­Quantitative Analysis of Dihedral Angle Variability in Diverse Protein Families

­Quantitative Analysis of Dihedral Angle Variability in Diverse Protein Families | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 28 January 2025 V1 Latest version Share on ­Quantitative Analysis of Dihedral Angle Variability in Diverse Protein Families Authors : Ishani Sharma , Payel Ghosh , and Shekhar Mande 0000-0002-6634-0820 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.173810161.10530080/v1 531 views 157 downloads Contents Abstract Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Proteins dynamically exhibit fluctuating conformations, primarily driven by variations in their main chain dihedral angles phi (ϕ) and psi (ψ). In general, these conformational ensembles are such that the overall three-dimensional structure of proteins remains invariant. The conservation of three-dimensional structure thus imposes limits on variations in individual residues’ ϕ and ψ, and moreover, compensatory mechanisms exist when one of these dihedral angles undergoes a significant change. This study attempts to quantify conformational variability in protein structures by analysing dihedral angle fluctuations across diverse protein families using Shannon entropy. We assessed 26 protein families, whose structures have been determined by X-ray crystallography, NMR spectroscopy, and for one structure on which molecular dynamics simulations have been carried out. Additionally, we included a family of protein structures predicted using AlphaFold and compared the results with those of experimentally determined structures. Our findings reveal consistent Shannon entropy values across most protein families with slight variability, suggesting a natural limit to dihedral angle fluctuations that balance structural integrity and flexibility. Significant local correlations in dihedral angle adjustments reveal sophisticated compensatory mechanisms essential for maintaining overall structural integrity. These insights enhance our understanding of the delicate balance between protein stability and flexibility and have significant implications for protein engineering, drug design, and the broader study of protein dynamics. ­Quantitative Analysis of Dihedral Angle Variability in Diverse Protein Families Ishani Sharma 1 , Payel Ghosh 1,* , Shekhar C. Mande 1,2,* Address: Bioinformatics Centre, Savitribai Phule Pune University, Pune- 411 007. National Centre for Cell Science, Ganeshkhind, Pune- 411 007. Address for correspondence: Payel Ghosh: Bioinformatics Centre, Savitribai Phule Pune University, Ganeshkhind, Pune- 411 007, INDIA. Email: [email protected] Shekhar C. Mande: Bioinformatics Centre, Savitribai Phule Pune University, Ganeshkhind, Pune- 411007, INDIA & National Centre for Cell Science, Ganeshkhind, Pune- 411 007, INDIA Email: [email protected] Acknowledgements: Authors thank the Bioinformatics Centre, Savitribai Phule Pune University for providing computational facilities. This work was supported by DBT Studentship to IS. SCM thanks Dr Anand Deshpande for generous philanthropic donation to the Savitribai Phule Pune University and JC Bose fellowship of the Department of Science and Technology, New Delhi. Conflicts of Interest The authors declare no conflicts of interest. Data Availability Statement The data generated by bioinformatics analyses are available in the Supplementary material of this article. Abstract Proteins dynamically exhibit fluctuating conformations, primarily driven by variations in their main chain dihedral angles phi (ϕ) and psi (ψ). In general, these conformational ensembles are such that the overall three-dimensional structure of proteins remains invariant. The conservation of three-dimensional structure thus imposes limits on variations in individual residues’ ϕ and ψ, and moreover, compensatory mechanisms exist when one of these dihedral angles undergoes a significant change. This study attempts to quantify conformational variability in protein structures by analysing dihedral angle fluctuations across diverse protein families using Shannon entropy. We assessed 26 protein families, whose structures have been determined by X-ray crystallography, NMR spectroscopy, and for one structure on which molecular dynamics simulations have been carried out. Additionally, we included a family of protein structures predicted using AlphaFold and compared the results with those of experimentally determined structures. Our findings reveal consistent Shannon entropy values across most protein families with slight variability, suggesting a natural limit to dihedral angle fluctuations that balance structural integrity and flexibility. Significant local correlations in dihedral angle adjustments reveal sophisticated compensatory mechanisms essential for maintaining overall structural integrity. These insights enhance our understanding of the delicate balance between protein stability and flexibility and have significant implications for protein engineering, drug design, and the broader study of protein dynamics. DD MMMM YYYY \acceptedDD MMMM YYYY Keywords Protein structure dynamics, Phi (φ) and Psi(ψ) dihedral angles, Conformational variability, Shannon entropy, Pearson correlation coefficient, Correlated motion, Structural compensation. Proteins perform a myriad of functions in biological systems, relying on their ability to adopt and transition between multiple conformational states. The overall conformation of a protein is fundamentally dictated by its main chain dihedral angles phi (ϕ) and psi (ψ), which are critical in defining the spatial configuration of the polypeptide chain. The backbone dihedral angles, ϕ and ψ, are influenced by the tetrahedral geometry around alpha carbon atoms and the planar peptide bonds. Anfinsen’s seminal work revealed that the amino acid sequence contains all the necessary information for a protein to fold into its native structure 1 . While the sequence determines the structure, it is believed that the specific dihedral angles at each residue, ultimately shape the final tertiary structure of a protein. The permissible ranges for these dihedral angles were established by the pioneering work of Ramachandran and Sasisekharan 2 , by considering both the hard and soft limits of interatomic contacts. These ranges ensure that, even though natural protein sequences exhibit a degree of randomness 3 , the structural constraints imposed by dihedral angles enable proteins to fold into stable and functional conformations. The two dihedral angles, (ϕ) and (ψ), also known as Ramachandran angles, describe the torsional rotation of an amino acid