Proc. Natl. Acad. Sci. USA
`Vol. 95, pp. 11158–11162, September 1998
`Biophysics
`
`Clustering of low-energy conformations near the native structures
`of small proteins
`DAVID SHORTLE*, KIM T. SIMONS, AND DAVID BAKER†
`Department of Biochemistry, University of Washington School of Medicine, Seattle, WA 98195
`
`Edited by Peter G. Wolynes, University of Illinois, Urbana, IL, and approved July 17, 1998 (received for review May 13, 1998)
`
`Recent experimental studies of the dena-
`ABSTRACT
`tured state and theoretical analyses of the folding landscape
`suggest that there are a large multiplicity of low-energy,
`partially folded conformations near the native state. In this
`report, we describe a strategy for predicting protein structure
`based on the working hypothesis that there are a greater
`number of low-energy conformations surrounding the correct
`fold than there are surrounding low-energy incorrect folds. To
`test this idea, 12 ensembles of 500 to 1,000 low-energy struc-
`tures for 10 small proteins were analyzed by calculating the
`rms deviation of the Ca coordinates between each conforma-
`tion and every other conformation in the ensemble. In all 12
`cases, the conformation with the greatest number of confor-
`mations within 4-Å rms deviation was closer to the native
`structure than were the majority of conformations in the
`ensemble, and in most cases it was among the closest 1 to 5%.
`These results suggest that, to fold efficiently and retain
`robustness to changes in amino acid sequence, proteins may
`have evolved a native structure situated within a broad basin
`of low-energy conformations, a feature which could facilitate
`the prediction of protein structure at low resolution.
`
`Prediction of the structures of proteins from their amino acid
`sequence traditionally has followed one approach. First, a
`candidate conformation is generated, either by a de novo
`conformational search method or by turning to the database of
`known protein structures. This conformation then is scored for
`the quality of the match between the sequence of the target
`protein and the spatial positions forced on the residues when
`placed in the candidate conformation. This process is contin-
`ued until practical limitations force termination of the search,
`at which point the conformation with the most favorable score
`is considered to be the best candidate for the structure of the
`target protein.
`A central assumption underlying this standard approach is
`that the native state is the conformation of lowest energy. The
`configurational entropy of the protein chain cannot be in-
`cluded in the scoring function because the focus is on finding
`one conformation. Consequently, this approach is not rigor-
`ously based on Anfinsen’s hypothesis that the native state lies
`at the global minimum in free energy (1). Interpreted literally,
`the Anfinsen hypothesis implies that, because the native state
`of a protein is an ensemble of many similar conformations, the
`target or goal of protein structure prediction should be this
`ensemble rather than just a single conformation. Because this
`ensemble of conformations is probably very narrowly distrib-
`uted around the mean, it is often considered a safe assumption
`to ignore this source of complexity and concentrate on one
`representative conformation, which in all likelihood would
`approximate the mean of the ensemble.
`
`Proteins participate in a second, much larger ensemble of
`conformations, usually referred to as the ‘‘denatured state’’ (2,
`3, 4). In the past few years, considerable attention has been
`given to experimental and theoretical characterization of this
`complex and structurally diverse ensemble. Although current
`physical methods do not provide as high a resolution descrip-
`tion of denatured states as they do for native states, the
`emerging picture is one of significant population of transient
`native-like local structures weakly coupled to each other (5, 6).
`An analysis of long range structure in an expanded denatured
`state of staphylococcal nuclease suggests that many of the
`global topological features of the native state are retained in
`the denatured state (7). In other words, the ensemble average
`structure of the denatured state resembles the native state,
`albeit at very low resolution.
`In addition to forming a much more diverse ensemble, the
`conformations in the denatured state may have their structure
`and dynamic behavior determined by a smaller number of
`energy terms. Several authors have argued that burial of
`hydrophobic surface is the dominant force shaping structure in
`the denatured state ensemble (3, 5). In addition, the highly
`dynamic character and the much lower density of atoms
`suggest that dispersion forces, hydrogen bonds, and salt bridges
`may contribute little to the properties of denatured proteins.
