J. Phys. Chem. 1994,98, 6851-6861
6851
Planck-Benzinger Thermal Work Function: Definition of Temperature-Invariant Enthalpy in Biological Systems? Paul W . Chun' Department of Biochemistry and Molecular Biology, College of Medicine, University of Florida, Gainesville. Florida 32610-0245 Received: November 12, 1993; In Final Form: March 16, 1 994a
In reexamining the thermodynamic parameters of a number of self-associating protein systems in the standard state near 300 K, we found that at the stable temperature ( Ts), the thermodynamic quantities AGo(Ts) and AW(TJ reach a minimum and maximum, respectively, while T U o (Ts)approaches zero. On the basis of the Planck-Benzinger thermal work function, AW(7') = A H O ( T 0 ) - AGo(T). Therefore W ( T 0 ) = AWD(T,) AGo(Ts)a t ( Ts).Values for W ( T 0 )a t ( Ts)for six self-associating protein systems were found to deviate by less than 0.05% from values for W ( T 0 )a t 0 K. We have demonstrated the universal applicability of the Planck-Benzinger thermal work function in the analysis of hydrophobic enhancement for protein folding, self-associating protein systems, micellization, and formation of biological membranes. Benzinger's definition is applied in measuring AH(To) for DNA unwinding and protein unfolding in the nonstandard state near 340 K. AH(7') = AH(T0) ACp(7') d T at the melting temperature, (Tm),where AH(7') and TAS(7')are of the same magnitude, A W ( b = AH(T0) and AG(7') approaches zero. Values for AH(T0) a t (Tm) for the proteins we examined deviated by less than 0.04% from values for AH( TO)a t 0 K. The heat of reaction of any biological system consists of two terms, the heat capacity integral between product and reactant, and the temperature-invariant chemical bond energy, AH( TO), which is a primary, indispensable source of the energy which allows life processes to proceed with quantitative precision. Failure to evaluate W (TO)or AH( TO)in assessing any biological system will thus give only a partial picture of the processes taking place within that system.
+
+ JF
In an examination of the heat of reaction, it could be said that it is a composite of two expressions which have fundamentally different origins. One of the terms deals with the difference in heat capacities between products and reactants, while the other has a direct relationshipto chemical bonding. In micromolecular reactions, differences in heat capacities could be considered negligibleand consequently, the heat of reaction consists primarily of the contributionfrom the chemical bonding term and, therefore, is a close approximation of the chemical bonding energy. However, in the case of macromolecular interactions such as those we have examined1-3.596J*J2the difference between the heat capacities of products and reactants may be substantial enough to totally obscure any differencebetween the chemical bond term and the heat integral. Thus the heat of reaction cannot be used to accurately represent the chemical bond forces associated with these systems. It was Benzinger2whorecognized that the enthalpy term AH(7') is a compositeterm representing both the chemical bonding energy and the heat of a given reaction in the nonstandard state, such that (1) AH(To),the chemical bond energy at 0 K, is a virtual quantity invariant with temperature and integrals of the differencein heat capacities between reactants and product together contributing to the Gibbs free energy function.' Therefore, the heat integral
-
-t Thisworkwassupportedby NSF GrantsPCM 79-25683 BMB 83- 12101 (02) and in part by a faculty development award, Division of Sponsored
Research, University of Florida. * Addreas correspondence to P. W. Chun, Box 100245, Health Science Center, Department of Biochemistry and Molecular Biology, University of Florida, Gainaville, FL 32610. E-mail:
[email protected]. Fax: (904) 392-2953. Abstract published in Aduance ACS Absrrucrs, June 1, 1994.
