BACKGROUND
despite several decades of arduous effort, mapping of protein primary sequences to their three dimensional folds, referred to as the second genetic code, remains an unsolved scientific problem.
what appears to be lacking is a comprehensive theory, integrating two factors which definitely condition the isomorphism between sequence and fold, namely  the pattern of hydrophobicities embedded in the polypeptide chain  <cit>  and  the packing of amino acid side chains to give densely packed  <cit>  protein interiors.
under the present circumstances, the more tractable approach is the 'inverse protein folding problem'  <cit> , that is to identify protein primary sequences  <cit>  consistent with and supportive of a given fold, an idea which has found considerable application in the de novo design of targeted protein structures  <cit> .
yet even here, it was realized earlier on, that in de novo design, attainment of dense, well-packed protein cores  was neither an automatic part of the design process nor acquired simply by chance  <cit> .
most often, it was observed  that design led to molten globules or complete unraveling of the structure  <cit> .
an instructive example was the repeated failure to design parallel  <dig> - tim barrel  <cit> , finally resolved successfully by offredi et al.
<cit> , where a term optimizing for side chain packing specificity was deliberately included in the computational process.
in fact, one indicator for successful computational design  <cit>  is the attainment of densely packed side chains in the interior of the targeted protein, experimentally characterized by the absence of ans binding  <cit> .
thus, a comprehensive theory with regard to the packing of side chain atoms within proteins would not only provide insight into protein structures, facilitate their prediction  <cit>  and would also be a valuable aid in the design of novel proteins.
traditionally, there have been two models of protein packing:  the 'jigsaw puzzle' and  the 'nuts and bolts' model which lie on the opposite ends of the spectrum.
the jigsaw puzzle model attributed to crick  <cit> , postulates the stereo specific interdigitation of amino acid side chains giving rise to densely packed protein interiors.
on the other hand, the nuts and bolts model  <cit>  does not require the association of side chains with specific geometry and asserts that the internal architecture of proteins arises simply due to the high compaction of side chain atoms within a constrained volume.
lately, another model referred to as the 'oil drop' model  <cit>  has been proposed in order to capture the dynamic fluctuations in protein cores.
possibly, all these models concentrate exclusively on some special features of interior packing.
however, using a surface complementarity function a previous report from this laboratory  <cit>  demonstrated that binary association between two hydrophobic side chains , with high surface fit and maximal overlap between their corresponding residue surfaces, did exhibit specific inter-residue geometry.
it was thus clear that at least for a subset of contacts  predictions of the jigsaw puzzle model were indeed valid.
one drawback from all such studies is that the inter-residue interactions which sustain a native fold are more accurately modeled as a network rather than a discrete assortment of binary interacting pairs.
several attempts have been made to view protein structures as contact networks  <cit>  wherein the amino acids have been designated as nodes and their mutual non covalent interactions as edges.
the character of these networks  exhibit variability depending on the cutoffs used to define inter-atomic contact.
by and large, most protein contact networks preserve 'small-world' character   <cit>  and display signatures of assortative mixing   <cit> .
however, degree distribution can be exponential, sigmoidal or dependent on a single exponent - as a function of the criteria used to define atomic interactions  <cit> .
it has also been noted that in certain aspects protein contact networks differ significantly from other real world networks, for example in the restricted number of edges a node can have.
apart from providing insights into protein structures, these networks have been used to identify residues implicated in folding nuclei  <cit>  and transition states  <cit> , identifying functional residues involved in the active site  <cit> , hubs stabilizing the packing of secondary structural elements  <cit> , rationalization of the difference in protein stabilities from thermophilic/mesophilic organisms  <cit>  and estimation of folding rates  <cit> .
the utility of the network view of the protein structure is thus fairly well established.
in this study, the distribution of such networks from a database of protein structures has been analyzed in order to identify specific topological patterns in side chain association within protein cores.
such an analysis led to the recognition that certain packing topologies defined as packing motifs were found preferably in proteins.
a limited region of the topological space was exhaustively mapped in terms of frequently occurring packing motifs, combinations of which could lead to networks of larger sizes.
it was found that indeed larger networks could be assembled out of a basis set of smaller ones.
one such frequently occurring motif namely the three residue clique received special attention with regard to its composition and geometry of associating residues.
central to pursuing the research objectives outlined above was the extension of the jigsaw puzzle model into protein contact networks.
thus protein contact networks have been defined primarily in terms of surfaces rather than distance between point atoms .
as mentioned previously, earlier studies  <cit>  had established quantitative measures  to identify those residue pairs whose interacting side chains exhibit specific geometry.
these measures have now been used to define 'surface contact networks' based only on those inter-residue interactions which severely constrain geometry and thus could play a predominant role in stabilizing a particular fold.
RESULTS
the primary objective of this study is to find, characterize and classify recurring patterns in the packing of side chain atoms within a protein which sustains its native fold.
in this task we have deliberately chosen those contacts which strongly and specifically condition the inter-residue geometry of association.
since the majority of atomic contacts inside a protein are contributed by side chain atoms, it is natural to represent such interior packing as a network, defined primarily in terms of contact between their corresponding van der waals surfaces .
in addition, point atom contact networks  have also been studied simultaneously , by way of comparison.
contact between any two surfaces can be characterized in terms of overlap  that is the extent to which two surfaces are conjoined and by their goodness-of-fit or surface complementarity  .
in a previous work, it was demonstrated that when surface association between two amino acid side chains were greater than equal to  <dig>  and  <dig>  in ov and sm respectively , angular distributions specifying inter-residue geometry exhibited significant deviations from a random distribution  <cit> .
for a corresponding van der waals surface, the values of sm were found to be marginally lower for the same binary interactions.
in contrast to point atoms, the definition of 'contact'  between two surfaces is not necessarily commutative .
in networks based on surface contact, nodes representing residues a and b have been connected with an edge only when  the contact between a and b is commutative and  their reciprocal sm and ov both are greater than equal to  <dig>  and  <dig>  respectively.
for strong association between two residue surfaces their contact is expected to be commutative, which also effectively simplifies the network to an undirected graph.
for both point atom and surface contact networks, inter-atomic distance and surface-overlap bear a strong positive linear correlation.
sm on the other hand appears to be an additional feature for the latter.
interestingly, the choice of  <dig>  Å as the interaction cut off for point atoms led to maximum resemblance between the two categories of networks.
the first step was to study the distribution of networks in the protein database on the basis of size i.e.
the number of constituent nodes.
networks of smaller size  dominated the distribution  with a rapid decay in frequency for larger networks .
the distributions were however characterized by a long tail such that networks with greater than  <dig> nodes were also found, though with highly diminished frequency.
the distributions for both point-atom and surface contact networks were very similar.
the characteristic shape of the distribution could be adequately described by a power law  = k.x-n, where x is the network size), though, surface contact networks did exhibit some deviations in the range of graphs with  <dig> to  <dig> nodes .
the exponent, n was found to be  <dig>  and  <dig>  for ascn and apcn respectively.
relaxation of the cutoffs on sm and ov did not appear to significantly alter the basic character of the distribution apart from decreasing the population for smaller networks thereby extending the tail for larger ones.
on the other hand more stringent cut offs  led to the disintegration of the larger graphs, consequently increasing the frequency of small  and medium  sized networks with a drastic curtailment in the number of larger graphs  .
thus, as has been previously observed  <cit> , there appears to be a very narrow margin in terms of more stringent contact criteria which can abruptly change the spread and extent of contact networks within proteins.
number of networks found in the database for different ranges of network size .
cutoffs in surface complementarity  and overlap  respectively are given in bold within parentheses .
results for the chosen set of cutoff values, used in the analysis  are highlighted in bold.
the same calculations repeated for polypeptide chains distributed in bins with 75- <dig>  151- <dig>  301- <dig> residues gave similar curves, though for bins of larger chain length, networks of larger size appeared, thereby extending the long tail of the distribution.
as expected, frequency distributions of polypeptide chains containing networks of a particular size gave a similar decaying trend with increasing network size; that is networks of smaller size were found embedded in polypeptide chains regardless of the chain length, whereas instances of larger graphs were progressively rare.
these distributions tend to indicate that  small  to medium  sized networks are found universally in all protein structures, whereas linkage and/or fusion of these smaller networks to form larger ones is protein specific and is thus context dependent.
very large networks  were found only in  <dig> proteins  almost all of which had chain length exceeding  <dig> residues with close packing between extended secondary structural elements .
overall, most of the very large networks were found in α|β proteins .
in a protein contact network, there is an obvious upper bound to the highest possible degree a node can have  due to the limited volume of the residues involved in packing.
for the present set of criteria, the highest degree of a node was found to be restricted to  <dig> and  <dig> for ascn and apcn respectively.
from a graph theory perspective, local cohesiveness  of side chains should lead to dense packing within protein cores.
to test this proposition, average unweighted  and weighted  clustering coefficients were calculated for contact networks of all sizes.
both these coefficients gave rise to almost identical measures.
in parallel, a statistically significant number of random graphs  were generated  for the direct calculation of their clustering coefficients.
in a log-log plot , average clustering coefficients of the contact networks decayed much less rapidly with increasing network size compared to corresponding random graphs.
since, these coefficients essentially determine the cliquishness of a typical neighborhood  <cit>  in terms of clustering of local triplets  <cit> , it follows that  triplet cliques serve as the units of  clustering.
in other words, a graph of whatever size or connectivity will result in zero clustering  if there is no 'closed triplet' found embedded within it.
it is to be noted that any higher order clique could be considered as an association of nested triplet cliques.
since, these results confirm the probability of finding closed triplets to be much higher in proteins than random graphs, 3-cliques as 'clustering units' have thus received detailed attention in terms of geometry and composition, to be discussed in later sections.
packing motif
one of the central concepts formulated in this study is that of a 'packing motif'.
to start with, a packing motif is defined as a graph with a limited number of nodes , consisting of unique topological connections, which can be found either in isolation or can appear as a component or an induced subgraph, embedded within a larger graph.
