Several protein properties embody symmetry of self-similarity. Symmetries observed in crystalline structures can be described with translations, rotations, and reflections; symmetry of self-similarity, on the other hand, manifests itself through scale invariance. Self-similar forms are composed of subunits that resemble the overall object. Many natural structures in nature - the branching trees, crinkly coastlines, mountain surfaces - demonstrate self-similarity[1]. If the Hausdorff dimension[2] of a self-similar object is found to be greater than its topological dimension, the object is called a fractal-object. What makes this information relevant in the field of Bioinformatics is the fact the several works of last three decades have established that proteins are fractal-objects.

Protein surfaces are fractal[3]. Mass [4, 5], hydrophobicity and polarizability distribution of protein interior are fractal. Many other protein properties (main-chain connectivity profile, potential energy profile, dipole moment distribution etc. [6, and references therein]) - are fractal too. However, a computational resource that calculates fractal properties of protein interior and exterior - is (apparently) impossible to find.

ProtFract, the free online server, calculates the fractal dimension of many of the protein properties enlisted above.

References:

[1] Mandelbrot, BB (1982). The Fractal Geometry of Nature, W.H. Freeman and Company.

[2] Hausdorff F (1919) Dimension undusseres Mass. Math Ann 79:157-179

[3] Lewis M, Rees DC (1985) Fractal Surfaces of Proteins. Science. 230:1163-1165.

[4] Enright MB, Leitner DM (2005) Mass fractal dimension and the compactness of proteins. Phys Rev E 71:011912

[5] Banerji A, Ghosh I (2009) Revisiting the myths of protein interior: studying proteins with mass-fractal hydrophobicity-fractal and polarizability-fractal dimensions. PLoS One 4(10):e7361

[6] Banerji A, Ghosh I (2011) Fractal symmetry of protein interior: what have we learned? Cell. Mol. Life Sci. DOI 10.1007/s00018-011-0722-6