choosing the parameters for the calculation of the fractal dimension of a discretized non - strictly self - symmetrical natural object is the first step in its surface roughness assessment .
there is a strict connection between the density of points of the discretized set , the fractal scales used for the calculation of d , and the d value itself ( figure 1 ) .
the choice of fractal scales may be summarized as follows : ( i ) choose exponential distances between the different scales ; ( ii ) select the smallest scale greater than the resolution of the discretization , otherwise d will be severely overestimated ( figure 1b ) , and the largest scale to not exceed the size of the smallest object inspected , otherwise it will include a large amount of empty space which will result in underestimated d values .
following these rules , we chose fractal scales   [ 0.4 , 0.8 , 1.6 , 3.2 ]  .
the largest one is about the size of a small ligand or amino acid side - chain diameter .
the next step in parameterizing the calculation of d is the selection of a suitable density of the points on the ses generated by msms ( figure 1a ) . while calculating d for different densities between one and 20 pp , we identified two intervals separated around 6 pp , in which d behaved differently .
the first interval harbors densities that are too low for the selected fractal scales , so that the hyperspheres for the smallest  can not reach any neighboring points , thus c(x,)=1/n , which leads to overestimating d. this interval has a large dispersion compared to the other one ( figure 1a ) , but the orderings of the elements appear to be stable ( figure 1d ) , meaning that a switch in density inside the interval would not drastically change the overall ordering of the set .
the same assumption also holds for the second interval containing densities greater than 6 pp .
it contains densities that are more favorable for a fractal dimension calculation , i.e. the production of stable regression lines , as well as the lower dispersion introduced by surface sampling effects .
naturally , a decrease in point density reduces the qualities of the discretization and results in a more crude approximation of d , but it also speeds up the computations , so we found a compromise at a density of 7 pp . note that choosing different fractal scales shifts the limit between the two intervals , depending on the choice of the lowest . discrimination ability of the fractal dimension d depending on the fractal scale and point density ( pp : points per  ) calculated for the 604 ligands of the main set of protein - ligand complexes .
a ) dispersion of d. a tendency of decreasing dispersion is apparent with increasing discretization density . a steep fall in d values as well as
b , c ) regression lines of the doubly logarithmic plot of the correlation sum for the ligand of the pdb i d : 1c5z complex for densities on both sides of the steep fall in d : b ) 5 pp , ( r=0.952 ) ; c ) 7 pp , ( r=0.999 ) .
d ) gray scale coded matrix containing the spearman s rank correlation of the vectors of d values of the ligands ( one 604-dimensional rank vector for each density ) .
each filled square represents the pairwise rank correlation between two corresponding vectors , illustrating how the change in density shuffles the relative positioning of the d values .
few changes in the ligands positions in the rank vector indicate stable discrimination ability ( strong correlation , dark color ) .
comprehensive definition of the model parameters : the analyzed spatial molecular data relied exclusively on prior crystallographic surveys on protein - ligand complexes , dna and rna . the pdb i d lists are provided as supplementary information .
the protein - ligand complexes used in this survey were selected from two publicly available databases : ( i ) the pdbbind ( www.pdbbind.org ) was accessed to create the main set of complexes , the results of which were verified on ( ii ) a subset of the sc - pdb , termed the verification set for reference .
a subset of 604 protein - ligand complexes based on the pdbbind database was selected according to the following criteria : ( i ) monomeric structures with predicted primary binding site volume not exceeding the volume of the corresponding ligand by more than 50 % ; ( ii ) structures for which the msms ses calculation failed due to software errors were excluded .
the main set ( supplementary listing 1 ) provided the following five sets of spatial data used in the survey at hand : ( i ) the proteins set contains spatial data gathered by stripping any atoms from the complexes that did not belong to a protein chain such as waters and ligands resulting in a set of 604 proteins ; ( ii ) the ligands set contains the ligands of the active sites of the protein - ligand - complexes resulting in a set of 604 small molecules ; ( iii ) the liganded pockets set contains the main pockets of the complexes resulting in a set of 604 active sites ; ( iv ) the protein surface clefts set consists of pockets selected by pocketpicker that are missing a ligand in the published crystallographic model ; ( v ) the background surface patches set comprises of protein points subsets extracted using msms at uniformly distributed random positions on the surfaces of the 604 proteins
a subset of 8639 protein - ligand complexes based on the sc - pdb database was selected according to the following criteria : ( i ) the complexes are not present in the main set ; ( ii ) very large structures ( more than 2.3510 surface points ) and structures for which the msms ses calculation failed due to software errors were excluded .