around the two bonds on both sides of an α-Carbon (C α ) atom. These angles define the overall spatial arrangement of a protein’s backbone. As the number of available protein structures has increased, the allowed and forbidden values for these angles have been revisited in several studies 4 . Dihedral angle analysis has been widely used in protein structure validation research, with tools primarily utilizing allowed limits of the backbone torsion angles (ϕ-ψ) in their scoring functions 5,6 . Additionally, efforts have been made to reconstruct the native three-dimensional structure of a protein from its approximate dihedral angles and to calculate energy terms to assess system stability 7 . Although side chains contain comprehensive structural information and play a crucial role in hydrophobic collapse 8 , it is the backbone that primarily dictates the tertiary structure 9 . Consequently, the backbone plays a pivotal role in determining the overall protein tertiary structure. Lovell et al. (2003) introduced the C α deviation as a robust metric to evaluate deviations in backbone dihedral angles, thereby enhancing our comprehension of protein geometry and stability 10 . Correlated motions, where different residues of a protein move in unison, are crucial for maintaining protein structure and optimizing biological function. The conformational flexibility exhibited by proteins is mainly harnessed through correlated motions, which enhance the efficiency of biological functions 11 . A key factor influencing protein structure and stability is the interplay between local and long-range interactions within the polypeptide chain. Steric clashes and excluded volume effects impose constraints on dihedral angles, leading to correlations between the angles of adjacent and even distant residues. These constraints arise because the spatial demands of atoms prevent certain conformations, forcing the polypeptide chain into specific dihedral angle configurations. Excluded volume effects, in particular, limit the conformational space available to a protein by restricting how closely atoms can approach one another, thus amplifying the interdependence of dihedral angles. These correlations challenge the traditional view of isolated pair interactions and highlight the complex, cooperative nature of protein folding 12,13,14 . Understanding these torsion angle correlations is vital, as they provide significant insights into protein folding and stability 14 . Moreover, proteins are not static structures; they undergo conformational changes, driven primarily by the backbone dihedral angles, as part of their function. Such torsion angle fluctuations are critical to understanding protein dynamics and function in structural biology. These fluctuations often indicate the flexibility or rigidity of different regions in the protein structure, giving insights into the dynamics and conformational changes that proteins undergo to fulfil their functions 15 . Additionally, backbone torsion angle preferences also appear to be influenced by the amino acid sequence of the polypeptide, as evidenced by strong inter- and intra-residue correlations between torsion angles derived from the Protein Data Bank (PDB) 16, 14 . These correlations favour native-state torsion angles but are context-dependent, shaped by the specific amino acid sequence rather than solely by excluded volume or steric clashes. Furthermore, the influence of local sequence context and structural constraints on dihedral angles has been well documented 17 . Therefore, understanding these dynamics is crucial, as conformational changes are essential to protein folding pathways and energy landscapes. During folding, proteins sample different conformations, necessitating accurate modeling of these transitions 18,19 . Studies on protein folding and misfolding have uncovered mechanisms critical for function and stability 20,21 . Proteins with homologous sequences often exhibit similar structures, reflecting their evolutionary conservation 22 . Upon superposition, these exhibit overall low Root Mean Square Deviation (RMSD) values, however, even in well-aligned regions, individual residues may not align or superimpose perfectly. This slight misalignment can be attributed to local conformational variations, such as peptide flips or critical transitions at the residue level. These transitions, including cis-trans isomerization and beta-turn interconversions, introduce subtle yet significant changes in dihedral angles that can affect protein stability and function 23,24 . This discrepancy raises intriguing questions about how local dihedral angle fluctuations at the residue level fit within the overall protein structure and how structural integrity is maintained. Furthermore, it is crucial to understand whether these conformational variations or fluctuations are compensated globally—through small changes traversing over all residues, akin to a ripple effect—or if the adjustments and compensation happen locally, in the immediate vicinity of the fluctuations. While it is recognized that fluctuations in backbone torsion angles must be limited to preserve overall tertiary structural integrity, this understanding has been predominantly qualitative and therefore there is a need to quantify permissible ranges of variability. Our objective is to elucidate the extent to which proteins can accommodate local fluctuations while maintaining their global tertiary structure, thereby identifying structural tolerance thresholds. Our study introduces a novel approach that leverages Shannon entropy to quantify conformational variability in dihedral angles. Shannon entropy, a well-established measure of uncertainty and information content, provides a robust and quantitative framework for capturing the diversity of conformational states that proteins can adopt. Correlations between the ϕ and ψ angles of adjacent residues, particularly within secondary structures, revealing interdependencies for maintaining protein structural integrity have been noted 25,26,15 . However, an extensive analysis of residue-wise pairwise correlations across diverse protein families has not yet been conducted. Our previous study showed that conformational differences among closely related polypeptides often arise from changes in the main chain dihedrals of a few residues, leading to significant structural variations, especially in regions experiencing large conformational shifts due to domain motions 27 . Pearson correlation analyses of local dihedral angle fluctuations uncovered significant correlations both within individual residues and between adjacent residues. These