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Modeling Hydrogen Bonds in Three Dimensions a
L. L. Van Zandt & W. K. Schroll
a
a
Department of Physics , Purdue University , W. Lafayette , Indiana , 47907 Published online: 21 May 2012.
To cite this article: L. L. Van Zandt & W. K. Schroll (1990) Modeling Hydrogen Bonds in Three Dimensions, Journal of Biomolecular Structure and Dynamics, 8:2, 431-438, DOI: 10.1080/07391102.1990.10507814 To link to this article: http://dx.doi.org/10.1080/07391102.1990.10507814
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is
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Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102 Volume 8, Issue Number 2 (1990), €!Adenine Press (1990).
Modeling Hydrogen Bonds in Three Dimensions* L.L. Van Zandt and W.K. Schroll Department of Physics Purdue University W. Lafayette Indiana 47907
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Abstract The effective potential between two hydrogen bonded atoms is calculated on the basis of the Lippencott-Schroeder bent bond model, taken to be a typical model interaction. We differ from other calculations in that the minimum energy configuration for the proton is treated adiabatically, its position being recomputed at each value of the larger atoms separation. We find the typical hard core to have been a consequence of an artificial restriction of the proton to a fixed angle with the larger atom axis, basically a one-dimensional assumption. Free to move in three dimensions, the proton is squeezed off the axis as the separation narrows, and the hard core feature is gone. Depending on the degree of bond bending, the anharmonicity of the bond may be diminished, eliminated, or even reversed.
Introduction Figure 1 shows a typical linear hydrogen bonded configuration. One proton, covalently bonded to a Nitrogen atom- the donor- experiences some small charge transfer and Coulomb interaction with a more distant Oxygen atom- the acceptor. Both Nand 0 are parts of some unshown additional complex structure. Very commonly, the proton is ignored and the whole assembly is more simply treated as some effective interaction between the N and 0. Other atoms besides N and 0 may play the roles of donor and acceptor. Certain features of this potential are widely accepted: at large separations the attractive force becomes vanishingly weak; at short range there is a hard repulsive core. Figure 2 shows a typical plot of potential vs. acceptor-donor distance. The potential is strongly anharmonic. The long range behavior is not an issue here. We will examine the requirements for the hard core. Double helical DNA is held together, at least in part, by such Hydrogen bonds; each base looks at its paired partner through two or three of these bonds. The secondary structure of proteins is likewise stabilized and determined in a large part by various hydrogen bonds between different parts of the polymer. When part of a larger molecule, a complete, first-principles, quantum mechanical, *Supported by ONR Contract N00014-87-K-0162 funded by SDIO.
431
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Thymine
Adenine
-- -- --
Figure 1: A typical hydrogen bond structure. A proton is covalently bonded to a Nitrogen atom, the donor. A nearby Oxygen atom can weakly couple to the proton. Both the Oxygen and Nitrogen atoms are part of some larger structures. 0.15.,---------------------------.,
0.10
0.05
V(eV) 0.00
-0.05
-0.10
-0.15·-b----~~-..-~---..---~~-........----....--.~~-.-~--l
'2.60
3.10
3.60
4.10
4.60
5.10
Figure 2: Morse potential vs. acceptor-donor separation distance. This is a widely used simplified model for the effective acceptor-donor interaction.
