1578
David R. Rosselnsky
the hydroxide ion to enter the cavity before being close enough to a monohydrogen phosphate group to pick up its hydrogen atom. If we divide the reaction in eq 2 into two steps +O,POH + +O,PO‘ t H+ (3) H++ OH--+ H,O (4)
-
we can estimate the entalpy change AH2in eq 2. AH for the dissociation HP0d2- H+ + Pod3- is about 15 kJ mol-‘.’‘ As the acidity of the monohydrogen phosphate group is larger in a-ZRP than in a “free” state, this is a maximum estimate of A”,. AH4 is about -56 kJ mol-’ and we find A H , = AH,t A H , < -41 kd mol-’
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As the number of particles in eq 2 is unchanged, a rough estimate of TAS2is zero resulting in AG2 < -41 kJ mol-l. The neutralization of the monohydrogen phosphate groups involves also the breaking and formation of hydrogen bonds inside the cavity. If we assume that the energy terms arising from these processes cancel each other, the energy AG2 is available to move the zirconium phosphate layers apart. There is one 0(7)*-0(9) and one 0(8)-.0(8) van der Waals contact per cavity and two monohydrogen phosphate groups. Neutralization of only one of these should provide sufficient energy to move the layers apart, thus creating a transition state where hydrated alkali metal ions can reach the ,03PO- groups formed. The hydroxide ion acts as a wedge for the alkali metal ion. In the final steps of the sorption the zirconium phosphate layers, the alkali metal ions, and the water molecules inside a-ZRP rearrange themselves to obtain the energetically most favorable configuration. Kinetic studies‘ on the sorption of potassium ions are
in general agreement with the model we propose. When a-ZRP is brought in contact with a potassium hydroxide solution, its layers move apart as shown by a broad reflexion a t low angle in the x-ray powder pattern6 This reflexion was not present after the system had reached “equilibrium” (15 days). The neutralization and accompanying expansion of the cavities (the sorption of the hydrated metal ions) are thus fast processes, while the rearrangement to the less hydrated equilibrium phase is slow, since it involves diffusion of water molecules in the solid, metal-ion-loaded ZRP. Acknowledgment. The Swedish Natural Science Research Council provided the financial support, which is hereby gratefully acknowledged. References and Notes (1) For No. 9 In this series,see J. Inorg. Nucl. Chem., 38, 2377 (1974). (2) (a) J. Albertsson, Acta Chem. Scand., 20, 1689 (1966); (b) A. Clearfield and J. A. Stynes, J. Inorg. Nucl. Chem., 28, 117 (1964). (3) A. Clearfield, L. Kullberg, and A. Oskarsson, J. Phys. Chem., 78, 1150 (1974). (4) A. Cleatfield, W. L. Deux, A. S.Medlna, G. D. Smb, and J. R. Thomas, J. Phys. Chem., 73, 3424 (1969). (5) A. Clearfleld, W. L. Duax, J. M. Garces, and A. S. Medlna, J. Inorg. Nucl. Chem., 34, 329 (1972). (6) A. Clearfield and R. Hunter, J. Inorg. Nucl. Chem., 38, 1085 (1978). (7) A. Clearfield and D. Tuhtar, J. Phys. Chem., 80, 1296 (1976). (8) G. Alberti, U. Costantino, S.Allulll, and M. A. Massuccl, J. Inorg. Nucl. Chem., 37, 1779 (1975). (9) A. Clearfield and D. Smith, Inorg. Chem., 8, 431 (1969). (10) R. Stedman, L. Almqvist, G. Raunb, and G. Nilsson, Rev. Scl. Insbwn., 40, 249 (1969). (1 1) J. 0. Thomas, Institute of Chemistry, University of Uppsala, Sweden, Publication No. UUIC-813-9 (1976). (12) H. M. Rietveld, J. Appl. Crystal/cgr.,2, 65 (1969). (13) 0. E. Bacon, Acta Crysfa//ogr., Sect. A, 28, 357 (1972). (14) G. Albert1 and U. Costantino, J . Chromtogr., 102, 5 (1974). (15) National Bureau of Standard Technical Note 2703, Washington, D.C. 1968.
