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Approach for comparing protein structures and origami models
Journal Pre-proof Approach for comparing protein structures and origami models
Hay Azulay, Aviv Lutaty, Nir Qvit PII:
S0005-2736(19)30280-9
DOI:
https://doi.org/10.1016/j.bbamem.2019.183132
Reference:
BBAMEM 183132
To appear in:
BBA - Biomembranes
Received date:
21 May 2019
Revised date:
5 October 2019
Accepted date:
28 October 2019
Please cite this article as: H. Azulay, A. Lutaty and N. Qvit, Approach for comparing protein structures and origami models, BBA - Biomembranes(2019), https://doi.org/ 10.1016/j.bbamem.2019.183132
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© 2019 Published by Elsevier.
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Approach for Comparing Protein Structures and Origami Models 1,2,*
Hay Azulay
, Aviv Lutaty
1,3
, Nir
1
Independent researcher
2
Koranit, Israel, 2018100
3
Kiryat Motzkin, Israel, 2641312
Qvit
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4
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The Azrieli Faculty of Medicine in the Galilee, Bar-Ilan University, Henrietta Szold St. 8, POB 1589, Safed, Israel.
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*Correspondence: [email protected], [email protected], [email protected]
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ABSTRACT
The research fields of proteins and origami have intersected in the study of folding and de-novo
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design of proteins. However, there is limited knowledge on the analogy between protein
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structures and origami models. We propose a general approach for comparing protein
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structures with origami models, and present a test case, comparing transmembrane β-barrel and α-helical barrel with the Yoshimura and Kresling origami models. While both shapes and
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structures may look similar, we demonstrated that the β-barrel and the α-helical barrel are in
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agreement only with the shape and structural characteristics of the Kresling model. Through the analogy it is explained how the structural characteristic can help the β-barrel and α-helical barrel to adjust length and diameter in response to changes in the membrane structure. However, such conformations only apply to the α-helical barrel, and the β-barrel, in spite of resembles to the Kresling model, remains stiff due to hydrogen bonds between the β-strands. Thus, our analysis suggests that there are similar patterns between protein structures and origami models and that this approach may provide an important insight on the role that the structure of a protein fulfils, and on preferred structural design of novel proteins with unique characteristics.
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Journal Pre-proof Keywords Protein, Origami, Structure, β-barrel, α-helical barrel, Kresling model, Yoshimura model.
INTRODUCTION Proteins and origami are generally studied by independent research fields. The functionality of a protein is determined by various factors including its structure, activity, and subcellular localization to name a few. Therefore many studies look into the anatomy, taxonomy [1], and mechanism [2] of proteins. Furthermore, studies of protein folding attempt to
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predict how a specific sequence of amino acids would fold, using data from recording of folding stages, force
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spectroscopy experiments [3], X-ray crystallography [4], Nuclear Magnetic Resonance (NMR) [5], and simulations [6].
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This research is essential for different applications, for example, it can help in developing a pharmacological intervention
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to fight viruses [7] and other diseases, as well as in studying biological structures, such as spider silk, that can be applied to design improved materials for engineering applications [8].
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In parallel to protein research the study of mathematical principles behind origami folding led to a dizzying rate of new
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structures that are applied in different areas, such as architecture, robotics [9], and aeronautics [10], which demonstrates how art can contribute to science [11]. Among other things, it has been proven that by applying specific
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mathematical conditions it is possible to design a set of creases that will result in flat folding of a three-dimensional (3D)
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model [12].
Origami is also interwoven into biological research: (i) Classic origami is the basis for the known folding art in which the model is folded from a single sheet of paper without cutting [13]. Folding principles of classic origami were applied to study the deployment mechanism of the wings of ladybird Beetle [14]. Protein folding process was compared with classic origami folding as in both cases the structure is created through a sequential folding into a 3D compact structure [15-16]. Moreover, Protein Data Bank (PDB) developed the PDB-101 website (https://pdb101.rcsb.org/), which provides resources, and teaching aids for teachers, such as folded 3D origami models to help beginners learn about proteins [17].
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Journal Pre-proof (ii) Modular origami is an origami branch in which the model is assembled from separate folded sheets of paper [18]. This principle is implemented in "Protein Origami", which is an approach for the de-novo design of proteins with new folds unknown in nature. To construct a new protein, folded DNA segments are assembled, in a process that is similar to the assembly of folded paper pieces in modular origami [19]. The approach can be also applied for the design of nanostructures for other applications aside from biology [20]. Modular origami models have been also folded for educational purposes in chemistry and biology, for example chemical molecules that can be assembled into chains were
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modeled by origami [21].
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Following the above, it seems that proteins and origami have a lot in common. However, in spite of the similarity, as far
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as we know, no significant attempt has been made to compare the shape and structural characteristics of proteins and origami models. In this paper, a general approach for comparing protein structures and origami models with similar
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shape is presented, and their characteristics are analyzed from a mechanical point of view, in an attempt to get insight
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on the role of the protein structure with respect to the chemical/physical properties of the protein and the function it fulfills. The approach is demonstrated in the comparison of the outer membrane protein A (OmpA) of Escherichia coli
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that has a β-barrel structure, and the transmembrane (TM) domain of the Large Conductance Mechanosensitive Ion
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Channel (MscL) that has a α-helical structure, with the Kresling and Yoshimura origami models.