the surface patches for the pockets and clefts were excised using a hard proximity margin of 3  around the output of pocketpicker .
for the sampling of the background surface patches 20 random atoms on the surface of each protein were selected . each single surface patch
was then defined by using spheres with 6  radius around each randomly selected surface atom .
the ppi set was extracted from the freely available dockground ppi database ( dockground.bioinformatics.ku.edu/ ) , benchmark 3.0 .
it contains 131 complexes with both bound and one available unbound structures for each entry , as well as 102 complexes , each with both bound and both unbound structures available and provided .
thus , we were able to inspect 233 complexes with a grand total of 466 ppis .
since we analyzed ppis of known bound complexes , we used the corresponding complex partners for the definition of the interface s position , and a hard proximity margin of 3.5  .
we chose this to be slightly higher than 3.0  , because at 3.0  the sampled patches for the ppi exhibited cut - off artifacts such as holes in the surfaces influencing the roughness estimation .
the protein , delivered in 20 mm acetate buffer , ph 5.0 with 200 mm nacl , 10 % ( v / v ) glycerol and 0.05 % 2-mercaptoethanol , was concentrated to a protein concentration of 3 mg / ml . 0.5 l of a 200 mm hiv protease inhibitor ( compound 7 ) solution in dmso was added to 80 l of the concentrated hiv-1 protease solution .
because it turned out that the inhibitor was poorly soluble in the resulting protein solution , an additional 2 l of a 50 mm inhibitor solution in peg400 was added to the last 20 l of the protein solution .
0.1 l of this latter protein solution was mixed with 0.1 l reservoir solution of 0.1 m nacitrate ph 3.5 and 3 m nacl and equilibrated at 20 c in a sitting drop setup .
20 % glycerol plus inhibitor was added to a crystal and the crystal was flash - frozen in liquid nitrogen .
data were collected on beam - line id-29 at the european synchrotron radiation facility ( esrf ) in grenoble .
data processing was done with xds and scaling with the ccp4 program scala , as implemented in the global phasing autoproc procedure .
the crystal structure was solved by molecular replacement using the structure of hiv-1 protease ( pdb i d : 5hpv ) as a model .
data collection and refinement statistics are listed in table 4 . we found a peptide fragment fitted by perryman et al .
( pdb i d : 4e43 ) in the active site of their 1.54  structure of 
, the peptide appeared to be in a large part cleaved by the protease and only the p1-p4 residues were fitted .
weaker density indicated that partially also intact peptide was bound . similarly to the 4e43 structure , an acetate ion and glycerol molecule were found bound in the exosite .
choosing the parameters for the calculation of the fractal dimension of a discretized non - strictly self - symmetrical natural object is the first step in its surface roughness assessment .
there is a strict connection between the density of points of the discretized set , the fractal scales used for the calculation of d , and the d value itself ( figure 1 ) .
the choice of fractal scales may be summarized as follows : ( i ) choose exponential distances between the different scales ; ( ii ) select the smallest scale greater than the resolution of the discretization , otherwise d will be severely overestimated ( figure 1b ) , and the largest scale to not exceed the size of the smallest object inspected , otherwise it will include a large amount of empty space which will result in underestimated d values .
following these rules , we chose fractal scales   [ 0.4 , 0.8 , 1.6 , 3.2 ]  .
the largest one is about the size of a small ligand or amino acid side - chain diameter .
the next step in parameterizing the calculation of d is the selection of a suitable density of the points on the ses generated by msms ( figure 1a ) . while calculating d for different densities between one and 20 pp , we identified two intervals separated around 6 pp , in which d behaved differently .
