Comparative Analysis of TM and Cytoplasmic β-barrel Conformations Using Joint Descriptor

Macroscopic descriptors have become valuable as coarse-grained features of complex proteins and are complementary to microscopic descriptors. Proteins macroscopic geometric features provide effective clues in the quantification of distant similarity and close dissimilarity searches for structural comparisons. In this study, we performed a systematic comparison of β-barrels, one of the important classes of protein folds in various transmembrane (TM) proteins against cytoplasmic barrels to estimate the conformational features using a joint-based descriptor. The approach uses joint coordinates and dihedral angles (β and γ) based on the β-strand joints and loops to determine the arrangements and propensities at the local and global levels. We then confirmed that there is a clear preference in the overall β and γ distribution, arrangements of β-strands and loops, signature patterns, and the number of strand effects between TM and cytoplasmic β-barrel geometries. As a robust and simple approach, we determine that the joint-based descriptor could provide a reliable static structural comparison aimed at macroscopic level between complex protein conformations.

between secondary structural elements (SSEs) and several essential parameters that govern the structure and constraints [26][27][28][29][30] . Elucidation of geometric features using detailed atomic and Cα residue representations is a common approach to the analysis of helical and β-sheet packing characterizations [31][32][33] . In addition, various levels of coarse-grained models have been suggested to reduce the complexity [34][35][36] . However, a basic model that relates to the structural arrangement of both β-barrels TM and β-barrels cytoplasm structures at the macroscopic level is required to define the connectivity similarities and dissimilarities. Because β-strand structural arrangements play a critical role in the function and organization of β-barrel proteins, models that utilize the β-strand conformations would be more useful. Furthermore, such model would facilitate the understanding of how β-barrel structural organizations endure both membrane and water-soluble environments while utilizing their β sheet arrangements, overall topology, and conformational heterogeneity. Therefore, methods that identify the macroscopic principles underlying β-barrel protein folding and design factors in the overall structure, function, dynamics, and interactions become promising.
In the present study, we attempted to perform a systematic comparison of β-barrels TM and β-barrels cytoplasm geometries using the joint-based description approach 37,38 . For comparison, structures of high-resolution, non-homologous β-barrels TM and β-barrels cytoplasm protein structures were obtained from the PDB and grouped based on their geometric analyses. We considered the understanding of both β-barrel types with the topological patterns that exhibit diverse structural and functional relationships. We believe that β-barrel structural organizations can be statistically quantified based on β-strands arrangements and conformational distributions. Thus, both β-barrels TM and β-barrels cytoplasm structures were analyzed in terms of two dihedral angles; β and γ. Our results demonstrate that TM β-strands are predominantly right-handed and anti-parallel as reported previously 39,40 , and they are comparatively like the conformational flexibility of TM helices in nature [40][41][42][43] . In contrast, structures of β-barrels cytoplasm are short, mixed and varied in their arrangements with one another. All the observed conformational distributions and adjacent frequencies were associated with the arrangement of β-strands and loops in the context of how β-barrel structures have adapted in both membrane and water-soluble environments respectively. Despite uniformity, we observed that there are some considerable differences between β-barrels TM and β-barrels cytoplasm that share an equal number of β-strands. The comparative analysis results of β-barrel proteins prove that the joint-based description approach is a powerful descriptor that can be used to boost the geometrical characterization of β-barrels at both local and global levels. Possibly, our joint-based descriptor approach will be an asset in β-barrels structural prediction, as well as in the in-silico design, modeling, and validation of topologies with the given β-strand and loop segments.