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METHOD Analysis approach
Proteins are made up from a primary sequence of amino acids that is folded into secondary and tertiary unique structures, and it is possible to analyze their molecular structure, interactions, and mechanical properties [22]. However, generally, it is easier to understand why a specific structure is selected for engineering application than to understand why a protein structure evolved to its current structure. Since the mechanical and physical design and the folding principles of origami models have been applied to study biological systems, including Coleopteron wing folding, and growth pattern of pine cone [23], and since in some cases there is analogy in the folding processes of proteins and Azulay
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Journal Pre-proof origami, proteins are analyzed here through a four-step approach that combines knowledge from origami models into protein research. The four steps of the approach are (Figure 1): (1) Screening – Choose a protein and use data from the PDB, which was derived, for example from X-ray spectrometry and/or NMR, and examine the protein structure. The PDB data of the protein is presented using software, such as Jmol [24], that can help identify secondary and tertiary structures. (2) Identifying – Identify origami models with shape/conformation that are similar to the selected protein structure. The
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analog origami models can be based on known folding patterns or new designs. Known origami models can be found in
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the literature [25], and new models can be obtained using dedicated design tools, that create computerized foldable
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origami models [26].
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(3) Comparing – Compare the similarity of the chosen protein structure and the identified origami models based on: (i) The number of domains, (ii) the number of chains in each domain, and (iii) the geometrical classification of the structure
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(for example, a type of polygon, cylinder, sphere, or a freeform shape). To increase the level of confidence in the
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equivalence, several views of the protein from different sides should be compared. For training purpose, it may be beneficial to "disconnect" chains and draw the spread structure arrangement on a two-dimensional (2D) plane to
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compare it with the origami "crease pattern", which is a 2D unfolded representation of the 3D origami model. The 2D
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model of the protein topology can be obtained from software application such as Pro-origami [27]. However, since in many cases, the orientations of the secondary structures are not detected, another sub-step is required to correct the diagram into drawing that incorporated this information. (4) Analyzing – Analyze the mechanical characteristics of the structure of the origami model and the protein structure. The analyses include mathematical tools, origami design principles, characteristic of known origami models, the folding/creation sequence of the origami model and Finite Element Analysis (FEA), which is an analysis that is often applied in mechanical studies to simulate the response of a structure to forces that are applied at specified locations. The analysis is performed while considering the functionality of the protein and the forces that it is required to
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Journal Pre-proof withstand. For proteins that are known to dynamically change conformation to achieve desired functionality, it is
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desired that different conformations of the protein and the folded/unfolded analog origami models are compared.
Figure 1. A workflow of the comparison approach of protein structure and origami models. EXAMPLES Example of comparison between protein structure and analogue origami models In the following example, the proteins of interest that we compare with an origami model are: (1) the outer membrane protein A (OmpA) of Escherichia coli, and (2) the TM domain of MscL.
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Journal Pre-proof The OmpA is a one-domain protein has a relatively simple β-barrel structure (Figure 2A). OmpA, like many other membrane proteins that functions as the gateway to the cell, has both structural and ion-permeable roles, which has several functions, such as, allowing bacterial survival during osmotic stress [28]. β-barrels are twisted β-sheets that are arranged to form a closed structure in which the first β-strand is bonded to the last β-strand. The structure is a hyperboloid of revolution, with minimal surface, and it is arranged with positively charged residues on one side of the pore and negative on the other (Figure 2A). The β-barrel proteins serve essential
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functions such as, cargo transport and signaling, and they create a transmembrane passage/valve, which can be found in
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chymotrypsin enzyme, TIM-barrels, streptavidin, and lipocalins [29]. Some of the residues of the β-strands are
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connected through hydrogen bonds, which are perpendicular and planar between adjacent antiparallel β-strands and distorted between adjacent parallel β-strands. As a result, the hydrogen bonds between antiparallel β-strands are
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stronger than hydrogen bonds between parallel β-strands. In addition, compared to β-sheets, β-barrels are constructed
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with more hydrogen bonds between the β-strands than required for the twist and the closer, which stiffens the
β-strand
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Hydrogen bonds
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structure. Therefore, β-barrels with antiparallel β-strands, such as the OmpA are stiff structures [30].
α
D Mountain fold
α TM1 TM2
Folds created between creases Valley fold
Figure 2. The protein structures and compared origami models: (A) The OmpA β-barrel protein structure (PDB ID: 1BXW). The thick lines indicate β-strands, α is the tilt angle of the β-strands with respect to the barrel axis and the dotted lines are the hydrogen bonds. (B) TM α-helical barrel domain of the MscL (PDB ID: 2OAR). The curly cylinders represent α-helices, where each color indicates a chain. (C) Yoshimura origami model, the dotted line indicates a crease that is tilted, similar to β-strands/α-helices. (D) Kresling origami model, the marked lines represent mountain creases that are tilted, similar to β-strands/α-helices. In addition, undesired folds that are created when folding the Kresling model are also indicated by arrows. Azulay
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The second protein that is presented is the TM α-helical barrel domain of the MscL. The MscL is a membrane valve that is designed to relief osmotic imbalance, allowing a quick and transient increase in compensatory solute flux out of the cell (Figure 2B). It is constructed from a TM domain that is combined from two circles of axisymmetric α-helices. Both the inner circle, which is labeled TM1, and the outer circle labeled TM2 have five α-helices, where a TM1 helix is connected to an adjacent TM2 helix forming a chain. Similar to the β-strands in β-barrels, the α-helices are tiled with
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respect to the barrel axis (Figure 2B). The TM domain is connected to the Cytoplasmic domain that is also constructed
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from α-helical barrel. Finally, since there are no hydrogen bonds between the chains the α-helical barrel is flexible [2]. In the second step, we identified cylindrical origami models that are analog to the shapes of β-barrel/α-helical barrel.
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The analogy in origami, to β-strands/α-helices that construct a barrel, is a rolled sheet with twist creases that form a
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cylinder. The characteristics of the barrel are the cylinder shape, the number of strands/helices, the angle between the
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strands/helices and the barrel axis, as well as the orientation of the hydrogen bonds. There are two known origami models with shapes that are similar to the barrel structures: The Yoshimura model (Figure
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2C), and the Kresling model (Figure 2D) [25]. Comparing the structures of β-barrel and α-helical barrel with two different
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origami models helps to identify the structural characteristics that: (i) Directly affect the functionality of the protein, and (ii) can imply on the forces that are applied on the protein.