findings highlight that compensatory adjustments of dihedral angles occur within the local environment to preserve overall structure and function amidst conformational variations or fluctuations. Materials and Method The workflow for this study is outlined in Figure 1. The pipeline involves a series of steps designed to analyse the conformational variability of protein structures. Selection of Protein Families Twenty diverse and representative protein families were selected from the Protein Data Bank (PDB) 16 (https://www.rcsb.org). The criteria for selection included resolution ranges from 0.5 to 2.0, R-free values ranging between 0.1 and 0.3, and a minimum chain length of 100 amino acids per chain. All selected families are represented by at least 50 structures. Additionally, four protein families determined by NMR, each with over 30 independent structural models, were included in the study. The NrdE subunit of ribonucleotide reductase (class Ib) was analyzed using 51 conformations from a 500 ns molecular dynamics simulation of a Cryo-EM structure (unpublished data). Furthermore, the protein family Albumin, comprising 35 sequences downloaded from UniProt 28 (http://www.uniprot.org ), was included in the analysis, and its protein structure was predicted using the AlphaFold3 29 server (https://alphafoldserver.com). For structures with multiple chains, only one chain was extracted from each protein structure using PyMOL software 30 . Structural and Sequence Alignment Structural alignment was performed using standalone TM-align 31 , aligning all selected protein structures to a single reference structure. Reference structures were chosen based on high resolution, completeness, and high data quality. This method established a consistent framework for uniform structural comparisons across subsequent analyses. Custom Python scripts processed TM-align output to generate residue mappings, identifying spatial correspondences between residues in the reference and aligned structures. For example, residue 1 in the reference structure aligns with residue 1 in the aligned structure, residue 2 aligns with residue 2, and so forth. These mappings were essential for calculating dihedral angle differences and assessing conformational variability. To compute the residue-wise average Root Mean Square Deviation (RMSD), all protein structures were aligned to a reference structure using PyMOL, and coordinates were exported. The focus was on the α-carbon (C α ) atom for each residue. A custom Python script calculated the pairwise Euclidean distances between the corresponding Cα atoms of the reference structure and each aligned structure. The RMSD for each residue was computed as the square root of the mean squared distances. This process was repeated for all residues across all aligned structures, and the average RMSD for each residue was calculated by taking the mean RMSD across all structures. Multiple sequence alignments were performed for all protein sequences within each family using the stand-alone Clustal Omega 32 program (v1.2.3). This alignment facilitated the understanding of sequence conservation and variability within the protein families. Sequence identity ranges were then computed with respect to the sequence of a reference protein chosen for each family. Dihedral Angle Calculation and Refinement Phi (ϕ) and psi (ψ) angles for each residue in all selected protein families were calculated using custom Python scripts based on the PDB files. Differences in phi (Δϕ) and psi (Δψ) angles between aligned residue pairs in the reference and target structures were computed using residue mappings, focusing on absolute values to highlight the extent of differences. The Python script was adapted to compute the minimum circular difference, addressing the ‘wrap-around’ effect (i.e. when the difference in dihedral angles exceeds 180°, the value is adjusted by subtracting it from 360° to ensure it falls within the correct [−180°,180°] range) while calculating Δϕ /Δψ values. Conformational Entropy Calculation Residue-wise conformational entropy of delta phi (Δϕ) and delta psi (Δψ) angles was computed for each protein family to identify regions of conformational variability. Shannon entropy (H) was calculated from histograms of Δϕ and Δψ values using custom Python scripts. The detailed methodology for determining optimal histogram bin sizes and constructing probability distributions is provided in the supplementary material. \begin{equation} H=-\sum_{i=1}^{k}{p_{\text{i\ }}\log_{2\ }p_{i}}\nonumber \\ \end{equation} where \(p_{i}\) is the probability of the i th bin, and k is the total number of bins. Secondary Structure, relative SASA & B-Factor Analysis Secondary structural states of each residue were evaluated using the standalone DSSP tool 33 , determining the proportion of residues in various secondary structures (e.g., % helix, % beta sheet, % random coil, % turn) based on a representative structure from each protein family (Table 1). From the DSSP output files, residue wise relative solvent accessible surface area (SASA) was also computed. Additionally, residue-wise B-factors were extracted from the PDB files using a custom Python script that leveraged the Bio.PDB module. Residue-wise and Global Correlation Analysis Delta phi (Δϕ) and delta psi (Δψ) values were aggregated for each protein family to facilitate a comprehensive analysis. Pearson correlation coefficients between Δϕ and Δψ for all residue pairs were computed using the ‘pearsonr’ function from the SciPy library (‘scipy.stats’ module), with p-values used to assess statistical significance. These pairwise correlations were visualized using heatmaps generated with Python libraries. To extend this analysis, we integrated the Δϕ and Δψ values for each protein family. Using the ‘pearsonr’ function from the SciPy library, we calculated Pearson correlation coefficients to examine the relationship between Δψ of the i th residue and Δϕ at various offsets (i+0 to i+5). The statistical significance of these correlations was evaluated using p-values, ensuring a robust examination of conformational variability and inter-residue dependencies within protein structures. Data Representation Line plots and heatmaps were created to visually represent the data, with the entire analytical process facilitated through Python, leveraging its diverse libraries for efficient data handling, robust statistical analysis, and effective graphical presentations. Results Figure 2 depicts the inherent geometric relationship between dihedral angles. Specifically, the relationship between the ψ angle of one residue and the ϕ angle