electron orbital theory of these bonds does not exist, and because of the many body nature of the interactions probably never will. Such calculations have been done for systems of two two heavy atoms and as many as six hydrogens (1). If, as in the case which interests us here, the heavier atoms were part of a more extended molecule, their electronic wave functions would be extensively correspondingly modified. The modified wave functions of the electrons involved in the covalent bonding to the larger molecular complex are the highest lying in energy and are thus precisely
Modeling Hydrogen Bonds in 3D
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those also involved in the hydrogen bonding. Thus the quantum mechanical hydrogen bonding problem, with multiple heavy atom electronic wave functions to treat, would have to be computed for each different compound in which an interesting hydrogen bond might occur. Even worse, however, since we have here to deal with a flexing, dynamical system, and since it is the stiffness against this flexing that is the matter at issue in this paper, the quantum mechanical calculations would have to be done multiple times to explore the configuration space of all the distortions the large molecule would suffer as it vibrates. Thus hydrogen bonds have been parametrized by simpler functional forms. There is every prospect that such parametrization will continue to be the most important theoretical approach to hydrogen bonding in the future. This will be particularly true when we are interested more in the larger dynamical system, rather than the hydrogen bond per se. The task of the theorist will be to find useful functions to perform this simplifying, approximating description. The literature shows this done in several ways. One of the earliest and rather more elaborate of these model forms is due to Lippencott & Schroeder (2) over 35 years ago. This model is successful in describing the bond properties over a wide range of compounds and structures where they are encountered. Hagler, Huler and Lifson (3) have used a different form, based primarily on a Lennard-Jones "six-twelve" function for hydrogen bonds in protein crystals. Levitt (4) has modified their function for use in a molecular dynamics study. The very widely used AMBER program (5) uses a "ten-twelve" function. There are others. For some applications, it suffices merely to replace the potential with a simple parabolic well; the coefficient of the quadratic form is then the Hooke's law spring constant linking the hydrogen bonded larger atoms. This simple parametrization suffices for analysis of molecular spectra. The reason is that only single excitations or vibration modes of a molecule are needed to characterize a spectrum, thereby requiring only very low amplitudes of motion about equilibrium. If higher amplitude oscillations are at issue, non-linear properties of the bonds are needed, and a more complex form is necessary, one which shows the long range short range asymmetry mentioned above. One commonly used substitution is the Morse potential.
[1] Peyrard and Bishop (6) have recently used a Morse potential in modeling a DNA polymer, looking for possible effects of solitons. Prohofsky and his students (7-10) have written a series of papers concerning the theory of melting of DNA based on the anharmonic shape of the Morse potential. The graph in Figure 2 is drawn for a Morse function. The hard core repulsion is a conspicuous feature of the Morse potential. The evidence for the existence of a hard core repulsion is actually weak. The work of Levitt (4) and of Hagler, Huler & Lifson (3) involves the X-ray structures of protein
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crystals, and reveals a hard core interaction between the acceptor atom and the proton. This is very reasonable. However, it does not necessarily translate into a hard core in the effective interaction between donor and acceptor. To be sure, the two atoms must eventually encounter a hard core at their respective van der Waals radii, but the model potentials discussed here produce a strong repulsion at much larger separations than direct donor-acceptor contact would require. The reason for a hard core at more distant separations in these model potentials is that the possibility for the proton to leave the donor-acceptor axis and move into a lower symmetry, lower energy position has not been allowed. Lippencott and Schroeder (2) were careful to emphasize the one-dimensional nature of their first model, appropriate for the compounds they considered; the acceptorproton hard core is naturally encountered when the proton is restricted to an intermediate position. In a later publication (11) they model the bent bond with an off-axis proton. In this second paper, L&S do not take up the question of the effective donor-acceptor spring constants or non-linearities. Nor, it appears, has anyone else, leaving the one-dimensional bond model to dominate theoretical thinking.
