BACKGROUND
for over  <dig> years  <cit>  contact maps have been used as an alternative representation of protein structures.
a contact map is a 2-dimensional representation of the residue interactions in a protein structure.
this 2-dimensional representation takes the form of a binary matrix.
a given cell  of the matrix can only take two values,  <dig> if the residues i and j are in contact or  <dig> otherwise.
the definition of interaction varies but it is usually based on some cut-off distance between the atoms of the two residues.
one can also see this description from another perspective as a residue interaction graph  with residues as nodes and the contacts as edges.
in this view the binary matrix is no more than the adjacency matrix representing the graph.
although they constitute a simple 2-dimensional representation of the molecule, contact maps still capture all important features of a protein fold.
as such they are an invaluable tool for the analysis of biological macromolecules.
they provide a computationally tractable representation of an otherwise complex problem, with the important advantage of being structural descriptors independent of the coordinate frame.
thus providing a sort of internal coordinates description, rotationally and translationally independent.
however the simplified representation loses on accuracy as compared to the original 3-dimensional model.
multiple applications can be found in the literature that make use of the concept.
contact maps have been used for development of structural alignment algorithms  <cit> , for automatic domain identification  <cit> , in structural modelling by the extraction of contact-based empirical potentials  <cit>  or for the identification of residues critical for folding  <cit> , stability  <cit>  and function  <cit> .
furthermore they have been used as a proxy for 3-dimensional structure prediction by means of machine learning techniques in order to predict residue contacts from sequence information  <cit> .
several methods have been proposed in the past for the reconstruction of contact maps.
most of them develop around the common mathematical theory of distance geometry first applied to chemistry by blumenthal  <cit> .
the theory took really off when crippen and havel  <cit>  applied it to the problem of protein structure determination by nmr methods.
in a typical nmr experiment distances between spatially close hydrogen atoms can be determined for a protein in solution through the detection of the nuclear overhausser effect   <cit> .
the noe data can be seen then as a set of distance ranges between some pairs of hydrogen atoms.
distance geometry deals with distances between points and their embedding in 3-dimensional space.
in principle given a proper metric matrix with all exact distances among a set of points an analytical solution to the embedding can be found easily.
the problem becomes more complicated when not all distances are given  and when only distance ranges rather than exact distances are known.
this is the case of the nmr experiments and equivalently of contact maps: we know some distance ranges between pairs of atoms for which we would like to find 3-dimensional coordinates.
a heuristic algorithm  to solve the problem was proposed by crippen and havel and has been applied extensively ever since.
other algorithms have been proposed such as the alternating projection algorithm by glunt et al.
<cit>  or the geometric build-up algorithm by wu and wu  <cit> .
however the problem of reconstructability of protein contact maps has not been fully addressed in the literature.
a few studies  <cit>  have tried to evaluate the accuracy of the existing methods but they all lack in completeness of the test set and thorough assessment of the different parameters or do not provide fully realistic protein models but only cα traces.
our aim here is twofold.
we would like to find what is the reconstruction accuracy for an average protein so that the limits of the utility of contact maps in protein structure prediction can be precisely assessed.
as a second aim we are looking for optimal criteria in the definition of a contact map decomposition model: atoms selected as interaction centres and distance cut-off.
by decomposing a representative set of pdb protein structures into residue interaction graphs and then reconstructing them based purely on the contact information we should be able to assess the accuracy and loss of information in the decomposition process by comparing to the original native structure .
if a specific contact map model that reconstructs optimally can be found, that would help direct efforts in prediction of contact maps.
previous work has looked at optimality of contact definition from very different points of view, mainly in relation to how well contacting pairs describe the residue propensities when discriminating decoys from native structures.
here we look at it in a purely geometrical way, we are intending to find out how much of the 3d geometrical topology is captured by the network of contacts.
additionally by introducing artificial noise in the contact maps we also look at the effect of inaccurate contact information in the 3-dimensional recovery, essential to the applicability of contacts for predictive purposes.
