Reviews in Computational Chemistry Volume 18

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126 Polarizability in Computer Simulations indicate cooperativity in the hydrogen-bond energies, 226 and dielectric measurements on polypeptide chains show an enhancement of the dipole moment of the peptide group in an a-helix. 227 Other quantum mechanical studies have addressed the importance of polarizability on protein folding, 228 enzyme catalysis, 229 DNA base pair stacking, 230 and nucleic acid interactions with ions. 231 Several polarizable models for proteins and the peptide group have been developed, using polarizable point dipoles, 32,44,45,232 electronegativity equalization models, 10,146 and the two-state empirical model. 183 Simulations using point polarizable dipole models by Warshel and co-workers 44,45 and by Wodak and co-workers 46 examined the role of polarizability on protein stability, dielectric properties, and enzymatic activity. For example, Van Belle et al. 46 found that the helix dipoles are enhanced, in agreement with the dielectric measurements of Wada, 227 and, further, the helix dipoles are enhanced not only through hydrogen bonds to the backbone, but also through association with side chain atoms. Polarization has also been shown to influence the folding time scales for small polypeptides. 183 For nucleic acids, a point polarizable dipole model was recently introduced. 232 Despite these studies and acknowledgment of the importance of polarizability from both electronic structure and experimental studies, not many simulations of proteins or nucleic acids using polarizable models have been done to date. An implication of resonance-assisted hydrogen bonding is that as the charges are polarized, through hydrogen bonds or other interactions, the hybridization of the atoms involved can change. For example, studies of crystal structures of formamide reveal that the C O bond length increases and the C N bond length decreases due to the formation of hydrogen bonded dimers. 233 Other crystal structures and ab initio quantum calculations on amides further validate the fact that hydrogen bonds can change those bond lengths. 234 The hydrogen bonds in these structures are in the amide plane and promote the double bond, zwitterionic state. On the other hand, the interactions in which the amino nitrogen serves as a hydrogen-bond acceptor would stabilize the single bond form. Partial sp 3 hybridization of the amino nitrogen leads to pyramidalization. Indeed, nonplanarities of some peptide bonds have been observed in atomic-resolution structures of proteins. 183,235,236 In addition, the planarity of the peptide bond is dependent on a protein’s secondary structure, with residues in an a-helix being more planar than elsewhere. 183 For nucleic acids, ab initio calculations indicate that the amino group can be pyramidalized through interactions with neighboring molecules or ions. 237 For both the peptide bond and nucleic acid bases, there is reason to believe that a significant degree of nonplanarity can be induced by the environment. To treat these effects, the polarization of the electrostatic degrees of freedom—charges or dipoles—would have to be coupled to the bonded interactions, as has been developed for the peptide bond. 183
Comparison of the Polarization Models 127 COMPARISON OF THE POLARIZATION MODELS Mechanical Polarization One important difference between the shell model and polarizable point dipole models is in the former’s ability to treat so-called mechanical polarization effects. In this context, mechanical polarization refers to any polarization of the electrostatic charges or dipoles that result from causes other than the electric field of neighboring atoms. In particular, mechanical interactions such as steric overlap with nearby molecules can induce polarization in the shell model, as further described below. These mechanical polarization effects are physically realistic and are quite important in some condensed-phase systems. As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles; in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipoleand EE-based models of molecular systems often use the Lennard–Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born–Mayer potentials, especially when ionic systems are being modeled. Regardless of the specific potential used, PPD- and EE-based models typically lack coupling between the short-range potential and the long-range electrostatic degrees of freedom. The dipoles and fluctuating charges respond solely to the local electric field (see Eq. [3]), with no regard for local shortrange interactions. In other words, the polarizability, a, of each point dipole in a PPD model is independent of the local environment. The situation is different for the shell-based models. Because the van der Waals and exchangerepulsion interactions being modeled by the short-range nonelectrostatic part of the potential are electron–electron interactions, the interaction sites are almost always taken to be coincident with the shell (electronic) charge, rather than the core (nuclear) charge or center of mass. The short-range interactions in the shell model couple with only one end of the finite dipole, rather than with both ‘‘ends’’ of the point dipole. Consequently, the shell model includes a coupling between the short-range interactions and the orientation of the dipole—a coupling that is not present in point dipole-based models. The coupling of short-range interactions and dipole orientations is in fact quite realistic physically, and the lack of such a coupling is a disadvantage of the PPD models. One way to better understand this coupling is to recognize that the shell models have two mechanisms for polarization: a purely electrostatic induction effect, governed by the fixed polarizability in Eq. [27], and a