it follows that no two distinct motifs are super-imposable onto each other.
in other words two motifs are identical  if there exists a one-to-one correspondence between their vertex sets which preserve adjacency  <cit> .
the same motif can be found in different proteins and since a node  does not conventionally represent any particular amino acid, it could stand for different sets of residues associated with diverse inter-residue geometries in the actual three dimensional assemblies.
thus a packing motif is a reduced representation of three dimensional residue clusters, rather analogous to super-secondary structural motifs where, for example, different combination of residues in unrelated proteins can fold into  a helix-turn-helix.
in order to aid numerical manipulations, each motif was uniquely represented by a linear array of numbers  which can be regarded as a complete set of invariants  <cit>  between any two isomorphic graphs.
initially each node of a given motif was assigned a string of numbers  where d is its degree) starting with its own degree; followed by the degrees of its direct neighbors sorted in descending order.
these numeric strings were collected as elements of an array and further sorted in descending order.
finally these sorted strings were concatenated, separated by a delimiter .
this identifier-string representation of each motif facilitated the computational detection, classification and clustering of motifs from the protein database.
one of the primary objectives of this study is to  identify recurrent motifs in smaller graphs and  to ascertain whether larger graphs can be constituted by an assembly of suitable motifs with appropriate topological connections.
firstly, all contact networks observed in the protein database were sorted according to their size .
for smaller graphs ranging from 3- <dig> nodes , each set  is expected to populate a limited number of motifs .
thus a motif could essentially be viewed as a prototype, while the actual networks observed in proteins as members belonging to a specific type of motif.
to estimate the maximum number of possible motifs in networks of a given size , a series of random graphs were generated, conditioned by the fact that all nodes had to be connected to at least one other node in the graph .
in a protein contact network, there is a definite upper bound to the maximum number of edges a node can have.
therefore the highest degree  was determined from the set of actual protein contact networks, and this number was used to constrain the maximum attainable degree for the corresponding random graphs.
good agreement between the actual number of motifs observed in the database and the possible number of motifs from random graphs  were found for n =  <dig>   <dig> with rapidly increasing divergence for n ≥  <dig> .
most probably this was due to systematic overestimation in the number of unique random graphs.
thus for graphs with n ranging from  <dig> to  <dig>  the number of possible motifs were recalculated by varying the maximum allowable degree from  <dig>  to the highest observed value in corresponding protein contact networks, which happened to be either  <dig> or  <dig> .
however despite lowering the cutoff on the permissible number of edges for a node it appeared that for n ≥  <dig> a diminishing number of possible motifs is actually being realized within proteins .
for a given network size, the number of networks observed in the database and the corresponding number of unique motifs have been tabulated.
for example,  <dig> motifs were observed for  <dig> networks  constituted of  <dig> nodes .
all networks were systematically searched for size of the maximal clique   which interestingly was found to be no more than  <dig> for embedded cliques  and not exceeding  <dig> for complete graphs .
in fact, the number of networks with a maximal clique of  <dig> and  <dig> nodes respectively, were found to be  <dig> and  <dig> in case of ascn .
since an n-clique should exactly have  diagonal edges, it follows that any possible closed-ring topology of n >  <dig> to be found in the database can have at the most  diagonal edges.
thus the possible network architectures spanning the space under study is expected to be restricted to a few basic topologies namely linear chains, closed triplets , closed quadruplets , higher order ring closures  with a restricted number of diagonal edges and possibly a series of non-planar graphs.
for n =  <dig> there were trivially only two possible motifs  the open linear chain  and  the isolated closed triplet clique .
both possibilities were found in protein contact networks, though with considerable difference in the number of their respective occurrences.
the overwhelming majority of these three-node graphs are found to be open linear chains  which offer greater flexibility unlike isolated closed triplet cliques  which can only occur, satisfying additional geometric constraints.
it could also be possible that triplet cliques once formed display an inherent tendency to evolve into larger networks given the fact that a significantly larger number of these cliques were found to be embedded as induced subgraphs in larger graphs  relative to isolated instances.
out of a total of  <dig> polypeptide chains in the database, embedded triplet cliques were found at least once in  <dig>   <dig> for ascn and apcn respectively whereas for isolated instances the corresponding numbers were  <dig>  and  <dig> .
it is a relatively simple task  to enumerate the possible number of motifs and then find their respective number of members  in the protein database.
it is however a more complex exercise to propose a sound classification scheme, which leads to the regular ordering of actually observed motifs.
to this end two additional concepts were defined namely family and path.
two motifs g and g'  are related by a path if the motif g' can be formed from g such that the node added to g is linked to only one pre-existing node by a single edge.
in other words the transformation g → g' is a path provided the newly added node has degree of one and the degree of one and only one pre-existing node in g increases by one.
again, all motifs which can be linked by successive paths: g → g' → g'' ... etc.
fall within the same family.
however, in case the intermediate g' was missing, g'' was still retained in the same family.
thus, essentially a path leads to linear branching about nodes belonging to a basic core topology.
it follows that a motif of larger size  can either belong to an already existing family provided it is appropriately linked by a path or belong to an entirely new family  which was found to be remarkably less frequent.
for n =  <dig>  there are six possible motifs, of which five  were found to have members.
two motifs  found to have the highest number of members  could be related by a path to open linear chains  or family: f <dig> .
a third motif  was included in the family: f <dig>  originating from closed triplet cliques .
the remaining two  motifs were closed four membered rings  and were put in distinct families  .
thus up to n =  <dig>  a total of  <dig> motifs with a total number of  <dig>  and  <dig>  members were organized into  <dig> families , with the overwhelming majority  of members incorporated into families f <dig> and f <dig>
more or less the same trend was preserved for n =  <dig> where new motifs with significant number of members were again placed in the families f <dig> and f <dig>  additional motifs with marginal membership were included in f3a and f3b, which were essentially branched four membered rings.
two more families  were created at this point, the former  originating from the closed pentagon  whereas the latter  includes the pentagon with a single diagonal edge .
other families at this point include topologies demonstrated by two or more closed triplets; fused along their edges  , connected at a node  or connected by an edge  .
once again, families other than f <dig> and f <dig> exhibited negligible memberships.
moving up levels n =  <dig>   <dig> led to the inclusion of only five more families:  linkage of two four membered rings through a node  ,  embedded quadruplet cliques with additional linear branching   non-planer graphs excluding quadruplet cliques  ,  closed six membered ring with or without diagonals edges   and  graphs where two non-adjacent nodes are connected by more than two sequences of successively connected nodes  .
the addition of nodes from n =  <dig> to n =  <dig>   <dig> primarily led to the addition of motifs in the pre-existing families by,  increasing the length and branching of the linear chain ,  increased linear branching about the triplet cliques ,  progressive branching and inclusion of diagonal edges of the higher order closed rings .
at this stage it became obvious that the initial definitions were leading to a proliferation of families with almost negligible membership.
thus to reduce the number of such families some exceptions were made.
for families originating from five membered rings , motifs with a closed triplet fused about any two vertices of the pre-existing pentagon   were included in the same respective families.
finally up to n =  <dig>   <dig>  and  <dig>  motifs with  <dig>  and  <dig>  members respectively were organized into  <dig> families.
the same procedure described above was performed separately for polypeptide chains in each individual protein class , in order to investigate any preference for specific motifs or families.
by and large no outstanding preference was observed  , though a somewhat reduced propensity was found for family f <dig>  in the case of all α proteins  with a relative increase in propensity for α|β .
the statistics was not robust for most families barring f <dig> and f <dig> due to their extremely low frequency of occurrence .
the trend regarding the distributions of motifs  was not radically changed for different cutoffs on sm and ov, other than the reduction of smaller  motifs on gradually relaxing the cutoffs.
notably, the population of f <dig>  was found to gradually approach that of f <dig>  upon systematic lowering of the cutoffs, allowing weaker links to close the cluster.
on the other hand, the application of more stringent cutoffs  led to an increase in the population of smaller motifs, predominantly in the f <dig> family.
most notable was the increase in frequencies of motifs with  <dig> nodes  probably due to the exclusion of a few weaker links leading to 'minimally connected' open linear chains .
since from n =  <dig> to n =  <dig>   <dig> a diminishing number of families  are added with negligible membership, it is highly likely that larger networks  will generate motifs either populating already existing families or will be assembled by joining pre-existing motifs following a defined set of rules.
since the same trend of preferential membership in the first two families were followed in networks of size n =  <dig>   <dig>   <dig>  it was decided to begin the construction of higher order graphs out of a motif basis set obtained from networks of sizes up to n =  <dig>  with the understanding that motifs of sizes greater than  <dig> nodes  would also be utilized depending on the context of a particular network.
variants of a motif with branching from nodes different from those originally observed  though preserving core topology would also be used in the resolution of larger graphs into subgraphs.
for n =  <dig>  the total number of motifs became comparable to the number of networks or members .
thus, the resolution of larger graphs in terms of the proposed basis set was attempted for n greater than  <dig>
generally, a graph can be resolved into either a degenerate subset of spanning subgraphs  and/or induced subgraphs   <cit> .
thus, analogous to a spanning subset, deleting a judiciously chosen set of specific edges of a graph should produce independent unconnected components.
since, in this study, such isolated components are treated as graphs  it should be possible to resolve a larger graph into a set of motifs  or their variants from pre-existing families, by deleting specific edges.
such edges, however, strictly exclude those being involved in a closed ring , so that the method does not trivially produce an arbitrary combination of motifs.
on the other hand, in an induced subgraph, there exists an identical topological relation between any two corresponding nodes to that of the original graph.
this one-to-one mapping serves as the basis for a computational search for motifs embedded as induced subgraphs in a larger graph.
these two fundamental concepts of graph-analysis were successfully implemented to test the hypothesis whether the motif space is by and large adequate in assembling larger graphs.
contact networks for n =  <dig>  were carefully examined using cytoscape  <cit>   and it was found that the majority of  networks could be resolved across one or more edges to produce isolated components which were invariably motifs belonging to pre-existing families .
