Secondary structure in polymorphic forms of alpha-synuclein amyloids

Numerous Alpha-synuclein amyloid structures available in PDB enable their comparative analysis. They are all characterized by a flat structure of each individual chain with an extensive network of inter-chain hydrogen bonds. The identification of such amyloid fibril structures requires determining the special conditions imposed on the torsion angles. Such conditions have already been formulated by the Authors resulting in the model of ide-alised amyloid. Here, we investigate the fit of this model in the group of A-Syn amyloid fibrils. We identify and describe the characteristic supersecondary structures in amyloids. Generally, the amyloid transformation is suggested to be the 3D to 2D transformation engaging mostly the loops linking Beta-structural fragments. The loop structure introducing the 3D organisation of Beta-sheet change to flat form (2D) introduces the mutual reorientation of Beta-strands enabling the large-scale H-bonds generation with the water molecules. Based on the model of idealised amyloid we postulate the hypothesis for amyloid fibril formation based on the shaking, an experimental procedure producing the amyloids.


INTRODUCTION
Alpha-synuclein (A-Syn) is a protein whose biological role and structural changes, despite advanced research, remain to be identified (Burré et al., 2018). The mutations in A-Syn are considered as risk conditions (Ottolini et al., 2017;Ghosh et al., 2016). A large spectrum of forms of A-Syn in the context of health and disease is discussed in the paper of Lashuel and others (Lashuel et al., 2013). Post-translational modifications identified in neurodegeneration in patients with Parkinson's disease concern the aggregation propensity of A-Syn (Beyer 2006). A significant contribution of chaperones in the process of A-Syn amyloid transformation is presented in the paper of Burmann (Burmann et al., 2020). There is also a certain degree of synergy between amyloid proteins, including A-Syn and Tau, especially concerning land others iquidliquid phase separation (Guiney et al., 2020;McDowall et al., 2016). Difficulties in designing therapy arise from the complexity of the amyloid formation mechanism (Dehay et al., 2015;Sivanesam & Andersen 2016). Despite the failure to recognize the amyloid transformation mechanism, analyses of the relationship between structure, function and toxicity are being conducted, focusing on the pathogenesis mechanisms of the synucleopathies phenomenon (Villar-Piqué et al., 2016).
The transformation of A-Syn (described as a 'natively unfolded' monomer adopting an α-helical secondary structure only under the condition of binding to the lipid vesicles) into a highly β-structural one in the amyloid fibril is studied using many experimental techniques: ultracentrifugation, in vitro cell-crosslinking and scanning transmission electron microscopy (Bartels et al., 2011). Adopting multivariate conformations identified under diverse conditions, including complexing to nanoscale surfaces, is treated as a potential way of targeting conformational plasticity (D'Onofrio et al., 2020). The A-Syn affinity to the membrane and its ease of interaction is considered a stimulus for therapy design (Bozelli et al., 2020;Kachappilly et al., 2022). The concept of amyloid transformation proposed for prion proteins is also being considered (Lassen et al., 2016. Concentration-dependent transformation conditions allow the identification of the minimum concentration for fibril production (Afitska et al., 2019). This observation is broadened by research on the impact of various types of interphases on the reaction process, including air-water and hydrophobic surface-water interphases (Ben-Amotz 2022; Schutzius et al., 2015;Yi-Jie & Qu 2014;Eremin & Fokin 2021;Kaplaneris et al., 2022;Ishiyama et al., 2022;Gupta et al., 2022;Cannalire et al., 2022). The possibility of transformation into an amyloid form -thus rich in β-structure -as well as proteins with a high helical structure presence, is surprising