ProtFract helps you to quantify various self-similar properties of proteins with fractal dimension-based constructs. If you want to know the details of the theory behind such constructs, refer to previous publications of our group, as specified in the section ‘Proteins and Fractals’.

If your interest is in finding the mass fractal dimension, and/or the hydrophobic fractal dimension, and/or the polarizability fractal dimension, and/or the correlation dimension amongst peptide dipoles, and/or the correlation dimension amongst all the charged residues, and/or correlation dimension amongst all the aromatic residues, and/or correlation dimension between aromatic residues and positively charged residues - you will have to opt for ProtFract services for protein interior investigation.

If instead, your interest lies in finding the corrugation profile of protein surface of your interest, and/or in finding the changing magnitudes of correlation dimension of surface atoms when being scanned by probe spheres of different radii - you will have to opt for ProtFract services for protein exterior investigation.

Results provided by ProtFract services for protein interior investigation basically involve two principal algorithms; one to calculate the mass-hydrophobicity-polarizability fractal dimension, the other to calculate correlation dimension to quantify the extent of symmetry in self-similar organization of various biophysical entities, - an outline of these algorithms are provided below. For an in-depth knowledge about these, however, refer to [1] and [2].

**Section 1 (Interior Algorithms):**

**Algorithm for Correlation Dimension Calculation:**

Correlation dimension is calculated using a log-log graph of the Correlation integral (C(r)) versus distance range (r). From the slope (of the steepest region) of this log-log plot the correlation dimension is calculated.
Denoting the suitably representative atom of the residue as a geometric point, the coordinate of it is considered. Choice of representative atom varies from case-to-case; thus, while calculating the correlation dimension amongst all active chiral centres within a protein, all the C-alpha coordinates and two C-beta coordinates (from THR and ILE) are considered; but while calculating the side-chain interactions, only the C-beta coordinates of residues were considered, etc.

Formula for calculating the correlation dimension is given by:
**C(r)=g/(N*(N-1)/2)**

where g: total number of pairs of points which have distance between them that is less than any specified threshold of distance, r. N: total number of points

**Algorithm for Mass/Hydrophobicity/Polarizability Fractal Dimension Calculation:**

For fractal dimension calculation, protein is radially partioned into shells. Cumulative mass/hydrophobicity/polarizability in consecutive shells are calculated. Fractal dimension for each shell is calculated as: log(total (mass/hydrophobicity/polarizability))/log(shell radius)
The log-log plot obtained from this result, is subjected to scanning operation employing an overlapping window of 5 items. Finally, magnitude of the middle point of the linear portion of the ordinate of the log-log plot, is considered as the corresponding fractal dimension.

**Section 2 (Exterior Algorithms):**

Results provided by ProtFract services for protein exterior investigation involve two algorithms; one to quantify the extent of self-similarity in the roughness profile of the protein surface under consideration, the other to quantify the symmetry in self-similar nature of dependencies amongst surface atoms when scanned through probe spheres of varying radii. Outlines of these algorithms are presented below. To calculate the FD magnitude characterizing the surface roughness, a table is prepared that enlists the various magnitudes of molecular surface area as calculated by probe spheres with varying radii. Then, by applying the formula **FD = 2 - [(log Ai)/(log Ri)] **, the surface FD is calculated. Interested readers can refer to [3] and [4] to obtain further details. FD magnitudes are presented for the entire protein surface to obtain a (global) idea about the corrugation profile of the protein concerned; and for any particular stretch of protein surface (in PDB format) to obtain the magnitude of corrugation profile in any local stretch of protein surface. To calculate the correlation dimension amongst surface atoms, the correlation integral[2] amongst surface atoms is evaluated. This analysis may help the user to obtain an idea about probe-sphere-dependent symmetry of interaction profiles amongst surface atoms.

Here, users should note two (related) points of considerable importance.
First : FD attempts to quantify the symmetry of statistical self-similarity in surface roughness. Hence, if the number of atoms in the definition of contact surface and/or re-entrant surface is found to be less than the statistical parametric limit, FD for contact surface and/or re-entrant surface (whichever applicable), cannot be calculated. Such a problem is often encountered when user inputs a particular patch of the surface (involving very few residues) and especially, when FD of re-entrant surface for this patch is being calculated. In such cases, it is found frequently that with increasing probe-radius, the number of atoms in re-entrant surface accessed by the probe sphere, drops below the required statistical limit. Thus in many cases, the re-entrant surface FD information won't be presented. Moreover, aforementioned inadequacy of statistically significant number of atoms, explains the reason behind unusually low or high magnitude of re-entrant surface, as are reported at times.

Second : FD calculation depends primarily on the definition of contact surface, re-entrant surface and total surface. Furthermore, FD calculation depends upon logarithmic descriptions of these surfaces with probe spheres of varying radii. Since the entire relationship has a logarithmic nature, linear results that would have resulted from linear relationship : Total surface FD = Contact surface FD + Re-entrant surface FD - should not be expected.

**References:**

[1] 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

[2] Banerji A, Ghosh I. 2011. Fractal symmetry of protein interior: what have we learned? Cell. Mol. Life. Sc. 68: 2711-2737.

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

[4] Pettit and Bowie, 1999: Pettit FK, Bowie JU. 1999. Protein surface roughness and small molecular binding sites. J Mol Biol 285: 1377-1382.