`If the energetics are less dependent on the high resolution
`details of chain–chain interactions, the ensemble-averaged
`properties of the denatured state might be easier to predict
`than those of the native state. The database-derived energyy
`scoring functions currently used for structure prediction are
`thought to model primarily hydrophobic interactions (8, 9, 10)
`and thus may be suitable for prediction of structure in the
`denatured state ensemble.
`If the ensemble-averaged topology (or low resolution struc-
`ture) of the denatured state is approximately the same as that
`of the native state, the basin or minimum in the energy
`landscape containing both native and denatured states must
`have a partition function that is much larger than any other
`ensemble of structurally similar conformations. This idea is the
`fundamental hypothesis underlying the approach to structure
`prediction described in this paper.
`
`RESULTS
`Current knowledge of the residual structure in the denatured
`state and its energetic basis is too limited to reach definitive
`conclusions about the appropriateness of this large, dynamic
`ensemble as a target for predicting structural features of folded
`proteins. Therefore, we consider arguments concerning the
`structural correspondence between the denatured state and
`the native fold only as a general point of departure for the
`analysis reported here. In this spirit, we present two conjec-
`
`The publication costs of this article were defrayed in part by page charge
`payment. This article must therefore be hereby marked ‘‘advertisement’’ in
`accordance with 18 U.S.C. §1734 solely to indicate this fact.
`© 1998 by The National Academy of Sciences 0027-8424y98y9511158-5$2.00y0
`PNAS is available online at www.pnas.org.
`
`This paper was submitted directly (Track II) to the Proceedings office.
`Abbreviation: rmsd, rms deviation.
`*Permanent Address: Department of Biological Chemistry, The Johns
`Hopkins University School of Medicine, Baltimore, MD 21205.
`†To whom reprint requests should be addressed. e-mail: baker@ben.
`bchem.washington.edu.
`
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`Biophysics: Shortle et al.
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`Proc. Natl. Acad. Sci. USA 95 (1998)
`
`11159
`
`FIG. 1. Schematic diagram of a hypothetical folding energy land-
`scape. The x axis corresponds to a generalized structure coordinate
`(17, 26). The solid line corresponds to the internal free energy (17),
`and the dashed line corresponds to the value of a database-derived
`scoring function such as the one used in this work. The scoring function
`follows the true potential because it is sensitive to hydrophobic burial
`but produces noise and fails to detect the sharp drop in energy of the
`native state because of inaccuracies in quantifying hydrogen bonds,
`electrostatic, and van der Waals interactions. However, the scoring
`function is able to detect the higher density of low-energy states in the
`broad region surrounding the native state.
`
`tures as working assumptions to be verified by future experi-
`mental and computational work rather than as established
`facts.
`The first assumption is that the minimum in which native-
`like conformations reside is broader than any other minimum
`at the lowest energy levels in the folding landscape. The second
`assumption is that the breadth of this minimum results from
`the long range character of hydrophobic interactions and
`consequently should be detectable by using database-derived
`energyyscoring functions, which capture some of the features
`of hydrophobic interactions. Together, these two assumptions
`are equivalent to the statement that effective burial of hydro-
`phobic residues can be retained throughout a larger range of
`
`FIG. 2. Histograms of the rmsd (Ca coordinates) from the native
`state to members of each of the Park–Levitt sets. An arrow marks the
`position of the center of the largest cluster of conformations by using
`a 4-Å rmsd cutoff. The bin intervals along the x axis are in 0.5-Å
`increments.
`
`structural perturbations of the native topology than of any
`other topology.
