ß-Sheet modeling by helical surfaces

Denis Znamenskiy, Khan Le Tuan, Anne Poupon, Jacques Chomilier1 and Jean-Paul Mornon

Equipe Systèmes Moléculaires et Biologie Structurale, LMCP, CNRS UMR7590, Universités Paris 6 et Paris 7, Case 115, 75252 Paris cedex 05, France


    Abstract
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
We present a topological description of a ß-sheet in terms of a piece of helical surface. It requires only two easy-to-handle parameters: the twist, i.e. the turn of the helical surface per residue, and the coiling, which is a curvature along the strands or in the direction perpendicular to the strands of the sheet. This method applies fairly well to three- and four-strand sheets, forming a too limited structure to be able to build a barrel. From an analysis of ß-sheets derived from a structural database, we show that this picture can even be reduced to the use of one main value, the twist angle. The dependence of ß-sheet twisting on the number of strands in a sheet, and also on the length and direction of strands, has been demonstrated. The applications of such a description may include the rapid modeling of 3D structures.

Keywords: helical surface/modeling/protein sheet/twist


    Introduction
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
Predicting the three-dimensional structure of proteins from their sequence, the so-called `folding problem', is still far from being solved. In the strategies developed to this end, the first step is to predict regular secondary structures, namely helices and strands. The second step is to determine the fold, i.e. the relative topology of these various elements, considered rigid as a first approximation. We are working in this field of folding where the number and nature of secondary structures are considered to be known from other methods. In this paper, we focus on the second step and in particular the prediction of the fold of domains composed of several strands gathered in ß-sheets. To predict the latter correctly, it is often necessary to introduce a parametrically variable geometry that is able to model the twist of the sheet. In the present study, we performed a statistical analysis of sheets of different sizes and strand architectures retrieved from the Protein Data Bank (PDB), in order to derive pertinent and simple models.

Studies of ß-sheets and in particular of ß-barrels have already been carried out by many workers (Cohen et al., 1980Go, 1982Go; Lasters et al., 1988Go; Lasters, 1990Go; Flower, 1994Go; King et al., 1994Go; Murzin et al., 1994Go; Wang et al., 1996Go). It was shown that ß-barrels could be fitted by hyperboloids (Lasters et al., 1988Go; Lasters, 1990Go; Flower, 1994Go; King et al., 1994Go; Murzin et al., 1994Go). The result is that barrels can be classified by means of two parameters: the number of strands in the ß-sheet, n, and the `shear number', S, which is a measure of the stagger of the strands in the ß-barrel (Murzin et al., 1994Go). Briefly, S is a measure of the inclination of the strands relative to the axis of the barrel. Only a small number of combinations of n and S parameters are observed in nature (Murzin et al., 1994Go).

In this work, we thoroughly investigated the topology of small ß-sheets, typically three or four strands with five to nine residues each. The influence of the number, the length and the direction of strands on the twist was analyzed. Strands were chosen as independent units, basically as rigid bodies and loops connecting them have been discarded in this approach (Efimov, 1993Go; Gellman, 1998Go).

The main chains of ß-sheets form a two-dimensional curved surface whose shape depends on interactions between strands. It should be noted that we were not actually able to find a plane sheet in our analysis, as curvature is very common. The geometry of a ß-sheet has been fully defined by two parameters determining the twist and the coiling (Salemme, 1983Go; Daffner, 1994Go). Different definitions of twist are given in the literature. It has been defined as the average twist of the strands, which is derived from the backbone dihedral angles {phi} and {Psi} (Salemme, 1981Go; Chou et al., 1982Go; Yang and Honig, 1995Go). Another definition uses the four C{alpha} atoms of the two terminal residues from each of the two most external strands of a sheet (Wang et al., 1996Go). Here the twist depends on particular positions of the sheet's four `corner' C{alpha} atoms. A right-handed twist is indeed an intrinsic feature of a ß-sheet (Wang et al., 1996Go) and coiling is defined as a curvature along the strands or in the direction perpendicular to the strands of a ß-sheet. The ß-sheet coiling depends on the nature of residues in the protein and in particular on their relationships with each other. Since the small sheets usually have weak constraints on the border strand's positions, compared with a barrel, there is no strong bending factor contributing to sheet coiling. Therefore, in the present study, coiling was neglected. Hereafter we shall consider twist as the main parameter to describe small sheets.