in the first stages of these studies (Jayawardena et al., 2017). Another example here may be A-Syn, which in 2/3 of the chain length in the WT form exhibits a helical structure, with the remainder being a loose section with random coil configuration (Ulmer et al., 2005;Tuttle et al., 2016). Considering the breadth of literature on experimental studies, the availability of an increasing number of amyloid structures in PDB resources makes it possible to analyse the structure of amyloid fibrils as such (Goedert et al., 2017;Eisen-Vol. 70, No 2/2023435-445 berg & Sawaya 2017Wei et al., 2020). Experiments conducted in different environmental conditions indicate a significant impact on the environment, and therefore, the impact of external conditions (Ben-Amotz 2022; Sawaya et al., 2021;Serpell 2000;Rana et al., 2008;Lambley et al., 2020).
Identification of amyloid fibril structures as consisting of chains with a planar structure requires determining the conditions that should be met in order to obtain just such a form, additionally involving almost all C=O and H-N peptide bond groups in the construction of the hydrogen bond network. Such conditions have already been defined when constructing a model of idealised amyloid (Roterman et al., 2022). The main aim of the present study is to investigate the fulfillment of the conditions formulated for idealised amyloid in the group of A-Syn amyloid fibrils. We also describe the specificity of the secondary structures in amyloids. The primary specificity of amyloid structures is the flatness (2D) of single-chain structures comprising fibrils with a high presence of the β-structure. The combination of several flat-structured chains in the β-form in a parallel chain arrangement makes it possible to engage almost all -N-H and -C=O groups of peptide bonds in the construction of inter-chain hydrogen bonds. Thus, it is possible that each amino acid -in the i-th chain -through these groups participates in the construction of two hydrogen bonds (-N-H atoms engaged in the i-1-th chain and -C=O group with i+1-th chain). It significantly stabilizes the fibril structure, which is observed in experiments. Such engagement is exhibited by amino acids in the helix structure -with H-bonds oriented parallelly and β-structure -with H-bonds oriented antiparallelly. However, globular proteins contain chain fragments in which H-bonds are not generated -like loops and twists in hair-pin structures (Roterman et al., 2022;Roterman 1995). Additionally, the helical form introduces the structure's spatiality (3D), while in theory, the β-structure has the potential of unlimited propagation in a linear form and, in the case of β-sheet, in 2D form as a flat plate. The β-sheet structure -can be generated by different sections of the same chain with different amino acid sequences, while in the amyloid, the β-sheet is formed as a result of a combination of sections of the same sequence originating from many adjacent chains in parallel mutual orientations.
It is shown in this paper that the flat structure of the chain can be reached for β-structural fragments representing special Phi, Psi angles with bends introduced by single R-or L-helical conformation. The identification of such an organisation in polymorphic forms of A-Syn is shown (Peng et al., 2018). Two reference non-amyloid proteins: all-helical and all-β-structural -are also present in the analysis to verify the proposed model.
The role of α-sheet in amyloid construction in comparison to Beta-structural-based amyloids is also discussed (Prosswimmer & Daggett, 2022;Lee et al., 2002;Trexler & Rhoades E. 2012;Balupuri et al. 2019, Milner-White et al., 2006, Hayward & Milner-White 2021. The hypothesis of the external force field of 2D form influence of flat structure construction is presented with the main role of loops linking Beta-structural fragments of chains in native structures of proteins is presented.
Moreover, based on the already proposed model of idealised amyloid we postulate the hypothesis for amyloid fibril formation.