other networks could not be resolved into pre-existing motifs by simply cutting across edges and in such instances the majority of possible induced subgraphs embedded in the graph were recognized as pre-existing in the motif basis set .
cases were also found where a larger graph was resolved into both components and induced subgraphs  .
it is to be noted that there can be more than one sequence of steps to assemble a graph from degenerate sets of subgraphs following either of these procedures.
as expected, for all cases, newly emerging motifs were restricted only to ring closures of greater than  <dig> nodes.
thus, predictably, for graphs with more than  <dig> nodes, new motifs should mostly be closed ring topologies with increasing number of nodes in the ring.
possibly, introduction of cofactors and metals into proteins could distort contact networks to give rise to novel topological architectures.
preliminary analyses of networks associated with metal binding sites  exhibit novel 'highly connected' motifs not found in ascn/apcn which show a greater tendency to form higher order cliques  provided the metal is also included in the contact network.
triplet clique
the classification of motifs into families reveals that the overwhelming majority of contact networks found in protein structures occur in the first two families  originating from core topologies of either open linear chains or closed triplet cliques.
although the simple rule governing the classification of motifs leads to about thirteen families in all, a significant proportion of these families have such negligible membership that they can be currently disregarded.
to investigate whether the most frequently occurring motifs exhibit any preference in their constituent amino acid residues and whether their side chains pack with specific geometry, closed triplet clique  was chosen for further investigation.
analysis of the relative frequencies of isolated and embedded triplet cliques appeared to suggest that isolated cliques  have an inbuilt tendency for further branching about the three constituent nodes resulting in their being embedded in larger graphs.
thus, to improve the statistics, both isolated triplet cliques and those embedded as induced subgraphs in larger graphs were pooled together.
further, since hydrophobic residues show greater propensity for burial and inclusion into contact networks, only the subspace of triplet cliques composed exclusively of hydrophobic residues  were considered.
the resultant number of triplet cliques thus reduced to  <dig>   <dig> out of a total of  <dig>   <dig> for ascn and apcn respectively.
interestingly, the number of such cliques was found to be significantly higher for ascn relative to apcn, and since surface contact networks have been defined with a view to identify side chain associations with specific inter-residue geometry, results from ascn alone are being discussed, which in any case should give superior statistics.
for a combination of three residues packed in the form of a closed triplet clique  three possibilities can be expected:  c1: all the three constituent residues are non-identical   c2: one residue unique, the other two being identical  and  c3: all three residues identical .
starting with the set of seven hydrophobic residues  the total number of possible combinations for each case are  <dig>   <dig> and  <dig> respectively.
for assemblies of three identical residues , the highest frequencies were observed for leu-leu-leu , followed by ile-ile-ile  and an almost equal proportion for phe-phe-phe and val-val-val  .
a negligible fraction of triplets was found to be composed exclusively of ala, trp and tyr.
in all probability an assembly of three leucines provides optimal conditions, in terms of shape and size for cohesive packing.
ala and trp represent the opposite ends of the spectrum with regard to volume and the association of tyrosines could be disfavored due to the partial charge of its terminal side chain oxygen .
a similar trend was observed for triplet cliques with one unique and two identical residues .
40% of all triplets in this category were composed of two leucines, with x-ile-ile, x-phe-phe and x-val-val exhibiting frequencies  <dig> %,  <dig> % and  <dig> % respectively.
predictably, x-ala-ala, x-trp-trp and x-tyr-tyr were rarely found.
for hydrophobic clusters with three non-identical residues , the most frequent composition was that of ile-leu-val .
it is notable that the most frequent triplet clique in this category can also be considered to be an exception as the overwhelming majority of triplets consist of at least one aromatic residue .
even here, occurrence of only one aromatic in the triplet clique appears to be preferred over two, whereas cliques composed exclusively of aromatics seldom occur.
thus, the data indicates that even though most of the possible residue combinations are realized in local closed triplets within proteins, there is a wide divergence in their respective frequencies.
some residue combinations definitely appear to be preferred over others.
moreover, since only a subspace has been studied, the compositional propensities appear to be fairly pronounced, rather than outstanding.
without the use of surfaces and careful classification of triplet cliques  these could well be overlooked.
the overall pattern in composition remained fairly unmodified upon changing cutoffs in sm and ov.
even then, the formation of well packed three residue cliques in proteins appears to be constrained in terms of the total volume occupied by the triplet and probably their inter-residue geometry.
in all probability, only some residue combinations optimally satisfy these constraints.
the question then is what are the geometrical constraints imposed on these three-residue cliques?
extending the methodology established by thornton and singh  <cit>   an internal right-handed cartesian frame of reference was defined  for each of the three residues constituting the triplet clique.
connecting the origins of the three internal frames of reference constructs a triangle, which can be considered to be a reduced geometric representation of the assembly.
a global frame of reference was then defined on the triangle plane , the global z axis being the normal to the plane and the origin being set at the centroid of the triangle .
however, there can always be two degenerate directions of the normal .
therefore in order to secure uniformity among all the reference frames the following conventions were adopted:
<dig>  for c <dig> , the three residues  were first sorted on the basis of their side chain volume r <dig> > r <dig> > r <dig>  let the vector directed from the origin of r <dig> to r <dig> be v <dig> and that from r <dig> to r <dig> be v <dig>  then the global z axis was defined as v <dig> × v <dig> and the global x axis as the unit vector directed from the global origin towards the origin of r <dig>
<dig>  for a given composition in c <dig>  one specific example was arbitrarily chosen whose unique residue was designated r <dig> and r <dig>  r <dig> were assigned such that the identical procedure outlined above  resulted in an acute angle being subtended between the global z and the internal z of r <dig>  all other triangles with the same composition were superposed onto this template.
the calculations were repeated starting from different templates to confirm that the results were not artifacts of this geometrical procedure.
<dig>  in case of c <dig> , a randomly chosen triplet was arbitrarily assigned r <dig>  r <dig>  r <dig> and the global frame was defined following procedure .
all other triangles of the same composition were superposed onto this template.
to select for the best possible superposition  in each case  <dig> combinatorial possibilities were checked.
similar to c <dig>  the calculation was repeated for different starting templates.
for almost all the compositions the lengths of the triangular edges and the internal angles were severely constrained, with standard deviations ranging from ~  <dig>  -  <dig>  Å and ~  <dig> - 10° in lengths and angles respectively.
in almost all the cases, the triangle approximates to being equilateral with the average of all the three sides lying between 5- <dig> Å and angles close to 60° .
inclusion of bulky residues in the triplet cliques ,  did not appear to significantly alter the overall trends observed in these triangular parameters.
the longest average lengths were observed for ile-leu-leu , leu-ile-ile , ile-val-val  and ile-phe-phe  .
average lengths of sides  and internal angles   of the triangle formed by joining the origins of the three internal frames corresponding to the constituent residues  are tabulated.
only those compositions have been given whose frequencies are greater than equal to  <dig>
relative geometries of the three constituent residues from the perspective of the abstract triangle defined above were analyzed by means of two more angles, namely tilt and swivel.
dot product of the global z axis defined on the triangle plane with z axes  of the internal frames of the three residues defines the tilt angle .
it essentially describes the orientation of the residue  plane  with respect to the triangle plane.
as is well known, the angular distribution of two randomly oriented vectors should fall of as a function of sin θ' dθ'/ <dig>   <cit>  and the deviation of an actual observed distribution from one which is random can be estimated by means of χ <dig>  examination of χ <dig> of θt shows that for triplet cliques composed of at least one aromatic, their corresponding tilt angles  exhibit significant deviation from randomness.
compositions such as phe-leu-leu  =  <dig> ), phe-leu-val , phe-ile-leu , tyr-leu-leu , tyr-phe-leu , tyr-ile-leu , phe-ile-val , tyr-leu-val  etc  indicates a preferred orientation of the aromatic ring plane  with respect to the global triangle plane.
the actual distribution of the angles  shows the angular bins 60-90°, 90-120° to be preferentially populated  in contrast to ranges 0-30°, 150-180°, 30-60°, 120-150°, which exhibit a corresponding depletion .
thus, both the angular distribution and visual inspection of the triplet cliques indicate that for bulky aromatics, their normals  tend to be perpendicular to the global z axis, as if the side chain tend to enclose the volume demarcated by the edges of the triangle, rather than penetrating into its perimeter .
the other residues  however did not exhibit any consistent specificity in their tilt.
χ <dig> of tilt angles  and swivel angles  of residues constituting the clique  for compositions showing significant deviation in θ1t from a random distribution.
χ <dig>  for three-bin and six-bin models are  <dig>  and  <dig> , respectively.
compositions which have a predicted frequency of less than  <dig> for any particular angular bin, assuming a random distribution are marked with an asterisk .
this minimal number  is  <dig> for a 3-bin and  <dig> for a 6-bin model for tilt  angles and  <dig> for a 6-bin model for swivel  angles.
angular distribution of θ1t in different angular bins  for compositions that have shown significant deviations from a random distribution:
in order to investigate the rotation of the residue planes  about an axis parallel to their own internal z, the component of the global z axis of the triangle was projected onto the respective planes and the orientation of this vector  with respect to the internal x axis  was computed.
since the angle φs is restricted to a plane, each quadrant is expected to be equally populated for a random distribution.