of its adjacent residue is influenced by the tetrahedral geometry around the alpha carbon atoms and the planar peptide geometry intervening them, making the ψ and the adjacent ϕ vectors essentially parallel. Table 1 lists all the structures chosen for the analysis, the methods of their structure determination, resolution ranges and other parameters. For trypsin, HIV-1 Protease and KRAS proteins, the structures originated from a single organism, resulting in nearly identical sequences within each group. Conversely, structures for the remaining proteins were evolutionarily related, originating from different organisms. As can be seen from Table 1, all the structures superpose with one reference structure with an overall root mean square deviation (RMSD) of less than 3 . It is also evident from the table that the proteins were selected in such a way that they represent diverse structural folds covering all four different SCOP classes 34 . To quantify the fluctuations in dihedral angles ϕ and ψ across each protein family, we computed the residue-wise Shannon entropy. An illustrative conformational Shannon entropy for 152 structures of HIV-1 Protease with 99 aligned residues is shown in Figure 3a, depicting variations in conformational entropy across its secondary structures. Significant variability in conformational entropy is observed among different secondary structures. For detailed values and further insights, see Supplementary Figure S1 and Table ST1. Comparison of Figures 3a, 3b, 3c, and 3d reveals distinct patterns in conformational entropy, based on the structural analysis methods used. Figure 3a illustrates the conformational entropy of HIV-1 Protease, showing lower entropy values in alpha helices, moderate values in beta strands, and higher values in coils and turns, indicating more rigid conformations in structured regions and greater flexibility in unstructured regions, as expected. Conversely, Figure 3b presents the conformational entropy of Titin (50 models), demonstrating greater variability across all secondary structures, with particularly elevated entropy in coils and turns, reflecting the dynamic and flexible nature of proteins in solution. Figure 3c displays the entropy of Albumin protein (35 sequences) whose structure was predicted using the AlphaFold 29 server (https://alphafoldserver.com/). The trends in entropy levels across secondary structural states are comparable to those observed in X-ray crystallography data, reflecting the ability of AlphaFold predictions to approximate experimentally determined structural parameters. Similarly, Figure 3d displays the entropy of the NrdE subunit (51 conformations) from a 500 ns molecular dynamics simulation. The trends in entropy levels across secondary structural states are consistent with previous observations, but the fluctuations in entropy values are more pronounced. This underscores the significant conformational flexibility of NrdE and highlights the capability of molecular dynamics simulations to capture time-dependent dynamic states. A comprehensive analysis was conducted to investigate the relationships between conformational entropy, B-factor, RMSD, and relative Solvent Accessible Surface Area (SASA) across all protein families studied, with detailed results provided in supplementary material, Figure S2. Figure 4(a) provides a detailed analysis of the HIV-1 Protease family, highlighting the intricate relationships between conformational flexibility and structural features. In Panel 4a, the scaled phi (ϕ) and psi (ψ) entropies are presented alongside the scaled relative SASA. Although the correlations between SASA and Phi entropy (r = 0.25) and Psi entropy (r = 0.36) are significant, they are less pronounced compared to other metrics. Panel 4b shows the B-factor, which reflects atomic displacement or flexibility, revealing positive correlations with Phi entropy (r = 0.39) and Psi entropy (r = 0.35). Panel 4c presents the scaled RMSD per residue, exhibiting strong correlations with both Phi entropy (r = 0.57) and Psi entropy (r = 0.56). In addition, the correlation between conformational entropy and RMSF was examined (Figure 4d). The analysis reveals regions where both RMSF and the entropies for Psi and Phi dihedral angles peak simultaneously, indicating that residues with high flexibility also exhibit significant variability in these dihedral angles. This is reflected in the statistically significant positive correlations, with Phi entropy showing a correlation of 0.47 and Psi entropy a correlation of 0.49 with RMSF. However, this correlation is not consistent across all residues. For instance, at residue positions 147 and 148, peaks in Psi and Phi entropy, respectively, are not mirrored by corresponding RMSF values. To explore the relationship between conformational variability and structural compensation, we calculated Pearson correlations between Δϕ and Δψ angles across all residue pairs. The heatmap in Figure 5a, which illustrates the correlation matrix for HIV-1-Protease, reveals a distinctive off-diagonal correlation pattern. A similar, yet more pronounced, off-diagonal pattern is observed in Titin (Figure 5b), suggesting that local adjustments in ϕ and ψ angles are closely linked. Specifically, changes in the ψ angle of one residue strongly correlate with changes in the ϕ angle of the adjacent residue. Similar heatmaps were generated for all protein families analyzed in this study, with the corresponding results provided in Supplementary Material, Figure S3. Correlations among neighboring dihedral angle fluctuations (Δϕ and Δψ) are observed across all analyzed protein families, as listed in Table 2 and illustrated in Figure 6. While the highest correlations are generally observed at the i+1 offset, indicating correlations between the Δϕ angle of one residue and the Δψ angle of the adjacent residue, several protein families also exhibit significant correlations at the i+0 offset, which refers to correlations between the Δϕ and Δψ angles within the same residue. The correlation values tend to diminish beyond the i+2 offset, indicating that compensatory adjustments primarily occur within the immediate local residue environment. 