Journal of Biomolecular NMR

, Volume 54, Issue 1, pp 97–114 | Cite as

Transforming between discrete and continuous angle distribution models: application to protein χ1 torsions

  • Jürgen M. Schmidt


Two commonly employed angular-mobility models for describing amino-acid side-chain χ1 torsion conformation, the staggered-rotamer jump and the normal probability density, are discussed and performance differences in applications to scalar-coupling data interpretation highlighted. Both models differ in their distinct statistical concepts, representing discrete and continuous angle distributions, respectively. Circular statistics, introduced for describing torsion-angle distributions by using a universal circular order parameter central to all models, suggest another distribution of the continuous class, here referred to as the elliptic model. Characteristic of the elliptic model is that order parameter and circular variance form complementary moduli. Transformations between the parameter sets that describe the probability density functions underlying the different models are provided. Numerical aspects of parameter optimization are considered. The issues are typified by using a set of χ1 related 3 J coupling constants available for FK506-binding protein. The discrete staggered-rotamer model is found generally to produce lower order parameters, implying elevated rotatory variability in the amino-acid side chains, whereas continuous models tend to give higher order parameters that suggest comparatively less variation in angle conformations. The differences perceived regarding angular mobility are attributed to conceptually different features inherent to the models.


Circular statistics Directional data Probability density Gaussian model Elliptic model Jinc function Bessel function Torsion angle conformation Rotamer equilibria Differential probability 3J Vicinal coupling constants Amino-acid side chain Protein structure FKBP 

Supplementary material

10858_2012_9653_MOESM1_ESM.pdf (22 kb)
Tables are provided showing selected values of F(R) in support of Figure 2, as well as the detailed fit results underpinning Figure 3 (PDF 21 kb)


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of BiosciencesUniversity of KentCanterbury, KentUK

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