Karplus relation

Angular restraints are another important source of structural information. Several empirical relationships between scalar couplings and dihedral angles have been found during the last decades. The most important one is certainly the Karplus relation for -couplings. Another, relaxation-based angular restraint is the so-called CCR between two dipolar vectors or between a dipolar vector and a CSA tensor. 

Fig. 9.9 pro-R and pro-S determination of side-chain methylene protons in the P position in peptides. (A) The different possible configurations and (B) the Karplus relations for the corresponding jHanp-. Jhpc- and )HpN[Pg.228]

Vicinal scalar coupling constants, 3J, between atoms separated by three covalent bonds from each other are related to the enclosed torsion angle, 6, by Karplus relations [45]. 

The magnitude of the long-range allylic coupling, (V ) is controlled by the dihedral angle between the C-H bond and the plane of the double bond in a relationship reminiscent of the Karplus relation. 

All of the many one, two and three-bond interactions in organic molecules, e. g., the various Karplus relations involving J X,Y) as a function of dihedral angle, or the interaction used to find fully substituted signals in Figure 1.1... 

In aliphatic and cycloaliphatic compounds, vicinal carbon-proton coupling constants 3Jch are related to the dihedral angle 0, as known from the Karplus-Conroy relation for iJ[iH. The Fermi-contact contribution to 3JCH as a function of the dihedral angle 0 calculated for propane [131] is displayed in Fig. 3.15, and the Karplus relation given by eq. (3.17) can be derived ... 

Three-bond carbon-phosphorus couplings JCP depend on configuration, as expected, transoid couplings being larger than cisoid ones, as demonstrated for cyclohexyl- and 2-norbornylphosphonates in Table 4.52. The Karplus relation (4.13) can be derived from the data for 2-norbornyl derivatives [376]. 

The use of a generalized Karplus relation was the basis for comparison between calculated and experimental vicinal proton-proton coupling con-... 

Molecular mechanics has been used in combination with NMR spectroscopy to solve structural problems. MM-NMR techniques have been extensively used to solve protein structures125,1221. The main NMR information used in the modeling process is based on Karplus relations and NOE effects. Recent applications also involved the simulation of paramagnetic shifts in proteins with metal centers such as co-balt(II)11231. In such systems, the fact that protons close to the metal centers have short relaxation times (T and T2) can be used to establish connectivity patterns1124-1261. Applications in the area of simple coordination compounds are quite raretl27-130] although these are of importance, especially in respect of the determination of solution structures of metalloproteins, where the modeling of the metal center can be one of the more serious problems (see also Chapter 12). 

The solid line in Fig. 5.2 is a plot of Eq. 5.3 with the parameters A = 7, B = — 1, and C = 5 Hz. It is apparent that large values of J are predicted for cis (9 = 0°) and tram (6 = 180°) conformations and small values for gauche (60° and 120°) conformations. These predictions have been amply verified, and the Karplus relation is of great practical utility in structure determinations. 

Karplus relating the coupling constant J to the projected valence-angle 0, namely,... 

C. A. G. Haasnoot, F. A. A. M. de Leeuw, H. P. M. de Leeuw, and C. Altona, Interpretation of vicinal proton-proton coupling constants by a generalized Karplus relation. Conformational analysis of the exocyclic C-4 -C-5 bond in nucleosides and nucleotides, Recueil, J. Roy. Netherlands Chem. Soc., 98 (1979) 576-577. 

Using the Karplus relation, it is relatively easy to discern anti versus gauche orientation of adjacent protons. 

Despite the extremely wide use of multi-dimensional NMR techniques for conformational studies of biological macromolecules, at present, only a small part of the information contained in the NMR spectrum can be used for structure determination [118,119]. The reason is that there are no well-established correlations between NMR chemical shifts and the structural parameters [118] and only a few useful correlations besides Karplus relations [120,121] for nuclear spin-spin coupling constants. As a consequence, a great deal of important information about the system is not available without turning to quantum chemical approaches for the theoretical interpretation. 

For the former they found a Karplus relation by correlating the ubiquitin values with the backbone 0 angles from its crystal structure, Eq. (62)... 

Figure 12.71 (a) Selected vicinal constants from Karplus relation for allylic systems y =Jjcos (HCCH) + 2.6... 

R. Briischweiler, D.A. Case, Adding harmonic motion to the Karplus relation for spin-spin coupbng, J. Am. Chem. Soc. 116 (1994) 11199—11200. 

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