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Reviews in Computational Chemistry
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Kenny B. Lipkowitz Department of Ch
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vi Preface three-dimensional struct
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viii Preface some descriptors and i
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Epilogue and Dedication My associat
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Contents 1. Clustering Methods and
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Contents xv Electron Transfer in Po
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Contributors John M. Barnard, Barna
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Contributors to Previous Volumes *
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Volume 3 (1992) Tamar Schlick, Opti
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Volume 7 (1996) Geoffrey M. Downs a
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Volume 11 (1997) Mark A. Murcko, Re
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T. Daniel Crawford* and Henry F. Sc
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Topics Covered in Volumes 1-18 * Ab
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Reviews in Computational Chemistry
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2 Clustering Methods and Their Uses
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10 Clustering Methods and Their Use
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42 The Use of Scoring Functions in
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Table 1 Reference List for the Most
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Page 117 and 118: CHAPTER 3 Potentials and Algorithms
Page 119 and 120: Vn (1 + cos(nω + γ)) 2 K θ (θ
Page 121 and 122: are modified by their environment w
Page 123 and 124: Table 1 Polarizability Parameters f
Page 125 and 126: The polarizable point dipole models
Page 127 and 128: two to three orders of magnitude sl
Page 129 and 130: M i z i + q i d i k i and shell cha
Page 131 and 132: Shell Models 103 on estimates of sh
Page 133 and 134: minimization can be replaced by mor
Page 135 and 136: The energy required to create a cha
Page 137 and 138: for all i ði:e:; 8 iÞ: @U @qi l
Page 139 and 140: where we have used q Cl ¼ qNa. The
Page 141 and 142: Electronegativity Equalization Mode
Page 143 and 144: Electronegativity Equalization Mode
Page 145 and 146: of N molecules is taken as a Hartre
Page 147 and 148: is treated using variable charges.
Page 149 and 150: water have been developed, includin
Page 151 and 152: Applications 123 classical and rigi
Page 153: developing polarizable models. A va
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Page 159 and 160: ecome significant at field strength
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Page 163 and 164: References 135 9. P. Cieplak and P.
Page 165 and 166: References 137 48. E. L. Pollock an
Page 167 and 168: References 139 Computational Chemis
Page 169 and 170: References 141 139. J. Hinze and H.
Page 171 and 172: References 143 178. J. J. P. Stewar
Page 173 and 174: References 145 216. M. W. Mahoney a
Page 175 and 176: CHAPTER 4 New Developments in the T
Page 177 and 178: Introduction 149 applications). For
Page 179 and 180: FCWD(∆E ) FCWD(0) ∆E = 0 ∆E
Page 181 and 182: Introduction 153 Equations [6]-[12]
Page 183 and 184: Paradigm of Free Energy Surfaces 15
Page 185 and 186: Paradigm of Free Energy Surfaces 15
Page 187 and 188: In Eqs. [18] and [19], F0i is the e
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Page 203 and 204: Table 1 Main Features of the Two-Pa
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and Hss ¼ U rep ss 1 2 X j;k ðmj
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λ i /eV 4 3 2 1 0 0 10 20 30 40 Bo
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the width/Stokes shift relation (Eq
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Table 3 Mapping of the Q Model on S
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This situation is of course not sat
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F ± (Y ad )/λ I 0.8 0.4 0 −0.4
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it by choosing the GMH basis set 7
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Anharmonic higher order terms gain
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of the radiation is the perturbatio
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individual vibrational excitations
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m 12 /D 6 5.5 5 4.5 4 3.5 17 18 19
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In Eq. [144], the coordinates Y km
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ε/M −1 cm −1 4000 2000 0 analy
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βσ 2 /10 3 cm −1 14 12 10 8 Opt
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0.2 0.1 0 12 16 20 24 28 32 The app
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References 207 7. M. D. Newton, Adv
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References 209 59. R. Kubo and Y. T
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Reviews in Computational Chemistry Volume 18

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