Calculations and Results We have coded the Schroeder-Lippencott formulas for H bond energy for the bent bond case of reference (9) in order to explore their properties. (For convenience, we reproduce the L&S functions in an Appendix, available upon request from the author or the journal office). In this task, we believe that the general behavior is not specific to the exact L&S forms, but would emerge approximately the same from any of the functions in the literature that did not include the restriction to one-dimensionality. We have departed from L&S in one important respect: as the donor and acceptor atoms move, we allow the proton to follow adiabatically. This difference is discussed elsewhere (12) in connection with the one-dimensional model. The L&S approach is a high frequency limit, whereas our departure treats a relatively lower frequency case. This pure L&S model is unrealistic in its implicit assumption that the donor-proton system is fully rotationally symmetric. Because the donor atom is part of the larger structure of the molecule, the proton direction will be constrained to some degree by other interactions that do not involve the hydrogen bond. The additional molecular structure shown in figure 1 is part of a base molecule of DNA polymer. The proton is constrained to the plane of the base and approximately the plane of the H-bond by torsional forces acting through the C-N single bond. Within the plane of the base, it is restrained by an angle bending force constant ofC-N-p. The torsional constant around the single bond is much the weaker; hence the proton tends easily to swing up out of the plane. Very little attention has been given to force constants associated with single hydrogen motions. The protons are hard to see in X-ray scattering, and many authors simply lump them together with their associated CorN. We have assigned a typical torsion force constant to this proton motion of 0.05 millidyne/A. (In some circumstances, much larger torsional constants than this are assigned, but these cases
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Modeling Hydrogen Bonds in 3D
involve larger groups of atoms swinging about some single connecting bond; the "torsional" forces in such a case include other interactions and steric forces besides pure torsions around the bond.) With this torsional force constant, we add another term to the hydrogen bond potential energy in the form
[2]
k. is the torsional force constant and 8 is the equilibrium proton angle, absent the 0
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acceptor atom influence.
1./)
c
w _J 0
z
-0.40
(J) c
-0.80 raton Bond Angle (radians)
>-
C)
0:::
w ~
U - 1.20 -t-r...,-,-...,-,-,..,..,..,..,rrrrrrrrrrrrr-r-1r-r-1-r-r-r-r-r-r...-r...-r...-r......,.......,....,..,..,..,.., .::.20
2.40
2.60
2.80
3.00
3.20
SIGMA m ANGSTROM Figure 3: Potential energies as a function of donor-acceptor separation for the dimensionally constrained and unconstrained modified Lippencott & Schroeder model potentials. For the unconstrained case, the angle between the Nitrogen-proton direction and the Nitrogen-Oxygen axis is also shown. Numerical "noise" on the proton angle curve reflects how very soft the proton restoring elastic forces are near the proton equilibrium position.
Figure 3 shows the computed potential energy of a L&S bond in which an off axis search for the energy minumum proton position has been carried out at every point. The assumed minimum potential angle for the proton was 0°. On the same graph for comparison purposes, we have included the corresponding one-dimensional bond energy, the uppermost curve. The figure also includes a curve showing the angle between the donor-acceptor axis and the donor-proton axis, that is the angle by which the proton swings out of the way. The energy is a much weaker function of the proton angle than of the separation cr, hence the angle curve is numerically somewhat
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VanZandt and Schroll
\ > Q)
Straight L&S
-0.05
c
>-
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C)
cr:: w
z w
-0.10
k(bend) = 0.05 mDyne/A
theto(equi) = 34•
-0.154,,.,.,.,.,~~rrrrrrrrrrrrrr~~TTTTTTTI~""",
2.20
2.40
2.60
2.80
3.00
3.20
SIGMA 1n ANGSTROM Figure 4: Potential energies as a function of donor-acceptor separation for the case of a wry hydrogen bond. The one-dimensional case is offset to higher energy by the energy necessary to move the proton onto the donor acceptor axis against the elastic potential. The preferred proton direction is assumed to lie 34 o off the denoracceptor axis.