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Initially, Yoshimura model is compared to the β-barrel. After detecting the similarity between the 3D structure of the protein and the shape of the origami model, we examined the β-barrel 2D spread state and the Yoshimura crease pattern. The 2D model of the protein topology is created in Pro-origami software (Figure 3A). However, to compare the protein structure with an origami model, we need to imagine that we disconnect two β-strands and open the β-barrel to a two-dimensional spread state (Figure 4B), such that the diagram captures the orientation of the β-strands with respect with the β-barrel axis (Figure 3B).
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β-strand
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Figure 3. Two-dimensional patterns of the proteins and origami models: (A) The 2D topology of the OmpA β-barrel, as obtained from the Pro-origami application [27], (B) The 2D drawing of the OmpA with tilted β-strands. The tilt is measured with respect to the β-barrel axis – the marked line represents a β-strand, the dash line represents a hydrogen bond, and δ (delta) is the angle between the β-strands and the hydrogen bonds. (C) The 2D topology of the MscL protein as obtained from the Pro-origami application [27], (D) The 2D drawing of the α-helices of the TM MscL α-helical barrel domain. (E) Yoshimura pattern – the marked lines are the creases of a single rhombus tile. The continues lines that tilt right can be analogue with the β-strands/α-helices, the creases that tilt left can be analogue with the hydrogen bonds, and δ is the angle between crossing creases of the rhombus. (F) Kresling pattern – the continuous marked line is a crease that can be analogue with the β-strand and TM2 in the α-helical barrel of the MscL and the dashed line with TM1. The spread state of a folded model is known in origami as "crease pattern", and it captures the pattern of fold lines. The abstract β-strand pattern is drawn from the unrolled pattern of β-barrel, with β-strands drawn as straight lines, and links that are created by hydrogen bonds between β-strands drawn as dashed lines (Figure 3B). Azulay
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Journal Pre-proof The Yoshimura model (Figure 3E) is assembled from a tessellation of rhombuses that are analog to the zig-zag pattern of β-strands and hydrogen bonds [31]. Yoshimura detected the rhombuses pattern in the post-buckling mode of thin metal cylinders that were subjected to axial loading, and explained that the rhombuses surfaces require minimal energy to maintain their shape under load [32]. B
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Figure 4. Spread process of the β-barrel/α-helical barrel proteins: (A) β-barrel folded state – the β-strands are twisted to create a barrel (PDB ID: 1BXW), (B) β-barrel half-folded state – the barrel is opened and the β-strands are arranged on a plane side by side, (C) TM α-helical barrel folded state – the α-helices are twisted to create a barrel (PDB ID: 2OAR), (D) α-helical barrel unfolded to chains.
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Buckling is the sudden change of a structure in response to a load that is higher than a critical force. The analytical analysis of cylinder buckling is outside of the scope of this paper (but is reviewed in [33-34]). However, we can learn on
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the parameters that are important for the axial buckling of a cylinder by analyzing the parameters in the formula of the
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critical buckling force for a long slender column (Figure 5):
𝐹𝐶 =
𝜋 2 EI (𝐾𝐿)2
(1)
where Fc is the critical force (vertical load on column), E is the modulus of elasticity, I is the smallest area moment of inertia of the cross-section of the column, K is the effective length factor of the column, whose value depends on the conditions of end support of the column (for example one fixed end and one free end as shown in Figure 5), and L is the unsupported length of the column.
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Figure 5. Buckling of a beam: (A) Before buckling, (B) after buckling. Fc is the critical force that leads to buckling, E is the modulus of elasticity, I is the smallest area moment of inertia of the cross-section of the column, K is the effective length factor of the column, whose value depends on the conditions of end support of the column (for example one fixed end and one free end), and L is the unsupported length of the column.
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Embedding Yoshimura pattern on thin-walled cylinders has been shown to guide the post-buckling configuration under
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axial force [35]. It is therefore not a surprise that the Yoshimura pattern was applied to metal drink cans to allow easily collapse to a folded state [36]. Some proteins also need to fold due to external loads that are applied by the membrane
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when the cell changes size, structure or potential [37], and it is possible that their structure evolved to fold easily. Since the β-barrel shape is similar to a cylinder with Yoshimura pattern, which is associated with axial buckling, FEA is applied
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to test the response of a cylinder to external forces, and to evaluate whether the features of the shape of the buckled
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cylinder are similar to the features of the shape of the β-barrel. Similarity of the features in the β-barrel shape and the Yoshimura pattern may indicate that the β-barrel structure evolved to respond to axial forces. Buckling mode depends on the relations between the structure dimensions. Following, the phenomenological model to test the post-buckling mode of the proteins due to axial load are a 1:1×106 scaled-up β-barrel that is modeled as a thin cylinder. The dimensions of the phenomenological model are specified in Table 1. The FEA is performed in the commercial ANSYS software. In the analysis, axial force is applied on the upper surface of the cylinder and its buckled mode shape is calculated. For the cylinder that mimics the β-barrel, along the circumference of buckled cylinder in the strain map there are 12 rhombuses (Figure 6A). Hence, there are 12 fold lines between the rhombuses compared to Azulay
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Journal Pre-proof eight β-strands in the analyzed β-barrel. The angle of the rhombus crease with the cylinder axis is ~10O (Figure 6A), and it was found in β-barrels analysis that the average tilt angle, made by β-strand axes, with the barrel axis is ~36O-40O [3839]. In addition, in anti-parallel β-barrel (Figure 3A), the angle δ, between β-strands and a hydrogen bond, is ~90O [29], and in the Yoshimura pattern of the buckled cylinder, the same angle between crossing creases of the rhombuses is ~45O (Figure 3B). Hence, the Yoshimura model and the β-barrel structure differ in shape features. Table 1: The dimensions of the scaled up (1:1×106) protein models in the FEA Height (mm)
Diameter (mm)
Thickness (mm)
OmpA β-barrel
30
18
1
MscL TM α-helical barrel domain [6]
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Model
We compare next the shape of the TM α-helical barrel and the Yoshimura origami model. Considering the comparison
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process of the β-barrel, it appears that the α-helical barrel and the Yoshimura origami model have tilted structure.
