This Web app considers a biological helical structure as the product of a continous helix and a set of parallel planes. Based on the covolution theory, the Fourier Transform (FT) of a helical structure would be the convolution of the FT of the continous helix and the FT of the planes. The FT of a continous helix consists of equally spaced layer planes (3D) or layerlines (2D projection) that can be described by Bessel functions of increasing orders (0, ±1, ±2, ...) from the Fourier origin (i.e. equator). The spacing between the layer planes/lines is determined by the helical pitch (i.e. the shift along the helical axis for a 360° turn of the helix). If the structure has additional cyclic symmetry (for example, C6) around the helical axis, only the layer plane/line orders of integer multiplier of the symmetry (e.g. 0, ±6, ±12, ...) are visible. The primary peaks of the layer lines in the power spectra form a pattern similar to a X symbol. The FT of the parallel planes consists of equally spaced points along the helical axis (i.e. meridian) with the spacing being determined by the helical rise. The convolution of these two components (X-shaped pattern of layer lines and points along the meridian) generates the layer line patterns seen in the power spectra of the projection images of helical structures. The helical indexing task is thus to identify the helical rise, pitch (or twist), and cyclic symmetry that would predict a layer line pattern to explain the observed layer lines in the power spectra. This Web app allows you to interactively change the helical parameters and superimpose the predicted layer liines on the power spectra to complete the helical indexing task. PS: power spectra; PD: phase differences across the meridian; YP: Y-axis power spectra profile; LL: layer lines; m: indices of the X-patterns along the meridian; Jn: Bessel order

HILL: Helical Indexing using Layer Lines