0022-3654/94/2098-685 1%04.50/0
must be incorporated into the original Gibbs-Helmholtz expression4J in order to relate the chemical bond energy to the other thermodynamic variables:
AG(7') = AH(T) - JT:ACp(7') d T - [TAS(7') -
A proper substitution of eq 1 and
Since the only manipulation of the Gibbs-Helmholtz expression was to add and subtract the same quantity, eq 3 is a general thermodynamic relationship not subject to any assumptions and a direct consequence of the first, second, and third laws of thermodynamics. [Abbreviations: K, Kelvin; ( T,),stable temperature at which TASo(T)= 0; (T,,,), melting temperature; (Th), temperature at which AGo(T) becomes zero; AH(To), temperature-invariant enthalpy or chemical bond energy at 0 K; AW( T ) , the Planck-Benzinger thermal work function; ( T w ) , temperature at which AW( T ) becomes zero; IMSL,International Mathematical Subroutine Library; S-CAM Apo A-11, S-carboxymethyl apolipoproteinA-11; GDH, bovine liver L-glutamate dehydrogenase; OPE, [p-(5,5-dimethylhexyl)phenoxy]ethanol; n-DTAB; n-dodecyltrimethylammoniumbromide; bP, base pair; 1 cal = 4.184 J.] 0 1994 American Chemical Society
6852 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
Chun
TABLE 1: Evaluation of A l P ( TO)and the Expansion Coefficients of the Planck-Benzinger Thermal Work Function in Self-Associating Protein System@ protein a,kcal/mol [AZP(To)] j3,kcal/(mol K2) y, kcal/(mol K3) a-chymotrypsin' S-CAM apo A-IIb GDH (boviney glucagond tubulin' (in the presence of Mg2+) colchicind(3 X lo-' M)
33.6002 f 0.86 50.7576 f 1.89 26.8310 f 1.29 29.7861 i 3.98 7 1 . 2 4 0 i 1.25 62.320 f 0.73
-1.3045 -1.8534 -1.0731 -1.3327 -2.3439 -1.8918
X X X X X X
lO-) lO-) lW3 lW 10-3
2.8808 X 4.0612 X 2.2872 X 3.1121 X 4.9295 X 3.7405 X
10-6 10-6 10-6 10-6 lo" 10-6
SD = 0.01178; R2 = 0.9993; PR > F = 0.0001. SD = 0.02757;R2 = 0.9951; PR > F = 0.0001. SD = 0.01380;R2 = 0.9988;PR > F 3:O.OOO1. SD = 0.20416; R2 = 0.91 17; PR > F = 0.0023. e SD = 0.03126; R2 = 0.9946; PR > F = 0.0002. fSD = 0.01400; RZ = 0.9988; PR > F = O.OOO1. Compiled using the general linear model (GLM model) of statistical analysis of an IMSL subroutine from which the coefficients AZP(To),4, and y for each protein were derived based on eq 8.11 d
TABLE 2: Evaluation of AH(TO)and Expansion Coefficients of the Planck-Benzinger Thermal Work Function at ( T P [(AH( Toll 8, Y? 9
protein a-chymotrypsin cytochrome c ribonuclease a-lactalbumin lysozyme
kcal/mol 14.7 i 0.8 9.1 f 0.4 17.5 i 1.0 6.9 & 0.3 17.1 f 0.6
kcal/(mol K2) 1.0006 X l C 7 1.9886 X l p 9.9886 X 10-6 9.6150 X l p 4.5091 X l V 7
kcal/(mol K3) 3.2165 X 10-2 3.3117 X 10-2 1.7070 X 1 t 2 2.9443 X 10-2 2.5989 X
a-chymotrypsin: sigma; a = 0.35252, j3 = 3.566 X 10-12, y = 1.143 X least-squares = 505040.7, PR > F = O.OOO1, residue squared = 0.9528 cytochrome c y = 4.366 X sigma; a = 0.35425, 4 = 2.662 X least-squares = 921904.3, PR > F = 0.0001, residue squared = 0.9994 ribonuclease sigma; a = 0.35338, j3 = 3.124 X 10-l0, y = 1.089 X least-squares = 212, 116.1, PR > F = 0.0001, residue squared = 0.9956 a-lactalbumin: sigma; a = 0.32992,P = 5.725 X y = 1.859 X least-squares = 483593.4, PR > F = 0.0001, residue squared = 0.9218 lysozyme: y = 1.204 X sigma; a = 0.33333, j3 = 1.676 X least-squares = 421622.4, PR > F = 0.0001, residue squared = 0.9968
lW7
lW7
lW7
invariant quantity, AH(To), is a primary source of the chemical bond energy essential for any reaction to proceed in an interacting system, and thus indispensable in the consideration of any biological interaction. It has, however, proven extremely difficult to measure AH(T0) at zero degrees Kelvin, although several attempts have been made. Proximate values have been extrapolated by the Nernst heat theorem.2JJfjJOJl In this communication, we describe a method of evaluating the temperature-invariant chemical bond energy, AW (TO), for selfassociating protein systems in the standard state, where TASO(7') = 0 at the stable temperature ( TJ near 300 K. It is thus possible to compare values of A H O ( T 0 ) at ( Ts)and at 0 K. Benzinger's definition, Equation (l), is then applied in measuring AH(T0) for DNA unwinding or protein unfolding in the nonstandard state at the melting temperature (T,) near 340 K, where AH( 7') and TAS(7') are of the same magnitude, AW( 7') = AH(T0) and AG(7') approaches zero.