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However, the tilt angle of the α-helices is ~30-40O (Figure 2B), compared to 20O tilt of the rhombuses in the axial postbuckling model (Figure 6C). In addition, the α-helical barrel is not structured with elements that are similar with the
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rhombuses that appear in the Yoshimura model. Hence, the TM α-helical barrel and the Yoshimura origami model differ
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in important shape characteristics.
The second origami model we compared the β-barrel/α-helical barrel to is the Kresling origami model (Figure 2D). The "chain saw" crease pattern (Figure 3D) of the Kresling model appears in the post-buckling mode when compressing a thin cylindrical shell while applying a twist load. The modified Kresling folding pattern was applied to the design of a stent, with a cylinder shape. The stent is inserted into the body folded, and when it reaches a desired location it unfolds to assure that the passage is kept open (i.e. veins) – similar to the way the β-barrel supports the cell membrane [40]. We applied a FEA to simulate the post-buckling structure of a cylinder, which is subject to twist forces that are associated with the Kresling pattern and to compare the buckled shape with the β-barrel/α-helical barrel shapes. The Azulay
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Journal Pre-proof cylinders, with a 1: 1×106 ratio scaled-up models of the OmpA β-barrel and the TM α-helical barrel of the MscL, are subjected to a combination of axial and twist forces. The dimensions of the phenomenological models in the FEA are specified in Table 1. Interestingly, in the FEA strain rate map result of the twist buckled cylinders, the number of folds in the twist postbuckling mode of a scaled up model of the β-barrel is identical to the number of β-strands in the β-barrel (Figure 6B). Similarly, the number of folds in the twist post-buckling mode of a scaled up FEA cylinder model of the α-helical barrel is
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identical to the number of α-helices in the TM domain of the MscL (Figure 6D).
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Figure 6. FEA results of the buckling mode of thin-walled cylindrical shells :(A) β-barrel cylinder model subjected to axial load that are indicated by horizontal arrows, α is the angle between the β-strand and the cylinder axis. (B) β-barrel cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (C) The MscL TM cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (D) The MscL TM cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (E) The top view of the MscL TM cylinder model after buckling and the top view of the α-helical TM barrel of the MscL.
The buckling mode-shape is a repeatable rotated pattern that looks like mountains and valleys [41], and the membrane compression is maximized at a specific angle to the axial direction which is determined by the geometry [33]. Hence, the valley/mountain lines in the buckled pattern take the same specific angle which appears to represent a minimum on the load-deflection curve. This means that the structure folds easily taking extra load at a reduced stiffness. The tilt angle of the folds with the cylinder axis in the twist post-buckling mode of the β-barrel model (Figure 6B) is ~30O, similar to angle between the β-strand axes and the OmpA barrel axis, which is ~36O-40O [38-39]. Furthermore, Azulay p. 12
Journal Pre-proof when folding the Kresling pattern from paper, additional fold lines appear between the creases. These new "undesired" creases follow a pattern, which is similar to the pattern of hydrogen bonds between the tilted strands in the β-barrels. For the FEA α-helical barrel (cylinder) model the tilt angle of the creases is ~30º and the tilt angle of the α-helices in the MscL is ~30º-40º. Hence, the tilt angle in the two cases is similar. In addition, the number of mountain creases in the outer circle and the valley creases in the inner circle are identical, and in correlation with the number of α-helices in TM2 and TM1 (Figure 6E).
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A summary of the comparison of the shape and structural features and characteristics of the OmpA β-barrel and MscL α-
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helical TM barrel protein structures as well as the Yosimura and the Kresling origami models is presented in Table 2.
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OmpA (β-barrel)
Yoshimura origami model *
Kresling origami model *
TM domain of MscL (α-helical barrel)
Yoshimura origami model **
Kresling origami model **
Geometrical classification of the structure
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Number of domains
One domain
A single pattern model
A single pattern model
One domain
A single pattern model
A single pattern model
Number of chains in each domain
Eight β-strands
12 tilted crease (FEA axial buckling analysis)
Eight tilted creases (FEA axial-twist buckling analysis)
10 α-helices arranged in two circles – five in TM1 and five in TM2
10 tilted creases (FEA axial buckling analysis)
10 tilted crease (FEA axial-twist buckling). Five mountains in the outer circle and five valleys in the inner circle
Shape feature
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Tilted β-strands
Tiled rhombuses crease pattern with tilted creases
Tilted mountain/valley creases
Tilted α-helices
Tiled rhombuses crease pattern with tilted creases
Tilted mountain/valley creases
Tilt angle of the βstrands and β-barrel axis is ~36O-40O
* Tilt angle of the diamond crease with the cylinder axis is ~10O
* Tilt angle of the crease is ~30O
Tilt angle of the αhelices and the αhelical barrel axis is ~30 O-40O
** Tilt angle of the diamond crease with the cylinder axis is ~20O
** Tilt angle of the creases in the FEA model of the α-helical cylinder is ~30O
Angle between the tilted creases and the hydrogen bonds is ~90O
Angle (δ) between the creases of the rhombuses is 45O
Angle (δ) between the tilted crease and undesired creases in the folded model are ~90O
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* The result is for a cylinder that is modeled as a thin cylinder with diameter/height that is proportional to the OmpA β-barrel ** The result is for a cylinder that is modeled as a thin cylinder with diameter/height that is proportional to the TM α-helical barrel domain of the MscL Azulay
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Journal Pre-proof In the last step, we analyze the protein structure based on the structural characteristics of the analog origami model the forces that may be applied on the protein and the function it fulfils in its environment. As a result of osmotic change the α-helical barrel adjusts by opening acting as a gate, while the β-barrel supports the membrane without changing its conformation. When the membrane lipid gets thicker, the α-helices angle with the α-helical barrel axis decreases stretching the barrel height and decreasing its radius (Figure 7B). When the membrane lipid gets thinner the α-helices angle with the α-helical barrel axis increases, hence, the α-helical barrel shortens and widens (Figure
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7C) [42]. Also, α-helical barrel should function under axial hydro pressure, which may result in bucking [43], and it is
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similar to pressure vessels that twist-buckle due to inside/outside pressure [44]. Considering the structures of the α-
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helical barrel and the Kresling model, axial, radial and tangential forces will cause the α-helices to tilt and change the
Force
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barrel diameter and height.