1. Tetrahedral Geometry and Dihedral Angle Compensation Protein conformational states are largely dictated by the main chain dihedral angles. Due to the near-planar geometry of the peptide bond, the dihedral angles ϕ and ψ primarily determine the final conformational state of a protein. It is well known that the values adopted by these angles are not static, but fluctuate around the observed mean positions. The extent of these fluctuations has not been comprehensively measured until now. Our study attempts to quantify the conformational fluctuations at the main chain dihedral angles. We further address the consequences of these fluctuations on the overall conformation, and compensatory changes in the dihedral angle variability at a global and local level. The correlation coefficients measured for the dihedral angle fluctuations (Δϕ and Δψ) of any aligned residue in a protein group compared with those of neighbouring residues allow us to understand and assess the compensatory mechanisms of structural fluctuations from a different perspective. As is well known, the ψ (psi) vector of the i th residue and the ϕ (phi) vector of the i+1 st residue are oriented approximately in the same direction due to their positioning within the same peptide bond plane. Consequently, the dihedral bonds associated with these angles function as effectively parallel vectors 35 . This geometric arrangement facilitates a coordinated orientation of these dihedral angles, suggesting that conformational variations in the psi angle of one residue may necessitate compensatory adjustments in the phi angle of adjacent residues to preserve the overall tertiary structure of the protein. Such interdependencies are crucial for maintaining the overall structure and function of the protein, highlighting a mechanism through which proteins can maintain their correct folded state despite local fluctuations. Shannon entropy, a measure of uncertainty or randomness, quantifies the complexity and information content of data 36 . In this study, we calculated the Shannon entropy of phi (ϕ) and psi (ψ) dihedral angles using a fixed binning approach of 60 bins across the 0- to 360-degree range to assess the magnitude of conformational variability across multiple protein families. Our findings indicate that alpha helices generally show lower entropy values, suggesting more restricted conformational space due to their stable secondary structure. Beta strands also exhibit relatively lower entropy, though with some variability depending on the specific protein family. In contrast, coils and turns display higher entropy values, reflecting greater conformational flexibility and dynamic nature. The lower entropy values in alpha helices and beta strands are expected due to their stable hydrogen bonding patterns and regular structures. Any deviations in these regions could indicate potential structural instability or involvement in functional conformational changes. For instance, the structured regions in HIV-1 Protease (Figure 3a) like alpha helices and beta strands are more rigid, providing the necessary stability for the enzyme’s function, while the higher entropy in coils and turns reflects the flexibility required for dynamic conformational changes during substrate binding and catalysis. The flaps of HIV-1 Protease, which are crucial for substrate access to the active site, demonstrate significant conformational variability. Similarly, Titin (Figure 3b) analyzed from 50 models shows high entropy in coils and turns, which corresponds to its role in muscle elasticity, allowing significant conformational changes during muscle stretching and contraction. The moderate to low entropy in alpha helices suggests a balance between stability and flexibility, maintaining structural integrity while enabling elasticity. This flexibility is essential for Titin’s function as a molecular spring within muscle fibers. Figure 3c demonstrates that the entropy trends for Albumin predicted by AlphaFold are comparable to those observed in X-ray crystallography. This agreement confirms that the dihedral angle variability quantified through Shannon entropy remains consistent across structural datasets, irrespective of their computational or experimental origin. Notably, the trends observed in AlphaFold-predicted structures further validate the robustness of our entropy-based approach in capturing dihedral angle fluctuations. In the NrdE subunit (Figure 3d), the high entropy values in coil and turn regions, often reaching values above 4, are indicative of their intrinsic flexibility and essential role in dynamic conformations required for protein function. The significant drop in entropy values in alpha helices, often falling below 2, suggests a more constrained conformational space with reduced variability in dihedral angles. Beta strands show entropy values between 2 and 3, reflecting a balance between stability and flexibility. These observations suggest that simulation data capture more dynamic and flexible states of the protein, which are often averaged out or not observable in X-ray crystallography structures. Simulations can explore a wider conformational space, particularly in regions like coils and turns, which might not be as well-defined in static X-ray structures. Furthermore, higher entropy values in regions such as active sites or allosteric sites indicate the conformational changes required to perform their biological roles. These observations can provide insights into regions that are more prone to conformational changes, potentially identifying targets for drug design or mutation studies. Interestingly, despite phi angles being more restricted in the Ramachandran plot with fewer allowed regions, and psi angles having a broader conformational space, the overall trend of variation shown by phi and psi entropy lines is remarkably similar across all three proteins. Both phi and psi entropy values peak in flexible regions like coils and turns while dropping in structured regions such as alpha helices and beta strands. This similarity underscores that even though phi angles have fewer permissible conformations, their entropy can still reflect substantial variability in regions requiring flexibility, mirroring the behaviour seen with psi angles. This consistent pattern across different proteins and structural contexts highlights the coordinated nature of conformational changes necessary to maintain both flexibility and stability in protein structures. The entropy trends observed in our study provide valuable insights into protein flexibility, suggesting inherent limits to dihedral angle fluctuations that balance local structural flexibility with overall stability. Our findings imply that proteins may maintain functional adaptability by operating within these defined entropy boundaries. This analysis underscores the delicate interplay between flexibility and structural stability that is essential for protein function. The results highlight the diverse structural and functional roles of these proteins, with higher entropy values potentially