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A Reversible Spherical Geometric Conversion of Protein Backbone Structure Coordinate Matrix to Three Independent Vectors of ρ, θ and φ

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Wei Li  *

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07 April 2024

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08 April 2024

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Abstract
Due to the vast conformational space proteins can adopt, accurate and efficient prediction of protein structure remains still a challenging task, coupled with the intricacies of interatomic interactions and the limitations of current computational models in effectively navigating this complex molecular landscape. Additionally, the lack of comprehensive experimental data for all protein structures further exacerbates the difficulty in reliable machine learning-based prediction of the three-dimensional conformations of the proteios building block of life. Geometrically, Cartesian coordinate system (CCS, X, Y and Z) and spherical coordinate system (SCS, ρ, θ and φ) are two interconvertible coordinate systems, and are like two sides of one coin. Since the beginning of Protein Data Bank (PDB) in 1971, CCS has been the default approach to specify atomic positions with X, Y and Z in PDB. In this manuscript, therefore, I present a novel method for the reversible spherical geometric conversion of protein backbone structure coordinate matrices to three independent vectors: ρ, θ and φ. This reversible conversion facilitates lossless extraction of essential structural features from protein backbone structural data, enabling the development of advanced novel algorithms for protein structure prediction in future. In short, this inter-atomic SCS approach offers a comprehensive yet efficient means of representing protein backbone geometry, leveraging spherical coordinates to capture spatial relationships in a compact and intuitive inter-atomic manner, and to provide a robust framework for reversible feature extraction for the ongoing efforts in advancing the field of protein structure prediction, the holy grail of computational structural biology.
Keywords: 
Subject: Biology and Life Sciences  -   Biophysics