Figure 7. Conformations of the α-helical barrel in membrane lipid: (A) Membrane and α-helical barrel have the same thickness; (B) Membrane thickens, and α-helical barrel adjusts by decreasing the tile angle of the α-helices with the barrel axis, and decreasing the diameter of the barrel. The red arrows indicate lateral pressure by membrane curvature strain. (C) Membrane becomes thinners, and the α-helical barrel adjusts to a decrease in external pressure by increasing the tilt angle of the α-helices and increasing the diameter of the barrel allowing molecules to enter. The red error indicate forces from external pressure, or from a decrease in the tension to the membrane cell, [42] [2]. Following the above comparison, initially the two origami models, the Yoshimura and the Kresling models, look similar to the β-barrel and α-helical barrel protein structures. However, further analysis demonstrated that the β-barrel/αhelical barrel and the Yoshimura model that appears in axial buckling differ in structural characteristics, and that βbarrel/α-helical barrel proteins structure and Kresling origami model that has a twist-buckling pattern are in agreement. Azulay p. 15
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CONCLUSIONS A new approach for the comparing of protein structures with origami models is presented. In the process, a protein of interest is selected, and analog origami model/s are identified. Based on the features of the analog origami models and the function that the protein performs, the structural characteristics of the protein are analyzed.
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As a proof of concept, a comparison and analysis of OmpA β-barrel and α-helical TM barrel of the MscL with the Yoshimura and Kresling origami models is performed. The function that the proteins fulfills as well as the unique
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mechanical characteristics of the origami models, shed light on the way that these protein structures are designed to
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fulfill their function. The Yoshimura model appears in the post-buckling mode of a thin wall cylinder that is subjected to
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axial force. The Kresling model appears in the post-buckling mode of a thin wall cylinder, that is subjected to a
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combination of axial and twist forces. It is shown that OmpA β-barrel and TM α-helical barrel of the MscL differ from the shape of the Yoshimura model and are analogue with the Kresling model.
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Based on the analogy and the characteristics of the Kresling model, it is suggested that these protein structures are specifically designed to adjust diameter and height by changing the β-strands/α-helices tilt in response to axial, radial or
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twist loads from changes in the lipid of the membrane. However, due to hydrogen bonds that link the β-strands, the
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OmpA β-barrel is stiff and does not change its diameter [45], and at this stage we can only explain how the structural characteristics of the Kresling model contribute to the function of the TM α-helical barrel of the MscL. Hence, given the function that the protein fulfils and the applied forces, the analogue origami models provide possible insight into why, through the course of evolution, the twisted structure of the α-helical barrel that is similar to Kresling origami model was selected. Following the above, the proposed approach should be applied to further analysis of proteins that can contribute to the understanding of protein structures and function. The mechanical-structural characteristics that will be identified in the
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Journal Pre-proof comparison of proteins and origami models can be applied in the design of de-novo proteins and in the design of nanomechanisms in non-biological applications.
AUTHOR CONTRIBUTIONS
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H.A. conceived the research. H.A. and A.L. performed analysis. N.Q. supervised the project. All authors provided critical
CONFLICT OF INTEREST
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The authors declare that they have no conflict of interest.
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feedback, helped shape the research, the analysis and wrote the manuscript.
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Graphical abstract
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Highlights Following are the highlights of the proposed paper: 1) A general approach, that combines knowledge from origami research, for comparing origami models and protein structures for studying the structural characteristics and function of proteins, is proposed. 2) A test case for the approach is given: β-barrel and α-helical barrel structures are compared with the Kresling and Yoshimura origami models. 3) It is shown that the structures of the β-barrel and α-helical barrel are in agreement with the shape and structural characteristics of the Kresling model. The resemblance of the protein structures to the Kresling model and the model characteristics indicate that the circular twist of the β-strands and α-helices in the barrels create a pattern that is analog with the post-buckling mode shape of a cylindrical shell. 4) Through the analogy it is explained how the structural characteristic can help the α-helical barrel to adjust length and diameter in response to changes in the membrane structure. However, such conformation only apply to the α-helical barrel, since the β-barrel, in spite of resembles to the Kresling model, remains stiff due to hydrogen bonds between the β-strands.
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Hay Azulay, Aviv Lutaty, Nir Qvit PII:
S0005-2736(19)30280-9
DOI:
https://doi.org/10.1016/j.bbamem.2019.183132
Reference:
BBAMEM 183132
To appear in:
BBA - Biomembranes
Received date:
21 May 2019
Revised date:
5 October 2019
Accepted date:
28 October 2019
Please cite this article as: H. Azulay, A. Lutaty and N. Qvit, Approach for comparing protein structures and origami models, BBA - Biomembranes(2019), https://doi.org/ 10.1016/j.bbamem.2019.183132
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© 2019 Published by Elsevier.
