Improved Assessment of Globularity of Protein Structures and the Ellipsoid Profile of the Biological Assemblies from the PDB.

In this paper, we present an update to the ellipsoid profile algorithm (EP), a simple technique for the measurement of the globularity of protein structures without the calculation of molecular surfaces. The globularity property is understood in this context as the ability of the molecule to fill a minimum volume enclosing ellipsoid (MVEE) that approximates its assumed globular shape. The more of the interior of this ellipsoid is occupied by the atoms of the protein, the better are its globularity metrics. These metrics are derived from the comparison of the volume of the voxelized representation of the atoms and the volume of all voxels that can fit inside that ellipsoid (a uniform unit Å cube lattice). The so-called ellipsoid profile shows how the globularity changes with the distance from the center. Two of its values, the so-called ellipsoid indexes, are used to classify the structure as globular, semi-globular or non-globular. Here, we enhance the workflow of the EP algorithm via an improved outlier detection subroutine based on principal component analysis. It is capable of robust distinguishing between the dense parts of the molecules and, for example, disordered chain fragments fully exposed to the solvent. The PCA-based method replaces the current approach based on kernel density estimation. The improved EP algorithm was tested on 2124 representatives of domain superfamilies from SCOP 2.08. The second part of this work is dedicated to the survey of globularity of 3594 representatives of biological assemblies from molecules currently deposited in the PDB and analyzed by the 3DComplex database (monomers and complexes up to 60 chains).

Structural Outlier Detection and Zernike-Canterakis Moments for Molecular Surface Meshes-Fast Implementation in Python.

Object retrieval systems measure the degree of similarity of the shape of 3D models. They search for the elements of the 3D model databases that resemble the query model. In structural bioinformatics, the query model is a protein tertiary/quaternary structure and the objective is to find similarly shaped molecules in the Protein Data Bank. With the ever-growing size of the PDB, a direct atomic coordinate comparison with all its members is impractical. To overcome this problem, the shape of the molecules can be encoded by fixed-length feature vectors. The distance of a protein to the entire PDB can be measured in this low-dimensional domain in linear time. The state-of-the-art approaches utilize Zernike-Canterakis moments for the shape encoding and supply the retrieval process with geometric data of the input structures. The BioZernike descriptors are a standard utility of the PDB since 2020. However, when trying to calculate the ZC moments locally, the issue of the deficiency of libraries readily available for use in custom programs (i.e., without relying on external binaries) is encountered, in particular programs written in Python. Here, a fast and well-documented Python implementation of the Pozo-Koehl algorithm is presented. In contrast to the more popular algorithm by Novotni and Klein, which is based on the voxelized volume, the PK algorithm produces ZC moments directly from the triangular surface meshes of 3D models. In particular, it can accept the molecular surfaces of proteins as its input. In the presented PK-Zernike library, owing to Numba's just-in-time compilation, a mesh with 50,000 facets is processed by a single thread in a second at the moment order 20. Since this is the first time the PK algorithm is used in structural bioinformatics, it is employed in a novel, simple, but efficient protein structure retrieval pipeline. The elimination of the outlying chain fragments via a fast PCA-based subroutine improves the discrimination ability, allowing for this pipeline to achieve an 0.961 area under the ROC curve in the BioZernike validation suite (0.997 for the assemblies). The correlation between the results of the proposed approach and of the 3D Surfer program attains values up to 0.99.

Solubility and Aggregation of Selected Proteins Interpreted on the Basis of Hydrophobicity Distribution.

Protein solubility is based on the compatibility of the specific protein surface with the polar aquatic environment. The exposure of polar residues to the protein surface promotes the protein's solubility in the polar environment. The aquatic environment also influences the folding process by favoring the centralization of hydrophobic residues with the simultaneous exposure to polar residues. The degree of compatibility of the residue distribution, with the model of the concentration of hydrophobic residues in the center of the molecule, with the simultaneous exposure of polar residues is determined by the sequence of amino acids in the chain. The fuzzy oil drop model enables the quantification of the degree of compatibility of the hydrophobicity distribution observed in the protein to a form fully consistent with the Gaussian 3D function, which expresses an idealized distribution that meets the preferences of the polar water environment. The varied degrees of compatibility of the distribution observed with the idealized one allow the prediction of preferences to interactions with molecules of different polarity, including water molecules in particular. This paper analyzes a set of proteins with different levels of hydrophobicity distribution in the context of the solubility of a given protein and the possibility of complex formation.