1. Introduction

Protein structure prediction is the computational task of determining the three-dimensional structure of a protein from its amino acid sequence. This process is crucial for understanding protein function, interactions, and designing novel therapeutics [1,2,3,4,5,6,7,8]. By definition, accurate prediction of protein structure from its amino acid sequence is a formidable challenge in computational structural biology, with profound implications for understanding biological function and designing novel therapeutics [9,10,11,12]. Over the past decade, numerous computational methods have been developed to tackle this problem, ranging from physics-based simulations to machine learning approaches. For instance, physics-based simulations utilize principles of molecular mechanics and dynamics to simulate folding pathways, while homology modeling (e.g., algorithms such as PSI-BLAST, HHblits, and HMMER) leverages evolutionary relationships between proteins to infer structures [13,14,15,16,17,18,19,20]. Of recent further interest, machine learning techniques, particularly deep learning, have emerged as powerful tools for predicting protein structures by learning patterns from large datasets [4,21,22,23,24,25,26,27,28,29,30]. Recent advancements in deep learning, exemplified by AlphaFold, have revolutionized protein structure prediction. Deep learning models, particularly convolutional neural networks (CNNs) and recurrent neural networks (RNNs), have shown remarkable performance in predicting protein structures directly from amino acid sequences. AlphaFold, for instance, integrates multiple sequence alignment (MSA) with a deep learning architecture to accurately predict protein structures, outperforming traditional methods in terms of accuracy and speed [31,32,33,34,35,36,37,38,39]. Moreover, hybrid approaches that integrate multiple methods are also gaining traction, aiming to combine the strengths of different approaches for improved accuracy and efficiency [40,41,42,43].
In particular, the key for machine learning-based protein structure prediction is the extraction of essential structural features from protein data, which serve as the basis for predicting the three-dimensional arrangement of atoms in a protein molecule [44,45,46,47,48,49,50,51,52,53]. Take AlphaFold for example, which is an artificial intelligence system developed by DeepMind, a subsidiary of Alphabet Inc. (Google’s parent company). Here’s a simplified explanation of how AlphaFold works:
  • Input data: AlphaFold takes the primary sequence of amino acids that make up a protein as input. This sequence is derived from the genetic code.
  • Homology detection: AlphaFold starts by comparing the target protein sequence to a large database of known protein structures to identify proteins with similar sequences. This process is known as homology detection. If similar structures are found, they can provide valuable clues about the structure of the target protein.
  • Multiple sequence alignment (MSA): After identifying similar sequences, AlphaFold aligns them with the target sequence to create a multiple sequence alignment (MSA). This step helps identify conserved regions in the protein sequence, which are likely to correspond to important structural elements.
  • Structure prediction: Using the MSA and other relevant data, AlphaFold employs deep learning techniques, particularly deep neural networks, to predict the 3D structure of the target protein. The neural network is trained on a diverse set of protein structures to learn the complex relationships between amino acid sequences and their corresponding 3D structures.
  • Model refinement: AlphaFold then refines the initial predicted structures through iterative optimization techniques to improve accuracy and consistency. This refinement process helps correct any errors and inconsistencies in the predicted structure.
  • Output: The final output of AlphaFold is a predicted 3D model of the protein’s structure, including the positions of individual atoms. This model provides valuable insights into the protein’s function, interactions with other molecules, and potential implications for drug discovery and other applications [54,55,56,57,58,59,60].
Overall, AlphaFold combines advanced machine learning algorithms with biological insights to accurately predict protein structures, significantly advancing our understanding of biology and opening new avenues for drug discovery and biotechnology. While current algorithms in protein structure prediction have made significant strides, particularly with the advent of machine learning techniques, enhanced sampling methods, and improved force fields [61,62,63,64,65,66,67,68], further improvement in protein structure prediction algorithms is necessary for several reasons:
  • Accuracy for large proteins and complexes [69,70,71]: while current algorithms perform reasonably well for small to medium-sized proteins, accurately predicting the structures of large proteins or protein complexes remains challenging. Improvements are needed to capture the intricacies of these larger systems, including domain-domain interactions and conformational changes.
  • Conformational flexibility [53,72,73]: Proteins exhibit conformational flexibility, adopting multiple states with different structural conformations. Current algorithms often struggle to accurately predict these flexible regions or transitions between states. Enhancements in sampling techniques and modeling approaches are needed to better capture and predict protein flexibility.
  • Computational Efficiency [74,75]: High-resolution protein structure prediction often requires substantial computational resources and time. Improvements in algorithm efficiency and scalability are necessary to enable broader application and faster turnaround times for protein structure prediction tasks [76,77,78,79,80,81,82,83].

2. Motivation

Protein is life’s proteios building block, experimental determination of protein structure is critical to understand how they perform their functions [21]. Since the establishment of PDB in 1971 [84,85,86,87,88], protein structure has been described using Cartesian coordinates, i.e., X, Y and Z, to specify atomic positions [89]. Geometrically, Cartesian coordinate system (CCS, X, Y and Z) and spherical coordinate system (SCS, ρ , θ and ϕ ) are two interconvertible coordinate systems (Figure 1), and are like two sides of one coin [89]. Therefore, in this manuscript, I put forward an inter-atomic spherical coordinate system (IASCS) approach to redefine protein structure with ρ , θ and ϕ [89] for the reversible spherical geometric conversion of protein backbone structure coordinate matrices into three independent vectors: ρ , θ , and ϕ .
Unlike traditional Cartesian coordinates, which represent a protein’s backbone geometry in terms of its x, y, and z coordinates, the approach here leverages spherical coordinates to capture the inherent curvature (i.e., structural features [90]) of protein backbone structures more intuitively. By transforming the Cartesian coordinate matrix into spherical coordinates, this approach effectively decouple the spatial information into radial distance ( ρ ), polar angle ( θ ), and azimuthal angle ( ϕ ), providing a more natural representation of the protein’s backbone conformation [91]. In light of the fact that protein backbone structure consists of a network of covalently bonded atoms, the radial distance ( ρ ) here represents the equilibrium atomic bond length, which is defined as the inter-nuclear distance at which the system energy minimum occurs [92,93,94].