RESULTS
we studied the reconstructability of a set of representative native pdb protein structures .
firstly we decomposed the native proteins into contact maps with different contact type definitions and for several distance cut-offs.
then we used our reconstruction software to recreate the 3d structures based solely in the information supplied by the contact maps.
to measure the accuracy we then proceed by evaluating the rmsd of the generated models with the original structure.
we measured the rmsd on the cα atoms over all residues, independent of whether the reconstructions were based on cα contact maps or not.
this seems to be a well-established way of measuring the similarity between two structures especially when they are closely related and should facilitate the comparison to other published work.
another well-established method for structure comparison, gdt  <cit> , was not deemed to be appropriate here as it is most useful in comparing structures over a broader range of dissimilarity as is the case in the casp experiment.
optimal cut-off
in figure  <dig> we present the accuracy of reconstruction as measured by rmsd vs. the distance cut-off for contact maps based on cα, cβ and cα + cβ contact-types .
the range of cut-offs chosen was based on values previously used in the literature keeping them within a biochemically sensible range: the minimum cut-off was 6Å as values below result in too sparse contact maps.
at the other end we chose 15Å since beyond that the contact map starts to lose in information content becoming fully connected.
the first interesting observation is the existence of an optimal cut-off for all the contact types.
this optimal value is not very precisely defined in most cases, it seems to span the cut-off distances from  <dig> to 11Å with higher cut-offs having only a marginal loss of accuracy.
however we consider of a more significant value the lower cut-offs.
first of all because of the biochemical meaning of the contacts.
it is in the region about the 8Å cut-off where our definition of contact lead to distances between atoms that are in the range of the van der waals interactions.
also the information content of the contacts should be taken into account.
as shown in figure 3a the practically unchanged accuracy values in the higher cut-off regions are accompanied by an increase in the total number of contacts .
thus we could see this as a loss of information content per contact i.e.
we are adding a lot more information that is simply redundant.
figure 3b illustrates this better by representing the gain in accuracy with respect to contacts added vs the distance cut-off.
the accuracy gain occurs only up to 8Å, after that there is no change as more contacts are added.
additionally no dependence on the protein length across all cut-offs could be observed .
the reconstruction process seems to work with the same accuracy as measured by rmsd regardless of the protein size.
this holds across all proteins tested  and is in agreement with what similar studies found  <cit> .
our rmsd vs distance cut-off plots show no further improvement in accuracy beyond the optimal cut-off region.
this is in clear disagreement with  <cit>  where the reconstruction quality is reported to further increase for cut-off values as big as 18Å.
this can be explained by the fundamentally different procedure of computing the reconstructed models: in our case an all atom approach with realistic regularization of the coordinates through a restraint-only harmonic potential was used for the construction of the models.
vassura et al.
on the other hand uses a simpler cα trace model, without a final refinement phase.
optimal threshold values found here are in agreement to some of the reported optimal values found in other studies.
there has been many attempts in the past to find an optimal contact map definition with respect to both distance cut-off and interaction centre.
the optimizations were based in different criteria according to what the focus was in the particular study.
some authors like gromiha et al.
<cit>  studied the correlation of relative contact order with folding rate, finding that from several cut-offs 8Å gave the best correlations for the cα contact type when considering long range interactions only.
karchin et al.
<cit>  found that residue burial expressed as contact counts performs best at fold recognition for cβ contact type with a cut-off of 14Å.
similarly benkert et al.
<cit>  used the same residue burial measure and surprisingly found that a cut-off of 9Å was optimal, possibly due to differences in normalisation procedures.
quite a few studies tried to find an optimal contact definition based on the discriminatory power of contact-based empirical potentials in distinguishing decoys from native structures.
bolser et al.
<cit>  found that the best performing two-body potential was that derived from cβ contact definition with a 12Å cut-off.
vendruscolo et al.
<cit>  found that for the cα contact type the best cut-off was at  <dig> Å for a two-body contact potential.
as contact maps are only meaningful in the context of obtaining 3d protein models the reconstructability criterium should not be neglected when considering a contact definition for instance in the prediction of contacts.
contacts containing more geometrical information will be more valuable when building 3-dimensional models.
this is of special importance if we consider that the reconstruction of contact maps seems to be possible even with sparser contact maps , which means that contacts even at optimal definitions still seem to contain redundant information.
optimal interaction centre
comparing the accuracy values between the cα, cβ and cα + cβ cases  it is apparent that cα + cβ performs better across the whole range of cut-offs tested, with cβ alone doing also better than cα.
figure  <dig> shows again this comparison for proteins divided into their respective scop classes.
the trend holds within each of the scop classes.
melo et al.