Methods and Procedures
having the dimension of entropy, not of heat. This function cannot serve as a term in an expression of chemical equilibrium. Those criteria are met by formulation of the Planck-Benzinger thermal work
(i) Computational Procedure for Self-Associating Protein Systems in the Standard State. The question of whether the thermodynamic functions must, by definition, exhibit zero slope at 0 K, in order to be consistent with the third law of thermodynamics, appears to be central to any discussion of the resulting data. It is also essential that AGo(T) and M(7') converge at AH(T0). We tested an exponential model, a log model, a four-term polynomial function, AGo(T) = a + BT yT2 + 6F,and a three-term polymonial function, eq 8, in order to find the best fit of the data for a number of self-associating protein systems. In our treatment, the Gibbs free energy data in the standard state were fitted to a three-term linear polynomial in the 2733 13 K temperature range, the range in which experiments had been conducted (a-chymotrypsin:13J4S C A M apo A-II;lSbovine liver L-glutamatedehydrogenase;16glucagon;17tubulin (Mg2+);1* tubulin (colchicine). The remaining thermodynamic parameters were determined by manipulation of the Gibbs polynomial function:
In terms of the Planck-Benzinger thermal work function, A W(T), the directional driving energy potentially available in any interacting system, is made up of the sum total of the chemical bond energy and the Gibbs free energy. Both terms are reversible heat fluxes. The thermal work function is merely a different viewpoint of eq. (3) in terms of the heat obtainable from the formation of chemical bonds between interacting units, a constant over all temperatures, and the heat potentially available in a chemical bond, released when the bond is broken:
where AHo ( TO)= a,using the general linear model procedure of IMSL'sI mathematical and statistical subroutines of Vax 11/ 750. Innumerable techniques of error-minimizing statistical analysis20were also applied to determine the polynomial function which would give the best possible fit in the maximum number of cases. Once evaluated as shown in Table 1, the coefficients a,8, and y were fitted to other thermodynamic polynomials. The goodness of fit of the experimental data was 99%or better in each case." Thermodynamic parameters AHo( T), ACpo( T), TASO(T), and A W ( T ) were then defined as follows:
le7
lo-'
E Compiled using the general nonlinear model procedure of statistical analysis20from which coefficients a,8, and y for each protein were derived based on eqs 10 and 1 1 .
Max Planckderived thecharacteristic function, = S -H/T,9 which may also be expressed as: $ = J(Cp/T) d T - l / T J C p dT
A W ( T ) = AH(T0) - AG(T)
(4)
+
(6)
AHO(T)=a-@P-2yP
This new state function clearly posits that the temperature-
ACpo(T) = -2OT- 6 7 p
Temperature-Invariant Chemical Bond Energy
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6853
TABLE 3: Comparison of AHO( TO)at ( E ) and AW( TO)at 0 K for Self-Associating Protein S y s t w self-assoc protein systems W (TO)at 0 kcal/mol AP(Tu) AGO (TJ W ( T u ) kcal/mol , ( T d ,K 26.83 i 1.24 GDH (bovine) 35.01 -8.13 26.87 0.53 313 50.76 i 1.89 57.38 -6.40 50.98 & 0.22 303 S-CAM-apoA I1 glucagon 29.79 i 3.98 36.25 -6.42 29.83 & 1.42 295 a-chymotrypsin 33.60 & 0.86 39.59 -6.02 33.59 0.03 303 71.24 1.25 tubulin (Mg2+) 78.51 -7.27 7 1.24 3.09 317 tubulin (colchicine) 62.32 0.73 70.50 -8.24 62.32 1.56 336.5 * Compiled using the general linear model procedure of statistical analysismfrom which coefficients a, @, and y for each protein were derived based on q s 7 and 8. Other model systems (4th GLM,log and exponential models as a function of temperature)6were tested, but the third linear polynomial model gave the best fit to the data.