Data
The structures of A-Syn amyloids (as available in PDB July 2022) analysed in this paper are provided in Table 1.
Two reference proteins are present in the analysis: the GPCR bovine rhodopsin (PDB ID 1GZM - (Li et al., 2004)) mainly Alpha Up-down bundle (CATH classification 1.20.1070.10) representing the structure dominated  (Tuttle et al., 2016) by helical form and a FeS cluster assembly protein SufD from Escherichia coli (PDB ID 2ZU0 (Wada et al., 2009)) The structure of the second protein is characterised by the presence of large domain build by almost all Betastructural form.

The torsion angle-based relationship characterizing the amyloid structures
The specificity of amyloid structures is, apart from a high presence of β structure, the flatness of every single chain, a component of the fibrils. The planarity of a polypeptide chain (i.e. its linear propagation) is provided by the rectilinear β-segments (satisfying the Psi=-Phi condition) together with single R-α or L-α helical conformations which introduce a local bend while maintaining a flat structure. Those single helical conformations are characterized by the Psi≈Phi condition. This is why those two characteristics can be put in the common expression |Psi|=|Phi| constituting the basis of the model of idealised amyloid (described in detail in (Roterman et al., 2022).

Quantification of the fit of the idealised amyloid model
The |Psi|=|Phi| relationship of idealised amyloid is assumed here as the hypothetical criterion for amyloid structure identification. The closer to this relationship is the distribution of (Phi, Psi) angles, the more probable the construction of a flat structure.
To enable the assessment of the degree of fit of the idealised amyloid model, the operation of replacing the Phi and Psi values with |Phi| and |Psi| respectively is performed. This leads to a situation where all three lines are in a common position (see the illustration in Figure  3 based on the amyloidogenic fragment (30-100) of the 2N0A).
The degree of fit of chain's (Phi, Psi) angles to the model of idealised amyloid is measured by a Pearson correlation coefficient: where x i stands for |Phi i | and y i stands for |Psi i |.
The correlation coefficient CC for the (|Phi i |,|Psi i |) angles of residues in an analysed chain was calculated in the following cases: For all residues in an analysed chain (coefficient marked as CC-0).
For the residues after elimination of outstanding points: i.e. by eliminating them until a value of 0.8 is reached (coefficient marked as CC-F).
For the residues participating in β-strands and satisfying the condition |Psi|=|Phi| with a tolerance of ± 10 deg distance from |Psi|=|Phi| line (coefficient marked as CC-β). Residues with a high CC-F coefficient value are identified as responsible for the stabilization based on a hydrogen bond network.
The described type of analysis will be used to identify the amyloid form providing the maximum proportion of residues with a conformation satisfying the condition |Psi|=|Phi|.

The polypeptide's geometry-based model
This model has already been described on many occasions (see for example Roterman 1995). Here, only the basic assumptions facilitating an interpretation of the discussed results will be recalled. The main idea is to rep-resent the structure of the polypeptide chain using two geometrical parameters. One of them is the curvature radius R which is low and well-defined for the helical structures and very large (theoretically infinitely large) for the β-structures (straight line for certain Phi, Psi angles). The value of the curvature radius is the result of the mutual orientation of two neighbour peptide bond planes expressed by the angle between two peptide bond planes versus the axis defined by virtual Cα-Cα bonds. This angle -called the V-angle -is close to zero (parallel mutual orientation) for helical forms and close to 180 deg for the β-structure (anti-parallel mutual orientation) (Fig. 1). There is one special β-structure with a V-angle=180 deg which produces an ideal straight line (Fig. 1B).

General characteristics of secondary structures in amyloids
In this section, the general characterization of amyloid secondary structures based on the proposed earlier model of idealised amyloid will be presented, The basic feature of amyloids is the flatness of the structure of single chains that make up the fibril. The planar amyloid structure of each polypeptide chain in a fibril is stabilized by a specific system of hydrogen bonds. In the β-structure with the Psi=-Phi conformation (corresponding to the idealised amyloid), each amino acid participates in the construction of two hydrogen bonds with the adjacent chains. The arrangement of the chains is parallel, which causes residues from corresponding positions from adjacent chains to interact with each other. Such a structure ensures the maximum use of hydrogen bonds, thus providing much stability. Rectilinear chain segments are terminated by single residues with R-or L-helical conformation. A single residue with such a helical conformation does not disturb the planar structure, changing only the turn by introducing a bend (Fig. 2B). The presence of two residues with such a conformation introduces a three-dimensional arrangement disturbing the planarity (Roterman et al., 2022).
Introducing such a single residue with helical conformation results in a change of hydrogen bond ordering. In contrast to a helical structure (which represents a parallel hydrogen bond system -see Fig. 1A), in the ideal β-structure, the arrangement of hydrogen bonds is antiparallel (Fig. 1B). A single residue with the R-or Lhelical conformation introduces a local perturbation to this arrangement changing it to a parallel one. The planarity of a polypeptide chain (i.e. its linear propagation) is provided by the rectilinear β-segments (satisfying approximately the Psi=-Phi condition) together with single helical conformations meeting the Psi≈Phi condition (Fig. 4).
In contrast to the above-described structure present in amyloids, changing the turn by means of hair-pins ( Fig. 2A) engages more than one residue (most frequently 4 residues) and introduces spatiality. Moreover, the mutual orientation of residues in the hair-pins does not allow the formation of a hydrogen bond system in such bends.
It must be emphasized that the beta-sheet supersecondary structure in amyloids is different from the known, classical structure in other proteins. The classical beta-sheet structures are characterized by a system of hydrogen bonds binding fragments of the same chain. This allows the presence of loops, including the betaturn (hair-pin) type, which just introduce spatiality. Betastrand fragments in amyloids remain in one plane due to  (Tuttle et al., 2016). Grey lines -mark the β conformation fulfilling the Psi=-Phi relation. Vertical lines denote positions of amino acids: red -with L-α conformation, cyan -with R-α conformation. The purpose of this picture is to visualise the differences between 30-100 aa -amyloid fragment to N-and C-terminal fragments, where the Phi, Psi angles relation is significantly different with respect to 30-100 aa fragment.