the first interval harbors densities that are too low for the selected fractal scales , so that the hyperspheres for the smallest  can not reach any neighboring points , thus c(x,)=1/n , which leads to overestimating d. this interval has a large dispersion compared to the other one ( figure 1a ) , but the orderings of the elements appear to be stable ( figure 1d ) , meaning that a switch in density inside the interval would not drastically change the overall ordering of the set .
the same assumption also holds for the second interval containing densities greater than 6 pp .
it contains densities that are more favorable for a fractal dimension calculation , i.e. the production of stable regression lines , as well as the lower dispersion introduced by surface sampling effects .
naturally , a decrease in point density reduces the qualities of the discretization and results in a more crude approximation of d , but it also speeds up the computations , so we found a compromise at a density of 7 pp . note that choosing different fractal scales shifts the limit between the two intervals , depending on the choice of the lowest . discrimination ability of the fractal dimension d depending on the fractal scale and point density ( pp : points per  ) calculated for the 604 ligands of the main set of protein - ligand complexes .
a ) dispersion of d. a tendency of decreasing dispersion is apparent with increasing discretization density . a steep fall in d values as well as
b , c ) regression lines of the doubly logarithmic plot of the correlation sum for the ligand of the pdb i d : 1c5z complex for densities on both sides of the steep fall in d : b ) 5 pp , ( r=0.952 ) ; c ) 7 pp , ( r=0.999 ) .
d ) gray scale coded matrix containing the spearman s rank correlation of the vectors of d values of the ligands ( one 604-dimensional rank vector for each density ) .
each filled square represents the pairwise rank correlation between two corresponding vectors , illustrating how the change in density shuffles the relative positioning of the d values .
few changes in the ligands positions in the rank vector indicate stable discrimination ability ( strong correlation , dark color ) .
the analyzed spatial molecular data relied exclusively on prior crystallographic surveys on protein - ligand complexes , dna and rna . the pdb i d lists are provided as supplementary information .
the protein - ligand complexes used in this survey were selected from two publicly available databases : ( i ) the pdbbind ( www.pdbbind.org ) was accessed to create the main set of complexes , the results of which were verified on ( ii ) a subset of the sc - pdb , termed the verification set for reference .
a subset of 604 protein - ligand complexes based on the pdbbind database was selected according to the following criteria : ( i ) monomeric structures with predicted primary binding site volume not exceeding the volume of the corresponding ligand by more than 50 % ; ( ii ) structures for which the msms ses calculation failed due to software errors were excluded .
the main set ( supplementary listing 1 ) provided the following five sets of spatial data used in the survey at hand : ( i ) the proteins set contains spatial data gathered by stripping any atoms from the complexes that did not belong to a protein chain such as waters and ligands resulting in a set of 604 proteins ; ( ii ) the ligands set contains the ligands of the active sites of the protein - ligand - complexes resulting in a set of 604 small molecules ; ( iii ) the liganded pockets set contains the main pockets of the complexes resulting in a set of 604 active sites ; ( iv ) the protein surface clefts set consists of pockets selected by pocketpicker that are missing a ligand in the published crystallographic model ; ( v ) the background surface patches set comprises of protein points subsets extracted using msms at uniformly distributed random positions on the surfaces of the 604 proteins . external test set .
a subset of 8639 protein - ligand complexes based on the sc - pdb database was selected according to the following criteria : ( i ) the complexes are not present in the main set ; ( ii ) very large structures ( more than 2.3510 surface points ) and structures for which the msms ses calculation failed due to software errors were excluded .
the surface patches for the pockets and clefts were excised using a hard proximity margin of 3  around the output of pocketpicker . for the sampling of the background surface patches
each single surface patch was then defined by using spheres with 6  radius around each randomly selected surface atom .
the ppi set was extracted from the freely available dockground ppi database ( dockground.bioinformatics.ku.edu/ ) , benchmark 3.0 .
it contains 131 complexes with both bound and one available unbound structures for each entry , as well as 102 complexes , each with both bound and both unbound structures available and provided .
thus , we were able to inspect 233 complexes with a grand total of 466 ppis .
since we analyzed ppis of known bound complexes , we used the corresponding complex partners for the definition of the interface s position , and a hard proximity margin of 3.5  .