Model of Environmental Membrane Field for Transmembrane Proteins.

The water environment determines the activity of biological processes. The role of such an environment interpreted in the form of an external field expressed by the 3D Gaussian distribution in the fuzzy oil drop model directs the folding process towards the generation of a centrally located hydrophobic core with the simultaneous exposure of polar residues on the surface. In addition to proteins soluble in the water environment, there is a significant group of membrane proteins that act as receptors or channels, including ion channels in particular. The change of the polar (water) environment into a highly hydrophobic (membrane) environment is quite radical, resulting in a different hydrophobicity distribution within the membrane protein. Modification of the notation of the force field expressing the presence of the hydrophobic environment has been proposed in this work. A modified fuzzy oil drop model with its adaptation to membrane proteins was used to interpret the structure of membrane proteins-mechanosensitive channel. The modified model was also used to describe the so-called negative cases-i.e., for water-soluble proteins with a clear distribution consistent with the fuzzy oil drop model.

Divergence Entropy-Based Evaluation of Hydrophobic Core in Aggressive and Resistant Forms of Transthyretin.

The two forms of transthyretin differing slightly in the tertiary structure, despite the presence of five mutations, show radically different properties in terms of susceptibility to the amyloid transformation process. These two forms of transthyretin are the object of analysis. The search for the sources of these differences was carried out by means of a comparative analysis of the structure of these molecules in their native and early intermediate stage forms in the folding process. The criterion for assessing the degree of similarity and differences is the status of the hydrophobic core. The comparison of the level of arrangement of the hydrophobic core and its initial stages is possible thanks to the application of divergence entropy for the early intermediate stage and for the final forms. It was shown that the minimal differences observed in the structure of the hydrophobic core of the forms available in PDB, turned out to be significantly different in the early stage (ES) structure in folding process. The determined values of divergence entropy for both ES forms indicate the presence of the seed of hydrophobic core only in the form resistant to amyloid transformation. In the form of aggressively undergoing amyloid transformation, the structure lacking such a seed is revealed, being a stretched one with a high content of ß-type structure. In the discussed case, the active presence of water in the structural transformation of proteins expressed in the fuzzy oil drop model (FOD) is of decisive importance for the generation of the final protein structure. It has been shown that the resistant form tends to generate a centric hydrophobic core with the possibility of creating a globular structure, i.e., a spherical micelle-like form. The aggressively transforming form reveals in the structure of its early intermediate, a tendency to form the ribbon-like micelle as observed in amyloid.

Downhill, Ultrafast and Fast Folding Proteins Revised.

Research on the protein folding problem differentiates the protein folding process with respect to the duration of this process. The current structure encoded in sequence dogma seems to be clearly justified, especially in the case of proteins referred to as fast-folding, ultra-fast-folding or downhill. In the present work, an attempt to determine the characteristics of this group of proteins using fuzzy oil drop model is undertaken. According to the fuzzy oil drop model, a protein is a specific micelle composed of bi-polar molecules such as amino acids. Protein folding is regarded as a spherical micelle formation process. The presence of covalent peptide bonds between amino acids eliminates the possibility of free mutual arrangement of neighbors. An example would be the construction of co-micelles composed of more than one type of bipolar molecules. In the case of fast folding proteins, the amino acid sequence represents the optimal bipolarity system to generate a spherical micelle. In order to achieve the native form, it is enough to have an external force field provided by the water environment which directs the folding process towards the generation of a centric hydrophobic core. The influence of the external field can be expressed using the 3D Gaussian function which is a mathematical model of the folding process orientation towards the concentration of hydrophobic residues in the center with polar residues exposed on the surface. The set of proteins under study reveals a hydrophobicity distribution compatible with a 3D Gaussian distribution, taken as representing an idealized micelle-like distribution. The structure of the present hydrophobic core is also discussed in relation to the distribution of hydrophobic residues in a partially unfolded form.