Figure 6. A-Syn structure in the amyloid form (2N0A):
A -the structure of a complete fibril (chain A marked in red); Btwo chains with distinguished residues representing the R-α (cyan) and L-α (red) conformation according to the identification provided in Fig. 5.

Figure 2. The β-structure propagation:
A -a hair-pin system, B -amyloid. The bend in the form of a hairpin engages four residues with N-H and O=C groups -H-bonds system locally absent and introduces spaciousness. The one helical residue in the β-structural chain introduces a bend maintaining the flat structure of the chain and additionally continuing the system of H-bonds. The fragment presented in B is the fragment 36-53 of the A-Syn chain in amyloid form (PDB ID 2N0A)   The eliminated residues (by CC-F calculations) are marked in red on both the plots and 3D presentation. The points marked in blue -are the positions for residues satisfying the condition |Psi|=|Phi| for β-structural forms. For 6UFR the residues representing the highly ordered β-structural fragments are in space-filling representation (blue). a turn introduced by a single R-or L-helical conformations as shown in Fig. 4. Such a radical turn allows the chain to propagate in one plane. Figure 4 visualises the real conformation present in 36-53 fragment of A-Syn which appears to be almost ideally accordant with the proposed model of idealised amyloid. Linear propagation of β-structural fragments with single residues representing helical conformation results in a flat form with well-defined bends. The orientation of hydrogen bonds in β-fragments is antiparallel with local, parallel orientation in single helical residues.

Analysis of the fit of the idealised amyloid model 2N0A protein
In this section, the procedure of fitting the idealised amyloid model will be presented based on the reference A-Syn amyloid structure (2N0A -section 36-53). In Fig. 4, the fragments of the chain (marked as navy blue) show a rectilinear system obtained using the Phi and Psi torsion angles satisfying the relation Psi=-Phi. The residues marked in red have an α-helix conformation resulting in a turn of the chain while maintaining its flatness. Table 2. Summary of CC-0 and CC-F values for the set of A-Syn amyloid forms analysed in this paper. Residue d numbers eliminated in CC-F calculation are given. The CC-β values are given to express the high fulfillment of |Phi| = |Psi| relation. The % of residues in β-structural form for which the CC-β was calculated is given. The torsion angle profiles of the mentioned fragment are shown in Fig. 5. The fulfillment of relations Psi=-Phi is visible for the amyloid part (positions 30-100 aa), which is marked in gray. Within the clearly ordered and constant values of the Phi and Psi angles for the β form, single positions with the Phi and Psi angles corresponding to the R (cyan) and L (red) helical conformations are visible. The visualization of structures In this analysis is shown in Fig. 6. Table 2 provides the CC-0 and CC-F values of the chain as described in the Materials and Methods section. The values of CC-β express the status of residues satisfying the condition |Psi|=|Phi| in relation solely to the β-structural form. The % of residues satisfying the condition for CC-β is also given to characterise the degree of the presence of such restrictive structure in a particular protein. Figure 9 shows examples with high values of CC-0 and CC-F arising from the location of the Phi and Psi angles in accordance with the Psi=-Phi relationship. In a 3D visualization, these sections are represented by a rectilinear arrangement. The highlighted residues -red points -on the plots of |Phi| and |Psi| angles are residues treated as outstanding, the elimination of which results in a CC-F value > 0.8 (Table 2).