ragged. As the acceptor moves in toward the minimum, the proton, at first on the axis, swings suddenly out of the way. The hard core of the effective potential is largely gone. In figure 4, we show the same potential energy as in figure 3 except that we have assumed a proton equilibrium angle of 34 o away from the bond axis. The offset of the one-dimensional bond plotted for comparison comes from forcing the proton into the on-axis position against the torsional restoring force. At large separations, the potential energy shape is indistinguishable between the unrestricted and the constrained proton cases. Near the energy minimum, however, the proton swings drastically off the axis and the characteristic hard core repulsion essentially vanishes. The effect is so severe, in fact, thatthe sign of the first anharmonic term in the expansion of the potential may even be reversed. Such a system would show a thermal shrinkage for moderate temperatures. Figure 5 repeats the calculation for an equilibrium angle of 10°, a quite realistic value for DNA bonds. We may say in general of these curves that if the proton is not artificially constrained to remain on the central axis, the hard core repulsion is substantially eliminated. We have prepared table I as a listing of the effective force constant near equilibrium,
437
Modeling Hydrogen Bonds in 3D
0 00
>
Flond
(]) -0.05
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c
>-
(_')
0:::
w z w
-0.10 Wry L&S Bond
k(bend) = 0.05 mDyne/A
theta(equi)
-Q.154,~~~~~rr~~rrrrrrrr,TTTTTTTTTTTT,,_,_,_,_,
2.20
2.40
2.50
2.80
3.00
3.20
SIGMA 1n ANGSTROM Figure 5: Potential energies as a function of donor-acceptor separation for the case of a wry hydrogen bond. The difference between this case and that of figure 4 is a smaller assumed value for the equilibrium Nirtogen-proton axis direction deviation. The preferred direction has here been taken as 10°.
kN-0 = a V!aci and the first anharmonic coefficient, "'N-O = a3y;acr 3. The harmonic elastic force is progressively weakened as the proton angle increases, dropping by about half for the severest bend. The first anharmonic term is weakened, and may be "weakened" so much as to be reversed. The equilibrium separation is also shifted for the bent bonds. The most dramatic effect, however is for large excursions, showing that at all angles the hard core is severely softened or essentially absent. 2
()min
11 D model I
a -- 2a2v {Jq2 0.168 0.30 0.37
0.36
b -- 6 aav ou3 -0.32 2.24 -0.88
-0.98
Table 1: Numerical values of the second and third derivatives of the effective Nitrogen-Oxygen interaction potential.
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These results do not depend on any peculiar feature of the Lippencott & Schroeder potential. They are instead a consequence of lifting an artificial constraint to onedimension on the motion of the proton. We see that the constraint is approximately harmless for donor-acceptor distances beyond the minimum potential position, but fails completely in the closer regions. The effective donor-acceptor potential is somewhat softer, and not nearly as anharmonic as it has been taken to be. Thus we are led to the position that any calculations depending in a significant way upon the conventional shape and parametrization of this potential are in need of substantial revision( B).
Supplementary Material Supplementary Material"Appendix for the Lippincott-Schroeder Bent Bond" can be obtained from the author LLVZ. See also references (2) and (11). References and Footnotes 1. A recent calculation along these lines has been given by J.E. DelBene,.!. Comp. Chern. 10,603 ( 1988). She treats complexes of two larger atoms together with as many as six hydrogens. A smaller exploration of ammonia-amonium coupling, a similar system was also given by N.C. Baird, Int. J. Quant. Chern.: Quant. Bio. Symp. 1, 49 (1974). The additional complexities we encounter in our examples here where hydrogen bonding Nand 0 atoms are bonded into larger structures, partially charged, and progressively deformed, have never been attempted on this fundamental level. 2. E.R. Lippencott & R. Schroeder,.!. Chern. Phys. 23, 1009 (1955). 3. AT. Hagler, E. Huler & S. Lifson,.T. Am. Chern. Soc. 96,5319 (1974). 4. M. Levitt,.!. Mol. Bioi. 168, 595 (1983). 5. P.K. Weiner & P.K. Kollman,.!. Comp. Chern. 2, 287 (1981). 6. M. Peyrard & AR. Bishop, Phys. Rev. Lett. 62,2672 (1989). 7. Y. Gao, KV. Devi-Prassad & E.W. Prohofsky,.T. Chern. Phys. 80,6291 (1984). 8. Y. Kim, K.V. Devi-Prassad & E.W. Prohofsky, Phys. Rev. B 32,5185 (1985). 9. V.V. Prabhu, L. Young & E.W. Prohofsky, Phys. Rev. B 39,5436 (1989). 10. KM. Awati, M. Techera & E.W. Prohofsky, Bull. Am. Phys. Soc. 35, 649 (1990). 11. R. Schroeder & E.R. Lippencott,.T. Am. Chern. Soc. 79,921 (1957). 12. W.K. Schroll & L.L. Van Zandt, Low-frequency Parametrization of Hydrogen Bonding, to be published. 13. This includes, besides all those mentioned above, one of the author's own, a simulation of soliton dynamics involving H-bond stretching anharmonicity. L.L. Van Zandt. Phys. Rev. A 40, 6134, (1989).