<cit>  studying distance dependent empirical potentials explored several interaction centres concluding that the cβ atom was the best performing atom centre.
this seems to be a widely accepted result as indicates the use of the cβ contact type for the contact prediction category at the critical assessment of protein structure prediction  experiment  <cit> .
our study, purely based on the 3d geometrical information content of the contacts, confirms the preference for cβ as the interaction centre of choice.
it seems natural that cβ is better in order to derive empirical potentials as it spans both the backbone and the side-chain.
but also it is a superior point of choice for embedding a 3d structure from interatomic distance restraints.
the interaction centre is able to capture geometrical information for the backbone positioning as well as for the orientation of the side-chain leading to a more precise 3d description.
also of interest is the fact that the combination of both cα and cβ contacts leads still to better reconstruction performance, indicating that there is some more backbone information not contained in the cβ restraints.
this suggests an approach in the homology modelling of proteins based on distance restraints : using two atoms per residue to restrain the geometry will lead to more precise models.
we also obtained better accuracy results  by choosing a backbone atom and a side-chain atom farther away from the cβ.
reconstructions for different scop classes
we then address the question of whether the reconstruction process is dependant of the type of protein.
in order to do so we separate our  <dig> proteins into the four scop classes to which they belong to, each of the classes containing  <dig> structures.
figure  <dig> shows the accuracy values for each of these four classes.
the results hold for other cutoffs.
it is striking that the accuracy and spread of the all-β group is significantly better than that of the other three.
interestingly the median values are not very far away for the  <dig> classes but the variances are hugely different especially for the all-beta case.
contrary to this result, in a similar study saitoh et al.
<cit>  stated that they did not encounter a dependency of the accuracy of reconstruction based on the scop class.
this might be explained by the much smaller test set used in that study,  <dig> proteins in total and only  <dig> in the all-β class.
vassura et al.
<cit>  did find some differences across different classes especially a lower accuracy for the all-a class, which we also observe here.
variability of the reconstruction ensembles
the reconstruction process inherently leads to a non-unique solution fully matching the contact map.
we studied the variance of the ensemble of reconstructed structures.
the average spread of the pairwise rmsd among the ensemble structures is in most cases below 2Å.
in table  <dig> we present the spread values for a  <dig> proteins subset .
an example ensemble can be seen in figure  <dig>
the  <dig> proteins subset with chain lengths and the average pairwise rmsd of the reconstruction ensembles, based on cβ contact maps with 8Å cut-off.
as seen in figure  <dig> the reconstruction ensemble is reminiscent of an nmr structure ensemble, not surprisingly as both are based on fitting 3d coordinates to distance restraints.
this shows another advantage of the contact map representation, namely that the conformational flexibility of the molecules is implicit in the model.
comparison to previous studies
for completeness of this work we compare our results to those of two previously published reconstruction methods  <cit> .
in figure  <dig> we present our results  for the set of  <dig> proteins used by vendruscolo et al.
and subsequently by vassura et al.
together with their results .
our rmsd values are higher in most cases.
remarkably the values of vassura et al.
are a lot lower.
however caution should be taken in this comparison as they do not report on the variability  of the result.
as their algorithm  is stochastic the evaluation of the variability across different runs is important to consider.
another important issue to take into account is that these two previous studies are using a simpler representation of proteins, namely one based on only the cα atoms.
in contrast here we are constructing full atom protein chains with realistic bonds and angles.
this leads to higher rmsd values as more geometrical constraints need to be fulfilled.
tolerance to missing contacts and noise
as a final part of the study we then address the question of reconstruction of contact maps in the more realistic scenario of incomplete or noisy maps, which is likely to be the case when the input is a predicted set of contacts.
to do this instead of using real predictions, for instance from homology or machine learning methods, we simulate incomplete and noisy contact maps to thoroughly explore the effect of noise in the process of reconstruction.
interestingly enough there seems to be a non-linear relationship in the information redundancy with respect to cut-off.
figure 6b represents as before the reconstruction rmsd versus the deletion of contacts but this time only for contact type cβ and different cut-offs.
the loss of accuracy with lower percentage sampled subsets seems to decrease with higher cut-offs.
thus for the same percentage deletion one can recreate the original structure better with contact maps of higher cut-offs, i.e.
the redundancy is higher.