Surface Tension and Internal Pressure. A Simple Model Davld R. Rosselnsky Department of Chemistry, The University, Exeter, England (Received March 2, 1977)
A sphere in continuum model, with an internal surface, is used to relate surface tension and internal pressure. The results support the previous use of this model for polar interactions. The agreement of theory and experiment is close to that obtained with a recent lattice model. The surface tension y has been related’ by means of a simple lattice model to the internal pressure (aU/aV)T,N and molecular number density n by
TABLE I: Surface Tension Parameters of Several Liquids
Lattice model’
For liquids with an f 6pair potential the main approximation would introduce an inherent 11%error. The Onsager model2 (polar molecules each assigned spherical volumes in continua having bulk-liquid permittivities) explains the perhaps surprising accord1 of ether, among several nonpolar liquids: for ethers the dipole-dielectric interaction is demonstrably3 almost negligible, giving an observed vapor pressure close to that of the isostructural hydrocarbon, as predi~ted.~ The Onsager model implies a surface between sphere and surrounding dielectric, and it is of interest to examine whether explicitly defining this surface incurs any hitherto latent and unresolved complications. The Journal of Physical Chemistry, Vol. 8 I , No. IS, 1977
8
n-Heptane
Benzene
333 303
Diethyl ether
286
7.8
Argon Nitrogen Oxygen Methane
87 84 80 105 315
8.8 11.8 8.2 9.2 8.4
Carbon tetrachloride
9.1 8.0
58.1 47.6 45.2 20.0 15.4 23.2 22.5 53.5
66.1 47.9 43.0 34.3 27.0 40.1 42.6 63.3
From n and (a Ula V)T,Nas in ref 1. a Median value.’ From’ J. J. Jasper, J. Phys. Chem. R e f . Data, 1, 841 (1972). Changes in V implied in ( ~ U / ~ V ) are T , Nassumed to occur essentially in the regions between molecules, each,
Long-Range Interactions in Proteins
1579
considered singly, filling a spherical volume l/n. Concomitant changes are imputed to the surface area A of the enclosing spherical cavity similarly envisaged. Then
where the factor 2 / a refers to a compressible sphere of radius a. Now y is dG/dA, which is closely equal to dU/dA - T(dS/dA), both derivatives here being assumed constant with T, and deemed approximately equal to the partial derivatives. It is convenient to write the approximations as y = - (1- )
au
a aA T,N where 1/a representing 1 - T(aS/aA)(aA/aU)T,Nis calculable from experimental values of y and T. Hence with a expressed in terms of the molecular volume l / n = 4na3/3
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1
Values of 3.22a, from y ( T ) for several liquids,l are compared in Table I with the lattice value 8, eq 1. The average of the Table I values is 8.9, and the individual deviations are tolerable. Correction of the lattice value by the 11% referred to introductorily would give yet closer agreement. The inclusion of a on the right-hand side of eq 3 merely facilitates comparison and involves no circularity of argument. It could be omitted and (aU/aA)~8from eq 2 compared directly with that for the experimental y ( T ) . The accord shown in the table is, as for y itself with the lattice value,l to within 18% of absolute values. Discrepancies here are clearly associated with small size, but the larger molecules conform relatively well. For such sizes no major error in the theory of vapor pressures3 should thus ensue directly from the surface tension of the molecular scale cavities implied in the Onsager model employed in that theory? The approximate agreement of the lattice and the sphere/continuum models, in providing similar phenomenological relations for y, is a further satisfactory result. References and Notes
T2J
(1) H. T. Davis and L. E. Scrlven, J. Phys. Chem., 80, 2805 (1978). (2) L. Onsager, J. Am. Chem. SOC.,58, 1486 (1938). (3) D. R. Rosseinsky, Nature(London),227,944 (1970); J. G. Kirkwood, J . Chem. Phys., 2, 354 (1934).
(3)
An Approximate Treatment of Long-Range Interactions in Proteins' Matthew R. PlncusZsand Harold A. Scheraga*2b Department of Chemistry, Corneii University, Ithaca, New York 14853 (Received December 28, 1976)
An approximate method, valid beyond a critical distance, is presented for evaluating the long-range interaction energies between the residues of a protein. This approximation results in large reductions in the computer time required to evaluate the energies, compared to the usual procedure of summing the interactions over all pairs of atoms in the interacting residues. Beyond a critical distance between convenient reference atoms in each residue, the long-range nonbonded (attractive) energy may be approximated by -&/(RiJe, where Rij is the distance between the above-mentioned reference atoms, and the long-range electrostatic energy may be evaluated as the Coulombic interaction energy between the centers of positive and negative charge on each residue. In this approximate procedure, the interactions between residues are computed in terms of properties of the residues instead of those of the individual atoms of each residue. Introduction Short3- and medium4-range interactions dominate in determining protein structure, and empirical algorithms (which neglect long-range interactions) provide a fairly good description of the approximate ranges of the backbone dihedral angles.6 However, in order to obtain a molecule with the proper size and shape, it is necessary to take the long-range interactions into accountM to obtain a more accurate specification of the backbone dihedral angles. Because of the large amount of computer time required to compute all pairwise interatomic interactions (including the long-range ones) in a protein, generally only a portion of the long-range interactions are computed. In particular, those beyond some arbitrary cutoff distance from any given atom are neglected.1° While the long-range interactions beyond the cutoff distance are, individually, very small, there are many of them in a protein molecule, and their sum need not be insignificant. Further, since the longrange nonbonded interactions beyond the cutoff distance are attractive, they could serve to compact the structure
of a globular protein, i.e., their neglect might lead to a more expanded computed structure than actually exists (unless, as in the refinement of X-ray data on proteins,1° the molecule is constrained to fit the X-ray coordinates as closely as possible). The long-range electrostatic interactions could be attractive or repulsive, depending on the configurations of the charges. The purpose of this paper is to introduce a rapid, approximate procedure for computing these hitherto-neglected long-range interactions. Beyond a critical distance of separation, two interacting residues are treated as spheres of given radii, instead of as individual atoms,'l for calculating the long-range nonbonded interactions. By considering such interactions between two spheres, instead of between all pairs of atoms, a considerable saving of computer time can be effected. Similarly, beyond a critical distance of separation, the charge distributions of two interacting residuedl may be replaced by two interacting dipoles for calculating the long-range electrostatic interactions. The interaction energy between two dipoles (represented in terms of monopoles) can be calculated The Journal of Physical Chemistry, Voi. 81, No. 16, 1977