χ <dig> in φs did not appear to show any significant preferences for any residue.
therefore, for a given tilt, the residue plane can adopt multiple orientations about an axis perpendicular to it.
in order to determine whether the formation of triplet cliques was due to either long or short range contacts, the position of residues along the polypeptide chain was examined.
the sequence separation of two residues  was termed local when they were separated by less than   <dig> contiguous residues and non-local when greater than  <dig>  overall, out of a total of  <dig> clique forming contacts,  <dig> were local and  <dig> non-local.
the same calculations repeated for polypeptide chains distributed in bins with 75- <dig>  151- <dig>  301- <dig> residues followed the same trends.
thus, although the majority of contacts were non-local, a non-negligible fraction  are between residues closely located along the polypeptide chain.
as has been mentioned previously , motifs from the f <dig> family were relatively favored in α|β proteins and disfavored in the all α class.
the effect was accentuated for the set of all closed triplet cliques  wherein the propensities for the classes were - all α , all β , α|β  and α+β .
calculations were also carried out to quantify local packing densities  in and around triplet cliques and also in their absence.
plots of packing density ) versus burial ratio   exhibited an almost identical correlation for all the residue types , decaying as a cubic polynomial  = a.x <dig> + b.x <dig> + c.x + d), demonstrating loose packing with higher exposure to the solvent.
networks were distributed into two major categories, those with triplet cliques and those devoid of them.
the former were further subcategorized into the set of clique-nodes alone and that of the other non-clique members.
it was evident from the results that the clique-nodes are predominantly completely buried  and thus on an average, more tightly packed than the other non-clique members whose average exposure to the solvent was consistently found to be higher .
these regions of high local packing densities occur at or near the cliques with gradual decrease at the periphery.
on the other hand, networks devoid of triplet cliques are on an average less tightly packed as a consequence of higher exposure to solvent.
utility of surface contact networks in fold recognition - a case study
finally, an attempt was made to probe the efficacy of surface contact networks in correctly identifying sequences  consistent with a given fold.
cyclophilin-like fold  was chosen as a test case and two decoy sets  random sequences and  spliced sequences from other folds, both of the same length as 2haq  were generated and threaded onto the 2haq backbone template .
each decoy set consisted of  <dig> sequences.
in addition to 2haq, two more sequences  having sequence identities of 37% and 11%  respectively were also included, the latter to ascertain whether the method worked for sequences with low identities, yet mapping to the same fold.
surface contact networks of 2haq and  <dig> other close homologues  were rigorously analyzed  to identify a subset of links in 2haq where each link was conserved in  the homologues and thus could be regarded as a characteristic signature, representative of the fold.
this fold specific subgraph, {lcyp} consisting of  <dig> links between  <dig> nodes  spans the three dimensional structure of the entire protein in terms of spatially connecting almost all the secondary structural elements .
they were also found to have appreciably high surface complementarity , min:  <dig> , max:  <dig> ) and overlap , min:  <dig> , max:  <dig> ).
in order to characterize the corresponding subgraph of {lcyp} in a threaded structure, all nodes in {lcyp} were mapped in that structure  and the corresponding nodes extracted along with all their inter-connecting links.
this  subgraph  was then compared with {lcyp}  by means of two complementary topological measures  defined  to quantify the compatibility of a sequence to a given fold.
snet essentially evaluates the fraction of links in {lcyp} that are conserved in the corresponding subgraph, whereas dnet estimates the dissimilarity in the topological pattern between {lcyp} and the corresponding subgraph in terms of an abstract distance measure.
thus, for the corresponding subgraph extracted from the 2haq native structure, snet will attain its highest possible value of  <dig>  and dnet will be very close to its lowest possible value of  <dig> .
statistical analysis involved computing snet, dnet and estimating their mean  and standard deviation  for the  <dig> decoys of each set.
for the sequences 2haq, 3bo7_b and 3kop_f, the same sequence was threaded  <dig> times , each time with a randomly selected set of side chain conformers .
the statistical parameters  for both the decoy sets were in very good agreement.
average values of snet were found to be fairly low for both the decoy sets  compared to the native sequences 2haq, 3bo7_b and 3kop_f, the difference in their means being about  <dig> -  <dig> times the value of σdecoy.
again the highest average snet was obtained for 2haq followed by 3bo7_b and 3kop_f.
the fact that 3kop_f was effectively discriminated from the decoys demonstrated the possible utility of the method in identifying such cases .
dnet was also able to differentiate between decoys and sequences native to the fold even though with marginally lower significance .
thus, even this limited calculation appeared to indicate the ability of surface contact networks to correctly identify native sequences from decoys.
all statistical parameters were found to be by and large stable over  <dig>   <dig>   <dig> and  <dig> files.
random
<dig> files from each category, two decoy sets and three sequences native to the cyclophilin-like fold were used in the calculations.
for all sequences, side chain conformers were assigned randomly.
3bo7_b and 3kop_f has 37% and 11% sequence identity to 2haq.
standard deviation for each measure is given in parenthesis.
apart from fold detection, these measures can also be used to study networks in general by actually quantifying their resemblance and/or dissimilarity.
in other words, both snet and dnet should precisely reflect and sort networks according to varying degrees of likeness compared to a template.
in order to benchmark these functions, {lcyp} was compared with corresponding subgraphs from seven other homologous structures from the cyclophilin-like fold , with gradually decreasing sequence identities ranging from 39% to 10% upon structural alignment with 2haq.
prior to computing the topological measures, each structure was aligned to 2haq to generate a one-to-one mapping of their residues.
these maps were utilized to extract the corresponding subgraphs of {lcyp} from the homologous structures.
in case of insertions-deletions or non-alignment, the rows and columns corresponding to the missing node  were zero-padded.
both the measures snet and dnet exhibited excellent correlations with respect to sequence identity and root mean square deviation  of the homologues upon structural alignment to 2haq .
these results appeared to validate the ability of these topological measures to correctly assess the resemblance of fold specific subgraphs.
<dig> crystal structures  with varying sequence identities  were superposed onto 2haq and the corresponding subgraphs of {lcyp} constructed, prior to the estimation of snet, dnet.
both the topological measures show very high correlations with sequence identities as well as root mean square deviation upon structural alignment with 2haq.
CONCLUSIONS
the work presented here is based on the confluence of two related though distinct ideas,  some network topologies are preferred within protein interiors, leading to the concept of packing motifs and  the 'jigsaw puzzle' model can be successfully extended into the domain of protein contact networks
the implementation of both these ideas depends partly on representing the internal architecture of proteins in terms of surfaces rather than point atoms.
it has been noted previously that use of surfaces improved the performance of a side chain  prediction test  <cit>  and provided simple well defined criteria to identify those contacts which definitely constrain inter-residue geometry of the associating amino acid side chains  <cit> .
presumably these set of interactions could be playing a more critical role in sustaining the native fold.
networks based on surface contacts  is in effect a straightforward extension of the jigsaw puzzle model.
in the search for compositional or geometrical bias, surface contact networks appear to be indispensable.
in particular, triplet cliques composed exclusively of hydrophobic residues had a frequency  <dig> fold higher in ascn than apcn starting from a comparable  number of triplet cliques.
further more, compositional preferences along with strong geometrical constraints were far better explored by surfaces than point atoms .
one feature which appears to be more or less conserved in surface contact networks  is the almost ubiquitous presence of smaller networks  in all proteins which probably coalesce to produce larger networks specific to the particular fold.
thus, the distribution in network sizes and topologies appear to favor a nucleation-condensation phenomenon  <cit>  in protein packing wherein open linear chains, closed triplet cliques and other closed ring topologies could serve as basic packing units which could either get linked or recruit neighboring residues to grow into networks of larger size.
this notion of packing units led to the definition of 'packing motifs', which could serve as a 'basis set' in the assembly of extended graphs.
since graphs ranging from  <dig> to more than  <dig> nodes have been detected in proteins, the concept of a 'basis set of motifs' should represent sets of similar topologies  rather than a rigid set of isolated unique graphs.
this was the rationale behind the organization of motifs into families  and it soon became clear that some families were overwhelmingly preferred in protein topological space.
these families emanated from the 'minimally connected' open linear chains and three residue cliques  and cast their dominant influence in frequency distribution of motifs.
other families occurred with such abysmally low frequencies that they could be considered oddities rather than the rule.
thus, in accord with the inductive approach of the current work, it was felt that larger graphs  would either fall into pre-existing families or could be assembled by known motifs or their variants.
this possibility was explored for networks of  <dig> nodes and the observations tended to support the hypothesis.
one major drawback of the present study is the lack of a computational method to analyze larger graphs.
although, starting from the novel numerical scheme  a suitable algorithm was able to identify all possible induced subgraphs of a given larger graph, yet manual intervention was indispensable to resolve the larger graph into an optimal set of constituent motifs.
despite its rigor the tedious visual method restricted the analysis to a small fraction of the network space.
however, even with this handicap, the general trend appears to be unmistakable.
it thus appears that the topological space available to protein contact networks is severely constrained with clearly defined preferences.
a unique constraint is the definite upper bound found in the size of the maximal clique  - a property rarely observed in real world networks  <cit> .
obviously, this is due to the atomic environment in protein interiors that restricts the permissible number of edges a node can have.
thus, although only a limited portion of the topological space has been actively explored, the conclusions are expected to be significantly in the right direction.
the next step was to enquire whether packing motifs exhibited any preferences in terms of their constituent residues and geometry.
for this, triplet cliques were selected due to their ubiquitous presence primarily as induced subgraphs embedded in larger graphs.
it soon became evident that in the sub-space of hydrophobic residues, regular trends of propensities favoring specific residues or their combination do indeed exist and certain geometrical features exhibit very strong constraints .
what is perhaps notable is that these compositional and geometric preferences stand a possibility of detection only when statistical analysis is performed subsequent to the precise classification of motifs and the appropriate partitioning of the topological space.
in other words, looking for preferences in case of a pooled set of three residue graphs or subgraphs without adequate classification/characterization is most likely to end in failure.
the most direct application of this study should be in the area of protein fold recognition which is to select a polypeptide chain belonging to a particular fold  <cit>  from a set of decoys.
the most challenging aspect of this problem is to identify those chains consistent with a fold , even though identity upon alignment with the sequence  is significantly low .
preliminary calculations show that topological measures  defined on surface contact networks are indeed able to discriminate sequences native to a particular fold  from decoy sets.
although the threading procedure was fairly straightforward and the decoy sets rather limited, yet all indications appeared to suggest the robustness of the fold prediction method.
most probably the scores could be improved by the adoption of more sophisticated threading procedures which actually solves for the most optimal side chain packing arrangement, rather than averaging over a large number of random side chain conformers.
however, the interesting fact is that the native sequences specific to a particular fold could be distinguished from the decoy sets, based on the statistics of the topological measures alone, despite having randomized side chain conformers.
the utility of these functions to sort networks based on topological resemblance  is also notable.
large scale improvement and application of these methods are currently being investigated.
another fruitful area of research could be to explore the possibility of introducing triplet cliques into designed proteins to stabilize their packing analogous to the engineering of disulfide bridges, in order to improve thermal stability.
a library of triplets exhaustively documenting their conformational, geometrical and topological features might be useful in this regard.
thus in conclusion, it appears that out of innumerable topological possibilities, only a finite number are actually realized in protein contact networks which either are themselves or could be assembled from a limited set of preferred motifs.
one such recurrent motif, the triplet clique, exhibits clear preferences in its constituent residues and very strong constraints with regard to certain geometrical features.