the second test that we perform intends to asses the robustness of the 3d recovery process with respect to the presence of noise, the case of a more realistic prediction with false positives.
figure 6c represents the reconstruction accuracy versus the percentage of noise added.
the behaviour here is totally different than before.
an addition of only 2% of random contacts severely affects the 3d recovery process.
the cβ definition behaves better at all levels of noise.
an existing application  <cit>  is reported to perform better with noisy contact maps, but this seems to be due to their pre-filtering based on finding well connected nodes, equivalent to finding contact clusters.
as the test is against randomly added contacts this is not a very realistic filtering.
in a real scenario a) one would not have all well-connected real contacts of the native map and b) the false positives would be very different from random noise.
thus we argue that the filtering used in ft-comar based in common neighbours is not realistic and so the reported tolerance to noise could not be extended to real situations.
in our case we have tested the robustness of the algorithm still against random noise  but we do not perform any pre-filtering.
we believe this to constitute a more realistic benchmark.
the tests performed here are based on randomly generated inaccurate contact maps which in principle differ significantly from ab-initio predictions.
however from our results here we could conclude that with adequately precise ab-initio contact predictions one could produce reasonable models.
in fact we applied successfully some of these ideas in the casp <dig> community-wide experiment for structure prediction  <cit> .
in that case we used template-based contact maps that led to 3d models comparable to those of established methods.
the non-random noise of the template-based maps did not seem to affect significantly the 3d recovery.
CONCLUSIONS
in this work we have studied the viability of computing 3d protein models from contact maps.
we assessed the performance of a reconstruction procedure based on the well known distance geometry protocol used extensively in nmr protein structure determination.
we perform a comprehensive evaluation covering a representative set of the pdb spanning the  <dig> scop classes.
we then explore several possible contact map definitions and evaluate the accuracy of the reconstructions based on rmsd to the available native structure.
we found that contacts based on the cβ atoms are a better description of the 3-dimensional model than those based on cα, confirming other studies that used one-body and two-body empirical contact-based potentials for fold recognition to find this optimum.
reconstruction accuracy can be further improved by using the two contact definitions together cα + cβ.
with regards to contact cut-offs we found that the optimal lies in the region from  <dig> to 11Å.
we do not observe, contrary to previous studies  <cit>  that the accuracy improves for higher cut-offs.
because of the increasing amount of contacts that higher cut-off contact maps yield, we preferred as an optimal threshold the lower end of the optimal range.
a contact map based on a 9Å cut-off achieves maximal geometrical information per contact.
interestingly the accuracy of the reconstruction seems to be different for different classes of proteins.
particularly the all-β scop class yields very good accuracies across all its members as compare to the other classes, leading to the conclusion that some topologies are more amenable to be described in terms of single atom distance restraints.
these results are particularly valuable for the contact prediction community.
as contact prediction ultimately aims at obtaining 3-dimensional models of protein structures the usage of our optimal contact definition findings should contribute to better accuracies of the predictions.
at the same time the results can be useful in the structural alignment of proteins through contact map overlap  <cit> .
these methods seek a 3d alignment by optimising a contact map overlap measure.
clearly contacts that contain better 3-dimensional information should lead to improved results in the final alignments.
further our 3d recovery procedure seems to perform also very well even if only a partial subset of the contacts is available.
with as little as 40% of the contacts reasonably good models can be produced.
on the contrary the method is very sensible to the presence of non-real contacts.
the introduction of restraints at random points in the chain is simply fatal for the recovery of the original structure.
this indicates that contact predictions should focus on accuracy rather than coverage.