    Materials and methods
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
To be able to define a ß-sheet twist, we first constructed an `ideal sheet', an approximation of a ß-sheet by a helical surface:



where r is the distance between a strand and the z-axis, 2{pi}H is the step of the helical surface and {theta} is the angle of rotation of a strand around the z-axis. If we fix H, we obtain a surface with r and {theta} as radial coordinates. The parameters of the surface were chosen to minimize the root mean square (r.m.s.) distance between C{alpha} atoms of the `ideal sheet' and those of the real one. This is a common representation for a sheet with no coiling, as represented in Figure 1Go.



View larger version (30K):
[in this window]
[in a new window]
 
Fig. 1. Helical surface construction: the z-axis represents the direction of the central strand(s) of a sheet for an odd number of strands or the mean between the two central ones in the case of an even number of strands. H is the step of the helical surface, i.e. the distance along z between two successive turns of the helical surface. In this figure, only one half turn has been represented. r and {theta} are radial coordinates of C{alpha} atoms. For each strand r is invariant. Therefore, for a given H, the position of each C{alpha} atom is determined only by the value of {theta}. Given this surface we place the C{alpha} atoms of a sheet alternately on either side of the surface at a 0.9 Å distance from it, fixing the distance between the consecutive C{alpha} atoms to 3.8 Å.

 
The z-axis is the direction of the central strand in the case of an odd number of strands in the sheet and it is located in between the two central ones for an even number of strands in the sheet. The C{alpha} atoms of a strand are alternately positioned on either side of the helical surface at a 0.9 Å distance from it, in order to mimic the conformation of a ß-strand and the distance between two consecutive C{alpha} atoms is fixed at 3.8 Å. The distance between two adjacent strands, defined as the distance between a pair of C{alpha} atoms in contact, is constant and set to 4.7 Å (Figure 1Go). Therefore, the radius r, characteristic of one strand, is incremented by 4.7 Å for the next adjacent strand in each direction relative to the center. In other words, for a three-stand sheet, the central strand (let us call it number 2) is merged with the z-axis and the locus of positions of all C{alpha} atoms of strands numbers 1 and 3 is a cylinder of radius 4.7 Å. Therefore, the central strand coincides with the z-axis (r2 = 0) and the other two strands (1 and 3) are equidistant from the z-axis (r1 = r3 = 4.7 Å), having the z-axis in between them ({theta}1 = {theta}3 + {pi}) (their shapes and positions resemble those of DNA). As we move along the first or the third strand we make a turn around the z-axis. In the case of a four-strand sheet there are two pairs of equidistant strands: the inner pair and the outer pair. The inner pair of strands is closer to the z-axis with r2, 3 = 2.35 Å, which is half of the inter-strand distance (4.7 Å), while the outer pair has r1, 4 = 7.05 Å.

Once this three-dimensional model, based on the helical surface, has been determined, it is then possible to fit C{alpha} atom positions of an experimental sheet with this ideal sheet by varying only H in order to adjust it to the actual conformation, as r is only a function of the strand number. These comparisons were performed by the FIT program (Lu Guogang, personal communication) and the H value giving the least r.m.s. was chosen. We shall call T the `twist' of a sheet as follows:

It represents an angle, expressed in degrees per ångstrom (°/Å), of the turn of the helical surface under consideration. As an example, a four-strand sheet from an immunoglobulin domain containing 39 residues, of PDB code 1mel (Berman et al., 2000Go), was approximated by the `ideal sheet' with a specific twist value of 3.42°/Å. It resulted in an r.m.s. deviation calculated for C{alpha} atoms between the experimental and the ideal structures of 0.94 Å.

