Karplus equation

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Graph of the Karplus relation JHH(ph) = 12 cosph - cosph+2 obtained for ethane derivatives Karplus.svg
Graph of the Karplus relation JHH(φ) = 12 cosφ - cosφ+2 obtained for ethane derivatives

The Karplus equation, named after Martin Karplus, describes the correlation between 3 J-coupling constants and dihedral torsion angles in nuclear magnetic resonance spectroscopy: [2]

where J is the 3J coupling constant, is the dihedral angle, and A, B, and C are empirically derived parameters whose values depend on the atoms and substituents involved. [3] The relationship may be expressed in a variety of equivalent ways e.g. involving cos2φ rather than cos 2φ —these lead to different numerical values of A, B, and C but do not change the nature of the relationship.

The relationship is used for 3JH,H coupling constants. The superscript "3" indicates that a 1H atom is coupled to another 1H atom three bonds away, via H-C-C-H bonds. (Such hydrogens bonded to neighbouring carbon atoms are termed vicinal). [4] The magnitude of these couplings are generally smallest when the torsion angle is close to 90° and largest at angles of 0 and 180°.

This relationship between local geometry and coupling constant is of great value throughout nuclear magnetic resonance spectroscopy and is particularly valuable for determining backbone torsion angles in protein NMR studies.

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PREDITOR is a freely available web-server for the prediction of protein torsion angles from chemical shifts. For many years it has been known that protein chemical shifts are sensitive to protein secondary structure, which in turn, is sensitive to backbone torsion angles. torsion angles are internal coordinates that can be used to describe the conformation of a polypeptide chain. They can also be used as constraints to help determine or refine protein structures via NMR spectroscopy. In proteins there are four major torsion angles of interest: phi, psi, omega and chi-1. Traditionally protein NMR spectroscopists have used vicinal J-coupling information and the Karplus relation to determine approximate backbone torsion angle constraints for phi and chi-1 angles. However, several studies in the early 1990s pointed out the strong relationship between 1H and 13C chemical shifts and torsion angles, especially with backbone phi and psi angles. Later a number of other papers pointed out additional chemical shift relationships with chi-1 and even omega angles. PREDITOR was designed to exploit these experimental observations and to help NMR spectroscopists easily predict protein torsion angles from chemical shift assignments. Specifically, PREDITOR accepts protein sequence and/or chemical shift data as input and generates torsion angle predictions for phi, psi, omega and chi-1 angles. The algorithm that PREDITOR uses combines sequence alignment, chemical shift alignment and a number of related chemical shift analysis techniques to predict torsion angles. PREDITOR is unusually fast and exhibits a very high level of accuracy. In a series of tests 88% of PREDITOR’s phi/psi predictions were within 30 degrees of the correct values, 84% of chi-1 predictions were correct and 99.97% of PREDITOR’s predicted omega angles were correct. PREDITOR also estimates the torsion angle errors so that its torsion angle constraints can be used with standard protein structure refinement software, such as CYANA, CNS, XPLOR and AMBER. PREDITOR also supports automated protein chemical shift re-referencing and the prediction of proline cis/trans states. PREDITOR is not the only torsion angle prediction software available. Several other computer programs including TALOS, TALOS+ and DANGLE have also been developed to predict backbone torsion angles from protein chemical shifts. These stand-alone programs exhibit similar prediction performance to PREDITOR but are substantially slower.

References

  1. Minch, M. J. (1994). "Orientational Dependence of Vicinal Proton-Proton NMR Coupling Constants: The Karplus Relationship". Concepts in Magnetic Resonance . 6: 41–56. doi:10.1002/cmr.1820060104.
  2. Dalton, Louisa (2003-12-22). "Karplus Equation". Chemical & Engineering News . 81 (51): 37. doi: 10.1021/cen-v081n036.p037 .
  3. Karplus, Martin (1959). "Contact Electron-Spin Coupling of Nuclear Magnetic Moments". J. Chem. Phys. 30 (1): 11–15. Bibcode:1959JChPh..30...11K. doi:10.1063/1.1729860.
  4. Karplus, Martin (1963). "Vicinal Proton Coupling in Nuclear Magnetic Resonance". J. Am. Chem. Soc. 85 (18): 2870–2871. doi:10.1021/ja00901a059.