Analysis of the structure of A-Syn amyloid forms
In the amyloid with PDB code 7NCG, the rectilinear β-structure sections are visible (Fig. 7B) and the corresponding locations of the |Phi|versus|Psi| plot (Fig. 7A). Items that do not satisfy the |Psi|=|Phi| condition specified by the model are marked in red. The number of residues that do not satisfy the model is low and they do not disturb the overall form of this amyloid leaving it close to the principles of the proposed model.
A similar arrangement is identified in the case of 6URF (Fig. 7). It should be noted that this structure is characterised only by a few residues with conformations not satisfying model conditions.
The significant portion of a chain in 6PEO amyloid meets the |Psi|=|Phi| condition, but a fairly significant scattering of several residues deviating from the model can also be observed.
The highest CC-0 value and the low number of residues deviating from the model are characteristic for amyloid 7LC9-F (Fig. 8). Eliminating only two residues results in a very high value of CC-F=0. 851.
Further examples are the structures present in 6CU8 and 7C1D exhibiting the lowest CC-0 values (Fig. 9). In these two examples, no rectilinear propagation, but rather an "undulating" form is observed. This means, that the chain propagation is composed of sections with a smaller radius R. This reduction of radius applies to short sections, which, with an alternating arrangement similar to the R-and L-helical conformations, does not result in a spatial form. In this case, the hydrogen bonding arrangement is not parallel, although the deflection from the perpendicularity to the chain's plane is compensated, resulting in maintaining the structure's flatness.

Analysis of the structure of reference examples
Two reference proteins (Table 3) represent mainly helical and β-structural proteins. As shown in Fig. 10 their structure is highly ordered. The FeS cluster assembly of SufD from Escherichia coli is a protein which for the most part shows the β-structure (Banach et al., 2020). The analysed fragment of this protein is limited to the betasheets (Fig. 10B). The similarity with respect to amyloid is The parallel orientation in the β-sheet (Fig. 10B) indicates a structural similarity to amyloids. However, the sequential β-strands represent a different sequence, while in amyloids the sequences in all neighbouring chains are identical.
The status expressed by CC-β of sequential β-strands and β-sheets reveals differences with respect to the amyloid forms discussed (Table 4). The fragments showing high accordance with the |Psi|=|Phi| criterion (having high CC values) are accompanied by the fragments Figure 8. Presentation of the 7LC9-F amyloid with the highest CC-0 value (and the lowest number of outstanding residues). A -distribution of points representing |Phi| and |Psi|. Only two residues are considered outstanding (red points). Their location in the chain is highlighted in red (space-filling) on the 3D representation (B).

Figure 9. Examples with low CC values:
A -CC-0 -all points; CC-F blue points; The circled dot is the only point satisfying the |Psi|=|Phi|. C -red points eliminated for CC-F calculation, gray points -CC-F calculation, blue points -CC-β calculation. B, D -3D presentation of 6CU8 and 7C1D respectively with red residues -residues eliminated for CC-F calculation with completely opposite characteristics (low CC values). The non-β-structural conformations are present in the selected fragments (values given in parentheses in Table 4). Visual analysis of two β-sheets suggests the perfect β-structural organisation (Fig. 11). However, the analysis of Phi and Psi angles reveals some residues in the lower left quarter of the Ramachandran map. Some β-strands having torsion angles typical for β-structure appear to be not organised as expected in amyloids. The differentiation of β-strands can be observed. The status of sequential β-strands in the amyloid is identical for all fragments belonging to a common β-sheet. The distribution of |Psi|, |Phi| angles (Fig. 12A) identifies the large number of points satisfying the |Psi|=|Phi| relation, however, it also shows their dispersion over the chain. The highest similarity is observed in the 181-192 fragment (marked in Fig. 12B). The mainly-helical protein is difficult to evaluate with the |Phi|=|Psi| criterion due to the very low presence of β-structural conformations. The distribution of Phi, Psi angles along the chain (Fig. 13) shows high accordance with the helical forms (dark lines on Fig. 13). The CC-0 for this protein is equal to 0.347, the CC-F=0.807 (the eliminated residues marked as blue dots - Fig. 13.). The number of eliminated residues constitutes 19.2% of the whole length of the chain. The linear relation between torsion angles is mostly due to the Psi≈Phi relation present in the helical conformation.
The eliminated amino acids (in the CC-F calculations) represent the unstructured fragments like loops particularly those external with respect to the surrounding membrane. The space filling presentation ( Fig. 12C  and 12D) distinguishes the segments satisfying to a large extent the |Psi|=|Phi| relation, however, they belong mostly to the helical region.