3. Materials and Methods

3.1. Redefining Protein Backbone Structure with ρ , θ and ϕ : Caenopore-5 as an Example

To illustrate the redefinition of protein backbone structure with ρ , θ and ϕ , this article takes Caenopore-5 as an example, whose three-dimensional structure has been experimentally determined by liquid-state NMR spectroscopy [95,96,97]. As is a member of the pore-forming toxin family found in the nematode Caenorhabditis elegans, Caenopore-5 is characterized by a primary amino acid sequence (Figure 2) rich in hydrophobic residues, facilitating its insertion into lipid membranes [95,96].
>2JSA_1|Chain A|Saposin-like protein family protein 5|Caenorhabditis elegans (6239) [95,96,98]
MSGSHHHHHHSSGIEGRGRSALSCQMCELVVKKYEGSADKDANVIKKDFDAECKKLFHTIPFGTRECDHYVNSKVDPIIHELEGGTAPKDVCTKLNECP
Structurally, Caenopore-5 typically adopts a helical conformation, forming oligomeric pores upon interaction with lipid bilayers, with five helices stabilized by three disulfide bridges between Cys6 and Cys80, and between Cys9 and Cys74, and between Cys35 and Cys49, as shown in Figure 3. Functionally, Caenopore-5 exhibits potent antimicrobial activity, playing a crucial role in the nematode’s innate immune defense against pathogens [95,96,98]. As the three-dimensional structure of Caenopore-5 was experimentally determined by liquid-state NMR spectroscopy [95,96], the structural model deposited in PDB [85] is an NMR ensemble of 15, instead of 1, structural models calculated using experimentally recorded NMR data. As a result, the redefinition of the backbone structure of Caenopore-5 with ρ , θ and ϕ is illustrated for all 15 structural models of Caenopore-5 [95,96,98].