 despite several decades of arduous effort, mapping of protein primary sequences to their three dimensional folds, referred to as the second genetic code, remains an unsolved scientific problem. what appears to be lacking is a comprehensive theory, integrating two factors which definitely condition the isomorphism between sequence and fold, namely the pattern of hydrophobicities embedded in the polypeptide chain and the packing of amino acid side chains to give densely packed protein interiors. under the present circumstances, the more tractable approach is the'inverse protein folding problem', that is to identify protein primary sequences consistent with and supportive of a given fold, an idea which has found considerable application in the de novo design of targeted protein structures. yet even here, it was realized earlier on, that in de novo design, attainment of dense, well-packed protein cores was neither an automatic part of the design process nor acquired simply by chance. most often, it was observed that design led to molten globules or complete unraveling of the structure. an instructive example was the repeated failure to design parallel - tim barrel, finally resolved successfully by offredi et al., where a term optimizing for side chain packing specificity was deliberately included in the computational process. in fact, one indicator for successful computational design is the attainment of densely packed side chains in the interior of the targeted protein, experimentally characterized by the absence of ans binding. thus, a comprehensive theory with regard to the packing of side chain atoms within proteins would not only provide insight into protein structures, facilitate their prediction and would also be a valuable aid in the design of novel proteins. traditionally, there have been two models of protein packing: the'jigsaw puzzle' and the'nuts and bolts' model which lie on the opposite ends of the spectrum. the jigsaw puzzle model attributed to crick, postulates the stereo specific interdigitation of amino acid side chains giving rise to densely packed protein interiors. on the other hand, the nuts and bolts model does not require the association of side chains with specific geometry and asserts that the internal architecture of proteins arises simply due to the high compaction of side chain atoms within a constrained volume. lately, another model referred to as the'oil drop' model has been proposed in order to capture the dynamic fluctuations in protein cores. possibly, all these models concentrate exclusively on some special features of interior packing. however, using a surface complementarity function a previous report from this laboratory demonstrated that binary association between two hydrophobic side chains, with high surface fit and maximal overlap between their corresponding residue surfaces, did exhibit specific inter-residue geometry. it was thus clear that at least for a subset of contacts predictions of the jigsaw puzzle model were indeed valid. one drawback from all such studies is that the inter-residue interactions which sustain a native fold are more accurately modeled as a network rather than a discrete assortment of binary interacting pairs. several attempts have been made to view protein structures as contact networks wherein the amino acids have been designated as nodes and their mutual non covalent interactions as edges. the character of these networks exhibit variability depending on the cutoffs used to define inter-atomic contact. by and large, most protein contact networks preserve'small-world' character and display signatures of assortative mixing. however, degree distribution can be exponential, sigmoidal or dependent on a single exponent - as a function of the criteria used to define atomic interactions. it has also been noted that in certain aspects protein contact networks differ significantly from other real world networks, for example in the restricted number of edges a node can have. apart from providing insights into protein structures, these networks have been used to identify residues implicated in folding nuclei and transition states, identifying functional residues involved in the active site, hubs stabilizing the packing of secondary structural elements, rationalization of the difference in protein stabilities from thermophilic/mesophilic organisms and estimation of folding rates. the utility of the network view of the protein structure is thus fairly well established. in this study, the distribution of such networks from a database of protein structures has been analyzed in order to identify specific topological patterns in side chain association within protein cores. such an analysis led to the recognition that certain packing topologies defined as packing motifs were found preferably in proteins. a limited region of the topological space was exhaustively mapped in terms of frequently occurring packing motifs, combinations of which could lead to networks of larger sizes. it was found that indeed larger networks could be assembled out of a basis set of smaller ones. one such frequently occurring motif namely the three residue clique received special attention with regard to its composition and geometry of associating residues. central to pursuing the research objectives outlined above was the extension of the jigsaw puzzle model into protein contact networks. thus protein contact networks have been defined primarily in terms of surfaces rather than distance between point atoms. as mentioned previously, earlier studies had established quantitative measures to identify those residue pairs whose interacting side chains exhibit specific geometry. these measures have now been used to define'surface contact networks' based only on those inter-residue interactions which severely constrain geometry and thus could play a predominant role in stabilizing a particular fold.  the primary objective of this study is to find, characterize and classify recurring patterns in the packing of side chain atoms within a protein which sustains its native fold. in this task we have deliberately chosen those contacts which strongly and specifically condition the inter-residue geometry of association. since the majority of atomic contacts inside a protein are contributed by side chain atoms, it is natural to represent such interior packing as a network, defined primarily in terms of contact between their corresponding van der waals surfaces. in addition, point atom contact networks have also been studied simultaneously, by way of comparison. contact between any two surfaces can be characterized in terms of overlap that is the extent to which two surfaces are conjoined and by their goodness-of-fit or surface complementarity. in a previous work, it was demonstrated that when surface association between two amino acid side chains were greater than equal to and in ov and sm respectively, angular distributions specifying inter-residue geometry exhibited significant deviations from a random distribution. for a corresponding van der waals surface, the values of sm were found to be marginally lower for the same binary interactions. in contrast to point atoms, the definition of'contact' between two surfaces is not necessarily commutative. in networks based on surface contact, nodes representing residues a and b have been connected with an edge only when the contact between a and b is commutative and their reciprocal sm and ov both are greater than equal to and respectively. for strong association between two residue surfaces their contact is expected to be commutative, which also effectively simplifies the network to an undirected graph. for both point atom and surface contact networks, inter-atomic distance and surface-overlap bear a strong positive linear correlation. sm on the other hand appears to be an additional feature for the latter. interestingly, the choice of Å as the interaction cut off for point atoms led to maximum resemblance between the two categories of networks. the first step was to study the distribution of networks in the protein database on the basis of size i.e. the number of constituent nodes. networks of smaller size dominated the distribution with a rapid decay in frequency for larger networks. the distributions were however characterized by a long tail such that networks with greater than nodes were also found, though with highly diminished frequency. the distributions for both point-atom and surface contact networks were very similar. the characteristic shape of the distribution could be adequately described by a power law = k.x-n, where x is the network size), though, surface contact networks did exhibit some deviations in the range of graphs with to nodes. the exponent, n was found to be and for ascn and apcn respectively. relaxation of the cutoffs on sm and ov did not appear to significantly alter the basic character of the distribution apart from decreasing the population for smaller networks thereby extending the tail for larger ones. on the other hand more stringent cut offs led to the disintegration of the larger graphs, consequently increasing the frequency of small and medium sized networks with a drastic curtailment in the number of larger graphs. thus, as has been previously observed, there appears to be a very narrow margin in terms of more stringent contact criteria which can abruptly change the spread and extent of contact networks within proteins. number of networks found in the database for different ranges of network size. cutoffs in surface complementarity and overlap respectively are given in bold within parentheses. results for the chosen set of cutoff values, used in the analysis are highlighted in bold. the same calculations repeated for polypeptide chains distributed in bins with 75- 151- 301- residues gave similar curves, though for bins of larger chain length, networks of larger size appeared, thereby extending the long tail of the distribution. as expected, frequency distributions of polypeptide chains containing networks of a particular size gave a similar decaying trend with increasing network size; that is networks of smaller size were found embedded in polypeptide chains regardless of the chain length, whereas instances of larger graphs were progressively rare. these distributions tend to indicate that small to medium sized networks are found universally in all protein structures, whereas linkage and/or fusion of these smaller networks to form larger ones is protein specific and is thus context dependent. very large networks were found only in proteins almost all of which had chain length exceeding residues with close packing between extended secondary structural elements. overall, most of the very large networks were found in α|β proteins. in a protein contact network, there is an obvious upper bound to the highest possible degree a node can have due to the limited volume of the residues involved in packing. for the present set of criteria, the highest degree of a node was found to be restricted to and for ascn and apcn respectively. from a graph theory perspective, local cohesiveness of side chains should lead to dense packing within protein cores. to test this proposition, average unweighted and weighted clustering coefficients were calculated for contact networks of all sizes. both these coefficients gave rise to almost identical measures. in parallel, a statistically significant number of random graphs were generated for the direct calculation of their clustering coefficients. in a log-log plot, average clustering coefficients of the contact networks decayed much less rapidly with increasing network size compared to corresponding random graphs. since, these coefficients essentially determine the cliquishness of a typical neighborhood in terms of clustering of local triplets, it follows that triplet cliques serve as the units of clustering. in other words, a graph of whatever size or connectivity will result in zero clustering if there is no'closed triplet' found embedded within it. it is to be noted that any higher order clique could be considered as an association of nested triplet cliques. since, these results confirm the probability of finding closed triplets to be much higher in proteins than random graphs, 3-cliques as'clustering units' have thus received detailed attention in terms of geometry and composition, to be discussed in later sections. packing motif one of the central concepts formulated in this study is that of a'packing motif'. to start with, a packing motif is defined as a graph with a limited number of nodes, consisting of unique topological connections, which can be found either in isolation or can appear as a component or an induced subgraph, embedded within a larger graph. it follows that no two distinct motifs are super-imposable onto each other. in other words two motifs are identical if there exists a one-to-one correspondence between their vertex sets which preserve adjacency. the same motif can be found in different proteins and since a node does not conventionally represent any particular amino acid, it could stand for different sets of residues associated with diverse inter-residue geometries in the actual three dimensional assemblies. thus a packing motif is a reduced representation of three dimensional residue clusters, rather analogous to super-secondary structural motifs where, for example, different combination of residues in unrelated proteins can fold into a helix-turn-helix. in order to aid numerical manipulations, each motif was uniquely represented by a