In order to provide a value of the twist for every possible combinatory position–direction of strands within this `ideal sheet' method, all sheets of three and four strands from the PDB were exhaustively analyzed. Two databases were derived from the PDB, with a maximum of 50 and 95% sequence identity among any pair of members, respectively, called 50% PDB and 95% PDB (Hobohm et al., 1992Go; Hobohm and Sander, 1994Go). The strands were first distinguished and attributed to sheets according to PDB header description. This was checked by DSSP (Kabsch and Sander, 1983Go) and in case of disagreement, DSSP results were preferred. Once retrieved from the databank, they were sorted according to their structural configurations defined by the position and direction of the strands within a sheet. We wished to determine the configurations of strands yielding the same value for the twist, thus establishing a taxonomy of strands based upon easy geometrical considerations. Originally, it was believed that the twist of a sheet should also depend on its size, and therefore on the number of residues it contains. This is the reason why several classes have been constituted according to the number of residues in the sheet. This number was further divided by the number of strands involved in this sheet, yielding an average number of residues per strand, therefore allowing comparison of sheets with different numbers of strands. The first class contains sheets having on average less than five residues per strand, the second class contains sheets from five to six residues and so on. The number of classes was determined in order to keep a sufficient population of sheets in each class, for reasons of statistical consistency.

For every experimental sheet, an ` ideal sheet' was derived. The twist of this model was adjusted to yield the smallest r.m.s. between the modeled and the real sheets. For each group, both mean value and the associated confidence interval were calculated for twist. Experimental sheets often suffer multiple distortions, owing to their environment and their functional constraints. For instance, sheets which form a barrel are coiled and they can be described simply by the hyperboloid model (Lasters et al., 1988Go; Lasters, 1990Go; Flower, 1994Go; King et al., 1994Go; Murzin et al., 1994Go). Therefore, we defined additional conditions to select sheets that can be described by the helical model. The first condition (i) relies on the pseudo-dihedral angles between consecutive C{alpha} atoms, limited to the range 180 ± 45° for ß assignment, eliminating overly coiled strands. In the program PSEA (Labesse et al., 1997Go), the range for ß assignment, derived from a statistical analysis, was 170 ± 45°, taking into account the fact that all experimental sheets are slightly coiled. In order to keep plane conformations to ß-strands, we chose 180° to avoid a bias in angles, but kept the same angular spread as that established by Labesse et al. Another condition (ii) on connectivity between strands of a same sheet helped in choosing the less bent sheets along the z-axis. We considered hydrogen bonds between strands of a sheet in the `neighborhood' of the central C{alpha} of the longest strand of a sheet (Figure 2Go). If there was a path with a hydrogen bond between adjacent residues, connecting all strands near the central residue, then the sheet was considered as connected. Otherwise, the sheets were defined as unconnected and were omitted from the analysis. However, no restriction on the length of strands in a sheet was imposed.



View larger version (51K):
[in this window]
[in a new window]
 
Fig. 2 . Connectivity condition: presence of a path which crosses all the strands of a sheet (passing by C{alpha} atoms) in the `neighborhood' of the `central' C{alpha} atom of the longest strand.

 
For every sheet from the database, the previously described procedure was applied to find the best T value, as long as conditions (i) and (ii) were satisfied. To preserve the precision of results, a third condition (iii) was introduced, to eliminate all modeled sheets with r.m.s. > 3 Å, compared with native structures.

The hypothesis of a normal distribution of twist was verified with the {chi}2 test for each class containing >30 elements. To perform this test we defined four subclasses and compared the observed values with expected values on the assumption of normality. If the normal distribution had unknown values of mean µ and standard deviation {sigma}, we calculated the mean m and standard deviation s of the sample. We defined our subclasses for the x variable as:




These subclasses are equiprobable in case of a normal distribution. The expected size of each class is therefore N/4, where N is the number of elements in the class. We applied the {chi}2 test with one degree of freedom, taking a level {alpha} = 0.05.