Date Received: Apri/13, 1990
Communicated by the Editor Ramaswamy H. Sarma
Journal of Biomolecular Structure and Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tbsd20
Modeling Hydrogen Bonds in Three Dimensions a
L. L. Van Zandt & W. K. Schroll
a
a
Department of Physics , Purdue University , W. Lafayette , Indiana , 47907 Published online: 21 May 2012.
To cite this article: L. L. Van Zandt & W. K. Schroll (1990) Modeling Hydrogen Bonds in Three Dimensions, Journal of Biomolecular Structure and Dynamics, 8:2, 431-438, DOI: 10.1080/07391102.1990.10507814 To link to this article: http://dx.doi.org/10.1080/07391102.1990.10507814
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is
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expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102 Volume 8, Issue Number 2 (1990), €!Adenine Press (1990).
Modeling Hydrogen Bonds in Three Dimensions* L.L. Van Zandt and W.K. Schroll Department of Physics Purdue University W. Lafayette Indiana 47907
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Abstract The effective potential between two hydrogen bonded atoms is calculated on the basis of the Lippencott-Schroeder bent bond model, taken to be a typical model interaction. We differ from other calculations in that the minimum energy configuration for the proton is treated adiabatically, its position being recomputed at each value of the larger atoms separation. We find the typical hard core to have been a consequence of an artificial restriction of the proton to a fixed angle with the larger atom axis, basically a one-dimensional assumption. Free to move in three dimensions, the proton is squeezed off the axis as the separation narrows, and the hard core feature is gone. Depending on the degree of bond bending, the anharmonicity of the bond may be diminished, eliminated, or even reversed.
Introduction Figure 1 shows a typical linear hydrogen bonded configuration. One proton, covalently bonded to a Nitrogen atom- the donor- experiences some small charge transfer and Coulomb interaction with a more distant Oxygen atom- the acceptor. Both Nand 0 are parts of some unshown additional complex structure. Very commonly, the proton is ignored and the whole assembly is more simply treated as some effective interaction between the N and 0. Other atoms besides N and 0 may play the roles of donor and acceptor. Certain features of this potential are widely accepted: at large separations the attractive force becomes vanishingly weak; at short range there is a hard repulsive core. Figure 2 shows a typical plot of potential vs. acceptor-donor distance. The potential is strongly anharmonic. The long range behavior is not an issue here. We will examine the requirements for the hard core. Double helical DNA is held together, at least in part, by such Hydrogen bonds; each base looks at its paired partner through two or three of these bonds. The secondary structure of proteins is likewise stabilized and determined in a large part by various hydrogen bonds between different parts of the polymer. When part of a larger molecule, a complete, first-principles, quantum mechanical, *Supported by ONR Contract N00014-87-K-0162 funded by SDIO.
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Thymine
Adenine
-- -- --
Figure 1: A typical hydrogen bond structure. A proton is covalently bonded to a Nitrogen atom, the donor. A nearby Oxygen atom can weakly couple to the proton. Both the Oxygen and Nitrogen atoms are part of some larger structures. 0.15.,---------------------------.,
0.10
0.05
V(eV) 0.00
-0.05
-0.10
-0.15·-b----~~-..-~---..---~~-........----....--.~~-.-~--l
'2.60
3.10
3.60
4.10
4.60
5.10
Figure 2: Morse potential vs. acceptor-donor separation distance. This is a widely used simplified model for the effective acceptor-donor interaction.