`These assumptions are illustrated in a schematic energy
`landscape shown in Fig. 1, which positions the native state in
`a deep, narrow well located near the center of a broad, shallow
`minimum (solid line) (11). In a search to find the structure of
`a protein of known sequence, a relatively coarse grid search
`may generate multiple conformations within this broad min-
`imum. Although an energy function that does not correctly
`quantify dispersion interactions, hydrogen bonds, and electro-
`static interactions (Fig. 1, dashed line) may miss the steep drop
`in energy for conformations that comprise the native state, it
`may succeed in detecting the broad minimum if hydrophobic
`interactions are more or less correctly modeled.
`To the extent that these two assumptions are correct, protein
`structure at low resolution may be predicted by carrying out a
`coarse-grained sampling of conformational space and choos-
`ing the low-energy conformation having the largest number of
`structurally related low-energy conformations. In a situation
`such as that depicted in Fig. 1, relatively uniform sampling of
`conformation space followed by identification of the largest
`cluster of structurally related low-energy conformations would
`be expected to find the region of conformation space that
`contains the native state.
`Two Sets of Computer-Generated ‘‘Decoy’’ Conformations
`for 10 Small Proteins. To test this idea, we examined large sets
`of structures generated by Park and Levitt (12) for eight small
`proteins—cro repressor (2cro), a fragment of ribosomal pro-
`tein L7yL12 (1ctf), the 434 repressor (1r69), calbindin (3icb),
`scorpion neurotoxin (1sn3), pancreatic trypsin inhibitor (4pti),
`ubiquitin (1ubq), and an electron transfer protein with an
`iron-sulfur center (4rxn). In brief, these structures were pro-
`duced by an exhaustive search in which the angular relation-
`ships between five or six segments of fixed secondary structure
`
`Table 1. Clustering by structural similarity of the 1,000 lowest
`energy conformations in the Park–Levitt sets
`rmsd
`center to
`native (rank in
`proximity to
`native state)
`4.7 (44)
`3.2
`6.7
`1.7 (2)
`2.9
`2.9
`3.3 (12)
`4.2
`3.9
`1.7 (1)
`2.0
`1.7
`8.1 (417)
`7.2
`6.9
`2.5 (10)
`5.0
`6.1
`2.0 (3)
`2.0
`3.7
`3.1 (13)
`3.2
`5.4
`
`rmsd lowest
`energy
`conformation
`5.6
`
`Mean rmsd
`of ensemble
`8.8
`
`2.0
`
`5.2
`
`4.7
`
`2.1
`
`10.0
`
`5.3
`
`8.4
`
`8.1
`
`8.0
`
`9.2
`
`8.4
`
`9.2
`
`9.2
`
`8.4
`
`Protein
`2cro
`
`1ctf
`
`1r69
`
`3icb
`
`1sn3
`
`4pti
`
`1ubq
`
`4rxn
`
`rmsd
`cutoff
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`
`Cluster
`size
`44
`85
`151
`69
`132
`247
`45
`129
`257
`51
`83
`137
`18
`40
`120
`22
`44
`100
`52
`94
`154
`36
`82
`153
`
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`Proc. Natl. Acad. Sci. USA 95 (1998)
`
`more realistic chain representation was obtained by fitting
`backbone atoms (N, CA, C, O, CB) with correct bond distances
`and angles to the virtual Catrace by using fragments of known
`proteins (K.T.S. and D.B., unpublished work).
`A second, more structurally diverse set of conformations for
`four small all-helical proteins—staphylococcal protein A
`(1fc2), homeodomain repressor (1hdd), cro repressor (2cro),
`and calbindin (4icb)— also was analyzed. In previous work
`from this laboratory, ensembles of protein-like structures were
`generated by a Monte Carlo simulating annealing procedure in
`which segments of structure from the protein database were
`recombined to generate compact composites that scored well
`on the basis of a knowledge-based scoring function (13). To
`obtain local secondary structure compatible with the local
`structure, the protein segments used in this construction
`process were selected on the basis of similarity in amino acid
`sequence between the source protein and the target protein.