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Approach for Comparing Protein Structures and Origami Models 1,2,*
Hay Azulay
, Aviv Lutaty
1,3
, Nir
1
Independent researcher
2
Koranit, Israel, 2018100
3
Kiryat Motzkin, Israel, 2641312
Qvit
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4
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The Azrieli Faculty of Medicine in the Galilee, Bar-Ilan University, Henrietta Szold St. 8, POB 1589, Safed, Israel.
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*Correspondence: [email protected], [email protected], [email protected]
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ABSTRACT
The research fields of proteins and origami have intersected in the study of folding and de-novo
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design of proteins. However, there is limited knowledge on the analogy between protein
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structures and origami models. We propose a general approach for comparing protein
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structures with origami models, and present a test case, comparing transmembrane β-barrel and α-helical barrel with the Yoshimura and Kresling origami models. While both shapes and
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structures may look similar, we demonstrated that the β-barrel and the α-helical barrel are in
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agreement only with the shape and structural characteristics of the Kresling model. Through the analogy it is explained how the structural characteristic can help the β-barrel and α-helical barrel to adjust length and diameter in response to changes in the membrane structure. However, such conformations only apply to the α-helical barrel, and the β-barrel, in spite of resembles to the Kresling model, remains stiff due to hydrogen bonds between the β-strands. Thus, our analysis suggests that there are similar patterns between protein structures and origami models and that this approach may provide an important insight on the role that the structure of a protein fulfils, and on preferred structural design of novel proteins with unique characteristics.
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Journal Pre-proof Keywords Protein, Origami, Structure, β-barrel, α-helical barrel, Kresling model, Yoshimura model.
INTRODUCTION Proteins and origami are generally studied by independent research fields. The functionality of a protein is determined by various factors including its structure, activity, and subcellular localization to name a few. Therefore many studies look into the anatomy, taxonomy [1], and mechanism [2] of proteins. Furthermore, studies of protein folding attempt to
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predict how a specific sequence of amino acids would fold, using data from recording of folding stages, force
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spectroscopy experiments [3], X-ray crystallography [4], Nuclear Magnetic Resonance (NMR) [5], and simulations [6].
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This research is essential for different applications, for example, it can help in developing a pharmacological intervention
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to fight viruses [7] and other diseases, as well as in studying biological structures, such as spider silk, that can be applied to design improved materials for engineering applications [8].
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In parallel to protein research the study of mathematical principles behind origami folding led to a dizzying rate of new
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structures that are applied in different areas, such as architecture, robotics [9], and aeronautics [10], which demonstrates how art can contribute to science [11]. Among other things, it has been proven that by applying specific
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mathematical conditions it is possible to design a set of creases that will result in flat folding of a three-dimensional (3D)
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model [12].
Origami is also interwoven into biological research: (i) Classic origami is the basis for the known folding art in which the model is folded from a single sheet of paper without cutting [13]. Folding principles of classic origami were applied to study the deployment mechanism of the wings of ladybird Beetle [14]. Protein folding process was compared with classic origami folding as in both cases the structure is created through a sequential folding into a 3D compact structure [15-16]. Moreover, Protein Data Bank (PDB) developed the PDB-101 website (https://pdb101.rcsb.org/), which provides resources, and teaching aids for teachers, such as folded 3D origami models to help beginners learn about proteins [17].
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Journal Pre-proof (ii) Modular origami is an origami branch in which the model is assembled from separate folded sheets of paper [18]. This principle is implemented in "Protein Origami", which is an approach for the de-novo design of proteins with new folds unknown in nature. To construct a new protein, folded DNA segments are assembled, in a process that is similar to the assembly of folded paper pieces in modular origami [19]. The approach can be also applied for the design of nanostructures for other applications aside from biology [20]. Modular origami models have been also folded for educational purposes in chemistry and biology, for example chemical molecules that can be assembled into chains were
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modeled by origami [21].
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Following the above, it seems that proteins and origami have a lot in common. However, in spite of the similarity, as far
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as we know, no significant attempt has been made to compare the shape and structural characteristics of proteins and origami models. In this paper, a general approach for comparing protein structures and origami models with similar
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shape is presented, and their characteristics are analyzed from a mechanical point of view, in an attempt to get insight
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on the role of the protein structure with respect to the chemical/physical properties of the protein and the function it fulfills. The approach is demonstrated in the comparison of the outer membrane protein A (OmpA) of Escherichia coli
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that has a β-barrel structure, and the transmembrane (TM) domain of the Large Conductance Mechanosensitive Ion
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Channel (MscL) that has a α-helical structure, with the Kresling and Yoshimura origami models.
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METHOD Analysis approach
Proteins are made up from a primary sequence of amino acids that is folded into secondary and tertiary unique structures, and it is possible to analyze their molecular structure, interactions, and mechanical properties [22]. However, generally, it is easier to understand why a specific structure is selected for engineering application than to understand why a protein structure evolved to its current structure. Since the mechanical and physical design and the folding principles of origami models have been applied to study biological systems, including Coleopteron wing folding, and growth pattern of pine cone [23], and since in some cases there is analogy in the folding processes of proteins and Azulay
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Journal Pre-proof origami, proteins are analyzed here through a four-step approach that combines knowledge from origami models into protein research. The four steps of the approach are (Figure 1): (1) Screening – Choose a protein and use data from the PDB, which was derived, for example from X-ray spectrometry and/or NMR, and examine the protein structure. The PDB data of the protein is presented using software, such as Jmol [24], that can help identify secondary and tertiary structures. (2) Identifying – Identify origami models with shape/conformation that are similar to the selected protein structure. The
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analog origami models can be based on known folding patterns or new designs. Known origami models can be found in
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the literature [25], and new models can be obtained using dedicated design tools, that create computerized foldable
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origami models [26].