3.2. Redefining Protein Backbone Structure with ρ , θ and ϕ : How?

In light of the geometric interconvertibility of CCS and SCS (Figure 1) [89], a local SCS (LSCS) approach was proposed in as an alternative to the default half-a-century-old CCS approach and a global SCS (GSCS) approach proposed a decade ago [89,100]. Since the LSCS approach takes into account the whole inter-atomic covalent bonding network of protein, it is referred to as an inter-atomic SCS (IASCS) approach below.
First, to define an inter-atomic spherical coordinate system (IASCS) approach to redefine protein structure with ρ , θ and ϕ [89] for the reversible spherical geometric conversion of protein backbone structure coordinate matrices into three independent vectors: ρ , θ , and ϕ , a short peptide with five amino acids (Figure 4) is used here as an example. Specifically, the whole inter-atomic covalent bonding network of the backbone of the short peptide (Figure 4) is defined sequentially as below:
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and three N-terminal amide hydrogen atoms (Atom_finals in the IASCS coordinate system) of the backbone of the first residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and carbon α(Cα, Atom_final in the IASCS coordinate system) of the backbone of the first residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and hydrogen α(Hα, Atom_final in the IASCS coordinate system) of the backbone of the first residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and carbonyl carbon (Atom_final in the IASCS coordinate system) of the backbone of the first residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and carbonyl oxygen (the double-bonded oxygen atom, Atom_final in the IASCS coordinate system) of the backbone of the first residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the first residue of the short peptide with five amino acids (Figure 4) and the amide nitrogen atom (Atom_final in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between the amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and the amide hydrogen atom (Atom_final in the IASCS coordinate system) of the backbone of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and carbon α(Cα, Atom_final in the IASCS coordinate system) of the backbone of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and hydrogen α(Hα, Atom_final in the IASCS coordinate system) of the backbone of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and carbonyl carbon (Atom_final in the IASCS coordinate system) of the backbone of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and carbonyl oxygen (the double-bonded oxygen atom, Atom_final in the IASCS coordinate system) of the backbone of the second residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the second residue of the short peptide with five amino acids (Figure 4) and the amide nitrogen atom (Atom_final in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between the amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and the amide hydrogen atom (Atom_final in the IASCS coordinate system) of the backbone of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and carbon α(Cα, Atom_final in the IASCS coordinate system) of the backbone of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and hydrogen α(Hα, Atom_final in the IASCS coordinate system) of the backbone of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and carbonyl carbon (Atom_final in the IASCS coordinate system) of the backbone of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and carbonyl oxygen (the double-bonded oxygen atom, Atom_final in the IASCS coordinate system) of the backbone of the third residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the third residue of the short peptide with five amino acids (Figure 4) and the amide nitrogen atom (Atom_final in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between the amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and the amide hydrogen atom (Atom_final in the IASCS coordinate system) of the backbone of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and carbon α(Cα, Atom_final in the IASCS coordinate system) of the backbone of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and hydrogen α(Hα, Atom_final in the IASCS coordinate system) of the backbone of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and carbonyl carbon (Atom_final in the IASCS coordinate system) of the backbone of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and carbonyl oxygen (the double-bonded oxygen atom, Atom_final in the IASCS coordinate system) of the backbone of the fourth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the fourth residue of the short peptide with five amino acids (Figure 4) and the amide nitrogen atom (Atom_final in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between the amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and the amide hydrogen atom (Atom_final in the IASCS coordinate system) of the backbone of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between N-terminal amide nitrogen atom (Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and carbon α(Cα, Atom_final in the IASCS coordinate system) of the backbone of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and hydrogen α(Hα, Atom_final in the IASCS coordinate system) of the backbone of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent bond between carbon α(Cα, Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and carbonyl carbon (Atom_final in the IASCS coordinate system) of the backbone of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent double bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and carbonyl oxygen (the double-bonded oxygen atom, Atom_final in the IASCS coordinate system) of the backbone of the fifth residue of the short peptide with five amino acids (Figure 4);
  • the covalent single bond between carbonyl carbon (Atom_initial in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4) and the carboxyl carboxyl hydroxyl oxygen atom (Atom_final in the IASCS coordinate system) of the fifth residue of the short peptide with five amino acids (Figure 4);
For further extraction of structural features from the backbones of proteins, this article also puts forward an approach to redefine the backbone peptide bond connecting pattern with ρ , θ and ϕ . As shown in Figure 5, a peptide bond is formed by dehydration, which is the removal of the carboxyl hydroxyl on one amino acid and one hydrogen atom on the amine end of another amino acid. Specifically, the whole inter-atomic peptide bonding network of the backbone of the short peptide (Figure 4) is defined sequentially as below:
  • the peptide bond (Figure 5) as formed between R1 and R2 of the short peptide with five amino acids (Figure 4);
  • the peptide bond (Figure 5) as formed between R2 and R3 of the short peptide with five amino acids (Figure 4);
  • the peptide bond (Figure 5) as formed between R3 and R4 of the short peptide with five amino acids (Figure 4);
  • the peptide bond (Figure 5) as formed between R4 and R5 of the short peptide with five amino acids (Figure 4);
Thus, with the IASCS coordinate system defined above for the backbone of protein structure in place, for CCS, GSCS and IASCS, their main differences are summarized below:
  • CCS requires three parameters (X, Y and Z) for all atoms to construct a protein backbone structural model.
  • GSCS requires three parameters (R, θ and ϕ ) for all atoms to construct a protein structural model, because GSCS takes the centroid of the protein molecule as origin, where R is used to signify the distance between the centroid (an imaginary point) and any atom of the protein.
  • IASCS requires two parameters ( θ and ϕ ) to construct a protein structural model, because
    (a)
    IASCS takes into account the fact that protein backbone structure essentially is a network of covalently bonded atoms.
    (b)
    IASCS defines ρ as the equilibrium inter-atomic covalent bond length [92,93].
    (c)
    For a protein with unknown structure, our knowledge of its X, Y and Z is zero, which is untrue of our knowledge of its ρ [92,93].
  • Neither CCS nor GSCS take into account the inter-atomic covalent bonding network of protein backbone structures.