The Structure of Amyloid Versus the Structure of Globular Proteins.

The issue of changing the structure of globular proteins into an amyloid form is in the focus of researchers' attention. Numerous experimental studies are carried out, and mathematical models to define the essence of amyloid transformation are sought. The present work focuses on the issue of the hydrophobic core structure in amyloids. The form of ordering the hydrophobic core in globular proteins is described by a 3D Gaussian distribution analog to the distribution of hydrophobicity in a spherical micelle. Amyloid fibril is a ribbon-like micelle made up of numerous individual chains, each representing a flat structure. The distribution of hydrophobicity within a single chain included in the fibril describes the 2D Gaussian distribution. Such a description expresses the location of polar residues on a circle with a center with a high level of hydrophobicity. The presence of this type of order in the amyloid forms available in Preotin Data Bank (PDB) (both in proto- and superfibrils) is demonstrated in the present work. In this system, it can be assumed that the amyloid transformation is a chain transition from 3D Gauss ordering to 2D Gauss ordering. This means changing the globular structure to a ribbon-like structure. This observation can provide a simple mathematical model for simulating the amyloid transformation of proteins.

Structure of the Hydrophobic Core Determines the 3D Protein Structure-Verification by Single Mutation Proteins.

Four de novo proteins differing in single mutation positions, with a chain length of 56 amino acids, represent diverse 3D structures: monomeric 3α and 4ß + α folds. The reason for this diversity is seen in the different structure of the hydrophobic core as a result of synergy leading to the generation of a system in which the polypeptide chain as a whole participates. On the basis of the fuzzy oil drop model, where the structure of the hydrophobic core is expressed by means of the hydrophobic distribution function in the form of a 3D Gaussian distribution, it has been shown that the composition of the hydrophobic core in these two structural forms is different. In addition, the use of a model to determine the structure of the early intermediate in the folding process allows to indicate differences in the polypeptide chain geometry, which, combined with the construction of a common hydrophobic nucleus as an effect of specific synergy, may indicate the reason for the diversity of the folding process of the polypeptide chain. The results indicate the need to take into account the presence of an external force field originating from the water environment and that its active impact on the formation of a hydrophobic core whose participation in the stabilization of the tertiary structure is fundamental.

Identification of tunnels as in potato hydrolases.

Enzymes with an active center hidden in the middle of the molecule in a tunnel-like cavity constitute an interesting object of analysis due to the highly specialized environment for the course of the catalytic reaction. Identifying the tunnel is a challenge in itself. Moreover, the structural conditioning for the course of the reaction provides information on the diversity of the environment, which must necessarily meet the conditions of high specificity. The use of a fuzzy oil drop model to identify residues constituting the walls of the tunnel located in the center of the protein seems highly justified. The fuzzy oil drop model, which assumes the highest concentration of hydrophobicity in the center of the molecule, in these enzymes shows a significant hydrophobicity deficit resulting from the absence of any residues in the central part of the molecule. Comparison of the expected distribution in consistent with the 3D Gaussian distribution where the observed distribution resulting from the interaction of residues in the protein shows significant differences precisely in the positions of residues located near the center of the molecule. The inside characteristics of the tunnel are the background for the enzymatic reaction. This environment additionally constitutes an external force field, which creates favorable conditions for carrying out the catalytic process. The use of fuzzy oil drop model has been verified using the potato (solanum tuberosum) epoxide hydrolase I. This forms the preliminary basis for testing the fuzzy oil drop model. The data presented here provides an impetus for a large scale analysis of all proteins containing tunnels in enzyme structures available in the Protein Data Bank (PDB).