* **
*
Bovine liver GODH urodation T.9.nkn-.I-+"lMn
Figure 1. (A, left) Thermodynamicplot of bovine liver L-glutamatedehydrogenaseisodesmic association in 0.2 M NaPO, buffer, pH 7.4, l e M EDTA, in the temperature range 0-350 K. The experimental data are as is the following table,16 where each data point was evaluated with extrapolation of F statistics. (B,right) Thermodynamic plot of bovine liver L-glutamate dehydrogenase isodesmic association as a function of temperature between 220 and 380 K. W ( T 0 ) values can be evaluated readily at ( T u ) .
temp, K AGO( T p ,kcal/mol AGO( T), theoretically reevaluated
283 -7.27 -7.27
288 -7.53 -7.54
298 -7.95 -7.94
303 -8.06 -8.06
308 -8.12 -8.16
318 -8.14 -8.13
the following equations:
TASo(T) = -2@p - 3rp
AW"(T) = - @ p - r p
293 -7.77 -7.76
(8)
assuming that the heat capacity integral upon which these expressions are based is a continuous function. The values for these thermodynamic quantities were further extrapolated as a function of temperature down to 0 K using eqs 7 and 8. Values of these thermodynamic parameters were regenerated from the fitted coefficients of a,8, and 7, as shown in Figures 1A-6,using IMSL's program in which each equation was iteratively executed in do loops in steps of 1 K, and the values were plotted and overlaid for each set of experimental conditions. Each data point was evaluated with extrapolation of F statistics. The linear polynomial function for AGo(T) provided a reasonably good fit for the self-associating protein systems. The AGo(T) data collected over the limited temperature range 278320 K were extrapolated to 220-350 K in order to evaluate the various thermodynamic parameters, as shown in Figures 1A-6. (ii) ComputationalProcedurefor DNA Unwinding and Rotein Unfolding in the Nonstandard State. We have applied a half dozen nonlinear polynomial functions to experimental data for DNA unwinding and protein unfolding, seeking the best possible fit. These theoretical, nonlinear models for determination of the temperature-invariant enthalpy in a non-standard state included
1.
M(T) = a
+ @PeTT
AG(T) = a - @TeYT(yT-l)/r2 2.
M(T)= a
+ @eTT(rT + 1)
AG(T) = a - B p e Y T 3.
M(T)= a
+ @?eTT(yT+ 1) + 6p
AG(T) = a - @ p e T T + 6 p 4.
M(T)= a
AG(T) = a 5.
+ @eYT(T-l/r)r2+ OpeTT
+ (@eYT/r3)(rT-1 - r2p)
M(T)= a
+ @ p e y T ( y T+ 1) + 6 T
AG(T) = a - @ p e y T - 6 T l o g T 6. others: fourth-term nonlinear polynomial functions Of all the models we tried, model 1 (eq 9) provided the best fit to the data. In our treatment, the Gibbs free energy data were fitted to a three-term nonlinear polynomial in the temperature range 273353 K, the range in which experiments are most frequently c ~ n d u c t e d . ~ The ' - ~ ~remaining thermodynamic parameters were
6854 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
Chun
mpo A-ll .woofrtion T a l l U u a ~ ~ d - ~ -
W-b
W d 4 87
-
I A
46
1a
W
*
(
A
-16
-sol
-11
-
- 74
1 100
I
4a
\
SO
--I
in
\
I
:a0
I,
:u
8 n
I
an
Figure 2. (A, left) Thermodynamic plot of S-carboxymethylated apo A-I1 protein dimerization in 0.01 M Nap04 buffer, pH 7.4, in the temperature range &350 K. The experimental data are as in the following table,I5where each data point was evaluated with extrapolation of Fstatistics. (B, right) Thermodynamic plot of S-carboxymethylated apo A-I1 protein dimerization as a function of temperature between 220 and 380 K. W ( T 0 )values can be evaluated readily at ( T6). temp, K
280
285
290
295
300
305
315
AGO( T ) O X P , kcal/mol AGO (T), theoretically reevaluated
-5.40 -5.40
-5.79 -5.78
-6.03 -6.07
-6.30 -6.28
-6.44 -6.40
-6.41 -6.43
-6.21 -6.22
Figure 3. (A, left) Thermodynamic plot of a-chymotrypsin dimerization in 0.178 M NaCl, 0.01 sodium acetate buffer, pH 4.3, over the temperature range 0-350 K. The experimental data are as in the following table,I3J4where each data point was evaluated with extrapolation of F statistics. (B, right) Thermodynamic plot of a-chymotrypsin dimerization as a function of temperature between 220 and 380 K. AW(T0) values can be evaluated readily at (T$). temp, K
