Multiscale Chiral Zeros in Protein Structures

Chirality, or mirror asymmetry, is central to the structure and function of proteins. Yet, difficulties with quantifying asymmetry have long hampered insight into how it shapes structure-property relationships in protein systems. Two commonly used chirality measures are the pseudoscalar Osipov-Pickup-Dunmur (OPD) index, which reports both a magnitude of asymmetry and a sign corresponding to handedness, and the Hausdorff Chirality Measure (HCM), which only reports a magnitude of asymmetry. Theoretical arguments have shown that OPD is expected to have “chiral zeros,” which occur when a chiral object is incorrectly assigned a chirality index value of zero. However, their existence remains theoretical and their abundance in real (bio)molecules has not previously been studied or methodologically addressed. We examine in detail the differences between OPD and HCM in several protein systems encompassing many scales of chirality and find that chiral zeros are prevalent at each scale. We also demonstrate a quantitative relationship between paired HCM values of interaction sites of interacting protein pairs and show that this correlation disappears when considering OPD instead. Thus, OPD is unsuitable for the quantification of chirality of complex molecular structures except in the simplest cases of helicoids. The need for new mathematical approaches to differentiate opposite handed structures and to quantify chirality of (bio)chemical objects across multiple scales is emphasized.