 for over years contact maps have been used as an alternative representation of protein structures. a contact map is a 2-dimensional representation of the residue interactions in a protein structure. this 2-dimensional representation takes the form of a binary matrix. a given cell of the matrix can only take two values, if the residues i and j are in contact or otherwise. the definition of interaction varies but it is usually based on some cut-off distance between the atoms of the two residues. one can also see this description from another perspective as a residue interaction graph with residues as nodes and the contacts as edges. in this view the binary matrix is no more than the adjacency matrix representing the graph. although they constitute a simple 2-dimensional representation of the molecule, contact maps still capture all important features of a protein fold. as such they are an invaluable tool for the analysis of biological macromolecules. they provide a computationally tractable representation of an otherwise complex problem, with the important advantage of being structural descriptors independent of the coordinate frame. thus providing a sort of internal coordinates description, rotationally and translationally independent. however the simplified representation loses on accuracy as compared to the original 3-dimensional model. multiple applications can be found in the literature that make use of the concept. contact maps have been used for development of structural alignment algorithms, for automatic domain identification, in structural modelling by the extraction of contact-based empirical potentials or for the identification of residues critical for folding, stability and function. furthermore they have been used as a proxy for 3-dimensional structure prediction by means of machine learning techniques in order to predict residue contacts from sequence information. several methods have been proposed in the past for the reconstruction of contact maps. most of them develop around the common mathematical theory of distance geometry first applied to chemistry by blumenthal. the theory took really off when crippen and havel applied it to the problem of protein structure determination by nmr methods. in a typical nmr experiment distances between spatially close hydrogen atoms can be determined for a protein in solution through the detection of the nuclear overhausser effect. the noe data can be seen then as a set of distance ranges between some pairs of hydrogen atoms. distance geometry deals with distances between points and their embedding in 3-dimensional space. in principle given a proper metric matrix with all exact distances among a set of points an analytical solution to the embedding can be found easily. the problem becomes more complicated when not all distances are given and when only distance ranges rather than exact distances are known. this is the case of the nmr experiments and equivalently of contact maps: we know some distance ranges between pairs of atoms for which we would like to find 3-dimensional coordinates. a heuristic algorithm to solve the problem was proposed by crippen and havel and has been applied extensively ever since. other algorithms have been proposed such as the alternating projection algorithm by glunt et al. or the geometric build-up algorithm by wu and wu. however the problem of reconstructability of protein contact maps has not been fully addressed in the literature. a few studies have tried to evaluate the accuracy of the existing methods but they all lack in completeness of the test set and thorough assessment of the different parameters or do not provide fully realistic protein models but only cα traces. our aim here is twofold. we would like to find what is the reconstruction accuracy for an average protein so that the limits of the utility of contact maps in protein structure prediction can be precisely assessed. as a second aim we are looking for optimal criteria in the definition of a contact map decomposition model: atoms selected as interaction centres and distance cut-off. by decomposing a representative set of pdb protein structures into residue interaction graphs and then reconstructing them based purely on the contact information we should be able to assess the accuracy and loss of information in the decomposition process by comparing to the original native structure. if a specific contact map model that reconstructs optimally can be found, that would help direct efforts in prediction of contact maps. previous work has looked at optimality of contact definition from very different points of view, mainly in relation to how well contacting pairs describe the residue propensities when discriminating decoys from native structures. here we look at it in a purely geometrical way, we are intending to find out how much of the 3d geometrical topology is captured by the network of contacts. additionally by introducing artificial noise in the contact maps we also look at the effect of inaccurate contact information in the 3-dimensional recovery, essential to the applicability of contacts for predictive purposes.  we studied the reconstructability of a set of representative native pdb protein structures. firstly we decomposed the native proteins into contact maps with different contact type definitions and for several distance cut-offs. then we used our reconstruction software to recreate the 3d structures based solely in the information supplied by the contact maps. to measure the accuracy we then proceed by evaluating the rmsd of the generated models with the original structure. we measured the rmsd on the cα atoms over all residues, independent of whether the reconstructions were based on cα contact maps or not. this seems to be a well-established way of measuring the similarity between two structures especially when they are closely related and should facilitate the comparison to other published work. another well-established method for structure comparison, gdt, was not deemed to be appropriate here as it is most useful in comparing structures over a broader range of dissimilarity as is the case in the casp experiment. optimal cut-off in figure we present the accuracy of