linear array of numbers which can be regarded as a complete set of invariants between any two isomorphic graphs. initially each node of a given motif was assigned a string of numbers where d is its degree) starting with its own degree; followed by the degrees of its direct neighbors sorted in descending order. these numeric strings were collected as elements of an array and further sorted in descending order. finally these sorted strings were concatenated, separated by a delimiter. this identifier-string representation of each motif facilitated the computational detection, classification and clustering of motifs from the protein database. one of the primary objectives of this study is to identify recurrent motifs in smaller graphs and to ascertain whether larger graphs can be constituted by an assembly of suitable motifs with appropriate topological connections. firstly, all contact networks observed in the protein database were sorted according to their size. for smaller graphs ranging from 3- nodes, each set is expected to populate a limited number of motifs. thus a motif could essentially be viewed as a prototype, while the actual networks observed in proteins as members belonging to a specific type of motif. to estimate the maximum number of possible motifs in networks of a given size, a series of random graphs were generated, conditioned by the fact that all nodes had to be connected to at least one other node in the graph. in a protein contact network, there is a definite upper bound to the maximum number of edges a node can have. therefore the highest degree was determined from the set of actual protein contact networks, and this number was used to constrain the maximum attainable degree for the corresponding random graphs. good agreement between the actual number of motifs observed in the database and the possible number of motifs from random graphs were found for n = with rapidly increasing divergence for n ≥. most probably this was due to systematic overestimation in the number of unique random graphs. thus for graphs with n ranging from to the number of possible motifs were recalculated by varying the maximum allowable degree from to the highest observed value in corresponding protein contact networks, which happened to be either or. however despite lowering the cutoff on the permissible number of edges for a node it appeared that for n ≥ a diminishing number of possible motifs is actually being realized within proteins. for a given network size, the number of networks observed in the database and the corresponding number of unique motifs have been tabulated. for example, motifs were observed for networks constituted of nodes. all networks were systematically searched for size of the maximal clique which interestingly was found to be no more than for embedded cliques and not exceeding for complete graphs. in fact, the number of networks with a maximal clique of and nodes respectively, were found to be and in case of ascn. since an n-clique should exactly have diagonal edges, it follows that any possible closed-ring topology of n > to be found in the database can have at the most diagonal edges. thus the possible network architectures spanning the space under study is expected to be restricted to a few basic topologies namely linear chains, closed triplets, closed quadruplets, higher order ring closures with a restricted number of diagonal edges and possibly a series of non-planar graphs. for n = there were trivially only two possible motifs the open linear chain and the isolated closed triplet clique. both possibilities were found in protein contact networks, though with considerable difference in the number of their respective occurrences. the overwhelming majority of these three-node graphs are found to be open linear chains which offer greater flexibility unlike isolated closed triplet cliques which can only occur, satisfying additional geometric constraints. it could also be possible that triplet cliques once formed display an inherent tendency to evolve into larger networks given the fact that a significantly larger number of these cliques were found to be embedded as induced subgraphs in larger graphs relative to isolated instances. out of a total of polypeptide chains in the database, embedded triplet cliques were found at least once in for ascn and apcn respectively whereas for isolated instances the corresponding numbers were and. it is a relatively simple task to enumerate the possible number of motifs and then find their respective number of members in the protein database. it is however a more complex exercise to propose a sound classification scheme, which leads to the regular ordering of actually observed motifs. to this end two additional concepts were defined namely family and path. two motifs g and g' are related by a path if the motif g' can be formed from g such that the node added to g is linked to only one pre-existing node by a single edge. in other words the transformation g → g' is a path provided the newly added node has degree of one and the degree of one and only one pre-existing node in g increases by one. again, all motifs which can be linked by successive paths: g → g' → g''... etc. fall within the same family. however, in case the intermediate g' was missing, g'' was still retained in the same family. thus, essentially a path leads to linear branching about nodes belonging to a basic core topology. it follows that a motif of larger size can either belong to an already existing family provided it is appropriately linked by a path or belong to an entirely new family which was found to be remarkably less frequent. for n = there are six possible motifs, of which five were found to have members. two motifs found to have the highest number of members could be related by a path to open linear chains or family: f. a third motif was included in the family: f originating from closed triplet cliques. the remaining two motifs were closed four membered rings and were put in distinct families. thus up to n = a total of motifs with a total number of and members were organized into families, with the overwhelming majority of members incorporated into families f and f more or less the same trend was preserved for n = where new motifs with significant number of members were again placed in the families f and f additional motifs with marginal membership were included in f3a and f3b, which were essentially branched four membered rings. two more families were created at this point, the former originating from the closed pentagon whereas the latter includes the pentagon with a single diagonal edge. other families at this point include topologies demonstrated by two or more closed triplets; fused along their edges, connected at a node or connected by an edge. once again, families other than f and f exhibited negligible memberships. moving up levels n = led to the inclusion of only five more families: linkage of two four membered rings through a node, embedded quadruplet cliques with additional linear branching non-planer graphs excluding quadruplet cliques, closed six membered ring with or without diagonals edges and graphs where two non-adjacent nodes are connected by more than two sequences of successively connected nodes. the addition of nodes from n = to n = primarily led to the addition of motifs in the pre-existing families by, increasing the length and branching of the linear chain, increased linear branching about the triplet cliques, progressive branching and inclusion of diagonal edges of the higher order closed rings. at this stage it became obvious that the initial definitions were leading to a proliferation of families with almost negligible membership. thus to reduce the number of such families some exceptions were made. for families originating from five membered rings, motifs with a closed triplet fused about any two vertices of the pre-existing pentagon were included in the same respective families. finally up to n = and motifs with and members respectively were organized into families. the same procedure described above was performed separately for polypeptide chains in each individual protein class, in order to investigate any preference for specific motifs or families. by and large no outstanding preference was observed, though a somewhat reduced propensity was found for family f in the case of all α proteins with a relative increase in propensity for α|β. the statistics was not robust for most families barring f and f due to their extremely low frequency of occurrence. the trend regarding the distributions of motifs was not radically changed for different cutoffs on sm and ov, other than the reduction of smaller motifs on gradually relaxing the cutoffs. notably, the population of f was found to gradually approach that of f upon systematic lowering of the cutoffs, allowing weaker links to close the cluster. on the other hand, the application of more stringent cutoffs led to an increase in the population of smaller motifs, predominantly in the f family. most notable was the increase in frequencies of motifs with nodes probably due to the exclusion of a few weaker links leading to'minimally connected' open linear chains. since from n = to n = a diminishing number of families are added with negligible membership, it is highly likely that larger networks will generate motifs either populating already existing families or will be assembled by joining pre-existing motifs following a defined set of rules. since the same trend of preferential membership in the first two families were followed in networks of size n = it was decided to begin the construction of higher order graphs out of a motif basis set obtained from networks of sizes up to n = with the understanding that motifs of sizes greater than nodes would also be utilized depending on the context of a particular network. variants of a motif with branching from nodes different from those originally observed though preserving core topology would also be used in the resolution of larger graphs into subgraphs. for n = the total number of motifs became comparable to the number of networks or members. thus, the resolution of larger graphs in terms of the proposed basis set was attempted for n greater than generally, a graph can be resolved into either a degenerate subset of spanning subgraphs and/or induced subgraphs. thus, analogous to a spanning subset, deleting a judiciously chosen set of specific edges of a graph should produce independent unconnected components. since, in this study, such isolated components are treated as graphs it should be possible to resolve a larger graph into a set of motifs or their variants from pre-existing families, by deleting specific edges. such edges, however, strictly exclude those being involved in a closed ring, so that the method does not trivially produce an arbitrary combination of motifs. on the other hand, in an induced subgraph, there exists an identical topological relation between any two corresponding nodes to that of the original graph. this one-to-one mapping serves as the basis for a computational search for motifs embedded as induced subgraphs in a larger graph. these two fundamental concepts of graph-analysis were successfully implemented to test the hypothesis whether the motif space is by and large adequate in assembling larger graphs. contact networks for n = were carefully examined using cytoscape and it was found that the majority of networks could be resolved across one or more edges to produce isolated components which were invariably motifs belonging to pre-existing families. other networks could not be resolved into pre-existing motifs by simply cutting across edges and in such instances the majority of possible induced subgraphs embedded in the graph were recognized as pre-existing in the motif basis set. cases were also found where a larger graph was resolved into both components and induced subgraphs. it is to be noted that there can be more than one sequence of steps to assemble a graph from degenerate sets of subgraphs following either of these procedures. as expected, for all cases, newly emerging motifs were restricted only to ring closures of greater than nodes. thus, predictably, for graphs with more than nodes, new motifs should mostly be closed ring topologies with increasing number of nodes in the ring. possibly, introduction of cofactors and metals into proteins could distort contact networks to give rise to novel topological architectures. preliminary analyses of networks associated with metal binding sites exhibit novel'highly connected' motifs not found in ascn/apcn which show a greater tendency to form higher order cliques provided the metal is also included in the contact network. triplet clique the classification of motifs into families reveals that the overwhelming majority of contact networks found in protein structures occur in the first two families originating from core topologies of either open linear chains or closed triplet cliques. although the simple rule governing the classification of motifs leads to about thirteen families in all, a significant proportion of these families have such negligible membership that they can be currently disregarded. to investigate whether the most frequently occurring motifs exhibit any preference in their constituent amino acid residues and whether their side chains pack with specific geometry, closed triplet clique was chosen for further investigation. analysis of the relative frequencies of isolated and embedded triplet cliques appeared to suggest that isolated cliques have an inbuilt tendency for further branching about the three constituent nodes resulting in their being embedded in larger graphs. thus, to improve the statistics, both isolated triplet cliques and those embedded as induced subgraphs in larger graphs were pooled together. further, since hydrophobic residues show greater propensity for burial and inclusion into contact networks, only the subspace of triplet cliques composed exclusively of hydrophobic residues were considered. the resultant number of triplet cliques thus reduced to out of a total of for ascn and apcn respectively. interestingly, the number of such cliques was found to be significantly higher for ascn relative to apcn, and since surface contact networks have been defined with a view to identify side chain associations with specific inter-residue geometry, results from ascn alone are being discussed, which in any case should give superior statistics. for a combination of three residues packed in the form of a closed triplet clique three possibilities can be expected: c1: all the three constituent residues are non-identical c2: one residue unique, the other two being identical and c3: all three residues identical. starting with the set of seven hydrophobic residues the total number of possible combinations for each case are and respectively. for assemblies of three identical residues, the highest frequencies were observed for leu-leu-leu, followed by ile-ile-ile and an almost equal proportion for phe-phe-phe and val-val-val. a negligible fraction of triplets was found to be composed exclusively of ala, trp and tyr. in all probability an assembly of three leucines provides optimal conditions, in terms of shape and size for cohesive packing. ala and trp represent the opposite ends of the spectrum with regard to volume and the association of tyrosines could be disfavored due to the partial charge of its terminal side chain oxygen. a similar trend was observed for triplet cliques with one unique and two identical residues. 