A moment invariant for evaluating the chirality of three-dimensional objects

Abstract

Chirality is an important feature of three-dimensional objects and a key concept in chemistry, biology and many other disciplines. However, it has been difficult to quantify, largely owing to computational complications. Here we present a general chirality measure, called the chiral invariant (CI), which is applicable to any three-dimensional object containing a large amount of data. The CI distinguishes the hand of the object and quantifies the degree of its handedness. It is invariant to the translation, rotation and scale of the object, and tolerant to a modest amount of noise in the experimental data. The invariant is expressed in terms of moments and can be computed in almost no time. Because of its universality and computational efficiency, the CI is suitable for a wide range of pattern-recognition problems. We demonstrate its applicability to molecular atomic models and their electron density maps. We show that the occurrence of the conformations of the macromolecular polypeptide backbone is related to the value of the CI of the constituting peptide fragments. We also illustrate how the CI can be used to assess the quality of a crystallographic electron density map.

Figure 1.

The CI for biphenyl as a function of the inter-ring torsion angle. (a) A chemical formula for the biphenyl molecule with inter-ring torsion angle θ indicated. (b) The CI of the biphenyl molecule as a function of θ (red curve). Unlike G₀ (green curve), the values of CI, obtained with equations (A 4) and (A 5), exhibit a perfectly sinusoidal dependence on the inter-ring torsion angle. (c) The effect of noise on the values of CI. The blue and purple curves show the CI when a normally distributed error of 0.4 Å and 1.0 Å, respectively, is applied to the coordinates of biphenyl after each rotation. For reference, the red curve is the undisturbed CI from (b).

Figure 2.

The CI for l-tartaric acid. (a) A tartaric acid molecule placed in a 10 × 10 × 10 ų cubic unit cell, viewed approximately along the z-axis. (b) The CI for the tartaric acid electron density (values higher than 1σ above the mean) as a function of the upper resolution limit, d_max. As a reference, the value of CI computed from the tartaric acid xyz coordinates weighted by the corresponding atomic numbers is 1.7. (c–f) Overlapped two-dimensional projections of the electron density along the z-axis. The contours are drawn at 1σ above the mean. The upper resolution limit, d_max, is indicated.

Figure 3.

The CI and the protein backbone conformations. (a) The structure of an alanine di-peptide fragment, Cα-CO-N-(Cα-Cβ)-CO-N-Cα, in the conformation with φ = −180°, ψ = −180°. (b) The CI for the alanine di-peptide fragment as a function of its backbone torsion angles φ and ψ. The value of CI is computed by taking ρ(r) equal to the atomic number at atomic centres, and zero elsewhere. The sign of the value of CI is represented by colour (blue is negative and red is positive) and the magnitude by the brightness of the colour. The inner contours include 90% of the non-glycine, non-proline and non-pre-proline residues in the Top500 database [35]; the outer contours cover 98% of the residues. The regions with high absolute values of the CI (bright colour in bottom left, top right and bottom right) do not normally occur in proteins.

Figure 4.

The CI for an electron density map. The map was computed for lysozyme measured to 1.54 Å resolution, using observed structure factor amplitudes and (a) phases obtained using anomalous dispersion from intrinsic sulphur atoms (red curve), (b) phases subsequently improved by density modification (blue curve) and (c) phases computed from the refined, final model (purple curve). The correlation coefficient to the map corresponding to the final model is 0.51, 0.81 and 1.0, respectively. (d) Distribution of the local value of the CI, computed from all non-negative density values within a sphere of 3 Å radius centred at each grid point of the map.