The proposed hypothetical mechanism of amyloid fibril formation
The proposed hypothesis of amyloid fibril formation presented here concerns the method and the reasons of the transformation of a 3D structure into a planar form characteristic to the amyloid structures.
We also want to refer here to the existing, different hypothesis ( (Prosswimmer & Daggett, 2022;Lee et al., 2002;Trexler & Rhoades, 2012;Balupuri et al., 2019) of the formation of the amyloid based on the α-sheet structure as an intermediate in the aggregation process.
Shaking an aqueous solution of proteins as a laboratory technique for obtaining amyloid forms of proteins consists in increasing the presence of air-water inter-phase in the solution. Such a shaking process can simulate the conditions in a cell of a living organism. The airwater interphase has a surface structure, so it introduces an external force field with 2D characteristics.
Occasionally, a protein undergoes some conformational changes resulting in aggregation competent polypeptide chain. Such a chain, avoiding contact with air strives for maximum contact with water, which is available Figure 10. 3D structure 2ZU0 A -chain A -with two β-sheets marked in red. B -two β-sheets marked in red and blue -N-terminal residue -marked with space filling. Table 4. Status (CC-β) of β-strands in β-sheets (as distinguished in Fig. 10B). The values in parentheses -denote the β-strand status for the residues representing non-β-structural Phi, Psi angles. The values given for β-sheets are calculated only for the residues belonging to the β-structural area on the Ramachandran map.