`To avoid biasing the generated set toward the wild-type
`conformation, all known structural homologues were removed
`from the set of proteins used as sources of structural segments.
`The 500 structures with the best overall score were saved for
`analysis. Unlike the conformations in the Park–Levitt sets, the
`conformations in the Simons sets showed considerable vari-
`ability in the exact position of some helices (13).
`The scoring functions used to evaluate the decoy structures
`were based on the decomposition
`P(structure(cid:239) sequence)’P(sequence(cid:239) structure)pP(structure),
`
`or Bayes’ rule, where P(x) is the a priori probability of the
`occurrence of x and P(y (cid:239) x) is the conditional probability of y,
`given the occurrence of x. The first term on the right hand side
`quantifies the fit between the sequence and the structure and
`consisted of a residue-environment term that depends primar-
`ily on the hydrophobic interaction and a specific pair interac-
`tion term that captures interactions such as salt bridges and
`disulfide bonds. The second term on the right hand side is the
`probability that a candidate conformation is a properly folded
`protein structure. For scoring the Park–Levitt sets, P(struc-
`ture) consisted of an excluded volume component plus a
`secondary structure packing term that is sensitive primarily to
`the relative orientation and packing of b strands (K.T.S. and
`D.B., unpublished work). For the Simons sets, this term only
`depended on excluded volume and the radius of gyration (13).
`Results of Cluster Analysis. Analysis of the eight Park–
`Levitt sets began with scoring each of the 200,000 confor-
`mations and saving the 1,000 with the best scores, which were
`defined as the low-energy ensemble for each protein se-
`quence. The rms deviation (rmsd) of the Cacoordinates was
`calculated for each pair of conformations within a set, and
`the results stored in a 1,000 3 1,000 ‘‘distance matrix.’’ For
`each of a series of distance cutoffs ranging from 4 to 6 Å, the
`conformation having the most neighboring conformations
`within the distance cutoff was selected as the most central
`conformation. These results are listed in Table 1.
`Fig. 2 shows the distribution of rmsd distances between the
`wild-type structure and each conformation in the set, along
`with the position of the center conformation for the largest
`cluster within 4-Å rmsd. As can be seen, in all cases but one
`(protein 1sn3, which has a very irregular structure held to-
`gether by four disulfide bonds), the center conformation is
`significantly more similar to the native structure than the
`average member of the low-energy ensemble. In addition, for
`six of these seven cases, the center conformation was in the
`closest 1.5% of conformations with regard to rmsd from the
`native state.
`More graphic displays of the structural similarities among
`the 1,000 conformations in the Park–Levitt sets for proteins
`4pti and 4rxn are shown in Fig. 3. By applying the statistical
`method of multidimensional scaling to the set of 500,000
`
`FIG. 3. Multidimensional scaling maps of the ensemble of confor-
`mations in the Park–Levitt sets of conformations for 4pti (Upper) and
`4rxn (Lower). The distance in rmsd between each pair of conforma-
`tions is projected onto two dimensions, retaining relative distance
`relationships so that two structurally similar conformations tend to be
`located near each other. The position of each conformation is indi-
`cated by a small white dot. The position of the native state is marked
`with a white diamond, and the three conformations with three lowest
`(best) energy scores are marked with white boxes. The gray scale value
`of each pixel is determined by the lowest energy conformation within
`that small region of the map, with black being the very lowest energies.
`
`were allowed to vary. After optimally fitting the native con-
`formation as a trace of virtual Caatoms with only four allowed
`torsion angles between residues, segments of the protein chain
`that closely followed a straight line (namely helices and
`strands) were identified, and residues between these straight
`segments became candidates for hinge angles (13). A total of
`10 moveable residue positions were introduced as adjacent
`pairs in four or five hinge regions in each protein. Because
`torsion angles only were allowed to assume one of four possible
`values, each starting structure could be converted to 410 –1
`alternative conformations by exhaustively enumerating all
`possible combinations of torsion angle values. After generating
`’1,000,000 conformations, the 80% with the greatest number
`of steric clashes were discarded, leaving a set of 200,000 decoy
`structures. After most remaining steric clashes were removed
`from these decoys by minimization in dihedral angle space, a
`
`Page 3
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`

`
`Biophysics: Shortle et al.