reconstruction as measured by rmsd vs. the distance cut-off for contact maps based on cα, cβ and cα + cβ contact-types. the range of cut-offs chosen was based on values previously used in the literature keeping them within a biochemically sensible range: the minimum cut-off was 6Å as values below result in too sparse contact maps. at the other end we chose 15Å since beyond that the contact map starts to lose in information content becoming fully connected. the first interesting observation is the existence of an optimal cut-off for all the contact types. this optimal value is not very precisely defined in most cases, it seems to span the cut-off distances from to 11Å with higher cut-offs having only a marginal loss of accuracy. however we consider of a more significant value the lower cut-offs. first of all because of the biochemical meaning of the contacts. it is in the region about the 8Å cut-off where our definition of contact lead to distances between atoms that are in the range of the van der waals interactions. also the information content of the contacts should be taken into account. as shown in figure 3a the practically unchanged accuracy values in the higher cut-off regions are accompanied by an increase in the total number of contacts. thus we could see this as a loss of information content per contact i.e. we are adding a lot more information that is simply redundant. figure 3b illustrates this better by representing the gain in accuracy with respect to contacts added vs the distance cut-off. the accuracy gain occurs only up to 8Å, after that there is no change as more contacts are added. additionally no dependence on the protein length across all cut-offs could be observed. the reconstruction process seems to work with the same accuracy as measured by rmsd regardless of the protein size. this holds across all proteins tested and is in agreement with what similar studies found. our rmsd vs distance cut-off plots show no further improvement in accuracy beyond the optimal cut-off region. this is in clear disagreement with where the reconstruction quality is reported to further increase for cut-off values as big as 18Å. this can be explained by the fundamentally different procedure of computing the reconstructed models: in our case an all atom approach with realistic regularization of the coordinates through a restraint-only harmonic potential was used for the construction of the models. vassura et al. on the other hand uses a simpler cα trace model, without a final refinement phase. optimal threshold values found here are in agreement to some of the reported optimal values found in other studies. there has been many attempts in the past to find an optimal contact map definition with respect to both distance cut-off and interaction centre. the optimizations were based in different criteria according to what the focus was in the particular study. some authors like gromiha et al. studied the correlation of relative contact order with folding rate, finding that from several cut-offs 8Å gave the best correlations for the cα contact type when considering long range interactions only. karchin et al. found that residue burial expressed as contact counts performs best at fold recognition for cβ contact type with a cut-off of 14Å. similarly benkert et al. used the same residue burial measure and surprisingly found that a cut-off of 9Å was optimal, possibly due to differences in normalisation procedures. quite a few studies tried to find an optimal contact definition based on the discriminatory power of contact-based empirical potentials in distinguishing decoys from native structures. bolser et al. found that the best performing two-body potential was that derived from cβ contact definition with a 12Å cut-off. vendruscolo et al. found that for the cα contact type the best cut-off was at Å for a two-body contact potential. as contact maps are only meaningful in the context of obtaining 3d protein models the reconstructability criterium should not be neglected when considering a contact definition for instance in the prediction of contacts. contacts containing more geometrical information will be more valuable when building 3-dimensional models. this is of special importance if we consider that the reconstruction of contact maps seems to be possible even with sparser contact maps, which means that contacts even at optimal definitions still seem to contain redundant information. optimal interaction centre comparing the accuracy values between the cα, cβ and cα + cβ cases it is apparent that cα + cβ performs better across the whole range of cut-offs tested, with cβ alone doing also better than cα. figure shows again this comparison for proteins divided into their respective scop classes. the trend holds within each of the scop classes. melo et al. studying distance dependent empirical potentials explored several interaction centres concluding that the cβ atom was the best performing atom centre. this seems to be a widely accepted result as indicates the use of the cβ contact type for the contact prediction category at the critical assessment of protein structure prediction experiment. our study, purely based on the 3d geometrical information content of the contacts, confirms the preference for cβ as the interaction centre of choice. it seems natural that cβ is better in order to derive empirical potentials as it spans both the backbone and the side-chain. but also it is a superior point of choice for embedding a 3d structure from interatomic distance restraints. the interaction centre is able to capture geometrical information for the backbone positioning as well as for the orientation of the side-chain leading to a more precise 3d description. also of interest is the fact that the combination of both cα and cβ contacts leads still to better reconstruction performance, indicating that there is some more backbone information not contained in the cβ restraints. this suggests an approach in the homology modelling of proteins based on distance restraints: using two atoms per residue to restrain the geometry will lead to more precise models. we also obtained better accuracy results by choosing a backbone atom and a side-chain atom