Figure 12. Characteristics of all-β and all-helical proteins.
A -|Phi|, |Psi| angles as they appear in the β-structural part of the protein -chain fragment selected as an aim-oriented example for the analysis of the beta-structural form of the chain. B -3D presentation with conformations satisfying the relation |Psi|=|Phi| -red. The space-filling -a fragment of the highest satisfaction of |Psi|=|Psi| condition. C -|Phi|, |Psi| angles as they appear in allhelical protein. D -3D structure with amino acids satisfying the |Psi|=|Phi| conditions distinguished by a space-filling presentation.
The red dots -all |Phi|, |Psi| angles to calculate CC-0, grey dots -CC-F and red -CC-β. The same colours are used in 3D presentation (red and blue).
only on one side. The most thermodynamically favorable situation is the formation of a system of hydrogen bonds with a planar chain where the polypeptide chain "lies" on the water surface. In this way, the backbone chain engages as many C=O and N-H groups of peptide bonds as possible in such an interaction. This explains the reason for the almost planar form of the chain in amyloid fibril. A planar polypeptide chain formed due to the shaking finds a thermodynamically favorable opportunity to interact with another planar chain and the process of attaching the next flat chains continues until the aggregation nucleus is formed. The next phase starting from the nucleus is characterized by a rapid formation of fibrils composed of beta-sheet structures.
Amyloid transformation mainly requires conformational changes within the loops that change from 3D orientation to 2D one.
The influence of the environment on amyloid transformation with an unchanged sequence is widely recognized (Serpell, 2000). The particular form of the environment in the form of mentioned 2D interphase seems to favour the formation of the flat structure of the chains -components of amyloid fibrils.
The proposed hypothesis for amyloid fibril formation is illustrated in Fig. 14. It refers to the transthyretin (Ueda, 2022) which undergoes amyloid transformation relatively easily. A special object is a fragment of a beta-sheet identified as A (11-19), G(104-112) and H (114-124). Both the WT form (PDB ID 1DVQ (Klabunde et al., 2000) and the amyloid form (PDB ID 6SDZ (Schmidt et al., 2019)) are available in PDB. Another reason is the possibility to compare with the hypothesis proposed in the works (Prosswimmer & Daggett, 2022, Armen et al., 2004 and based on α-sheet structure as intermediate. Fragment beta-strand A(11-19) remains in this form in range (13-15), fragment G (104-112) changes its form to a disordered linearly propagating segment, while fragment H (114-124) remains in beta form on (118-124) fragment. The shortening of beta-stand segments can be observed.
The water-directed beta-structural fragment 104-111 can form hydrogen bonds with water molecules. The shape of the loop directs the second part of the betastrand above the water surface, i.e. to the air zone. This is due to the orientation imposed by the loop (113-115) present in the native form of the transthyretin (green fragment in Fig. 14A and C). The same fragment after changing the conformation of the loop results in a planar structure of the entire fragment 104-124 (green fragment in Fig. 14B and D).
It should be noted that the native Beta-structure segment (104-112) changes its form to a disordered pattern, while the native-form beta-structure segment (114-124) is limited to the amyloid segment (118)(119)(120)(121)(122)(123)(124). This means that the conformational change within the loop (green fragment in Fig. 14A-D) also applies to a short fragment of the Beta-ordered segment. As a result of these changes, the structure of the entire fragment 104-124 becomes planar, allowing the generation of hydrogen bonds with water molecules along its entire length ( Fig. 14E     and H), where a change in conformation within the loop (red fragment Fig. 14G and H) allows backbone contact (groups -C=O and -N-H of peptide bonds) with water molecules leading to the formation of hydrogen bonds with water molecules. The exact conformation changes within the discussed loops are shown in Fig. 15.
The proposed amyloid formation hypothesis relies on the 3D to 2D transformation engaging mostly the loops linking beta-structural fragments. The loops structure introducing the 3D organisation of beta-sheet change to a flat form introducing the mutual reorientation of betastrands enabling the generation of large-scale hydrogen bonds initially with water molecules to change into an inter-chain H-bonds system in the next step.
The α-sheet structure, widely discussed in the context of amyloids (Prosswimmer & Daggett, 2022;Lee et al., 2002;Trexler & Rhoades, 2012;Balupuri et al., 2019) is the structure formed by amino acids with the R-and L-helical conformations alternating in the polypeptide chain. Similarly to the Beta-sheet structure, the α-sheet structure is characterized by the rectilinear form ( Fig. 2 in Prosswimmer & Daggett, 2022). The distribution of Phi and Psi angles in α-sheet structures discussed in Prosswimmer & Daggett (2022) is consistent with the model assumed in this work. The difference is only quantitative. In the case of the α-sheet, the concentration of points on the Ramachandran map is significantly greater in the positions corresponding to the R-and Lhelix with the presence of the Phi and Psi angles representing the β-structure in a linear relationship. In the current work, for the β forms, a much greater concentration of points on the Ramachandran map is observed. The main point is concentrated on the 3D to 2D transition. The structure discussed in (Prosswimmer & Daggett, 2022) introduces the 3D structuralisation which is not present in amyloid formation. The linear propagation of the chain in the form of Beta is the most frequently observed secondary form. The linear propagation of the chain is also possible in the form of R-followed by Lhelix by turns. Such a system allows the generation of an H-bonds network. The mutual orientation of chains shall be parallel to allow propagation in fibril formation. However, the flat construction for individual chain is the most important one.