`
`Proc. Natl. Acad. Sci. USA 95 (1998)
`
`11161
`
`Protein
`2cro
`
`Mean rmsd
`of ensemble
`8.2
`
`Table 2. Clustering by structural similarity of the 1,000 most
`compact conformations in the Park–Levitt sets
`rmsd
`Cluster
`rmsd center to
`cutoff
`size
`native
`4 Å
`68
`3.0
`5 Å
`123
`2.3
`6 Å
`210
`4.1
`4 Å
`33
`8.6
`5 Å
`43
`11.4
`6 Å
`84
`7.2
`4 Å
`18
`9.7
`5 Å
`31
`3.7
`6 Å
`87
`2.9
`4 Å
`16
`6.1
`5 Å
`28
`6.1
`6 Å
`53
`4.6
`4 Å
`18
`9.0
`5 Å
`61
`9.0
`6 Å
`149
`8.1
`4 Å
`19
`10.3
`5 Å
`32
`10.4
`6 Å
`68
`10.4
`4 Å
`23
`10.1
`5 Å
`37
`10.1
`6 Å
`74
`11.9
`4 Å
`27
`10.0
`5 Å
`47
`9.4
`6 Å
`102
`10.4
`
`1ctf
`
`1r69
`
`3icb
`
`1sn3
`
`4pti
`
`1ubq
`
`4rxn
`
`9.6
`
`8.8
`
`10.0
`
`9.4
`
`9.3
`
`9.8
`
`9.0
`
`pairwise distances, a map can be generated that represents an
`optimal solution of separating conformations in two dimen-
`sions in relationship to their distance in rmsd. The resulting
`physical distances between points on the map are not related
`linearly; only local rank ordering of distances is preserved by
`this scaling method. As can be seen, the conformations of very
`lowest energy are distributed fairly randomly over the maps of
`both proteins. Yet, in both cases, there is a significantly higher
`number density of low-energy conformations in a region near
`the native state.
`To demonstrate that low energy plays an important role in
`these observed clusters of native-like conformations, the
`200,000 conformations in each of the Park–Levitt sets were
`sorted by compactness, and the 1,000 most compact confor-
`mations were selected. Clustering this ensemble of conforma-
`tions in the same manner gave the results seen in Table 2. For
`five of the proteins, the centers of the largest clusters are no
`more similar to the native structure than an average confor-
`mation within the ensemble.
`
`FIG. 4. Histograms of the rmsd (Ca coordinates) from the native
`state to members of each of the Simons sets. An arrow marks the
`position of the center of the largest cluster of conformations by using
`a 4-Å rmsd cutoff. The bin intervals along the x axis are in 0.5-Å
`increments.
`
`Table 3. Clustering by structural similarity of the Simons sets of
`500 conformations
`
`Protein
`1fc2
`
`1hdd
`
`2cro
`
`4icb
`
`rmsd
`cutoff
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`4 Å
`5 Å
`6 Å
`
`Cluster
`size
`410
`419
`431
`209
`296
`348
`16
`37
`90
`43
`89
`143
`
`rmsd center to
`native (rank in
`proximity to
`native state)
`4.0 (193)
`4.2
`5.3
`3.5 (17)
`4.5
`4.8
`4.4 (3)
`7.2
`4.9
`6.5 (82)
`7.0
`5.9
`
`rmsd of lowest
`energy
`conformation
`3.8
`
`Mean rmsd
`of ensemble
`4.9
`
`5.2
`
`7.9
`
`5.8
`
`6.8
`
`8.7
`
`9.4
`
`Surprisingly, for the three all-helical proteins, 2cro, 1r69,
`and 3icb, the cluster centers are considerably closer to the
`native structure than are the majority of configuration in the
`ensemble, although this trend may not be significant for 3icb.
