Protein Engineering, Vol. 15, No. 2, 79-89,
February 2002
© 2002 Oxford University Press
A Fourier analysis of symmetry in protein structure
Division of Mathematical Biology, National Institute for Medical Research, The Ridgeway, Mill Hill, London NW7 1AA, UK and 2 Mathematics Department, Stanford University, Palo Alto, CA, USA
The score matrix from a structure comparison program (SAP) was used to search for repeated structures using a Fourier analysis. When tested with artificial data, a simple Fourier transform of the smoothed matrix provided a clear signal of the repeat periodicity that could be used to extract the repeating units with the SAP program. The strength of the Fourier signal was calibrated against the signal from model proteins. The most useful of these was the novel random-walk approach employed to generate realistic ‘fake’ structures. On the basis of these it was possible to conclude that only a small proportion of protein structures have an unexpected degree of symmetry. Artificially generated ‘ideal’ folds provided an upper limit on the strength of signal that could be expected from a ‘perfectly’ repeating compact structure. Unexpectedly, some of the very regular ß-propellor folds attained the same strength but the majority of symmetric structures lay below this region. When native proteins were ranked by the power of their spectrum a wide variety of fold types were seen to score highly. In the ß
class, these included the globular ß proteins and the more repetitive leucine-rich ß folds. In the all-ß class; ß-propellors, ß-prisms and ß-helices were found as well as the more globular 
-crystalin domains. When this ranked list was filtered to remove proteins that contained detectable internal sequence similarity (using the program REPRO), the list became exclusively composed of just globular ß class proteins and in the top 50 re-ranked proteins, only a single 4-fold propellor structure remained.