274
280
285
290
295
305
308
AGO( T ) O X P , kcal/mol AGO (T), theoretically reevaluated
-5.10 -5.07
-5.43 -5.43
-5.67 -5.67
-5.85 -5.85
-5.96 -5.96
-6.01 -6.01
-5.98 -5.98
determined using the Gibbs polynomial function:
AG(T) = a - @TeYr(yT- I)/?*
(9)
using the general nonlinear model procedure of statistical analysis.20 Innumerable techniques of error-minimizing statistical analysis were also applied to determine the nonlinear polynomial function which would give the best possible fit in the maximum
number of cases. Once evaluated, the coefficients a,8, and y were fitted to other thermodynamic polynomials, as shown in Table 2. The goodness of fit of the experimental data was 96% or better in each case. The mean residue square varied from 0.92 to 0.99. Thermodynamic parameters AH(T), T U ( T ) ,ACp(T) and Aw(T) could then be defined as follows~
Temperature-Invariant Chemical Bond Energy G1u-n
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6855
urwlatioo
-*ny.d-dw--LM.
WrL W. ddu ~~
T
40
20
AW' : Delta Y = b T I d c T > + < c T - i > / e 2
AH (To)= 1500 cal I mole
i .i8r+04
I
-72.000
I
28.800
329.600
230.400
331.200
432.000
Tmnperrturddefjrrer Kelvin)
Figure 14. Thermodynamic plot of poly(A)-Poly(U) unwinding as a function of temperature between 0 and 340 K with data taken from Newmann and Ackerman.'O (Tmrepresents the melting temperature; bp, base pair. AH(T0) at ( Tm),the melting point of DNA, 1.5 f 0.5 kcal/mol.) temp, K AH,cal/mol of bp AG, cal/mol of bp temp, K AH,cal/mol of bp AG, cal/mol of bp 330.0 40.98 6290 283 32.22 331.0 8700 6270 298 313 -11-1
329.5
771n
47.18
."
1-21 L
27.22
332.0
21.85
1
I I&"
Temperature-Invariant Chemical Bond Energy ascertaining the structural integrity of proteins as well as determining the catalytic efficiency of enzymes, permitting a sitedirected, mutagenic approach to the examinationofstructurefunction problems in proteins. Acknowledgment. We are grateful to Dr. T. H. Benzinger, formerly of the National Bureau of Standards, for his continued advice and suggestions in thecourseof our thermodynamic studies of biological systems. Thanks also to Richard J. Holl, James Q. Oeswein, Angel J. Espinosa,and Kwangsue Chung for preliminary calculations. References and Notes (1) Chun, P. W. Inr. J. Quunrum Chemistry: Quuntum Biol. Symp.
1988, IS, 247-258.
(2) Benzinger,T.H. Nurure(bndon)1971,229.10&103. Dr.Thdore H. Benzinger first formulated the thermal work function in 1971, but its potential was largely ignored, due in part to the fact that it had not been clearlydefined. Subsequently, we hampreciselydcfmed this new state function, AWO(T), which we have designated the Planck-Benzinger thermal work function. Gibbs free energy for an ideal diatomic molecule in the standard state: God- = (AVa - EOo) = - NkT In f / e where Goais the thermodynamic potential. What counts is the effective minimum values at chemical equilibrium obtainable from the partition function,J32 The Gibbs free energy function was defined as ( G o ~ - E o o ) / Tand , subsequentlydefined byGiauqueas(GOT-HOo)/T. Letting ( G o ~ - P ~ ) / T = - # O /then T , (GoT - PO) = -v. For an ideal gas, this function would be analogous to the Planck-Benzinger thermal work function. (3) Chun, P. W. Biophys. J. 1994,66, NO 2, W-pOS230. Marc Lewis, of the Biomedical EngineeringInstrumentation Program, NIH, suggeststhat AGo(T) = a + BF + y P be used to determine the lower limit of the temperature-invariant enthalpy. (4) Lewis, G. N.; Randall, M. Thermodynumics, 2nd ed., revised by Pitzer, K. S., Brewer, L.; McGraw-Hill; New York, 1961; pp 164-182, Appendix 665-668. (5) Chun, P. W. Fed. Proc. ASBC 1987.46, Abst NO 2047. (6) Chun, P. W. Approximationof the Planck-Bcnzinger Thermal Work Function in Self-Associating Protein Systems; 27th Sanibel Symposia, Intl. S p p . on Quantum Biology and Quantum Pharmacology, 1987; Whitney Laboratory and Marineland, March 12-14. (7) Gibb, J. W. Am. J . Sci. 1878, (3 series) 16, 441-458, via Truns. Conn. Acud. Sci. 1878, 3, 228.