electron orbital theory of these bonds does not exist, and because of the many body nature of the interactions probably never will. Such calculations have been done for systems of two two heavy atoms and as many as six hydrogens (1). If, as in the case which interests us here, the heavier atoms were part of a more extended molecule, their electronic wave functions would be extensively correspondingly modified. The modified wave functions of the electrons involved in the covalent bonding to the larger molecular complex are the highest lying in energy and are thus precisely
Modeling Hydrogen Bonds in 3D
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those also involved in the hydrogen bonding. Thus the quantum mechanical hydrogen bonding problem, with multiple heavy atom electronic wave functions to treat, would have to be computed for each different compound in which an interesting hydrogen bond might occur. Even worse, however, since we have here to deal with a flexing, dynamical system, and since it is the stiffness against this flexing that is the matter at issue in this paper, the quantum mechanical calculations would have to be done multiple times to explore the configuration space of all the distortions the large molecule would suffer as it vibrates. Thus hydrogen bonds have been parametrized by simpler functional forms. There is every prospect that such parametrization will continue to be the most important theoretical approach to hydrogen bonding in the future. This will be particularly true when we are interested more in the larger dynamical system, rather than the hydrogen bond per se. The task of the theorist will be to find useful functions to perform this simplifying, approximating description. The literature shows this done in several ways. One of the earliest and rather more elaborate of these model forms is due to Lippencott & Schroeder (2) over 35 years ago. This model is successful in describing the bond properties over a wide range of compounds and structures where they are encountered. Hagler, Huler and Lifson (3) have used a different form, based primarily on a Lennard-Jones "six-twelve" function for hydrogen bonds in protein crystals. Levitt (4) has modified their function for use in a molecular dynamics study. The very widely used AMBER program (5) uses a "ten-twelve" function. There are others. For some applications, it suffices merely to replace the potential with a simple parabolic well; the coefficient of the quadratic form is then the Hooke's law spring constant linking the hydrogen bonded larger atoms. This simple parametrization suffices for analysis of molecular spectra. The reason is that only single excitations or vibration modes of a molecule are needed to characterize a spectrum, thereby requiring only very low amplitudes of motion about equilibrium. If higher amplitude oscillations are at issue, non-linear properties of the bonds are needed, and a more complex form is necessary, one which shows the long range short range asymmetry mentioned above. One commonly used substitution is the Morse potential.
[1] Peyrard and Bishop (6) have recently used a Morse potential in modeling a DNA polymer, looking for possible effects of solitons. Prohofsky and his students (7-10) have written a series of papers concerning the theory of melting of DNA based on the anharmonic shape of the Morse potential. The graph in Figure 2 is drawn for a Morse function. The hard core repulsion is a conspicuous feature of the Morse potential. The evidence for the existence of a hard core repulsion is actually weak. The work of Levitt (4) and of Hagler, Huler & Lifson (3) involves the X-ray structures of protein
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crystals, and reveals a hard core interaction between the acceptor atom and the proton. This is very reasonable. However, it does not necessarily translate into a hard core in the effective interaction between donor and acceptor. To be sure, the two atoms must eventually encounter a hard core at their respective van der Waals radii, but the model potentials discussed here produce a strong repulsion at much larger separations than direct donor-acceptor contact would require. The reason for a hard core at more distant separations in these model potentials is that the possibility for the proton to leave the donor-acceptor axis and move into a lower symmetry, lower energy position has not been allowed. Lippencott and Schroeder (2) were careful to emphasize the one-dimensional nature of their first model, appropriate for the compounds they considered; the acceptorproton hard core is naturally encountered when the proton is restricted to an intermediate position. In a later publication (11) they model the bent bond with an off-axis proton. In this second paper, L&S do not take up the question of the effective donor-acceptor spring constants or non-linearities. Nor, it appears, has anyone else, leaving the one-dimensional bond model to dominate theoretical thinking.