The hypothesis of equivalence of two populations was verified by the Wilcoxon test on the ranks (Wonnacott and Wonnacott, 1990Go). We will use this test in order to determine whether the twist depends on the size of the ß-sheet and precisely if there is a range of ß-sheet sizes where the twist is conserved. For this we compare the mean values of twist in several classes of strands. The Wilcoxon test provides us with a reliable method to answer these questions.


    Results
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
In this study, we analyzed the 95% and 50% PDB databases, gathering 2947 and 1370 entries containing ß-sheets of three and four strands, respectively. As the 95% PDB database was highly redundant, its comparison with the 50% PDB database appeared to be necessary and, interestingly, both showed qualitatively the same results. Less redundant databases were not considered since they provided sample sizes too small to perform a statistical analysis. Therefore, the 50% PDB database was chosen as a compromise between sample size and redundancy.

Four-strand sheets

While exploring the 50% PDB database, among the 1058 sheets containing four strands, we found 780 sheets satisfying conditions (i), (ii) and (iii) (dihedral conformation, connectivity, limited r.m.s.). Thus 26% of the four-strand sheets were not treated in our study. All possible combinatorial positions of strands were found among the selected sheets but, as shown in Figure 3Go, their occurrences are far from being equal. For the 50% PDB, 73.5% of the selected sheets are antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}), 12.8% are parallel ({uparrow}{uparrow}{uparrow}{uparrow}), 3% are sheets with one inverted strand in the center ({uparrow}{downarrow}{uparrow}{uparrow}), 2.3% are sheets with two inverted strands in the center ({uparrow}{downarrow}{downarrow}{uparrow}) and 0.3% have two pairs of parallel ({uparrow}{uparrow}{downarrow}{downarrow}) strands (see Figure 3Go). Statistical studies were performed only on antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}), parallel ({uparrow}{uparrow}{uparrow}{uparrow}) and mixed ({uparrow}{downarrow}{uparrow}{uparrow}) sheets, since the other groups had fewer than 20 representatives. The redundancy of the 95% PDB increases the amount of the most populated group, the antiparallel one.



View larger version (57K):
[in this window]
[in a new window]
 
Fig. 3. Occurrences of strand configurations for four- and three-stranded sheets.

 
All groups of sheets were separated into classes according to their size, i.e. the average number of residues per strand, as seen in Figure 4Go. The antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) group was divided into five classes, the parallel ({uparrow}{uparrow}{uparrow}{uparrow}) group into three classes and the ({uparrow}{downarrow}{uparrow}{uparrow}) group into two classes, as a function of their size. With only 18 entries, the ({uparrow}{downarrow}{downarrow}{uparrow}) group could not be split into classes and the average value of twist for this group is 4.8 ± 0.7°/Å. The number of classes has been chosen such that it results in sufficient population of the classes to perform pertinent statistical analysis.



View larger version (32K):
[in this window]
[in a new window]
 
Fig. 4. Dependence of average and mean values of twist on the number of residues for different four-stranded sheet topologies with the 50% database. The {chi}2 test for normality was applied only to samples containing >30 specimens. YES in a table means that the hypothesis of normal distribution of twist could be accepted and NO means that it could not be accepted at a level {alpha} = 0.05. The Wilcoxon test for average values of the twist for parallel sheets showed that the hypothesis of similarity of twist values for all groups of sheets could be accepted at a level {alpha} = 0.05.

 
It was found that the twist of a sheet depends both on its size, i.e. the number of residues it contains, and on its topology, i.e. the relative orientation of strands. These results are presented in Figure 4Go. The average and median twist values calculated for the antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) and mixed ({uparrow}{downarrow}{uparrow}{uparrow}) sheet groups follow similar distributions: the larger the sheet, the smaller is the twist. Short sheets containing fewer than 23 residues have a significantly higher degree of twist than larger sheets. On the other hand, the degree of twist for the parallel sheet group ({uparrow}{uparrow}{uparrow}{uparrow}) does not significantly change when the size of a sheet increases. In Figure 4Go, YES means that the hypothesis of normal distribution of the twist could be accepted and NO that it could not be accepted at this level. Although the distributions of sheets in each class are bell-like, the {chi}2 test for normality of distribution of T at a level {alpha} = 0.05 was not always positive. This may partly be a consequence of the sample size. Increasing the size of the class, if performed by increasing the redundancy from 50 to 95%, is useless. For instance, the {chi}2 test of the antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) sheet group with <24 residues per sheet is positive for the 50% database but negative for the 95% database.

