerably small as compared with that of pure water.6 Therefore, it is probable that the volume change due to hydration, AV,,, which is negative at low pressure, becomes positive under the conditions of high pressure and low temperature, where a water molecule is highly condensed. Thus, the second term on the right-hand side of eq 7 would be negative to compensate for the first t e r n (cavity effect), resulting in a negative dao/dT value in the high-pressure and low-temperature regions. The fact that protein denaturation occurs along the elliptical pressure-temperature ~ l a n e ~ might l - ~ ~ be essentially explained by these interdependent effects of temperature and pressure on the partial specific volume. As shown above, the temperature dependence of compressibility allows the prediction of some interesting physical features of protein molecules, although our P-DO-T diagram and its interpretation might have to be regarded as tentative since some assumptions were used for the simulation. For more detailed P-DO-T analysis of protein molecules, it is necessary to accumulate partial specific volume and X-ray diffraction data directly measured at various temperatures and hydrostatic pressures. At present, such experiments are technically possible, as demonstrated by some recent investigations.1j2s'2 The combination of thermodynamic and structural analyses would be fruitful to obtain an insight into the real image of protein dynamics.
Acknowledgment. This work was supported by a Grant-in-Aid for Scientific Research (No. 60580218) from the Ministry of Education, Science, and Culture. We thank Dr. Shigeru Hayakawa for his technical assistance with the computer graphics. Registry No. Lysozyme, 9001-63-2. (30)Kell, G.S.In Water; Franks, F., Ed.; Plenum: New York, 1972;Vol. 1, Chapter 10. (31) Brandts, J. F.; Oliveira, R. J.; Westort, C . Biochemistry 1970,9,1038. (32) Hawley, S.A.Biochemistry 1971,10, 2436. (33) Zipp, A.; Kauzmann, W. Biochemistry 1973,12, 4217.
