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.
Highlights
Deviations from what is often perceived as harmonious symmetry have long fascinated philosophers, artists and scientists
The chiral invariant (CI) presented here combines two important properties: its sign indicates the handedness of the object, and the magnitude represents the degree of its chirality
The CI is robust towards experimental uncertainties present in the data
Summary
Deviations from what is often perceived as harmonious symmetry have long fascinated philosophers, artists and scientists. Chirality is an important feature of many objects in. Three-dimensional objects, such as the left or right hand, are called chiral if they cannot be superimposed on their mirror image. Objects possessing mirror planes, inversion axes or a centre of symmetry are identical to their mirror image and are said to be achiral [1]. Chirality is a key concept and an intuitive and discriminative descriptor for many objects, in the biological realm. A chiral arrangement of atoms behaves differently from its mirror image, or enantiomer. Some enantiomers are distinguished by smell [5] or taste [6], and their perception may induce completely different responses. Several barbiturates show a depressant activity in one chiral form but are excitatory in the other [8]. Because there is no reason to assume that the enantiomer of a therapeutically active component is free from undesirable effects [10], development of new stereoisomeric drugs has been tightly regulated [11]
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