40% of all triplets in this category were composed of two leucines, with x-ile-ile, x-phe-phe and x-val-val exhibiting frequencies %, % and % respectively. predictably, x-ala-ala, x-trp-trp and x-tyr-tyr were rarely found. for hydrophobic clusters with three non-identical residues, the most frequent composition was that of ile-leu-val. it is notable that the most frequent triplet clique in this category can also be considered to be an exception as the overwhelming majority of triplets consist of at least one aromatic residue. even here, occurrence of only one aromatic in the triplet clique appears to be preferred over two, whereas cliques composed exclusively of aromatics seldom occur. thus, the data indicates that even though most of the possible residue combinations are realized in local closed triplets within proteins, there is a wide divergence in their respective frequencies. some residue combinations definitely appear to be preferred over others. moreover, since only a subspace has been studied, the compositional propensities appear to be fairly pronounced, rather than outstanding. without the use of surfaces and careful classification of triplet cliques these could well be overlooked. the overall pattern in composition remained fairly unmodified upon changing cutoffs in sm and ov. even then, the formation of well packed three residue cliques in proteins appears to be constrained in terms of the total volume occupied by the triplet and probably their inter-residue geometry. in all probability, only some residue combinations optimally satisfy these constraints. the question then is what are the geometrical constraints imposed on these three-residue cliques? extending the methodology established by thornton and singh an internal right-handed cartesian frame of reference was defined for each of the three residues constituting the triplet clique. connecting the origins of the three internal frames of reference constructs a triangle, which can be considered to be a reduced geometric representation of the assembly. a global frame of reference was then defined on the triangle plane, the global z axis being the normal to the plane and the origin being set at the centroid of the triangle. however, there can always be two degenerate directions of the normal. therefore in order to secure uniformity among all the reference frames the following conventions were adopted: for c, the three residues were first sorted on the basis of their side chain volume r > r > r let the vector directed from the origin of r to r be v and that from r to r be v then the global z axis was defined as v × v and the global x axis as the unit vector directed from the global origin towards the origin of r for a given composition in c one specific example was arbitrarily chosen whose unique residue was designated r and r r were assigned such that the identical procedure outlined above resulted in an acute angle being subtended between the global z and the internal z of r all other triangles with the same composition were superposed onto this template. the calculations were repeated starting from different templates to confirm that the results were not artifacts of this geometrical procedure. in case of c, a randomly chosen triplet was arbitrarily assigned r r r and the global frame was defined following procedure. all other triangles of the same composition were superposed onto this template. to select for the best possible superposition in each case combinatorial possibilities were checked. similar to c the calculation was repeated for different starting templates. for almost all the compositions the lengths of the triangular edges and the internal angles were severely constrained, with standard deviations ranging from ~ - Å and ~ - 10° in lengths and angles respectively. in almost all the cases, the triangle approximates to being equilateral with the average of all the three sides lying between 5- Å and angles close to 60°. inclusion of bulky residues in the triplet cliques, did not appear to significantly alter the overall trends observed in these triangular parameters. the longest average lengths were observed for ile-leu-leu, leu-ile-ile, ile-val-val and ile-phe-phe. average lengths of sides and internal angles of the triangle formed by joining the origins of the three internal frames corresponding to the constituent residues are tabulated. only those compositions have been given whose frequencies are greater than equal to relative geometries of the three constituent residues from the perspective of the abstract triangle defined above were analyzed by means of two more angles, namely tilt and swivel. dot product of the global z axis defined on the triangle plane with z axes of the internal frames of the three residues defines the tilt angle. it essentially describes the orientation of the residue plane with respect to the triangle plane. as is well known, the angular distribution of two randomly oriented vectors should fall of as a function of sin θ' dθ'/ and the deviation of an actual observed distribution from one which is random can be estimated by means of χ examination of χ of θt shows that for triplet cliques composed of at least one aromatic, their corresponding tilt angles exhibit significant deviation from randomness. compositions such as phe-leu-leu = ), phe-leu-val, phe-ile-leu, tyr-leu-leu, tyr-phe-leu, tyr-ile-leu, phe-ile-val, tyr-leu-val etc indicates a preferred orientation of the aromatic ring plane with respect to the global triangle plane. the actual distribution of the angles shows the angular bins 60-90°, 90-120° to be preferentially populated in contrast to ranges 0-30°, 150-180°, 30-60°, 120-150°, which exhibit a corresponding depletion. thus, both the angular distribution and visual inspection of the triplet cliques indicate that for bulky aromatics, their normals tend to be perpendicular to the global z axis, as if the side chain tend to enclose the volume demarcated by the edges of the triangle, rather than penetrating into its perimeter. the other residues however did not exhibit any consistent specificity in their tilt. χ of tilt angles and swivel angles of residues constituting the clique for compositions showing significant deviation in θ1t from a random distribution. χ for three-bin and six-bin models are and, respectively. compositions which have a predicted frequency of less than for any particular angular bin, assuming a random distribution are marked with an asterisk. this minimal number is for a 3-bin and for a 6-bin model for tilt angles and for a 6-bin model for swivel angles. angular distribution of θ1t in different angular bins for compositions that have shown significant deviations from a random distribution: in order to investigate the rotation of the residue planes about an axis parallel to their own internal z, the component of the global z axis of the triangle was projected onto the respective planes and the orientation of this vector with respect to the internal x axis was computed. since the angle φs is restricted to a plane, each quadrant is expected to be equally populated for a random distribution. χ in φs did not appear to show any significant preferences for any residue. therefore, for a given tilt, the residue plane can adopt multiple orientations about an axis perpendicular to it. in order to determine whether the formation of triplet cliques was due to either long or short range contacts, the position of residues along the polypeptide chain was examined. the sequence separation of two residues was termed local when they were separated by less than contiguous residues and non-local when greater than overall, out of a total of clique forming contacts, were local and non-local. the same calculations repeated for polypeptide chains distributed in bins with 75- 151- 301- residues followed the same trends. thus, although the majority of contacts were non-local, a non-negligible fraction are between residues closely located along the polypeptide chain. as has been mentioned previously, motifs from the f family were relatively favored in α|β proteins and disfavored in the all α class. the effect was accentuated for the set of all closed triplet cliques wherein the propensities for the classes were - all α, all β, α|β and α+β. calculations were also carried out to quantify local packing densities in and around triplet cliques and also in their absence. plots of packing density ) versus burial ratio exhibited an almost identical correlation for all the residue types, decaying as a cubic polynomial = a.x + b.x + c.x + d), demonstrating loose packing with higher exposure to the solvent. networks were distributed into two major categories, those with triplet cliques and those devoid of them. the former were further subcategorized into the set of clique-nodes alone and that of the other non-clique members. it was evident from the results that the clique-nodes are predominantly completely buried and thus on an average, more tightly packed than the other non-clique members whose average exposure to the solvent was consistently found to be higher. these regions of high local packing densities occur at or near the cliques with gradual decrease at the periphery. on the other hand, networks devoid of triplet cliques are on an average less tightly packed as a consequence of higher exposure to solvent. utility of surface contact networks in fold recognition - a case study finally, an attempt was made to probe the efficacy of surface contact networks in correctly identifying sequences consistent with a given fold. cyclophilin-like fold was chosen as a test case and two decoy sets random sequences and spliced sequences from other folds, both of the same length as 2haq were generated and threaded onto the 2haq backbone template. each decoy set consisted of sequences. in addition to 2haq, two more sequences having sequence identities of 37% and 11% respectively were also included, the latter to ascertain whether the method worked for sequences with low identities, yet mapping to the same fold. surface contact networks of 2haq and other close homologues were rigorously analyzed to identify a subset of links in 2haq where each link was conserved in the homologues and thus could be regarded as a characteristic signature, representative of the fold. this fold specific subgraph, {lcyp} consisting of links between nodes spans the three dimensional structure of the entire protein in terms of spatially connecting almost all the secondary structural elements. they were also found to have appreciably high surface complementarity, min:, max: ) and overlap, min:, max: ). in order to characterize the corresponding subgraph of {lcyp} in a threaded structure, all nodes in {lcyp} were mapped in that structure and the corresponding nodes extracted along with all their inter-connecting links. this subgraph was then compared with {lcyp} by means of two complementary topological measures defined to quantify the compatibility of a sequence to a given fold. snet essentially evaluates the fraction of links in {lcyp} that are conserved in the corresponding subgraph, whereas dnet estimates the dissimilarity in the topological pattern between {lcyp} and the corresponding subgraph in terms of an abstract distance measure. thus, for the corresponding subgraph extracted from the 2haq native structure, snet will attain its highest possible value of and dnet will be very close to its lowest possible value of. statistical analysis involved computing snet, dnet and estimating their mean and standard deviation for the decoys of each set. for the sequences 2haq, 3bo7_b and 3kop_f, the same sequence was threaded times, each time with a randomly selected set of side chain conformers. the statistical parameters for both the decoy sets were in very good agreement. average values of snet were found to be fairly low for both the decoy sets compared to the native sequences 2haq, 3bo7_b and 3kop_f, the difference in their means being about - times the value of σdecoy. again the highest average snet was obtained for 2haq followed by 3bo7_b and 3kop_f. the fact that 3kop_f was effectively discriminated from the decoys demonstrated the possible utility of the method in identifying such cases. dnet was also able to differentiate between decoys and sequences native to the fold even though with marginally lower significance. thus, even this limited calculation appeared to indicate the ability of surface contact networks to correctly identify native sequences from decoys. all statistical parameters were found to be by and large stable over and files. random files from each category, two decoy sets and three sequences native to the cyclophilin-like fold were used in the calculations. for all sequences, side chain conformers were assigned randomly. 