Alternative Structures of α-Synuclein.

The object of our analysis is the structure of alpha-synuclein (ASyn), which, under in vivo conditions, associates with presynaptic vesicles. Misfolding of ASyn is known to be implicated in Parkinson's disease. The availability of structural information for both the micelle-bound and amyloid form of ASyn enables us to speculate on the specific mechanism of amyloid transformation. This analysis is all the more interesting given the fact that-Unlike in Aß(1-42) amyloids-only the central fragment (30-100) of ASyn has a fibrillar structure, whereas, its N- and C-terminal fragments (1-30 and 100-140, respectively) are described as random coils. Our work addresses the following question: Can the ASyn chain-as well as the aforementioned individual fragments-adopt globular conformations? In order to provide an answer, we subjected the corresponding sequences to simulations carried out using Robetta and I-Tasser, both of which are regarded as accurate protein structure predictors. In addition, we also applied the fuzzy oil drop (FOD) model, which, in addition to optimizing the protein's internal free energy, acknowledges the presence of an external force field contributed by the aqueous solvent. This field directs hydrophobic residues to congregate near the center of the protein body while exposing hydrophilic residues on its surface. Comparative analysis of the obtained models suggests that fragments which do not participate in forming the amyloid fibril (i.e., 1-30 and 100-140) can indeed attain globular conformations. We also explain the influence of mutations observed in vivo upon the susceptibility of ASyn to undergo amyloid transformation. In particular, the 30-100 fragment (which adopts a fibrillar structure in PDB) is not predicted to produce a centralized hydrophobic core by any of the applied toolkits (Robetta, I-Tasser, and FOD). This means that in order to minimize the entropically disadvantageous contact between hydrophobic residues and the polar solvent, ASyn adopts the form of a ribbonlike micelle (rather than a spherical one). In other words, the ribbonlike micelle represents a synergy between the conformational preferences of the protein chain and the influence of its environment.

The Amyloid as a Ribbon-Like Micelle in Contrast to Spherical Micelles Represented by Globular Proteins.

Selected amyloid structures available in the Protein Data Bank have been subjected to a comparative analysis. Classification is based on the distribution of hydrophobicity in amyloids that differ with respect to sequence, chain length, the distribution of beta folds, protofibril structure, and the arrangement of protofibrils in each superfibril. The study set includes the following amyloids: Aß (1-42), which is listed as Aß (15-40) and carries the D23N mutation, and Aß (11-42) and Aß (1-40), both of which carry the E22Δ mutation, tau amyloid, and α-synuclein. Based on the fuzzy oil drop model (FOD), we determined that, despite their conformational diversity, all presented amyloids adopt a similar structural pattern that can be described as a ribbon-like micelle. The same model, when applied to globular proteins, results in structures referred to as "globular micelles," emerging as a result of interactions between the proteins' constituent residues and the aqueous solvent. Due to their composition, amyloids are unable to attain entropically favorable globular forms and instead attempt to limit contact between hydrophobic residues and water by producing elongated structures. Such structures typically contain quasi hydrophobic cores that stretch along the fibril's long axis. Similar properties are commonly found in ribbon-like micelles, with alternating bands of high and low hydrophobicity emerging as the fibrils increase in length. Thus, while globular proteins are generally consistent with a 3D Gaussian distribution of hydrophobicity, the distribution instead conforms to a 2D Gaussian distribution in amyloid fibrils.

Internal force field in selected proteins.

The fuzzy oil drop model suggests that the tertiary conformation of a protein - particularly a globular one - can be likened to a spherical micelle. During the folding process, hydrophilic residues are exposed on the surface, while hydrophobic residues are retained inside the protein. The resulting hydrophobicity distribution can be mathematically modeled as a 3D Gaussian. The fuzzy oil drop model is strikingly effective in explaining the properties of type II antifreeze proteins and fast-folding proteins, as well as a vast majority of autonomous protein domains. This work aims to determine whether similar mechanisms apply to other types of nonbonding interactions. Our analysis indicates that electrostatic and van der Waals forces do not conform to the Gaussian pattern. The study involves a reference protein (titin) which shows a high agreement between the observed distribution of hydrophobicity and the theoretical (Gaussian) distribution, a selection of amyloid structures derived from the Protein Data Bank, as well as transthyretin - a protein known for its susceptibility to amyloid transformation.