Results
Macroscopic descriptor for β-barrel structures. The present study deals in relation to the recently developed joint-based description of protein structures. To understand the macroscopic geometric features of both TM and cytoplasmic β-barrel proteins such as their conformational distribution and the arrangements of β-strands, we utilize the joint constraint principles that were previously prepared to represent TM helices 37 . These β-barrel proteins always show consecutive elements of β-strands and loops linked one another as described in Fig. 1. Like TM helical proteins, a set of joints connecting the individual β-strands and loops were selected to describe a β-barrel geometry based on the joint-based description approach. Specifically, the Cα carbon coordinates of the first and last residues of each β-strand were considered as structural joining points as they represent the basic elements of protein structures 37,38 . The spatial arrangement of such joint points was measured by the dihedral angles between them. For example, for a protein composed of eight β-strands (S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , S 7 , and S 8 ) and seven loops (L 1 , L 2 , L 3 , L 4 , L 5 , L 6 and L 7 ), a group of 16 joints (P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 , P 9 , P 10 , P 11 , P 12 , P 13 , P 14 , P 15 , and P 16 ) can be assigned (Fig. 1). As a result, dihedral angle involving first four joints P 1 , P 2 , P 3 and P 4 can be determined by measuring the angle between two planes made by P 1 , P 2 , P 3 and P 2 , P 3 , P 4 . Then, the second dihedral angle can be found by relating the structural points P 2 , P 3 , P 4 , and P 5 , and P 3 , P 4 , P 5 and P 6 joints are used to determine the third, fourth and so on. Here, we propose two new types of dihedral angles: β and γ for β strands and loops. Again, the first and third dihedral angles correspond to the type β. They are denoted as β 1 and β 2 respectively. In a similar way, the second and fourth dihedral angles correspond to the type γ, denoted as γ 1 and γ 2 , respectively. Thus, the β-strand conformations of both β-barrels TM and β-barrels cytoplasm proteins can be represented by the set of joints (P 1 , P 2 , P 3 …) and two types of dihedral angles (β 1 , γ 1 , β 2 , γ 2 , β 3 …) at the macroscopic level. Clockwise and counter-clockwise angle signs were specified based on a positive value (from 0 to 180 degrees) or a negative value (from −180 to 0 degrees) respectively (Fig. 1).
Dataset collection of targets β-barrel proteins. For β-barrels TM , 29 proteins with an even number from 4 to 26 β-strands and odd number 19 TM β-strands protein were collected as an initial dataset. For β-barrels cytoplasm , a total of 51 proteins from several superfamilies were chosen. The total number of both TM and cytoplasmic β-barrel proteins with various topologies, β-strands, and loop numbers were listed in Table 1 and Table 2 respectively. These datasets were directly obtained from Protein Data Bank (PDB) by grouping into β-barrels TM and β-barrels cytoplasm proteins. The dataset collection procedure is cited in the Methods section. Concisely in both cases, i) We selected 30% sequence criteria and filtered the high-resolution X-ray crystal structures from PDB, ii) We have separated the unique monomer β-barrel proteins, and iii) The non-homologous monomeric chains with respect to different number of β-strand was grouped as final target proteins as shown in Fig. 2(a,b). In the end, the target dataset of 29 and 51 β-barrel structures was analyzed and all such analyzed β and γ angles are grouped and classified (Supporting  Information SI Tables (1, 2, 3 and 4)).
Comparison of β and γ distributions and overall arrangements of both β-barrels TM and β-barrels cytoplasm . In this section, we compare both the β-γ plots for β-barrels TM and β-barrels cytoplasm at the macroscopic level in analogy with Ramachandran φ-ψ plot at the atomistic level to find out the distribution density of dihedral angles. Our β-γ plot reflects the Ramachandran map of peptide distribution, such as favorable and unfavorable regions for β-strands and loops as given in Fig. 3(a,b). The overall distribution can possibly assist to understand the similarities and dissimilarities in their structural arrangement patterns. Overall β and γ distributions are restricted to certain regions for β-barrels TM indicating they show conformationally similar arrangements that are observed in Ω-λ plot for TM helical proteins 37 . On the other hand, β and γ distributions for β-barrels cytoplasm are sparsely