we chose this to be slightly higher than 3.0  , because at 3.0  the sampled patches for the ppi exhibited cut - off artifacts such as holes in the surfaces influencing the roughness estimation .
the surface patches for the pockets and clefts were excised using a hard proximity margin of 3  around the output of pocketpicker .
for the sampling of the background surface patches 20 random atoms on the surface of each protein were selected .
each single surface patch was then defined by using spheres with 6  radius around each randomly selected surface atom .
the ppi set was extracted from the freely available dockground ppi database ( dockground.bioinformatics.ku.edu/ ) , benchmark 3.0 .
it contains 131 complexes with both bound and one available unbound structures for each entry , as well as 102 complexes , each with both bound and both unbound structures available and provided .
thus , we were able to inspect 233 complexes with a grand total of 466 ppis .
since we analyzed ppis of known bound complexes , we used the corresponding complex partners for the definition of the interface s position , and a hard proximity margin of 3.5  .
we chose this to be slightly higher than 3.0  , because at 3.0  the sampled patches for the ppi exhibited cut - off artifacts such as holes in the surfaces influencing the roughness estimation .
the protein , delivered in 20 mm acetate buffer , ph 5.0 with 200 mm nacl , 10 % ( v / v ) glycerol and 0.05 % 2-mercaptoethanol , was concentrated to a protein concentration of 3 mg / ml . 0.5 l of a 200 mm hiv protease inhibitor ( compound 7 ) solution in dmso was added to 80 l of the concentrated hiv-1 protease solution .
because it turned out that the inhibitor was poorly soluble in the resulting protein solution , an additional 2 l of a 50 mm inhibitor solution in peg400 was added to the last 20 l of the protein solution .
0.1 l of this latter protein solution was mixed with 0.1 l reservoir solution of 0.1 m nacitrate ph 3.5 and 3 m nacl and equilibrated at 20 c in a sitting drop setup .
, 20 % glycerol plus inhibitor was added to a crystal and the crystal was flash - frozen in liquid nitrogen .
data were collected on beam - line id-29 at the european synchrotron radiation facility ( esrf ) in grenoble .
data processing was done with xds and scaling with the ccp4 program scala , as implemented in the global phasing autoproc procedure .
the crystal structure was solved by molecular replacement using the structure of hiv-1 protease ( pdb i d : 5hpv ) as a model .
data collection and refinement statistics are listed in table 4 . we found a peptide fragment fitted by perryman et al .
( pdb i d : 4e43 ) in the active site of their 1.54  structure of 
, the peptide appeared to be in a large part cleaved by the protease and only the p1-p4 residues were fitted .
weaker density indicated that partially also intact peptide was bound . similarly to the 4e43 structure , an acetate ion and glycerol molecule were found bound in the exosite .
g. s. is a co - founder of insili.com llc , zrich , and a consultant in the pharmaceutical and chemical industry .
 quantifying the properties of macromolecules is a prerequisite for understanding their roles in biochemical processes . 
 one of the less - explored geometric features of macromolecules is molecular surface irregularity , or  
 roughness , which can be measured in terms of fractal dimension ( d ) . in this study 
 , we demonstrate that surface roughness correlates with ligand binding potential . 
 we quantified the surface roughnesses of biological macromolecules in a large - scale survey that revealed d values between 2.0 and 2.4 . 
 the results of our study imply that surface patches involved in molecular interactions , such as ligand - binding pockets and protein - protein interfaces , exhibit greater local fluctuations in their fractal dimensions than  
 inert surface areas . 
 we expect approximately 22 % of a protein s surface outside of the crystallographically known ligand binding sites to be ligandable . 
 these findings provide a fresh perspective on macromolecular structure and have considerable implications for drug design as well as chemical and systems biology . 
4 Materials and Methods
4.1 Fractal Scales and Density
4.1.1 Local Roughness Model
4.2 Data Sets
4.2.1 Pockets and Clefts
4.2.2 Random Surface Patches
4.2.3 ProteinProtein Interfaces
4.3 X-Ray Structure Analysis
Conflict of Interest Statement