Different Synergy in Amyloids and Biologically Active Forms of Proteins.

Protein structure is the result of the high synergy of all amino acids present in the protein. This synergy is the result of an overall strategy for adapting a specific protein structure. It is a compromise between two trends: The optimization of non-binding interactions and the directing of the folding process by an external force field, whose source is the water environment. The geometric parameters of the structural form of the polypeptide chain in the form of a local radius of curvature that is dependent on the orientation of adjacent peptide bond planes (result of the respective Phi and Psi rotation) allow for a comparative analysis of protein structures. Certain levels of their geometry are the criteria for comparison. In particular, they can be used to assess the differences between the structural form of biologically active proteins and their amyloid forms. On the other hand, the application of the fuzzy oil drop model allows the assessment of the role of amino acids in the construction of tertiary structure through their participation in the construction of a hydrophobic core. The combination of these two models-the geometric structure of the backbone and the determining of the participation in the construction of the tertiary structure that is applied for the comparative analysis of biologically active and amyloid forms-is presented.

Structural analysis of the Aß(11-42) amyloid fibril based on hydrophobicity distribution.

The structure of the Aß(11-42) amyloid available in PDB makes possible the molecular analysis of the specificity of amyloid formation. This molecule (PDB ID 2MVX) is the object of analysis. This work presents the outcome of in silico experiments involving various alternative conformations of the Aß(11-42) sequence, providing clues as to the amylodogenecity of its constituent fragments. The reference structure (PDB) has been compared with folds generated using I-Tasser and Robetta-the strongest contenders in the CASP challenge. Additionally, a polypeptide which matches the Aß(11-42) sequence has been subjected to folding simulations based on the fuzzy oil drop model, which favors the production of a monocentric hydrophobic core. Computer simulations yielded 15 distinct structural forma (five per software package), which, when compared to the experimentally determined structure, allow us to study the role of structural elements which cause an otherwise globular protein to transform into an amyloid. The unusual positions of hydrophilic residues disrupting the expected hydrophobic core and propagating linearly along the long axis of fibril is recognized as the seed for amyloidogenic transformation in this polypeptide. This paper discusses the structure of the Aß(11-42) amyloid fibril, listed in PDB under ID 2MXU (fragment od Aß(1-42) amyloid).

Secondary and Supersecondary Structure of Proteins in Light of the Structure of Hydrophobic Cores.

The traditional classification of protein structures (with regard to their supersecondary and tertiary conformation) is based on an assessment of conformational similarities between various polypeptide chains and particularly on the presence of specific secondary structural motifs. Mutual relations between secondary folds determine the overall shape of the protein and may be used to assign proteins to specific families (such as the immunoglobulin-like family). An alternative means of conducting structural assessment focuses on the structure of the protein's hydrophobic core. In this case, the protein is treated as a quasi-micelle, which exposes hydrophilic residues on its surface while internalizing hydrophobic residues. The accordance between the actual distribution of hydrophobicity in a protein and its corresponding theoretical ("idealized") distribution can be determined quantitatively, which, in turn, enables comparative analysis of structures regarded as geometrically similar (as well as geometrically divergent structures which are nevertheless regarded as similar in the sense of the fuzzy oil drop model). In this scope, the protein may be compared to an "intelligent micelle," where local disorder is often intentional and related to biological function-unlike traditional surfactant micelles which remain highly symmetrical throughout and do not carry any encoded information.

The aqueous environment as an active participant in the protein folding process.