distributed over greater regions signifying dissimilar patterns in their arrangements than the membrane counterparts. Conventionally, the relative conformation of adjacent structural elements i.e. neighboring elements both at residues and secondary structures level have influence on each other and that can be identified by the distribution of their dihedral angles [44][45][46] . Our new joint-derived dihedral angles such as β and γ distributions were related to the structural arrangement of β-stands for both β-barrels TM and β-barrels cytoplasm architecture just as previously demonstrated with TM helical proteins using Ω and λ dihedral angles 37 . The dihedral angle between the joints is not only associated with the arrangements of the individual β-strands and loops (a) Assignment of the β type and γ type dihedral angles. S 1 to S 8 are β-strands, L 1 to L 7 are loops, and P 1 to P 16 are joint points. Type β dihedral angles, such as β 1 , are defined by the four joint points in the Strand-Loop-Strand, such as P 1 , P 2 , P 3 , and P 4 . The γ-type dihedral angles are defined by the four joint points in the Loop-Strand-Loop, such as P 2 , P 3 , P 4 , and P 5 . (b) Assignment of the positive and negative signs for dihedral angles. The positive (+) sign and negative (−) signs represent the clockwise and counter-clockwise angles, respectively, in the projections for the dihedral angles. The figures present the projections for β 1 and γ 1 . but also involves in the dependency of adjacent secondary structural element orientations. If β-strands and loops are simplified as shown in Fig. 1, β-strands arrangements can be explained by β type dihedral angles between the i th β-strand (S i ) and its neighboring i + 1 th β-strand (S i+1 ). Similarly, the type γ can represent the arrangement of the loop segments between the i th loop (L i ) and its neighboring i + 1 th loop (L i+1 ). In addition, type γ dihedral angle also provides the relative arrangement between the i th β-strand (S i ) and i + 2 th β-strand (S i+2 ). Figure 4 show some specific relation between the dihedral angles and β-strands (or loops) arrangements. When the dihedral angle β i is close to 0 degrees, β-strand S i is antiparallel with the adjacent β-strand S i+1 ( Fig. 4(a)). In a similar way, when the dihedral angle β i is close to ±180 degrees, β-strand S i is parallel with the adjacent β-strand S i+1 ( Fig. 4(b)). When the dihedral angle γ i is close to 0 degrees, loop L i is antiparallel with the adjacent loop L i+1 , and β-strand S i and β-strand S i+2 are on the same side with respect to β-strand S i+1 (Fig. 4(c)). Meanwhile, when the dihedral angle γ i is close to ±180 degrees, loop L i is parallel with the adjacent loop L i+1 , and β-strand S i and β-strand S i+2 are on the opposite side respect to β-strand S i+1 ( Fig. 4(d)). Here, we attempt to compare both β-barrels TM and β-barrels cytoplasm geometric characteristics through consecutive β-strands and loops arrangements by investigating the distribution of β and γ dihedral angles. The overall distribution plots for all β and γ angles that belong to both β-barrels TM (29) and β-barrels cytoplasm (51) non-homologous protein structures was shown in Fig. 5. In both cases, it was evident that the β type dihedral angles were mostly confined to the range of −30° to +30°. Figure 5(a) shows more than 80% of β distribution between 0° to +30° for β-barrels TM , where the frequency in a clockwise (positive angles) orientation is evidently dominant than that of anti-clockwise (negative angles). We observed the most favored dihedral angle distribution regions between +10° and +20° corresponding to their restricted distribution space for TM β dihedral angles. Meanwhile, β type distributions for β-barrels cytoplasm were observed in an almost entire range of −150° to +150° with a strong preference towards 0° to +30° as given in Fig. 5(b). In contrast, type γ dihedral angles  2  6  4   5N  1NB9A, 1WHIA, 1PV4A, 1GCPA, 1HK9A, 1R6JA, 1O6AA, & 1G31A 8  32  24   6N  2JDID, 1WJXA, 1AGJA, 1DFUP, 1S98A, 1EFTA, 2D9RA, 2IMLA,  1BCOA, 1TS9A, 2BLNA & 2F1LA  12  60  48   7N  1WPOA, 1IK9A, 1ORUA, 1JEYA, 1RQPA, 1O70A, 1J0WA, 2GUJA,  & 1QZ8A  9  54  45   8N  1EAR, 1UE0A, 1A1XA, 1NYCA, 1OEWA, 1XE1A, 1T2WA, 1PQHA,  2CPLA, 1Y12A, 2Q03A, 2F9HA, Table 2. Selected non-homologous β-barrels cytoplasm protein structures (51), their PDB IDs, and the total number of β types and γ type dihedral angles used in this study. All protein names and their classifications used in this work are described in the Supplementary Information Table S10.