4. Results

In short, this article puts forward a reversible spherical geometric conversion method, which offers a promising approach for extracting essential structural features from protein backbone coordinate matrices by transforming Cartesian coordinates into spherical coordinates represented by radial distance ( ρ ), polar angle ( θ ), and azimuthal angle ( ϕ ). Specifically, the extraction of essential structural features from the backbone of all 15 structural models of Caenopore-5 [95,96,98] inculde:
  • the distribution pattern of the lengths ( ρ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 6;
  • the distribution pattern of the azimuthal angle ( θ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 7;
  • the distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 8;
  • the distribution pattern of the lengths ( ρ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 9;
  • the distribution pattern of the azimuthal angle ( θ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 10;
  • the distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98], as shown by Figure 11;
In the broader context, this work contributes to the ongoing efforts in advancing the field of computational biology and protein structure prediction as the proposed IASCS (i.e., the local ρ - θ - ϕ approach) opens up new avenues for research in feature extraction and protein structure prediction algorithm design, paving the way for more accurate and efficient prediction of protein structures:
  • the reversible nature of the spherical geometric conversion ensures that no information is lost during the geometric coordinate transformation process, preserving the fidelity of the original protein backbone structure and ensuring that it is able to be reconstructed using the extracted IASCS features (local ρ , θ and ϕ values). This feature is crucial for maintaining prediction accuracy and enabling further analysis of the extracted structural features for its future application in protein structure prediction [13,14,15,16,17,18,101].
  • the compact representation offered by spherical coordinates with Caenopore-5 as an example enhances computational efficiency, making our method suitable for large-scale (for both PDB and AlphaFoldDB) protein structure prediction tasks for the entire protein molecular space [47,102,103,104,105,106,107,108,109].
  • lastly, the intuitive nature of spherical coordinates also aids in the interpretation of structural features, potentially offering insights into the underlying principles governing protein folding and function [16,110,111,112,113,114,115,116,117,118].

5. Conclusion

In conclusion, the reversible spherical geometric conversion method presented in this manuscript offers a promising approach for extracting essential structural features [119] from protein backbone coordinate matrices [120]. By transforming Cartesian coordinates into spherical coordinates represented by radial distance ( ρ ), polar angle ( θ ), and azimuthal angle ( ϕ ), our method provides a more intuitive representation of protein geometry. This approach facilitates the design of algorithms for protein structure prediction by enabling the extraction of meaningful structural features [10,121,122,123,124,125,126].

6. Discussion

To date, while ∼ 217,966 experimental structures and ∼ 1,068,577 computed structures represent a big fraction of the entire protein molecular space [21,102,127], accurate computational approaches are still needed to address this gap and to enable large-scale structural bioinformatics [21]. Three years ago, with the advent of AlphaFold2 and RosettaFold [21,22,23,128,129], 2021 saw a big step forward in the development of protein structure prediction (PSP) [130,131,132,133]. While the progresses in PSP have ebbed and flowed historically, the past two years saw dramatic advances driven by the increasing neuralization [22] of PSP algorithms, whereby computations previously based on energy models and sampling procedures are replaced by neural networks [134,135,136,137,138,139,140,141,142,143,144,145,146].
As discussed previously in [89], distance is the data which is plugged into neuralized [22] algorithms to build a structural model of a protein. To improve the quality of the model, algorithm and data are like two sides of one coin. Geometrically, CCS and SCS are like two sides of one coin, too. Focusing on the improvement of algorithms (e.g., neuralization [22]) alone is probably not enough, unless we flip the coin over and take a look at the other side, i.e., data and IASCS, which allow us to extract two spherical structural features ( θ and ϕ ) from any protein with experimentally determined structure [147,148,149,150,151]. With the redefinition of protein backbone structure with ρ , θ and ϕ here, future work may involve exploring extensions of the spherical geometric conversion method to incorporate additional structural information, such as side-chain interactions and solvent accessibility, including the design of side chain placement algorithms with improved performance. Additionally, further validation on experimentally determined protein structures and comparison with existing methods will help assess the robustness and generalizability of the ρ - θ - ϕ approach [89,152].