farther away from the cβ. reconstructions for different scop classes we then address the question of whether the reconstruction process is dependant of the type of protein. in order to do so we separate our proteins into the four scop classes to which they belong to, each of the classes containing structures. figure shows the accuracy values for each of these four classes. the results hold for other cutoffs. it is striking that the accuracy and spread of the all-β group is significantly better than that of the other three. interestingly the median values are not very far away for the classes but the variances are hugely different especially for the all-beta case. contrary to this result, in a similar study saitoh et al. stated that they did not encounter a dependency of the accuracy of reconstruction based on the scop class. this might be explained by the much smaller test set used in that study, proteins in total and only in the all-β class. vassura et al. did find some differences across different classes especially a lower accuracy for the all-a class, which we also observe here. variability of the reconstruction ensembles the reconstruction process inherently leads to a non-unique solution fully matching the contact map. we studied the variance of the ensemble of reconstructed structures. the average spread of the pairwise rmsd among the ensemble structures is in most cases below 2Å. in table we present the spread values for a proteins subset. an example ensemble can be seen in figure the proteins subset with chain lengths and the average pairwise rmsd of the reconstruction ensembles, based on cβ contact maps with 8Å cut-off. as seen in figure the reconstruction ensemble is reminiscent of an nmr structure ensemble, not surprisingly as both are based on fitting 3d coordinates to distance restraints. this shows another advantage of the contact map representation, namely that the conformational flexibility of the molecules is implicit in the model. comparison to previous studies for completeness of this work we compare our results to those of two previously published reconstruction methods. in figure we present our results for the set of proteins used by vendruscolo et al. and subsequently by vassura et al. together with their results. our rmsd values are higher in most cases. remarkably the values of vassura et al. are a lot lower. however caution should be taken in this comparison as they do not report on the variability of the result. as their algorithm is stochastic the evaluation of the variability across different runs is important to consider. another important issue to take into account is that these two previous studies are using a simpler representation of proteins, namely one based on only the cα atoms. in contrast here we are constructing full atom protein chains with realistic bonds and angles. this leads to higher rmsd values as more geometrical constraints need to be fulfilled. tolerance to missing contacts and noise as a final part of the study we then address the question of reconstruction of contact maps in the more realistic scenario of incomplete or noisy maps, which is likely to be the case when the input is a predicted set of contacts. to do this instead of using real predictions, for instance from homology or machine learning methods, we simulate incomplete and noisy contact maps to thoroughly explore the effect of noise in the process of reconstruction. interestingly enough there seems to be a non-linear relationship in the information redundancy with respect to cut-off. figure 6b represents as before the reconstruction rmsd versus the deletion of contacts but this time only for contact type cβ and different cut-offs. the loss of accuracy with lower percentage sampled subsets seems to decrease with higher cut-offs. thus for the same percentage deletion one can recreate the original structure better with contact maps of higher cut-offs, i.e. the redundancy is higher. the second test that we perform intends to asses the robustness of the 3d recovery process with respect to the presence of noise, the case of a more realistic prediction with false positives. figure 6c represents the reconstruction accuracy versus the percentage of noise added. the behaviour here is totally different than before. an addition of only 2% of random contacts severely affects the 3d recovery process. the cβ definition behaves better at all levels of noise. an existing application is reported to perform better with noisy contact maps, but this seems to be due to their pre-filtering based on finding well connected nodes, equivalent to finding contact clusters. as the test is against randomly added contacts this is not a very realistic filtering. in a real scenario a) one would not have all well-connected real contacts of the native map and b) the false positives would be very different from random noise. thus we argue that the filtering used in ft-comar based in common neighbours is not realistic and so the reported tolerance to noise could not be extended to real situations. in our case we have tested the robustness of the algorithm still against random noise but we do not perform any pre-filtering. we believe this to constitute a more realistic benchmark. the tests performed here are based on randomly generated inaccurate contact maps which in principle differ significantly from ab-initio predictions. however from our results here we could conclude that with adequately precise ab-initio contact predictions one could produce reasonable models. in fact we applied successfully some of these ideas in the casp community-wide experiment for structure prediction. in that case we used template-based contact maps that led to 3d models comparable to those of established methods. the non-random noise of the template-based maps did not seem to affect significantly the 3d recovery.  