were distributed in the entire possible region between −180° to +180° for both β-barrels TM and β-barrels cytoplasm proteins. However, TM γ distribution shows dominant peaks between +90° to +180° region ( Fig. 5(c)). Possibly, γ preference of TM structures observed to be very dominance between +100° to +150° considering the overall distribution region. While type γ dihedral angles for β-barrels cytoplasm are distributed widely with equal preference (Fig. 5(d)). Interestingly, the frequencies of TM γ type dihedral angles in clockwise orientations are apparent than counter-clockwise, suggesting a preferable arrangement of adjacent β-strands tend to be right-handed orientation. Together, TM β-strands topological arrangements differ from cytoplasmic β-strands in overall β-barrel architecture, i.e. both β and γ dihedral angle distributions vary with one another. Membrane β-strands incline to pack within smaller conformational space, whereas cytoplasmic β-strand arrangements were relatively different. The propensities for antiparallel and right-handed orientation properties of β-strands for TM and soluble proteins are reported in several proteins packing studies 47,48 . Certainly, both β-barrels TM and β-barrels cytoplasm share related conformational distribution space with significant variations observed in the cytoplasmic structures.  In terms of arrangements, Fig. 5(a,b) show β type dihedral angles indicated the preference in the narrow range as a main accessible region that two neighboring β-strand (S i and S i+1 ) most likely to be arranged in antiparallel as depicted in Fig. 4. It should also be noted that cytoplasmic β-strand arrangements show antiparallel, parallel and mixed arrangements in the β-barrel proteins. Despite the sparse distribution in the all possible range of −180° to +180°, it is observed that TM γ-type dihedral angles were predominantly distributed between 30° to +180° region according to Fig. 4(c). However, the overall orientations of L i and L i +1 indicate that β-barrels TM neighboring loops are appeared to exist in both anti-parallel and parallel arrangements. On the other hand, cytoplasmic γ-type dihedral angles do not have a strong preference towards any region supporting the all possible antiparallel, parallel and mixed arrangements. Thus, cytoplasmic β-strand S i+2 are freely arranged between the same side and opposite side to β-strand S i .
Comparison of local patterns. A comparison of local patterns of TM β-barrels with cytoplasmic β-barrel geometries was attempted by analyzing the nearest neighbor frequencies of β and γ-type dihedral angles. The measurement of dihedral angle β i enables the prediction of the arrangement of neighboring β-strands (S i and S i+1 ) and the relative positions of two β-strands S i and S i+2 can be determined by measuring dihedral angle γ i . These indicate that the measurements of the continuous dihedral angles can allow us to predict how the β-strands in β-barrel proteins are continuously arranged and to which extent they are twisted and varied. For instance, the information of β i and β i+1 can determine the arrangement of S i , S i+1 , and S i+2 , and the information of γ i and γ i+1 may allow the prediction of the relative positions of S i+2 and S i+3 to S i and S i+1 . Here, to study the β-strands arrangements in both β-barrels TM and β-barrels cytoplasm systems, we examined the local patterns of continuous dihedral angle clusters such as Most of the β-strand structures in β-barrels TM were observed with (+, +) and (+, +, +) patterns for β i − β i+1 , γ i − γ i+1 , and the same trend have been traced with signature patterns of β-barrels cytoplasm as well. Interestingly, (+, +) signature pattern means a strong bias toward right-handed topology.
The twisted property of β-strands can be explained by both the conformation of individual β-strands and the relative orientations of adjacent β-strands using β and γ type dihedral angles. Figure 9 shows a conventional image of β-barrel proteins with several β-strands that have previously used to portray adjacent TM helical arrangements. Figure 9(a,b) show the schematic arrangement of three consecutive β-strands, i.e. S i , S i+1 , and S i+2 , depending on the pattern of β i − β i+1 cluster, observed in the front and side view respectively. As recorded in Fig. 4, four different (+, +), (+, −), (−, +) and (−, −) patterns determine four different types of arrangement between S i and S i+2 in parallel ( Fig. 9(b)). The dominance of (+, +) and (+, +, +) patterns in β i − β i+1 and β i − β i+1 − β i+2 cluster indicates that β-barrel favor twisted pattern. Figure 9(c) shows a schematic picture of β-barrel proteins with several β-strands in top view. It shows how the β-strands were extended depending on the pattern of γ i − γ i+1 cluster. The (−, −) and (+, +) patterns suggest that β-strands are set in one direction with a distorted pattern in such a way it forms a twisted topology. The (+, −) and (−, +) patterns, however, show that β-strands are arranged such that β-strands are tightly packed. The clear preference of (+, +) and (+, +, +) pattern for γ i − γ i+1 and γ i − γ i+1 − γ i+2 clusters suggests that β-strands in the membrane proteins favor the arrangements with a twisted pattern in the right-hand direction. Similarly, the dominant dihedral angular distribution of (90° to 180°, 90° to 180°) in the 16 possible patterns of γ i − γ i+1 cluster (Fig. 8) implies that there is also some angle preference in the twisted type arrangement of TM β-strands than cytoplasmic ones. These dihedral angle preferences and the right-handed properties of the β-strands are strongly interrelated with the overall β-barrel architectures. The specific conformations of β i − β i+1 and γ i − γ i+1 cluster confirms that adjacent positive angles may provide continued stability, thus β-strands twisting possibly control the conformational space.