Existing computational models applied in the protein structure prediction process do not sufficiently account for the presence of the aqueous solvent. The solvent is usually represented by a predetermined number of H2O molecules in the bounding box which contains the target chain. The fuzzy oil drop (FOD) model, presented in this paper, follows an alternative approach, with the solvent assuming the form of a continuous external hydrophobic force field, with a Gaussian distribution. The effect of this force field is to guide hydrophobic residues towards the center of the protein body, while promoting exposure of hydrophilic residues on its surface. This work focuses on the following sample proteins: Engrailed homeodomain (RCSB: 1enh), Chicken villin subdomain hp-35, n68h (RCSB: 1yrf), Chicken villin subdomain hp-35, k65(nle), n68h, k70(nle) (RCSB: 2f4k), Thermostable subdomain from chicken villin headpiece (RCSB: 1vii), de novo designed single chain three-helix bundle (a3d) (RCSB: 2a3d), albumin-binding domain (RCSB: 1prb) and lambda repressor-operator complex (RCSB: 1lmb).

Structural analysis of the Aß(15-40) amyloid fibril based on hydrophobicity distribution.

The Aß42 amyloid is the causative factor behind various neurodegenerative processes. It forms elongated fibrils which cause structural devastation in brain tissue. The structure of an amyloid seems to be a contradiction of protein folding principles. Our work focuses on the Aß(15-40) amyloid containing the D23N mutation (also known as the "Iowa mutation"), upon which an in silico experiment is based. Models generated using I-Tasser software as well as the fuzzy oil drop model - regarded as alternatives to the amyloid conformation - are compared in terms of their respective distributions of hydrophobicity (i.e. the existence of a hydrophobic core). In this process, fuzzy oil drop model parameters are applied in assessing the propensity of selected fragments for undergoing amyloid transformation.

Filamentous Aggregates of Tau Proteins Fulfil Standard Amyloid Criteria Provided by the Fuzzy Oil Drop (FOD) Model.

Abnormal filamentous aggregates that are formed by tangled tau protein turn out to be classic amyloid fibrils, meeting all the criteria defined under the fuzzy oil drop model in the context of amyloid characterization. The model recognizes amyloids as linear structures where local hydrophobicity minima and maxima propagate in an alternating manner along the fibril's long axis. This distribution of hydrophobicity differs greatly from the classic monocentric hydrophobic core observed in globular proteins. Rather than becoming a globule, the amyloid instead forms a ribbonlike (or cylindrical) structure.

Towards the design of anti-amyloid short peptide helices.

A set of short peptide sequences susceptible to fibrillar aggregation produces sequneces capable of arresting elongation of amyloid fibrils. The "stop" signals are short helices customized for each individual target. Such a helix should exhibit high amphiphilicity, with differing conditions present on each side (one side should be highly hydrophilic to enable water to interact with the aggregate, while the other side must retain a local distribution of hydrophobicity which matches that of the terminal portion of the fibril). The emergence and elongation of fibrillary forms resulting from linear propagation of local hydrophobicity peaks is shown using the fuzzy oil drop model.

Propagation of Fibrillar Structural Forms in Proteins Stopped by Naturally Occurring Short Polypeptide Chain Fragments.

Amyloids characterized by unbounded growth of fibrillar structures cause many pathological processes. Such unbounded propagation is due to the presence of a propagating hydrophobicity field around the fibril's main axis, preventing its closure (unlike in globular proteins). Interestingly, similar fragments, commonly referred to as solenoids, are present in many naturally occurring proteins, where their propagation is arrested by suitably located "stopper" fragments. In this work, we analyze the distribution of hydrophobicity in solenoids and in their corresponding "stoppers" from the point of view of the fuzzy oil drop model (called FOD in this paper). This model characterizes the unique linear propagation of local hydrophobicity in the solenoid fragment and allows us to pinpoint "stopper" sequences, where local hydrophobicity quite closely resembles conditions encountered in globular proteins. Consequently, such fragments perform their function by mediating entropically advantageous contact with the water environment. We discuss examples of amyloid-like structures in solenoids, with particular attention to "stop" segments present in properly folded proteins found in living organisms.