Calculations and Results We have coded the Schroeder-Lippencott formulas for H bond energy for the bent bond case of reference (9) in order to explore their properties. (For convenience, we reproduce the L&S functions in an Appendix, available upon request from the author or the journal office). In this task, we believe that the general behavior is not specific to the exact L&S forms, but would emerge approximately the same from any of the functions in the literature that did not include the restriction to one-dimensionality. We have departed from L&S in one important respect: as the donor and acceptor atoms move, we allow the proton to follow adiabatically. This difference is discussed elsewhere (12) in connection with the one-dimensional model. The L&S approach is a high frequency limit, whereas our departure treats a relatively lower frequency case. This pure L&S model is unrealistic in its implicit assumption that the donor-proton system is fully rotationally symmetric. Because the donor atom is part of the larger structure of the molecule, the proton direction will be constrained to some degree by other interactions that do not involve the hydrogen bond. The additional molecular structure shown in figure 1 is part of a base molecule of DNA polymer. The proton is constrained to the plane of the base and approximately the plane of the H-bond by torsional forces acting through the C-N single bond. Within the plane of the base, it is restrained by an angle bending force constant ofC-N-p. The torsional constant around the single bond is much the weaker; hence the proton tends easily to swing up out of the plane. Very little attention has been given to force constants associated with single hydrogen motions. The protons are hard to see in X-ray scattering, and many authors simply lump them together with their associated CorN. We have assigned a typical torsion force constant to this proton motion of 0.05 millidyne/A. (In some circumstances, much larger torsional constants than this are assigned, but these cases
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Modeling Hydrogen Bonds in 3D
involve larger groups of atoms swinging about some single connecting bond; the "torsional" forces in such a case include other interactions and steric forces besides pure torsions around the bond.) With this torsional force constant, we add another term to the hydrogen bond potential energy in the form
[2]
k. is the torsional force constant and 8 is the equilibrium proton angle, absent the 0
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acceptor atom influence.
1./)
c
w _J 0
z
-0.40
(J) c
-0.80 raton Bond Angle (radians)
>-
C)
0:::
w ~
U - 1.20 -t-r...,-,-...,-,-,..,..,..,..,rrrrrrrrrrrrr-r-1r-r-1-r-r-r-r-r-r...-r...-r...-r......,.......,....,..,..,..,.., .::.20
2.40
2.60
2.80
3.00
3.20
SIGMA m ANGSTROM Figure 3: Potential energies as a function of donor-acceptor separation for the dimensionally constrained and unconstrained modified Lippencott & Schroeder model potentials. For the unconstrained case, the angle between the Nitrogen-proton direction and the Nitrogen-Oxygen axis is also shown. Numerical "noise" on the proton angle curve reflects how very soft the proton restoring elastic forces are near the proton equilibrium position.
Figure 3 shows the computed potential energy of a L&S bond in which an off axis search for the energy minumum proton position has been carried out at every point. The assumed minimum potential angle for the proton was 0°. On the same graph for comparison purposes, we have included the corresponding one-dimensional bond energy, the uppermost curve. The figure also includes a curve showing the angle between the donor-acceptor axis and the donor-proton axis, that is the angle by which the proton swings out of the way. The energy is a much weaker function of the proton angle than of the separation cr, hence the angle curve is numerically somewhat
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SIGMA 1n ANGSTROM Figure 4: Potential energies as a function of donor-acceptor separation for the case of a wry hydrogen bond. The one-dimensional case is offset to higher energy by the energy necessary to move the proton onto the donor acceptor axis against the elastic potential. The preferred proton direction is assumed to lie 34 o off the denoracceptor axis.