The right-handed orientation and antiparallel arrangement of β-barrels TM can be related to the distribution of β type and γ-type dihedral angles found between +10° to +20° and +100° to +150° respectively. Few TM γ-type dihedral angles show a trend of relatively small dihedral value for short intracellular loops and larger dihedral value for long extracellular loops in their distributions locally, whereas no such bias has observed with β-barrels cytoplasm proteins. In general, conformational energetics strongly favors the antiparallel β-strand arrangements which have an advantage over the stability in terms of the packing interaction 49 . It is also known that the antiparallel barrel folds are particularly more stable when the strand orders are a consecutively right-handed conformation 50 . The combined orientations of individual strands and adjacent strands adopt an overall closed barrel or partially opened or distorted barrels. Structurally, the right-handed connectivity is common for all β-strands embedded in the finite hydrophobic membranes that prefer the antiparallel arrangements. Meanwhile, no specific trends have been found with β-barrels cytoplasm proteins in terms of their γ-type. Both TM and cytoplasmic adjacent loop orientations were observed as they can be found opposite side with no direct interaction between them. Though loops are arranged in a random orientation that can lead to maximization of entropy, they adopt  few recurrent conformations (adjacent loops orientations can be specific for protein families). Besides, we subsequently quantified the respective dihedral angle distributions in effect with the number and position of β-strands for both β-barrels TM and β-barrels cytoplasm proteins.

Comparison of β-strand numbers and positions effect in both β-barrels.
We also examined how the similarities and dissimilarities for both β-barrels TM and β-barrels cytoplasm in terms of the number and position of β-strands. As the numbers of β-strands vary, the arrangements of β-strands and loops might be changed due to change of interaction energy between β-strands and entropy of the flexible loop region. The distributions of dihedral angles for β and γ types were plotted according to their relative positions in β-strands for TM fold ( Fig. 10(a,c)) and cytoplasmic fold ( Fig. 10(b,d)) respectively. It seems β-barrels TM proteins are relatively larger than β-barrels cytoplasm proteins, and so the numbers of β-strands are observed less in cytoplasmic folds. According to the number of β-strands, the β and γ type dihedral distribution were plotted for TM configurations ( Fig. 11(a,c)) and cytoplasmic configurations ( Fig. 11(b,d)) respectively. For β-barrels TM proteins with 4, 10, 12, 14 and 19 β-strands obviously suggest that γ type distribution is also restricted to specific accessible regions and on the other hand, rather sparse distribution noted with 16, 18, 22 and 26 β-strands (Fig. 11(c)). For β-barrels cytoplasm proteins with 5, 6, 7 and 8 β-strands were found to be the major groups determining their distributions ( Fig. 11(b)). In both cases, the distribution of β and γ type dihedral angles was quite different compared to those of overall distributions as shown in Fig. 5. These results suggest that the dihedral angles are significantly affected by the relative position of β-strands and loops as the number of the β-strands fluctuate in both groups. Further, the patterns of the distribution of β i − β i+1 and γ i − γ i+1 cluster were analyzed according to three different numbers of groups for both cases.