reflecting greater complexity and adaptability within their conformational landscapes. A detailed summary of the mean phi and psi entropy values, along with their standard deviations, is provided in Supplementary Material, Table S1. Correlations Between Conformational Entropy and Structural Metrics across Proteins Families The analysis of correlations between conformational entropy and structural metrics across various protein families reveals distinct and insightful trends. Notably, correlations with relative SASA are generally lower compared to B-factor and RMSD, indicating that solvent accessibility exerts a more limited influence on dihedral angle fluctuations. For example, while proteins like KRAS (Figure S2i) (r = 0.43 for ϕ, r = 0.54 for ψ) and Trypsin (Figure S2s) (r = 0.38 for ϕ, r = 0.46 for ψ) show moderate correlations with SASA. In contrast, stronger correlations with B-factor underscore a more robust link between atomic displacement and conformational variability. Phospholipase A2 (Figure S2d), with high correlations (r = 0.72 for ϕ, r = 0.69 for ψ), exemplifies how regions with greater thermal motion tend to exhibit significant dihedral angle variability, reflecting their dynamic character. Conversely, proteins like Cytochrome-C (Figure S2e), with lower correlations (r = 0.25 for ϕ, r = 0.26 for ψ), suggest that in some proteins, atomic displacement may not always correlate strongly with dihedral angle variability, possibly reflecting regions where structural stability is maintained despite thermal motion. RMSD correlations were also strong across various proteins, particularly in HSP-90 (Figure S2g) (r = 0.69 for ϕ, r = 0.69 for ψ), where larger deviations from the average structure are closely associated with increased conformational variability. Top of Form Bottom of Form The correlation matrices generated for each protein family, representing Pearson correlations between Δϕ and Δψ angles for every residue pair, provide a comprehensive view of the underlying structural dynamics (Figures 5a, 5b, S3). These matrices reveal strong local compensatory adjustments, as indicated by the off-diagonal correlation patterns immediately adjacent to the main diagonal. This suggests that dihedral angle fluctuations are not isolated but show a dependency on adjacent or nearby residues. While the main diagonal shows subtle self-correlation, this aligns with the expectation that changes in Δϕ would correlate with changes in Δψ within the same residue due to the geometric linkage between these angles 37 . A particularly pronounced off-diagonal is quite evident in the heatmap of Titin (Figure 5b), derived from NMR data, which captures the protein in various conformational states, reflecting its dynamic nature. Titin’s role in muscle elasticity likely relies on coordinated movements between neighbouring residues, which are necessary for achieving the flexibility required in its function. Such findings point to a mechanism where dihedral angle adjustments are coordinated to preserve overall protein structure amidst local fluctuations. This highlights a dynamic equilibrium within the protein’s backbone, allowing it to adapt to changes while minimizing disruptions to its global conformation. These mechanisms are likely pivotal in maintaining the protein’s structural integrity and functional adaptability, as they enable proteins to accommodate local fluctuations without compromising their overall architecture. Scattered blocks of correlation (both positive and negative) away from the diagonal suggest interactions between residues that are not directly adjacent in sequence but still influence each other’s conformational states. This could be due to tertiary interactions or longer-range structural constraints. Interestingly, correlation patterns do not strictly follow secondary structural states. Although strong correlations exist within these regions, they are not as uniformly distributed as one might expect, suggesting that these correlations might depend more on the functional or structural context of the residues rather than their secondary structural classification alone. Furthermore, the observed correlations may also relate to the functional motions of the protein, where specific residues need to move in a coordinated manner to facilitate the protein’s activity. The heatmap corresponding to Azurin (Figure S3a, supplementary material) reveals significant off-diagonal correlations blocks far from the diagonal, indicating interactions between residues that are distant in sequence but close in three-dimensional space. These correlations span various secondary structures, including β-strands, α-helices, turns, and coils. Particularly within the BlueCu_1 domain of Azurin, identified through InterPro 38 and crucial for copper ion binding and electron transfer, the observed strong correlation patterns indicate that residues involved in copper coordination are finely tuned to maintain structural integrity during electron transfer. This is consistent with findings that even minimal changes in copper coordination can significantly influence electron transfer efficiency. The protein fold plays a central role in this process by reducing reorganization energy, thereby optimizing electron tunneling rates. High-resolution studies of Azurin have shown that the reduction in reorganization energy, coupled with the precise positioning of copper-binding residues, is essential for maintaining efficient electron transfer. These dynamics are critical for the functionality of blue copper proteins, where the delicate structural arrangement of copper-binding sites ensures the integrity of the electron transfer pathway, a key feature underpinning their biological roles 39 . Furthermore, the presence of strong correlations within turn and coil regions suggests that these flexible areas are not merely passive connectors between more rigid structural elements but actively contribute to the protein’s overall dynamics. This is supported by studies like those of Long and Tian 26 , which identify loop regions as critical facilitators of long-range correlations in proteins. These regions allow for significant interactions between non-adjacent residues, captured in the heatmap as off-diagonal correlation blocks. Additionally, the high correlation values observed in β-strands and α-helices are critical for maintaining Azurin’s structural integrity, especially around its copper-binding site, which is essential for electron transfer. In β-sheets, as Lange et al. demonstrated, strong correlations result from the tight hydrogen-bond coupling between neighboring strands, facilitating