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6861 (8) Wutrich, K.;Wagner, G.; Richart, R.; Braun, W. Biophys. J . 1980, 32, 549-560. (9) Planck, M. Translated by Ogg, A. 1927, Vorlesungm-Uber thermodywmic, 7th ed., As Treutise on Thermodynumics;3rd ed.; Dover: 1965, New York; P2/8 quoted. ( 10) Benzinger,T. H.; Hammer, C. Current Topicsin Cellular Regulution, Academic Press: New York, 1981; Vol. 18, Chapter 27, pp 475484. (11) Chun, P. W.; Wwein, J. Biophys. J . 1982, 37, 398a. (12) Chun, P. W. Approximationof the Planck-Bcnzinger Thermal Work Function in a Micellar System: n-Dodecyltrimethylammonium Bromide; [p(5,5-dimethylhexyl)octylphenoxyethoxy-cthanols(OPE13,klo) and n-doda cy1 trimethylammonium bromide (N-DTAB), unpublished work. (13) Aune, K. C.; Goldsmith, L. C.; Timasheff, S.N. Biochemistry 1971, 10, 1617-1622. (14) Aune, K. C.; Timasheff, S. N. Biochemistry 1971.10, 1609-1617. (15) Osborne, J. C., Jr.; Palumbo, G.; Brewer, H. B., Jr.; Edelhoch, H. Biochemistry 1976, I S , 317-320. (16) Reisler, E.; Eisenberg, H. Biochemistry 1971, 10, 2549-2663. (17) Formisano,S.;Johnson, M. L.; Edelhoch,H. 1977,Proc. Nurl. Acud. Sci. 1978, 74, 3340-3344; Biochemistry 1978, 17, 1468-1473. (18) Lee, J. C.; Timasheff. S.N. Biochemistry 1977.16. 1754-1764. (19) Andrew, J. W.; Wagenknecht, T.; Timaiheff, S. N: Biochemistry 1983, 22, 1556-1566. (20) Barr, A. J.; Godnight, J. H.; Sall, J. P.; Helwig, J. T. 1979, SAS. GLM127-GLM131, Statistical analysis system, University of Florida, 1979. (21) Privalov, P. L.; Khechenashvili, N. N. J. Mol. Biol. 1974,86,665684. (22) Privalov, P. L.; Potekhin, S . A. In Methods Enzymol. 1986, 131, 4-51. (23) Pfiel, W. Biophys. Chem. 1981, 13, 181-186. (24) Crook, E. H.; Trebbi, G. F.; Fordyce, D. B. J . Phys. Chem. 1964,68, 3592-3595. (25) Emerson, M. F.; Holtzer, A. J . Phys. Chem. 1967, 71,3320-3330. (26) Boje, L.; Hvidt, J. J. Chem. Thermodyn. 1971, 3, 663-667. (27) Corbett, R.; Roche, R. S . Biochemistry 1984, 23, 1888-1894. (28) Lumry, R.;Gregory, R. B. Free-Energy Management in Protein Reactions. In The Nucruuting Enzyme; Welch, C. R., Ed.; Wiley. Interscience: New York, 1986; Vol. 5, pp 1-212. (29) Privalov, P. Adv. Protein Chem. 1979,33, 167-241. (30) Newmann, E.; Ackermann, T. J. Phys. Chem. 1%9,73,2170-2178. (31) Jou, W. S.;Chun, P. W. J . Mol. Graphics 1991,9,237-240,243246. (32) Moelywyn-Hughes, E. A. Physical Chemisrry;Pergamon Press, New York, 1957; p 458.