Discussion
The present study provides a systematic comparison of macroscopic features of β-barrels TM and β-barrels cytoplasm proteins, providing a more simplistic way to understand the structural organization of complex proteins. Furthermore, our studies help to differentiate the specificity through various β-strands arrangements by analyzing the overall conformational space, signature patterns, positions and numbers of β-strands. For the comparative analysis of TM β-barrels with cytoplasmic structures, the joint-derived dihedral angles of β-strands were used. The β-γ dihedral angles are macroscopic descriptions of protein structures based on joint-based approach 37,38 . In both cases, specific allowed and disallowed conformational accessible spaces were found. Here, we used non-homologous datasets of β-barrels TM and β-barrels cytoplasm proteins for comparison and the approach can be extended to compare the conformational heterogeneities and geometric features of other polymers. It is well known that the adjacent β-strands in the membrane were mostly arranged in antiparallel due to a confinement inside membrane favoring stability and conformational energetics 49,50 . The overall dihedral angle distribution analyses for the β and γ-type suggest that the neighboring two β-strands would be antiparallel orientation as shown in Fig. 3(a) for TM proteins. On the other hand, cytoplasmic β-strands with greater conformational flexibility can possibly have all the antiparallel, parallel and mixed arrangements. In addition, the analyses of adjacent dihedral signature pattern analysis, such as β i − β i+1 and γ i − γ i+1 intensely supports the TM β-strands favored to be right-handed orientations with (+, +) pattern being clearly dominant. For cytoplasmic proteins, the same (+, +) signature patterns are clearly abundant preferring the right-handed orientations. Commonly, the (+, +) patterns were dominant over other signatures for β-strands arrangements. Though γ-type overall distribution is different in both the cases, they have common features of (+, +) patterns for loop-loop arrangements. Based on the results, we envisage that right-handed orientation is profoundly relevant to the stereochemistry of interaction Figure 12. Frequencies of the patterns for two consecutive β or γ type dihedral angles depending on the β-barrels TM proteins with a different number of β-strands. All the 29 non-homologous β-barrels TM proteins were categorized into three groups, i.e., proteins with 4 to 10 TM β-strands, 12 to 16 TM β-strands, and 18 to 26 TM β-strands. The bar diagrams show the observed numbers of (a) four different patterns of two consecutive β type angles, β i − β i+1 , in each group, and (b) four different patterns of two consecutive γ type angles, γ i − γ i+1 , in each group. between β-strands at the atomic level. Presumably, the successive combination of (+, +) pattern for β i − β i+1 (orientations of β-strands) and γ i − γ i+1 (orientations of loops) has an advantage for TM β-strands packing. With relatively greater conformational space, cytoplasmic proteins tend to have slightly loosely packed β-strands. Together, these results prove the packing similarities and dissimilarities between β-barrels TM and β-barrels cytoplasm proteins.
The difference in the overall dihedral angle distributions according to the number of β-strands for both the β and γ-type reveals additional specifics. We support the fact that the β-strands positioning of β-barrel proteins could be significantly restricted more inside membrane lipid bilayer than the cytoplasmic environment as the number of TM β-strands changes. Our results could aid to comprehend the differences in their overall structure as it is reflected in the variability of the dihedral angle distributions of β and γ-type at the macroscopic level. This infers the local arrangement of β-strands and loops can be affected and varied by the number of β-strands increases. The γ i − γ i+1 cluster analysis also shows a clear preference of a "right-handed" conformation as the number of β-strands fluctuates, implying a right-handed orientation is optimized form required for the efficient TM β-strand packing geometry. In contrast, (+, −) pattern appear marginally higher than (+, +) pattern of the cytoplasmic proteins with 4-6N β-strands. It was clearly observed that the arrangements of β-strand orientations are diverse depending on the β-strand positions in cytoplasmic proteins. Taken together, our results indicate that the β-barrel structure in the lipid membrane environment is more significantly controlled than the structures of cytoplasmic β-strand structures at local and global arrangements.
All such macroscopic geometric analyses provide significant clues for better understanding to improve in-silico protein structure and function prediction, modeling, and validation. The relationship between the dihedral angles and β-strand arrangements provide more focus on β-barrels engineering and designing principles. In general, any rule that is related to such structural property supports to compare and classify the protein functions. Here, the antiparallel arrangements and right-handed orientations are common for most of the β-strand structures, while the mixed loop arrangement is also essential for β-barrel structural formation. Perhaps, non-adjacent and various combinations of β and γ-type dihedral angles would be distinguished to locate interaction and non-interaction part of the β-strands. In future, the combination of macroscopic joint description of protein structures as a coarse-grain level with all-atom or Cα residue level can help us to improve our understanding towards large structural rearrangements and dynamic behavioral studies of complex structures.