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(3) Comparing – Compare the similarity of the chosen protein structure and the identified origami models based on: (i) The number of domains, (ii) the number of chains in each domain, and (iii) the geometrical classification of the structure
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(for example, a type of polygon, cylinder, sphere, or a freeform shape). To increase the level of confidence in the
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equivalence, several views of the protein from different sides should be compared. For training purpose, it may be beneficial to "disconnect" chains and draw the spread structure arrangement on a two-dimensional (2D) plane to
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compare it with the origami "crease pattern", which is a 2D unfolded representation of the 3D origami model. The 2D
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model of the protein topology can be obtained from software application such as Pro-origami [27]. However, since in many cases, the orientations of the secondary structures are not detected, another sub-step is required to correct the diagram into drawing that incorporated this information. (4) Analyzing – Analyze the mechanical characteristics of the structure of the origami model and the protein structure. The analyses include mathematical tools, origami design principles, characteristic of known origami models, the folding/creation sequence of the origami model and Finite Element Analysis (FEA), which is an analysis that is often applied in mechanical studies to simulate the response of a structure to forces that are applied at specified locations. The analysis is performed while considering the functionality of the protein and the forces that it is required to
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Journal Pre-proof withstand. For proteins that are known to dynamically change conformation to achieve desired functionality, it is
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desired that different conformations of the protein and the folded/unfolded analog origami models are compared.
Figure 1. A workflow of the comparison approach of protein structure and origami models. EXAMPLES Example of comparison between protein structure and analogue origami models In the following example, the proteins of interest that we compare with an origami model are: (1) the outer membrane protein A (OmpA) of Escherichia coli, and (2) the TM domain of MscL.
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Journal Pre-proof The OmpA is a one-domain protein has a relatively simple β-barrel structure (Figure 2A). OmpA, like many other membrane proteins that functions as the gateway to the cell, has both structural and ion-permeable roles, which has several functions, such as, allowing bacterial survival during osmotic stress [28]. β-barrels are twisted β-sheets that are arranged to form a closed structure in which the first β-strand is bonded to the last β-strand. The structure is a hyperboloid of revolution, with minimal surface, and it is arranged with positively charged residues on one side of the pore and negative on the other (Figure 2A). The β-barrel proteins serve essential
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functions such as, cargo transport and signaling, and they create a transmembrane passage/valve, which can be found in
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chymotrypsin enzyme, TIM-barrels, streptavidin, and lipocalins [29]. Some of the residues of the β-strands are
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connected through hydrogen bonds, which are perpendicular and planar between adjacent antiparallel β-strands and distorted between adjacent parallel β-strands. As a result, the hydrogen bonds between antiparallel β-strands are
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stronger than hydrogen bonds between parallel β-strands. In addition, compared to β-sheets, β-barrels are constructed
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with more hydrogen bonds between the β-strands than required for the twist and the closer, which stiffens the
β-strand
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Hydrogen bonds
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structure. Therefore, β-barrels with antiparallel β-strands, such as the OmpA are stiff structures [30].
α
D Mountain fold
α TM1 TM2
Folds created between creases Valley fold
Figure 2. The protein structures and compared origami models: (A) The OmpA β-barrel protein structure (PDB ID: 1BXW). The thick lines indicate β-strands, α is the tilt angle of the β-strands with respect to the barrel axis and the dotted lines are the hydrogen bonds. (B) TM α-helical barrel domain of the MscL (PDB ID: 2OAR). The curly cylinders represent α-helices, where each color indicates a chain. (C) Yoshimura origami model, the dotted line indicates a crease that is tilted, similar to β-strands/α-helices. (D) Kresling origami model, the marked lines represent mountain creases that are tilted, similar to β-strands/α-helices. In addition, undesired folds that are created when folding the Kresling model are also indicated by arrows. Azulay
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The second protein that is presented is the TM α-helical barrel domain of the MscL. The MscL is a membrane valve that is designed to relief osmotic imbalance, allowing a quick and transient increase in compensatory solute flux out of the cell (Figure 2B). It is constructed from a TM domain that is combined from two circles of axisymmetric α-helices. Both the inner circle, which is labeled TM1, and the outer circle labeled TM2 have five α-helices, where a TM1 helix is connected to an adjacent TM2 helix forming a chain. Similar to the β-strands in β-barrels, the α-helices are tiled with
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respect to the barrel axis (Figure 2B). The TM domain is connected to the Cytoplasmic domain that is also constructed
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from α-helical barrel. Finally, since there are no hydrogen bonds between the chains the α-helical barrel is flexible [2]. In the second step, we identified cylindrical origami models that are analog to the shapes of β-barrel/α-helical barrel.
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The analogy in origami, to β-strands/α-helices that construct a barrel, is a rolled sheet with twist creases that form a
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cylinder. The characteristics of the barrel are the cylinder shape, the number of strands/helices, the angle between the
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strands/helices and the barrel axis, as well as the orientation of the hydrogen bonds. There are two known origami models with shapes that are similar to the barrel structures: The Yoshimura model (Figure
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2C), and the Kresling model (Figure 2D) [25]. Comparing the structures of β-barrel and α-helical barrel with two different
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origami models helps to identify the structural characteristics that: (i) Directly affect the functionality of the protein, and (ii) can imply on the forces that are applied on the protein.
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Initially, Yoshimura model is compared to the β-barrel. After detecting the similarity between the 3D structure of the protein and the shape of the origami model, we examined the β-barrel 2D spread state and the Yoshimura crease pattern. The 2D model of the protein topology is created in Pro-origami software (Figure 3A). However, to compare the protein structure with an origami model, we need to imagine that we disconnect two β-strands and open the β-barrel to a two-dimensional spread state (Figure 4B), such that the diagram captures the orientation of the β-strands with respect with the β-barrel axis (Figure 3B).