DISCUSSION
The amyloid structure of the flat form of each polypeptide chain is stabilized by a specific hydrogen bond arrangement. Each amino acid in an idealized form contributes to the construction of two hydrogen bonds with adjacent chains. The chain arrangement is parallel, causing identical residues to interact with each other in adjacent chains. The maximum use of C=O and H-N groups for the construction of hydrogen bonds is ensured by the β-structure with the Psi=-Phi conformation. A rectilinear form with C=O groups facing in the opposite direction to the H-N group of the same amino acid is an arrangement that makes maximum use of the possibilities of the hydrogen bonding structure. The rectilinear sections are terminated by single residues with R-α or L-α conformations. A single residue with such conformation does not disturb the flat structure, only changing the direction (introduction of a turn/bend). In this case, the form of arranging hydrogen bonds is changed. In a structure with the Psi=-Phi conformation, the hydrogen bond arrangement perpendicular to the plane of the chain represents an anti-parallel arrangement (if a turn is applied to the hydrogen bond).
Numerous amyloid forms of A-Syn allow tracking the structural variation of these amyloids. Among the structures satisfying the |Psi|=|Phi| condition, there are structures showing some deviations from this condition. The deviation also provides fibrillar structures.
On the Ramachandran map presenting the distribution of radius of curvature R and V-angle (Roterman et al., 2022;Roterman, 1995) the area fulfilling the Psi=-Phi condition represents the one with the highest value of R (i.e. a straight chain section). A reduction of the curvature radius R is observed moving up and down versus this line gradually introducing the arc form. If this procedure is performed stepwise (i.e. one above the Psi=-Phi line followed by one representing Phi, Psi angles below that line), this introduces the structural form as shown in Fig. 9 for 7C1D (symmetrically located |Psi|, |Phi| angles). The result of such Phi, Psi angle combination is the sinusoidal arrangement of the chain as can be seen in 7C1D (Fig. 9) and others. The local introduction of an arc-like form in one direction is compensated by an ark form oriented in the opposite direction. In consequence, the linear propagation of the flat form can be continued. Nevertheless, this symmetrical (with respect to |Phi|=|Psi| line) set of altered Phi and Psi angles generates a flat arrangement, but probably with reduced stabilization derived from the hydrogen bonds (change of angle between the C=O and H-N groups in the hydrogen bonding arrangement of peptide bonds derived from adjacent polypeptides).
The condition of the presence of an R-or L-helical conformation at the turn of the chain can be satisfied in a less perfect form by a conformation with a low curvature radius R. Cyan and red lines in Fig. 5 and Fig. 6 illustrate the area with low values of V-angle and the R-radius of curvature. The presence of Phi and Psi angles from this area satisfies the condition for obtaining a turn, although the hydrogen bond arrangement is not as perfectly ordered as in the case of Phi and Psi angles accurately representing the R-α-or L-α-helical conformation.
This analysis complements the detailed structural analysis of A-Syn fibrils (Roterman et al., 2023) as well as the transthyretin (Roterman et al., 2022), where the characteristics of fibrils were considered from the perspective of the tertiary structure and characteristics of those fibrils. Microenvironmental conditions were also indicated as the cause of structural changes in A-Syn (Candelise et al., 2020;Stephens et al., 2019).
The problem of describing the field in the form of 3D analyzed on a large-scale provides many forms of mathematical notation to express its presence (Cheng et al., 2023).

CONCLUSIONS
The flat structure of single chains -components of amyloid fibril -is a characteristic of amyloid forms. This flatness is achieved through a β-type structure satisfying the Psi=-Phi condition resulting in a rectilinear chain propagation. Turns in chain arrangement are obtained by single residues with R-α or L-α helical conformation (satisfying Psi≈Phi condition). The combination of such β-structures and single conformations allows the introduction of the general model expressed by |Psi|=|Phi|. The hypothesis introduced in this paper is to treat the presence of conformations satisfying this relation as a criterion for amyloid identification and description. The stabilization of such a form is achieved through the special hydrogen bond arrangement. In β-structured sections, the hydrogen bond arrangement is antiparallel. Each individual residue of the L-α or R-α conformation introduces a local arrangement of these bonds as parallel. Access to multiple structural forms of A-Syn amyloids reveals variation in the form of this ordering. This variation is likely a result of different environmental conditions (Roterman et al., 2023). Particularly, the impact of the presence of air-water interphase (increasing this presence during shaking) is most likely indicated as a factor affecting the structuring, including the form of arranging the hydrogen bond network in amyloids.
External conditions in the form of air-water interphase also direct protein structuring to the planar form, which can be stabilized by a significantly high number of hydrogen bonds of -N-H and -C=O groups of peptide bonds. The accessibility of these groups for hydrogen bonds also includes segments that in their native form represent loops introducing 3D structuring (e.g. two parallel β-sheets) resulting in a flat structure for the entire polypeptide chain (Fig. 14G and H).

Acknowledgements. Many thanks to Anna
Śmietańska and Zdzisław Wiśniowski for technical support.
Conflicts of Interest. The authors declare no conflict of interest