3bo7_b and 3kop_f has 37% and 11% sequence identity to 2haq. standard deviation for each measure is given in parenthesis. apart from fold detection, these measures can also be used to study networks in general by actually quantifying their resemblance and/or dissimilarity. in other words, both snet and dnet should precisely reflect and sort networks according to varying degrees of likeness compared to a template. in order to benchmark these functions, {lcyp} was compared with corresponding subgraphs from seven other homologous structures from the cyclophilin-like fold, with gradually decreasing sequence identities ranging from 39% to 10% upon structural alignment with 2haq. prior to computing the topological measures, each structure was aligned to 2haq to generate a one-to-one mapping of their residues. these maps were utilized to extract the corresponding subgraphs of {lcyp} from the homologous structures. in case of insertions-deletions or non-alignment, the rows and columns corresponding to the missing node were zero-padded. both the measures snet and dnet exhibited excellent correlations with respect to sequence identity and root mean square deviation of the homologues upon structural alignment to 2haq. these results appeared to validate the ability of these topological measures to correctly assess the resemblance of fold specific subgraphs. crystal structures with varying sequence identities were superposed onto 2haq and the corresponding subgraphs of {lcyp} constructed, prior to the estimation of snet, dnet. both the topological measures show very high correlations with sequence identities as well as root mean square deviation upon structural alignment with 2haq.  the work presented here is based on the confluence of two related though distinct ideas, some network topologies are preferred within protein interiors, leading to the concept of packing motifs and the'jigsaw puzzle' model can be successfully extended into the domain of protein contact networks the implementation of both these ideas depends partly on representing the internal architecture of proteins in terms of surfaces rather than point atoms. it has been noted previously that use of surfaces improved the performance of a side chain prediction test and provided simple well defined criteria to identify those contacts which definitely constrain inter-residue geometry of the associating amino acid side chains. presumably these set of interactions could be playing a more critical role in sustaining the native fold. networks based on surface contacts is in effect a straightforward extension of the jigsaw puzzle model. in the search for compositional or geometrical bias, surface contact networks appear to be indispensable. in particular, triplet cliques composed exclusively of hydrophobic residues had a frequency fold higher in ascn than apcn starting from a comparable number of triplet cliques. further more, compositional preferences along with strong geometrical constraints were far better explored by surfaces than point atoms. one feature which appears to be more or less conserved in surface contact networks is the almost ubiquitous presence of smaller networks in all proteins which probably coalesce to produce larger networks specific to the particular fold. thus, the distribution in network sizes and topologies appear to favor a nucleation-condensation phenomenon in protein packing wherein open linear chains, closed triplet cliques and other closed ring topologies could serve as basic packing units which could either get linked or recruit neighboring residues to grow into networks of larger size. this notion of packing units led to the definition of'packing motifs', which could serve as a'basis set' in the assembly of extended graphs. since graphs ranging from to more than nodes have been detected in proteins, the concept of a'basis set of motifs' should represent sets of similar topologies rather than a rigid set of isolated unique graphs. this was the rationale behind the organization of motifs into families and it soon became clear that some families were overwhelmingly preferred in protein topological space. these families emanated from the'minimally connected' open linear chains and three residue cliques and cast their dominant influence in frequency distribution of motifs. other families occurred with such abysmally low frequencies that they could be considered oddities rather than the rule. thus, in accord with the inductive approach of the current work, it was felt that larger graphs would either fall into pre-existing families or could be assembled by known motifs or their variants. this possibility was explored for networks of nodes and the observations tended to support the hypothesis. one major drawback of the present study is the lack of a computational method to analyze larger graphs. although, starting from the novel numerical scheme a suitable algorithm was able to identify all possible induced subgraphs of a given larger graph, yet manual intervention was indispensable to resolve the larger graph into an optimal set of constituent motifs. despite its rigor the tedious visual method restricted the analysis to a small fraction of the network space. however, even with this handicap, the general trend appears to be unmistakable. it thus appears that the topological space available to protein contact networks is severely constrained with clearly defined preferences. a unique constraint is the definite upper bound found in the size of the maximal clique - a property rarely observed in real world networks. obviously, this is due to the atomic environment in protein interiors that restricts the permissible number of edges a node can have. thus, although only a limited portion of the topological space has been actively explored, the conclusions are expected to be significantly in the right direction. the next step was to enquire whether packing motifs exhibited any preferences in terms of their constituent residues and geometry. for this, triplet cliques were selected due to their ubiquitous presence primarily as induced subgraphs embedded in larger graphs. it soon became evident that in the sub-space of hydrophobic residues, regular trends of propensities favoring specific residues or their combination do indeed exist and certain geometrical features exhibit very strong constraints. what is perhaps notable is that these compositional and geometric preferences stand a possibility of detection only when statistical analysis is performed subsequent to the precise classification of motifs and the appropriate partitioning of the topological space. in other words, looking for preferences in case of a pooled set of three residue graphs or subgraphs without adequate classification/characterization is most likely to end in failure. the most direct application of this study should be in the area of protein fold recognition which is to select a polypeptide chain belonging to a particular fold from a set of decoys. the most challenging aspect of this problem is to identify those chains consistent with a fold, even though identity upon alignment with the sequence is significantly low. preliminary calculations show that topological measures defined on surface contact networks are indeed able to discriminate sequences native to a particular fold from decoy sets. although the threading procedure was fairly straightforward and the decoy sets rather limited, yet all indications appeared to suggest the robustness of the fold prediction method. most probably the scores could be improved by the adoption of more sophisticated threading procedures which actually solves for the most optimal side chain packing arrangement, rather than averaging over a large number of random side chain conformers. however, the interesting fact is that the native sequences specific to a particular fold could be distinguished from the decoy sets, based on the statistics of the topological measures alone, despite having randomized side chain conformers. the utility of these functions to sort networks based on topological resemblance is also notable. large scale improvement and application of these methods are currently being investigated. another fruitful area of research could be to explore the possibility of introducing triplet cliques into designed proteins to stabilize their packing analogous to the engineering of disulfide bridges, in order to improve thermal stability. a library of triplets exhaustively documenting their conformational, geometrical and topological features might be useful in this regard. thus in conclusion, it appears that out of innumerable topological possibilities, only a finite number are actually realized in protein contact networks which either are themselves or could be assembled from a limited set of preferred motifs. one such recurrent motif, the triplet clique, exhibits clear preferences in its constituent residues and very strong constraints with regard to certain geometrical features. 
a crucial part of the problem concerns the geometrical specificity in side chain association leading to densely packed protein cores, a hallmark of correctly folded native structures. the interaction of side chain atoms within the protein core has been represented as a contact network based on the surface complementarity and overlap between associating side chain surfaces. finally, topological measures based on surface contact networks appeared to be effective in discriminating sequences native to a specific fold amongst a set of decoys. one such frequently occurring motif, also envisaged as the unit of clustering, the three residue clique was invariably found in regions of dense packing. mapping protein primary sequences to their three dimensional folds referred to as the'second genetic code' remains an unsolved scientific problem. in this study an attempt has been made to find, characterize and classify recurring patterns in the packing of side chain atoms within a protein which sustains its native fold. study of the distribution of these motifs reveals the ubiquitous presence of typical smaller graphs, which appear to get linked or coalesce to give larger graphs, reminiscent of the nucleation-condensation model in protein folding. some network topologies definitely appear to be preferred and they have been termed'packing motifs', analogous to super secondary structures in proteins. out of innumerable topological possibilities, only a finite number of specific packing motifs are actually realized in proteins.
 mapping protein primary sequences to their three dimensional folds referred to as the'second genetic code' remains an unsolved scientific problem. a crucial part of the problem concerns the geometrical specificity in side chain association leading to densely packed protein cores, a hallmark of correctly folded native structures. thus, any model of packing within proteins should constitute an indispensable component of protein folding and design.  in this study an attempt has been made to find, characterize and classify recurring patterns in the packing of side chain atoms within a protein which sustains its native fold. the interaction of side chain atoms within the protein core has been represented as a contact network based on the surface complementarity and overlap between associating side chain surfaces. some network topologies definitely appear to be preferred and they have been termed'packing motifs', analogous to super secondary structures in proteins. study of the distribution of these motifs reveals the ubiquitous presence of typical smaller graphs, which appear to get linked or coalesce to give larger graphs, reminiscent of the nucleation-condensation model in protein folding. one such frequently occurring motif, also envisaged as the unit of clustering, the three residue clique was invariably found in regions of dense packing. finally, topological measures based on surface contact networks appeared to be effective in discriminating sequences native to a specific fold amongst a set of decoys.  out of innumerable topological possibilities, only a finite number of specific packing motifs are actually realized in proteins. this small number of motifs could serve as a basis set in the construction of larger networks. of these, the triplet clique exhibits distinct preference both in terms of composition and geometry. 