long-range communication within the protein 40 . Moreover, as Fenwick et al. highlighted, long-range correlations also exist between non-neighboring strands, leading to collective changes in β-sheets. They further described the ”β-lever” mechanism, where concerted crankshaft motions of peptide planes preserve these bonds, with stronger correlations between φ torsion angles than ψ angles 11 . In contrast, α-helices, while still exhibiting significant correlations, show slightly weaker interactions due to the alternating coupling of residues to different partners 40 . Moreover, theoretical and experimental studies suggest that side chains can also play a crucial role in propagating conformational changes over long distances, further influencing the structural dynamics of proteins 41,42 . The generalized correlation analysis also reveals subtle but significant interactions between distinct secondary structures, highlighting the sophisticated interplay of motions that support Azurin’s functional architecture. The heatmap analysis of Chain A in HSP-90α (Figure S3h, supplementary material), derived from X-ray crystallography, reveals significant structural dynamics within this essential chaperone. InterPro 38 and CDD 43 analyses identify the Hsp90_N domain, particularly the critical ATPase domain (HATPase_Hsp90-like) spanning residues 12-198, as vital for ATP binding and hydrolysis, which drive the conformational changes necessary for chaperone activity 44,45,46 . The heatmap shows highly organized and repetitive blocks of strong correlation within the ATPase domain, suggesting a well-structured network of interactions essential for ATP hydrolysis. These strong positive correlations indicate that residues within this domain move in a highly coordinated manner, ensuring the effective coupling of ATPase activity with the conformational changes required for client protein stabilization and folding 45,47 . Furthermore, the identification of multiple HEATSHOCK90 domains by InterPro underscores HSP-90α’s role in stabilizing client proteins and preventing aggregation. The high correlation observed in these regions suggests that their structural rigidity is crucial for both ATP hydrolysis and client protein interactions 47,46 . Finally, the periodic correlation patterns observed within the ATPase domain imply the presence of structural features essential for the allosteric regulation of HSP-90α’s activity, emphasizing the modular nature of these domains in coordinating the chaperone cycle 48 . The global correlation analysis between dihedral angle fluctuations (Δϕ and Δψ) and residue offset provides crucial insights into the structural coordination mechanisms that govern protein stability. As demonstrated in Figure 6 and detailed in Table 2, the Pearson correlations observed at the i+0 offset across various protein families underscore the intrinsic coupling between φ (phi) and ψ (psi) angles within individual residues. This coupling, rooted in the covalent bonding of the peptide backbone, ensures that alterations in one dihedral angle are often accompanied by compensatory changes in the other, thereby preserving local structural integrity—a relationship well-documented in the literature 37 . The i+1 offset correlations, frequently surpassing those at i+0, highlight the pivotal role of interactions between adjacent residues in maintaining the coherence of secondary structures. This enhanced correlation is not merely a function of spatial proximity but also of the geometric alignment between the ψ vector of the i th residue and the φ vector of the i+1 st residue within the same peptide bond plane. This near-parallel orientation facilitates a highly coordinated response to conformational changes, where shifts in the ψ angle of one residue necessitate compensatory adjustments in the φ angle of its neighbor. Such a mechanism is vital for preserving the integrity of local structural motifs, particularly in regions that are critical to protein function, such as active sites or binding. The observed decline in correlation values at larger residue offsets (i+2 and beyond) suggests that the influence of dihedral angle changes diminishes with increasing distance from the original residue, reinforcing the concept that structural compensation is primarily a local phenomenon. This trend underscores the importance of short-range interactions in stabilizing the protein’s tertiary structure, as distant residues are less likely to significantly contribute to local conformational adjustments. Interestingly, while the strongest correlations often occur between neighbouring residues, there are instances where correlations within the i th residue itself are also significant, sometimes nearly as strong as those between adjacent residues. This suggests that intra-residue adjustments play an important role in preserving structural stability. These findings indicate that proteins employ a multifaceted approach to structural compensation, relying on both interactions between adjacent residues and internal adjustments within residues to maintain overall stability and flexibility. This nuanced balance between flexibility and stability is crucial for proteins to adapt to changing environments while preserving their functional integrity. An analysis of structural features and their associated dihedral angle correlations across diverse protein families revealed significant variability, with some proteins displaying pronounced intra-residue and inter-residue coordination. However, no consistent pattern emerged across all proteins, highlighting the context-dependent nature of protein dynamics. These findings suggest that the observed correlations are shaped by specific structural and functional demands unique to each protein. The comprehensive results of this analysis are detailed in Supplementary Table S2. When evaluating the variability of intrinsically disordered proteins (IDPs), it is expected that stable conformational correlations will be absent. Unlike structured proteins, which adopt stable tertiary structures, IDPs are characterized by significant conformational flexibility and structural plasticity due to their lack of a fixed structure. This flexibility allows IDPs to adopt a wide range of conformations. Uversky (2013) emphasized that IDPs differ fundamentally from ordered proteins in terms of the amplitude, time scales, and coordination of their structural dynamics 49 . IDPs exist as ensembles of conformations and often include transiently populated secondary structures that facilitate efficient binding kinetics. This inherent heterogeneity and fluidity prevent the formation