Methods
Datasets as target β-barrel proteins used in this study. At first, we selected and filtered all membrane proteins from the Protein Data Bank by using the key word "Membrane proteins" and then, with the help of selection mode, β-barrel proteins X-ray crystal structures were separated. It was found that 837 structures belong to TM β-barrel proteins. The dataset of 592 β-barrel structures having sequence identity less than 90%, were identified to be unique proteins containing both homologous and non-homologous protein chains. From the above Figure 13. Frequencies of the patterns for two consecutive β or γ type dihedral angles depending on the β-barrels cytoplasm proteins with a different number of β-strands. All the 51 non-homologous β-barrels cytoplasm proteins were categorized into three groups, i.e., proteins with 4 to 6N β-strands, 7 to 8N β-strands, and 10 to 14N β-strands. The bar diagrams show the observed numbers of (a) four different patterns of two consecutive β type angles, β i − β i+1 , in each group, and (b) four different patterns of two consecutive γ type angles, γ i − γ i+1 , in each group. numbers, the search was made for a non-homologous dataset with the following criteria (1) sequence identity less than 30%, (2) ≤3.0 Å resolution structures, (3) selection of monomeric protein chains were chosen manually from homomeric structures when more than a single chain is available, (4) removal of remote homologous i.e. when more than one structure is available for the superfamily, to avoid the structural redundancy we removed all structures that share same superfamily and kept single structure to represent the superfamily. Finally, a total of 29 proteins satisfying all the above criteria were grouped as non-homologous membrane proteins. The above selection processes were done with the help of PDB and PISCES server 51,52 . Obtained TM β-barrel proteins with TM β-strands from 4TM to 26TM were classified using PPM server with OPM 53 and PDBTM 54 databases. Then, the Protein Data Bank database was used to acquire a list of all structures available of cytoplasmic β-barrels. From all beta proteins, only closed or partly opened β-barrel structures were chosen using CATH AND SCOP annotations [55][56][57] . To make certain the same analyses, structures with X-ray resolution lower than 3.0 Å with sequence identity less than 30% were considered with the help of PISCES server. From the list of 117 structures of β-barrel proteins, we removed the remote homologous and we obtained 51 structures as a final cytoplasmic dataset. We extracted the secondary structure such as β-strands and loops assignments from the obtained structures using STRIDE server 58 . The same were cross-referenced with the SHEET records in the PDB header information for cytoplasmic β-barrel proteins.
Joint-based structural joint points and β and γ-type dihedral measurements. The first and last residue Cα atoms of each TM and cytoplasmic secondary structural (SS) segment were projected as joint structural points for the dihedral calculations. Structural joint selections and β-strands boundaries were prepared based on the annotations given in OPM and STRIDE server respectively. The previously developed in-house program was used for the joint-derived dihedral angle measurements. It should be noted that the total number of β and γ-type dihedral angles for a protein structure entirely depends on the total number of TM β-strands and loops present in the structure i.e. the total number of SS segments and joint coordinates. While forming all linkage of the joint points, a macroscopic description of the overall protein structure was portrayed.
Statistical analyses dihedral angle patterns. To find out the preferred relative orientations, the conformational analysis was performed based on the dihedral angles obtained and the signature patterns dominance was revealed among various combinations of consecutive dihedral angles. For the selected 29 and 51 structures for both TM and cytoplasmic cases presented by the β n -γ n -β n+1 -γ n+1 dihedral angle set, as summarized in Supplementary Information Tables (S1, S2, S3 and S4); N stands for β strand numbers in the protein structures. The calculated dihedral angles were converted to positive (+ve) and negative (−ve) signatures to represent the conformations, as given in Supplementary Information Tables (S5, S6, S7 and S8). A consecutive β-β pattern was selected for each fold as β n -β n+1 . Grouped β n -β n+1 should be a consecutive, adjacent set, and no fixed order, whereas non-consecutive β n -β n+2 were not considered. For example, the β-β pattern angles were selected from β 1 -γ 1 -β 2 -γ 2 -β 3 -γ 3 to β n -γ n as any consecutive β n -β n . To make more defined distribution patterns, the consecutive β n -β n+1 and β n -β n+1 -β n+2 were also tested.