Ethical statement:

No ethical approval is required.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the author used OpenAI’s ChatGPT in order to improve the readability of the manuscript, and to make it as concise and short as possible. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the content of the publication.

Author Contributions

Conceptualization, W.L.; methodology, W.L.; software, W.L.; validation, W.L.; formal analysis, W.L.; investigation, W.L.; resources, W.L.; data duration, W.L.; writing–original draft preparation, W.L.; writing–review and editing, W.L.; visualization, W.L.; supervision, W.L.; project administration, W.L.; funding acquisition, not applicable.

Funding

This research received no external funding.

Acknowledgments

I thank the whole structural biology community, whose continued contribution is priceless to our continued understanding, both structural and functional, of the proteios building block of life.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1. Geometrically, CCS (X, Y and Z) and SCS ( ρ , θ and ϕ ) are like two sides of one coin, and are both applicable in the specifications of atomic positions to define protein structure [89].
Figure 1. Geometrically, CCS (X, Y and Z) and SCS ( ρ , θ and ϕ ) are like two sides of one coin, and are both applicable in the specifications of atomic positions to define protein structure [89].
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Figure 2. Amino acid sequence and secondary structure of Caenopore-5 [95,96,98].
Figure 2. Amino acid sequence and secondary structure of Caenopore-5 [95,96,98].
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Figure 3. The three-dimensional NMR structure of Caenopore-5 (PDB ID: 2JSA) stabilized by three disulfide bonds (red sticks) [95,96,98]. This figure is prepared by VMD [99].
Figure 3. The three-dimensional NMR structure of Caenopore-5 (PDB ID: 2JSA) stabilized by three disulfide bonds (red sticks) [95,96,98]. This figure is prepared by VMD [99].
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Figure 4. Chemical structure of a short peptide with five amino acids as an example to illustrate the redefinition of protein structure with ρ , θ and ϕ [89] for the reversible spherical geometric conversion of protein backbone structure coordinate matrices into three independent vectors: ρ , θ , and ϕ .
Figure 4. Chemical structure of a short peptide with five amino acids as an example to illustrate the redefinition of protein structure with ρ , θ and ϕ [89] for the reversible spherical geometric conversion of protein backbone structure coordinate matrices into three independent vectors: ρ , θ , and ϕ .
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Figure 5. The formation of a peptide bond, which is the bond between the carbonyl carbon and the nitrogen in the amide fragment.
Figure 5. The formation of a peptide bond, which is the bond between the carbonyl carbon and the nitrogen in the amide fragment.
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Figure 6. The distribution pattern of the lengths ( ρ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 6. The distribution pattern of the lengths ( ρ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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Figure 7. The distribution pattern of the azimuthal angle ( θ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 7. The distribution pattern of the azimuthal angle ( θ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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Figure 8. The distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 8. The distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 477 covalent bonds of the backbone of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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Figure 9. The distribution pattern of the lengths ( ρ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 9. The distribution pattern of the lengths ( ρ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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Figure 10. The distribution pattern of the azimuthal angle ( θ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 10. The distribution pattern of the azimuthal angle ( θ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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Figure 11. The distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
Figure 11. The distribution pattern of the polar angle (also known as the zenith angle, ϕ ) of 80 peptide bonds (Figure 5) of Caenopore-5 [95,96,98] as reversibly extracted from the 15 NMR structural models of the three-dimensional NMR ensemble of Caenopore-5 (PDB ID: 2JSA) [95,96,98].
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