of stable dihedral angle correlations typically seen in structured proteins. Wright and Dyson (2015) further described IDPs as forming ’fuzzy’ complexes, where disordered regions serve as flexible linkers, with some areas remaining disordered even after binding to their targets 50 . This results in dynamically heterogeneous complexes that integrate both structured and disordered regions, further highlighting the absence of stable conformational correlations in IDPs. Conclusion This study introduces a novel approach to understanding protein structural dynamics by quantifying the extent of conformational variability in ϕ (phi) and ψ (psi) dihedral angles using Shannon entropy—a method not previously employed for this purpose. Through rigorous quantitative analysis across a diverse array of protein families, we demonstrated that compensatory mechanisms in protein structures are fundamentally local phenomena, occurring primarily in the immediate vicinity of a fluctuation. Our findings reveal that the significant correlations observed at the i+1 residue offset, driven by the geometric alignment of ψ and ϕ angles within the peptide bond plane, are crucial for maintaining structural integrity. These coordinated adjustments between neighbouring residues effectively manage conformational variability, preserving the stability and three-dimensional architecture of protein structures while allowing necessary flexibility. This coordinated response to conformational changes highlights a finely tuned compensatory mechanism that allows proteins to maintain their three-dimensional architecture while accommodating necessary flexibility. Unlike previous studies, which have primarily offered qualitative insights into protein flexibility, our work provides a quantitative framework for assessing dihedral angle variability. 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Intrinsically disordered proteins in cellular signalling and regulation. Nat Rev Mol Cell Biol . 2015;16(1):18-29. doi:10.1038/nrm3920 Figure 1 Computational pipeline used to analyze conformational variability in protein structures. Components marked with an asterisk (*) represent external programs. All other codes, developed in-house, are available on GitHub at https://github.com/Ishani-2312/Research-pipeline.git. Figure 2 : Illustration of the backbone torsion angles ϕ (phi) and ψ (psi) along a pentapeptide. These dihedral angles are critical in defining the spatial configuration of the polypeptide chain. For ease of understanding and representation, only the main chain backbone atoms have been shown in this illustration. The Illustration has been generated using PyMOL software. Figure 3: Residue-wise conformational entropy and its relation to the secondary structures . Different secondary structural segments are indicated by red for α-helices, purple for β-strands, green for coils, and yellow for turns. Red lines represent ψ entropy and blue lines represent ϕ entropy. (a): Conformational entropy plot for HIV-1-Protease, using PDB ID: 1DIF as the reference structure. (b): Conformational entropy plot for Titin (NMR structure, PDB ID: 1BPV), (c): Conformational entropy plot for Albumin (AlphaFold-predicted structure, Reference UniProt ID: P02768) (d): Conformational entropy plot for the NrdE subunit from RNR class Ib, based on 51 analysed conformations, correlated with secondary structural states. The 51 conformations were obtained through a 500 ns MD simulation. Figure 4: Interplay between conformational flexibility and structural features in the HIV-1-Protease. Each metric has been scaled to allow direct comparison across the different datasets. (a) The top plot shows the scaled Phi (ϕ) entropy in blue and Psi (ψ) entropy in red for each residue, overlaid with the scaled Relative Solvent Accessible Surface Area (SASA) depicted by a purple dashed line. The bottom plot visualizes the secondary structure elements along the sequence: red rectangles represent helices, blue arrowheads indicate beta strands, and green underscores denote coils or turns. (b) The top plot similarly displays the scaled Phi and Psi entropies, with the B-factor represented by a yellow dashed line, indicating atomic displacement or flexibility. (c) In this panel, the top plot again features the scaled Phi and Psi entropies, alongside a green dashed line representing the scaled average Root Mean Square Deviation (RMSD) per residue. The secondary structure elements are shown as in the previous panels. Figure 5: Heatmap depicting the correlation between Δϕ and Δψ values across all residue pairs in HIV-1-Protease and Titin. (a) Heatmap of Pearson correlation coefficients between Δϕ and Δψ values for all residue pairs in HIV-1-Protease. The x-axis (columns) represents residue numbers for Δϕ values, while the y-axis (rows) represents residue numbers for Δψ values. The color intensity indicates the strength and direction of the correlation, with red hues showing positive correlations and blue hues indicating negative correlations. Secondary structural states are also indicated, with blue markers representing beta strands, red markers representing helices, green markers representing coils, and yellow markers representing turns. (b) Heatmap of Pearson correlation coefficients between Δϕ and Δψ values for all residue pairs in Titin. The structure and legend are identical to panel (a). Figure 6: Line plot of Pearson correlation coefficients for each protein family, illustrating how the correlation between Δψ at residue i and Δϕ at subsequent residues varies with increasing offset. The legend on the right indicates each protein family by different colors. Information & Authors Information Version history V1 Version 1 28 January 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords conformational variability correlated motion pearson correlation coefficient phi (φ) and psi(ψ) dihedral angles protein structure dynamics shannon entropy structural compensation Authors Affiliations Ishani Sharma Savitribai Phule Pune University Bioinformatics Centre View all articles by this author Payel Ghosh Savitribai Phule Pune University Bioinformatics Centre View all articles by this author Shekhar Mande 0000-0002-6634-0820 [email protected] Savitribai Phule Pune University Bioinformatics Centre View all articles by this author Metrics & Citations Metrics Article Usage 531 views 157 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Ishani Sharma, Payel Ghosh, Shekhar Mande. ­Quantitative Analysis of Dihedral Angle Variability in Diverse Protein Families. Authorea . 28 January 2025. 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