The Wilcoxon test (Wonnacott and Wonnacott, 1990Go) was performed to verify the hypothesis of equivalence of average twists for parallel sheet groups ({uparrow}{uparrow}{uparrow}{uparrow}) of different sizes. This hypothesis of equivalence could be accepted at a level {alpha} = 0.05. Therefore, an average value of twist for all parallel sheets ({uparrow}{uparrow}{uparrow}{uparrow}) was calculated, yielding 4.27 ± 0.21°/Å, for the 95% PDB database and 4.20 ± 0.26°/Å for the 50% PDB database.

A study of the more redundant 95% PDB database, in which 1739 sheets satisfying conditions (i), (ii) and (iii) were selected from 2332 sheets, yielded results close to those found for the 50% PDB database. The same dependence of average and mean twist values on the number of residues for different sheet configurations is found in the more redundant database.

Three-strand sheets

The same procedure was applied to explore the properties of sheets containing three strands. For the 50% PDB database, 766 sheets containing three strands and satisfying conditions (i), (ii) and (iii) (dihedral conformation, connectivity and small r.m.s.) were selected among 940 unconditioned ones. Thus 19% of the three-strand sheets were out of consideration in this study. All possible sheet topologies were also found, as shown in Figure 3Go. As in the case of four-strand sheets, the antiparallel ({uparrow}{downarrow}{uparrow}) sheets represent the majority of the sample (78.1% in the 50% PDB). The parallel ({uparrow}{uparrow}{uparrow}) and mixed ({uparrow}{downarrow}{downarrow}) groups are distributed almost equally and represent 10.3 and 11.6% of the 50% PDB, respectively. Owing to only three possible different topologies for three-strand sheets (antiparallel, parallel and mixed), all groups contain enough members for confident statistics. The group of antiparallel ({uparrow}{downarrow}{uparrow}) sheets was divided into seven classes, the mixed ({uparrow}{downarrow}{downarrow}) group into three classes and the group of parallel ({uparrow}{uparrow}{uparrow}) sheets into two classes.

As for the four-strand sheets, it was also found that the twist of a sheet depends on its size in terms of mean number of residues and on its topology, as reported in Figure 5Go. This dependence is similar to that of the four-strand sheets, i.e. the twist decreases as the size of the sheet increases, except for the parallel sheet, but it has a higher variance.



View larger version (34K):
[in this window]
[in a new window]
 
Fig. 5. Dependence of average and mean values of twist on the number of residues for different three-stranded sheet topologies with the 50% database. Legend is the same as Figure 4Go. The Wilcoxon test for average values of the twist for parallel sheets showed that the hypothesis of similarity of twist values for all groups of sheets could be accepted at a level {alpha} = 0.05.

 
The average twist of the short antiparallel ({uparrow}{downarrow}{uparrow}) sheets, fewer than five residues per strand on average, exceeds 6.5°/Å, which is significantly higher relative to longer sheets. It can be seen that in the antiparallel ({uparrow}{downarrow}{uparrow}) sheet group, the dependence of the average twist on the number of residues in the sheet is almost linear. Nevertheless, as for the antiparallel and mixed four-strand groups, the twist is correlated with the reverse of the number of residues. For the parallel ({uparrow}{uparrow}{uparrow}) sheet group, the correlation between twist and size is similar to that in the parallel group for four-strand sheets, i.e. the twist is fairly constant over the sizes. To verify the hypothesis of equivalent average twist for parallel ({uparrow}{uparrow}{uparrow}) sheet groups of different sizes, the Wilcoxon test was performed. At a level {alpha} = 0.05, this test showed that the hypothesis of similarity can be accepted for the 50% database. Therefore, an average twist value for parallel ({uparrow}{uparrow}{uparrow}) sheets was calculated for this database, yielding 4.2 ± 0.3°/Å.

