# Transforming between discrete and continuous angle distribution models: application to protein χ_{1} torsions

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## Abstract

Two commonly employed angular-mobility models for describing amino-acid side-chain χ_{1} 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 χ_{1} 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.

## Keywords

Circular statistics Directional data Probability density Gaussian model Elliptic model Jinc function Bessel function Torsion angle conformation Rotamer equilibria Differential probability^{3}

*J*Vicinal coupling constants Amino-acid side chain Protein structure FKBP

## Supplementary material

## References

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