ragged. As the acceptor moves in toward the minimum, the proton, at first on the axis, swings suddenly out of the way. The hard core of the effective potential is largely gone. In figure 4, we show the same potential energy as in figure 3 except that we have assumed a proton equilibrium angle of 34 o away from the bond axis. The offset of the one-dimensional bond plotted for comparison comes from forcing the proton into the on-axis position against the torsional restoring force. At large separations, the potential energy shape is indistinguishable between the unrestricted and the constrained proton cases. Near the energy minimum, however, the proton swings drastically off the axis and the characteristic hard core repulsion essentially vanishes. The effect is so severe, in fact, thatthe sign of the first anharmonic term in the expansion of the potential may even be reversed. Such a system would show a thermal shrinkage for moderate temperatures. Figure 5 repeats the calculation for an equilibrium angle of 10°, a quite realistic value for DNA bonds. We may say in general of these curves that if the proton is not artificially constrained to remain on the central axis, the hard core repulsion is substantially eliminated. We have prepared table I as a listing of the effective force constant near equilibrium,
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kN-0 = a V!aci and the first anharmonic coefficient, "'N-O = a3y;acr 3. The harmonic elastic force is progressively weakened as the proton angle increases, dropping by about half for the severest bend. The first anharmonic term is weakened, and may be "weakened" so much as to be reversed. The equilibrium separation is also shifted for the bent bonds. The most dramatic effect, however is for large excursions, showing that at all angles the hard core is severely softened or essentially absent. 2
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These results do not depend on any peculiar feature of the Lippencott & Schroeder potential. They are instead a consequence of lifting an artificial constraint to onedimension on the motion of the proton. We see that the constraint is approximately harmless for donor-acceptor distances beyond the minimum potential position, but fails completely in the closer regions. The effective donor-acceptor potential is somewhat softer, and not nearly as anharmonic as it has been taken to be. Thus we are led to the position that any calculations depending in a significant way upon the conventional shape and parametrization of this potential are in need of substantial revision( B).
Supplementary Material Supplementary Material"Appendix for the Lippincott-Schroeder Bent Bond" can be obtained from the author LLVZ. See also references (2) and (11). References and Footnotes 1. A recent calculation along these lines has been given by J.E. DelBene,.!. Comp. Chern. 10,603 ( 1988). She treats complexes of two larger atoms together with as many as six hydrogens. A smaller exploration of ammonia-amonium coupling, a similar system was also given by N.C. Baird, Int. J. Quant. Chern.: Quant. Bio. Symp. 1, 49 (1974). The additional complexities we encounter in our examples here where hydrogen bonding Nand 0 atoms are bonded into larger structures, partially charged, and progressively deformed, have never been attempted on this fundamental level. 2. E.R. Lippencott & R. Schroeder,.!. Chern. Phys. 23, 1009 (1955). 3. AT. Hagler, E. Huler & S. Lifson,.T. Am. Chern. Soc. 96,5319 (1974). 4. M. Levitt,.!. Mol. Bioi. 168, 595 (1983). 5. P.K. Weiner & P.K. Kollman,.!. Comp. Chern. 2, 287 (1981). 6. M. Peyrard & AR. Bishop, Phys. Rev. Lett. 62,2672 (1989). 7. Y. Gao, KV. Devi-Prassad & E.W. Prohofsky,.T. Chern. Phys. 80,6291 (1984). 8. Y. Kim, K.V. Devi-Prassad & E.W. Prohofsky, Phys. Rev. B 32,5185 (1985). 9. V.V. Prabhu, L. Young & E.W. Prohofsky, Phys. Rev. B 39,5436 (1989). 10. KM. Awati, M. Techera & E.W. Prohofsky, Bull. Am. Phys. Soc. 35, 649 (1990). 11. R. Schroeder & E.R. Lippencott,.T. Am. Chern. Soc. 79,921 (1957). 12. W.K. Schroll & L.L. Van Zandt, Low-frequency Parametrization of Hydrogen Bonding, to be published. 13. This includes, besides all those mentioned above, one of the author's own, a simulation of soliton dynamics involving H-bond stretching anharmonicity. L.L. Van Zandt. Phys. Rev. A 40, 6134, (1989).
Date Received: Apri/13, 1990
Communicated by the Editor Ramaswamy H. Sarma