The 95% PDB database, reduced to 1581 sheets out of 1846 after applying the conditions of selection, gave similar results for antiparallel and mixed groups. For the parallel sheet group, the hypothesis of similarity of the distribution of twist upon size was rejected at a level of 5%.


    Discussion
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
In the present work we explored two databases of protein structures extracted from the PDB, at two levels of sequence identity (95 and 50%) between any pair of entries. Databases with a smaller redundancy could not be considered because the number of samples would be too small to perform statistical studies. From these two databases, sheets of three and four strands were collected and classes built according to the mean number of residues per strand.

By simple geometric considerations, we proposed to model one sheet as a piece of helical surface, the C{alpha} atoms of the backbone lying on this surface. This has the main advantage of being very rapid to compute and easy to handle. Although it was not created in order to describe all available sheets, this helical model has proven to be potentially applicable to the majority of three- and four-strand sheets. It was found that among all considered sheets, antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) sheet topologies represent more than 70% of the set, while the number of sheets satisfying the three conditions necessary to be able to apply this method represents more than 80% of all three- and four-strand sheets found in the PDB databases. Therefore, it is important to emphasize that the results presented in this study concern the `small' and `regular' sheets which do not belong to a barrel.

This observation also illustrates the fact that the antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) topologies are more energetically favorable, which is also true for the three-strand sheets. The short antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) sheets of four strands, fewer than six residues per strand on average, present the highest twist. This may also be a consequence of a lower energy state compared with the parallel ({uparrow}{uparrow}{uparrow}{uparrow}) sheets. The low-energy state of the antiparallel ({uparrow}{downarrow}{uparrow}{downarrow}) sheets may favor greater flexibility.

The Wilcoxon test, which is used to decide if the mean twist of several classes can be considered as equal, was applied to the parallel sheets. For the four-strand sheets ({uparrow}{uparrow}{uparrow}{uparrow}) it was found that the twist values for parallel sheet groups ({uparrow}{uparrow}{uparrow}{uparrow}) of different sizes have the same distribution, at a level {alpha} = 0.05. This result is true for both the 95% and 50% PDB databases. For the three-strand sheets ({uparrow}{uparrow}{uparrow}) the same result was obtained with the 50% PDB database. However, it was not true for the 95% PDB database, where the average twists for the sheets with <24 and those with >24 residues per sheet could not be accepted as equivalent. It can also be a consequence of a high degree of redundancy of the 95% database. These results demonstrate that the value of the parallel sheet twist can be considered as independent of the size of the stands that constitute the sheet.

The results of the study performed on the 50% PDB database are consistent with those of the 95% PDB database. The study of the two databases was necessary for a comprehensive analysis of the variation of twist of ß-sheets. The results obtained from the 50% PDB database, being less influenced by the redundancy of the database, were favored over those of the of 95% PDB database. Hence we obtained a set of values of twist for every structural configuration of three- and four-strand ß-sheets.

One targeted output of this study was to obtain a simple but accurate model of small ß-sheets in order to perform ab initio predictions of protein 3D folds. Indeed, together with helices modeled by rigid cylinders, the ß-sheet helical surfaces proposed here will be used to generate a tight pack of regular secondary structures through an appropriate procedure that will be presented elsewhere.

Conclusion

The aim of this work was to estimate the dependence of the twist of a ß-sheet on the topology of strands and on their mean length. It was found that the twist of a ß-sheet is strongly connected to its strand architecture and to the size of the ß-sheet. Parallel sheets, of both four and three stands, present a merely constant twist, while all other kinds of ß-sheets behave in another way: the smaller the sheet, the more it is twisted. Therefore, parallel ß-sheet twists are less dependent on sheet size than the ß-sheets with other strand topologies. The most comprehensive study was performed on the antiparallel sheets because they represent the majority (>70%) of ß-sheets found in the PDB and, therefore, offer the largest sample size. The results of this study will be used in the algorithm RUSSIA (Rigid Secondary Structure Iterative Assembling), to be presented elsewhere.