`For 2cro, the centers of the 4-, 5-, and 6-Å groupings are closer
`to native than the cluster centers from the corresponding
`lowest energy ensemble. This may be a consequence of the fact
`that there are a limited number of ways to arrange helices to
`achieve a compact, self-avoiding configuration (15), and there
`are greater number of such well packed configurations around
`the native fold than other topological arrangements accessible
`in this ensemble.
`To analyze a second set of decoy structures constructed in
`an entirely different manner, the 500 conformations in the
`Simons sets also were clustered on the basis of structural
`similarity as measured by rmsd of the Ca coordinates (Table
`3). As shown in Fig. 4, these four sets of proteins contained
`very few conformations within 3-Å rmsd of the native state.
`Nevertheless, for the two proteins 1hdd and 2cro, the center of
`the largest 4-Å cluster was among the top 5% in rmsd. For the
`4icb set, which contained no member closer than 4-Å from the
`native state, the center of the largest 4-Å cluster had an rmsd
`from native of 6.2-Å, placing it only in the closest 20% of
`conformations. The fourth protein, 1fc2 or staphylococcal
`protein A, consists of a three-helical bundle. Not surprisingly,
`the level of structural diversity in the starting ensemble was
`relatively small. The bimodal distribution seen in Fig. 4 reflects
`the fact that there are only two topologies for packing three-
`helical bundles with very short connecting loops. The center of
`the largest cluster was only average in structural similarity to
`the native state, yet it did have the third helix on the correct
`side of the plane defined by the first two helices.
`
`DISCUSSION
`We describe a strategy for predicting protein structure at low
`resolution that goes beyond the standard approach of search-
`ing for the single lowest energy conformation. Instead of
`focusing on the lowest energy conformation, we search for the
`largest cluster of structurally related low-energy conforma-
`tions. In all 12 sets of low-energy conformations studied, the
`conformation with the most other conformations within 4-Å
`rmsd was much more similar to the native structure than the
`majority of the conformations, and, in 9 of the 12 cases, this
`conformation was more similar to the native structure than the
`lowest energy conformation in the set.
`Because the conformations in the Park–Levitt sets are
`rigidly fixed in secondary structure and have only four or five
`degrees of freedom for repositioning helices and strands, they
`correspond to a very limited search of conformation space. On
`
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`
`Proc. Natl. Acad. Sci. USA 95 (1998)
`
`the other hand, the algorithm used to generate the conforma-
`tions in the Simons sets explores many more degrees of
`freedom. In this case, the type and position of secondary
`structures are constrained only by similarity in sequence
`between short segments of the target protein under construc-
`tion and the template proteins from which structural segments
`were obtained. Thus, these sets represent a more realistic
`attempt to predict the structure of protein from sequence
`information alone. Overall, clustering of the Park–Levitt sets
`gave cluster centers closer to the native structure than did the
`Simons sets. Presumably, this is a consequence of the larger
`number of degrees of freedom used to generate the Simons
`sets. It will be important to determine in future work how
`readily the native minimum can be identified by using still
`more diverse conformational sampling strategies.
`The higher density of low-energy conformations near the
`native structure is not an artifact built into these sets of
`conformations by the algorithms used to generate them. That
`the native state does not occupy a unique position in an
`ensemble is fairly obvious for the Simons sets. In this case, only
`the sequence of the target protein was used in the build-up
`process. Because the structural segments used in this process
`were derived from a subset of proteins that did not include
`known homologues, there should be no intrinsic bias toward
`over-representation of the tertiary structure of the native state
`among the conformations generated.