in this work we have studied the viability of computing 3d protein models from contact maps. we assessed the performance of a reconstruction procedure based on the well known distance geometry protocol used extensively in nmr protein structure determination. we perform a comprehensive evaluation covering a representative set of the pdb spanning the scop classes. we then explore several possible contact map definitions and evaluate the accuracy of the reconstructions based on rmsd to the available native structure. we found that contacts based on the cβ atoms are a better description of the 3-dimensional model than those based on cα, confirming other studies that used one-body and two-body empirical contact-based potentials for fold recognition to find this optimum. reconstruction accuracy can be further improved by using the two contact definitions together cα + cβ. with regards to contact cut-offs we found that the optimal lies in the region from to 11Å. we do not observe, contrary to previous studies that the accuracy improves for higher cut-offs. because of the increasing amount of contacts that higher cut-off contact maps yield, we preferred as an optimal threshold the lower end of the optimal range. a contact map based on a 9Å cut-off achieves maximal geometrical information per contact. interestingly the accuracy of the reconstruction seems to be different for different classes of proteins. particularly the all-β scop class yields very good accuracies across all its members as compare to the other classes, leading to the conclusion that some topologies are more amenable to be described in terms of single atom distance restraints. these results are particularly valuable for the contact prediction community. as contact prediction ultimately aims at obtaining 3-dimensional models of protein structures the usage of our optimal contact definition findings should contribute to better accuracies of the predictions. at the same time the results can be useful in the structural alignment of proteins through contact map overlap. these methods seek a 3d alignment by optimising a contact map overlap measure. clearly contacts that contain better 3-dimensional information should lead to improved results in the final alignments. further our 3d recovery procedure seems to perform also very well even if only a partial subset of the contacts is available. with as little as 40% of the contacts reasonably good models can be produced. on the contrary the method is very sensible to the presence of non-real contacts. the introduction of restraints at random points in the chain is simply fatal for the recovery of the original structure. this indicates that contact predictions should focus on accuracy rather than coverage. 
we use an implementation of the well-known distance geometry protocol to build realistic protein 3-dimensional models from contact maps, performing an extensive exploration of many of the parameters involved in the reconstruction process. the reconstruction process does not provide a single solution to the problem but rather an ensemble of conformations that are within 2Å rmsd of the crystal structure and with lower values for the pairwise average ensemble rmsd. we try to address the questions: a) to what accuracy does a contact map represent its corresponding 3d structure, b) what is the best contact map representation with regard to reconstructability and c) what is the effect of partial or inaccurate contact information on the 3d structure recovery. contact maps have been extensively used as a simplified representation of protein structures. however a contact map's biological interest is subject to the availability of reliable methods for the 3-dimensional reconstruction of the structure. our results suggest that contact maps derived from the application of a distance cutoff of to 11Å around the cβ atoms constitute the most accurate representation of the 3d structure. thus contact maps represent a valid approximation to the structures with an accuracy comparable to that of experimental methods. the optimal contact definitions constitute key guidelines for methods based on contact maps such as structure prediction through contacts and structural alignments based on maximum contact map overlap. inspired by the model's simplicity many groups have dedicated a considerable amount of effort towards contact prediction as a proxy for protein structure prediction. they capture most important features of a protein's fold, being preferred by a number of researchers for the description and study of protein structures.
 contact maps have been extensively used as a simplified representation of protein structures. they capture most important features of a protein's fold, being preferred by a number of researchers for the description and study of protein structures. inspired by the model's simplicity many groups have dedicated a considerable amount of effort towards contact prediction as a proxy for protein structure prediction. however a contact map's biological interest is subject to the availability of reliable methods for the 3-dimensional reconstruction of the structure.  we use an implementation of the well-known distance geometry protocol to build realistic protein 3-dimensional models from contact maps, performing an extensive exploration of many of the parameters involved in the reconstruction process. we try to address the questions: a) to what accuracy does a contact map represent its corresponding 3d structure, b) what is the best contact map representation with regard to reconstructability and c) what is the effect of partial or inaccurate contact information on the 3d structure recovery. our results suggest that contact maps derived from the application of a distance cutoff of to 11Å around the cβ atoms constitute the most accurate representation of the 3d structure. the reconstruction process does not provide a single solution to the problem but rather an ensemble of conformations that are within 2Å rmsd of the crystal structure and with lower values for the pairwise average ensemble rmsd. interestingly it is still possible to recover a structure with partial contact information, although wrong contacts can lead to dramatic loss in reconstruction fidelity.  thus contact maps represent a valid approximation to the structures with an accuracy comparable to that of experimental methods. the optimal contact definitions constitute key guidelines for methods based on contact maps such as structure prediction through contacts and structural alignments based on maximum contact map overlap. 