    Notes
 
1 To whom correspondence should be addressed. E-mail: chomilie{at}lmcp.jussieu.fr Back


    Acknowledgments
 
The Genome program of the CNRS is acknowledged for funding. We thank Lu Guogang, of the Karolinska Institutet, for providing the FIT program.


    References
 Top
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 References
 
Berman,H.M., Westbrook,J., Feng,Z., Gilliland,G., Bhat,T.N., Weissig,H., Shindyalov,I.N. and Bourne,P.E. (2000) Nucleic Acids Res., 28, 235–242.[Abstract/Free Full Text]

Chou,K.C., Pottle,M., Nemethy,G., Ueda,Y. and Sheraga,H.A. (1982) J. Mol. Biol., 162, 89–112.[ISI][Medline]

Cohen,F.E., Sternberg,M.J.E. and Taylor,W.R. (1980) Nature, 285, 378–382.[ISI][Medline]

Cohen,F.E., Sternberg,M.J.E. and Taylor,W.R. (1982) J. Mol. Biol., 156, 821–860.[ISI][Medline]

Daffner,C. (1994) Protein Sci., 3, 876–882.[Abstract/Free Full Text]

Efimov,A.V. (1993) FEBS Lett., 334, 253–256.[ISI][Medline]

Flower,D.R. (1994) Protein Engng, 7, 1305–1310.[Abstract]

Gellman,S.H. (1998) Curr. Opin. Chem. Biol., 2, 717–725.[ISI][Medline]

Hobohm,U., Sander,C. (1994) Protein Sci., 3, 522.[Abstract/Free Full Text]

Hobohm,U., Scharf,M., Schneider,R. and Sander,C. (1992) Protein Sci., 1, 409–417.[Abstract/Free Full Text]

Kabsch,W. and Sander,C. (1983) Biopolymers, 22, 2577–2637.[ISI][Medline]

King,R.D., Clark,D.A., Shirazi,J. and Sternberg,J.E. (1994) Protein Engng, 7, 1295–1303.[Abstract]

Labesse,G., Colloc'h,N., Pothier,J. and Mornon,J.-P. (1997) CABIOS, 13, 291–295.[Abstract]

Lasters,I. (1990) Protein Engng, 4, 133–135.[Abstract]

Lasters,I., Wodak,S.J., Philippe,A. and van Cutsem,E. (1988) Proc. Natl Acad. Sci. USA, 85, 3338–3342.[Abstract]

Murzin,A.G., Lesk,A.M. and Chothia,C. (1994) J. Mol. Biol., 236, 1369–1381.[ISI][Medline]

Salemme,F.R. (1981) J. Mol. Biol., 146, 143–156.[ISI][Medline]

Salemme,F.R. (1983) Prog. Biophys. Mol. Biol., 42, 95–133.[ISI][Medline]

Wang,L., O'Connell,T., Tropsha,A. and Hermans,J. (1996) J. Mol. Biol., 262, 283–293.[ISI][Medline]

Wonnacott,T.H. and Wonnacott,R.J. (1990) Introductory Statistics for Business and Economics. 4th edn. Wiley, New York.

Yang,A.S. and Honig,B. (1995) J. Mol. Biol., 252, 366–376.[ISI][Medline]

Received December 17, 1999; revised March 9, 2000; accepted March 14, 2000.





This Article
Abstract
FREE Full Text (PDF)
Alert me when this article is cited
Alert me if a correction is posted
Services
Email this article to a friend
Similar articles in this journal
Similar articles in ISI Web of Science
Similar articles in PubMed
Alert me to new issues of the journal
Add to My Personal Archive
Download to citation manager
Search for citing articles in:
ISI Web of Science (2)
Request Permissions
Google Scholar
Articles by Znamenskiy, D.
Articles by Mornon, J.-P.
PubMed
PubMed Citation
Articles by Znamenskiy, D.
Articles by Mornon, J.-P.