`The conformations in the Park–Levitt sets, on the other
`hand, were derived from the native structure of the target
`protein, after it had been configured as a discreet state virtual
`Ca chain, by varying four or five hinge angles between fixed
`secondary structural segments. Because this construction pro-
`cess searched all allowed values of these angles, the resulting
`set of conformations is independent of the starting conforma-
`tion. In other words, if one picked the lowest energy member
`of the ensemble and repeated the construction algorithm,
`exactly the same conformational set would be regenerated.
`Thus, there is no bias toward the more native-like members of
`the ensemble.
`Why does clustering identify conformations considerably
`closer to the native structure than the conformation of lowest
`energy? One explanation is that the native topology provides
`the most robust arrangement of the chain for burying hydro-
`phobic residues, in the sense that large structural perturbations
`can be tolerated without steric clashes and with relatively small
`increases in hydrophobic exposure. For example, in a four-
`helix bundle protein, relatively large translations of the helices
`relative to one another plus moderate rotations of the helices
`preserve hydrophobic burial. Similarly, in ayb sandwich pro-
`teins, the two layers may undergo rotations and translations
`relative to one another without exposing large amounts of
`hydrophobic surface. From the standpoint of the “new view”
`of protein folding (15–17), the greater breadth of the native
`minimum is a consequence of the assumption that native
`interactions are stronger on average than nonnative interac-
`tions, which results in a lowering of the energy of conforma-
`tions with some native interactions formed. Our strategy also
`may be viewed as a type of signal averaging to compensate for
`noisy scoring functions, in which repeated independent at-
`tempts to find the native state are combined by picking the
`most common topology (the mode) rather than the lowest
`energy conformation.
`The structural elements in native structures would be robust
`to displacement if they (i) often have sufficient local interac-
`tions to be low in energy in isolation; (ii) minimally restrict the
`ability of the remainder of the chain to form structural
`elements low in energy; and (iii) readily combine with other
`low-energy elements to form conformations that are low in
`energy. These features are consistent with the known modu-
`larity of structure in partially folded states of proteins—
`
`synthetic peptides, large protein fragments, and denatured
`proteins. Structural characterization of these types of systems
`have demonstrated that segments of a protein chain frequently
`have a high propensity in isolation to form local structures
`similar to those formed in the native protein (18–20).
`Though limited to a very small sample, these results are
`encouraging and suggest that proteins in general may conform
`to some of the conditions we postulate might permit the
`prediction of structure at low resolution. If these results should
`prove to be general, they support the hypothesis that the native
`structures of proteins are in some sense surrounded by a large
`ensemble of low-energy conformations. In ascribing physical
`reality to this ensemble, we consider it most probable that it
`corresponds predominantly to the denatured state but also
`includes some high-energy forms of the native state involving
`large scale vibrational modes (21) plus partially unfolded states
`(22, 23).
`Recently, the claim has been made that structures of natu-
`rally occurring proteins are selected by evolution because they
`have a high ‘‘designability,’’ i.e., a large tolerance to changes in
`amino acid sequence (a high sequence entropy). One plausible
`mechanism for such designability observed in simple lattice
`models is negative in character: minimization of the likelihood
`of favorable interactions in alternative structural states (24,
`25). The results presented here suggest that a high tolerance of
`structural perturbation (high structural entropy) may be an
`additional, positive mechanism underlying tolerance of se-
`quence perturbations.
`
`The authors thank Ingo Ruczinski for the multidimensional scaling
`analysis shown in Fig. 3, Enoch Huang for helpful discussions and Britt
`Park and Michael Levitt for their decoy set. This work was supported
`by National Institutes of Health Grants GM34171 (to D.S.) and young
`investigator grants to D.B. from the National Science Foundation and
`the Packard Foundation. K.T.S. was supported by National Institutes
`of Health Training Grant PHS NRSA T32 GM07270.
`
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