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β-strand
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Figure 3. Two-dimensional patterns of the proteins and origami models: (A) The 2D topology of the OmpA β-barrel, as obtained from the Pro-origami application [27], (B) The 2D drawing of the OmpA with tilted β-strands. The tilt is measured with respect to the β-barrel axis – the marked line represents a β-strand, the dash line represents a hydrogen bond, and δ (delta) is the angle between the β-strands and the hydrogen bonds. (C) The 2D topology of the MscL protein as obtained from the Pro-origami application [27], (D) The 2D drawing of the α-helices of the TM MscL α-helical barrel domain. (E) Yoshimura pattern – the marked lines are the creases of a single rhombus tile. The continues lines that tilt right can be analogue with the β-strands/α-helices, the creases that tilt left can be analogue with the hydrogen bonds, and δ is the angle between crossing creases of the rhombus. (F) Kresling pattern – the continuous marked line is a crease that can be analogue with the β-strand and TM2 in the α-helical barrel of the MscL and the dashed line with TM1. The spread state of a folded model is known in origami as "crease pattern", and it captures the pattern of fold lines. The abstract β-strand pattern is drawn from the unrolled pattern of β-barrel, with β-strands drawn as straight lines, and links that are created by hydrogen bonds between β-strands drawn as dashed lines (Figure 3B). Azulay
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Journal Pre-proof The Yoshimura model (Figure 3E) is assembled from a tessellation of rhombuses that are analog to the zig-zag pattern of β-strands and hydrogen bonds [31]. Yoshimura detected the rhombuses pattern in the post-buckling mode of thin metal cylinders that were subjected to axial loading, and explained that the rhombuses surfaces require minimal energy to maintain their shape under load [32]. B
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Figure 4. Spread process of the β-barrel/α-helical barrel proteins: (A) β-barrel folded state – the β-strands are twisted to create a barrel (PDB ID: 1BXW), (B) β-barrel half-folded state – the barrel is opened and the β-strands are arranged on a plane side by side, (C) TM α-helical barrel folded state – the α-helices are twisted to create a barrel (PDB ID: 2OAR), (D) α-helical barrel unfolded to chains.
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Buckling is the sudden change of a structure in response to a load that is higher than a critical force. The analytical analysis of cylinder buckling is outside of the scope of this paper (but is reviewed in [33-34]). However, we can learn on
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the parameters that are important for the axial buckling of a cylinder by analyzing the parameters in the formula of the
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critical buckling force for a long slender column (Figure 5):
𝐹𝐶 =
𝜋 2 EI (𝐾𝐿)2
(1)
where Fc is the critical force (vertical load on column), E is the modulus of elasticity, I is the smallest area moment of inertia of the cross-section of the column, K is the effective length factor of the column, whose value depends on the conditions of end support of the column (for example one fixed end and one free end as shown in Figure 5), and L is the unsupported length of the column.
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Figure 5. Buckling of a beam: (A) Before buckling, (B) after buckling. Fc is the critical force that leads to buckling, E is the modulus of elasticity, I is the smallest area moment of inertia of the cross-section of the column, K is the effective length factor of the column, whose value depends on the conditions of end support of the column (for example one fixed end and one free end), and L is the unsupported length of the column.
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Embedding Yoshimura pattern on thin-walled cylinders has been shown to guide the post-buckling configuration under
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axial force [35]. It is therefore not a surprise that the Yoshimura pattern was applied to metal drink cans to allow easily collapse to a folded state [36]. Some proteins also need to fold due to external loads that are applied by the membrane
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when the cell changes size, structure or potential [37], and it is possible that their structure evolved to fold easily. Since the β-barrel shape is similar to a cylinder with Yoshimura pattern, which is associated with axial buckling, FEA is applied
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to test the response of a cylinder to external forces, and to evaluate whether the features of the shape of the buckled
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cylinder are similar to the features of the shape of the β-barrel. Similarity of the features in the β-barrel shape and the Yoshimura pattern may indicate that the β-barrel structure evolved to respond to axial forces. Buckling mode depends on the relations between the structure dimensions. Following, the phenomenological model to test the post-buckling mode of the proteins due to axial load are a 1:1×106 scaled-up β-barrel that is modeled as a thin cylinder. The dimensions of the phenomenological model are specified in Table 1. The FEA is performed in the commercial ANSYS software. In the analysis, axial force is applied on the upper surface of the cylinder and its buckled mode shape is calculated. For the cylinder that mimics the β-barrel, along the circumference of buckled cylinder in the strain map there are 12 rhombuses (Figure 6A). Hence, there are 12 fold lines between the rhombuses compared to Azulay
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Journal Pre-proof eight β-strands in the analyzed β-barrel. The angle of the rhombus crease with the cylinder axis is ~10O (Figure 6A), and it was found in β-barrels analysis that the average tilt angle, made by β-strand axes, with the barrel axis is ~36O-40O [3839]. In addition, in anti-parallel β-barrel (Figure 3A), the angle δ, between β-strands and a hydrogen bond, is ~90O [29], and in the Yoshimura pattern of the buckled cylinder, the same angle between crossing creases of the rhombuses is ~45O (Figure 3B). Hence, the Yoshimura model and the β-barrel structure differ in shape features. Table 1: The dimensions of the scaled up (1:1×106) protein models in the FEA Height (mm)
Diameter (mm)
Thickness (mm)
OmpA β-barrel
30
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1
MscL TM α-helical barrel domain [6]
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We compare next the shape of the TM α-helical barrel and the Yoshimura origami model. Considering the comparison
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process of the β-barrel, it appears that the α-helical barrel and the Yoshimura origami model have tilted structure.
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However, the tilt angle of the α-helices is ~30-40O (Figure 2B), compared to 20O tilt of the rhombuses in the axial postbuckling model (Figure 6C). In addition, the α-helical barrel is not structured with elements that are similar with the
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rhombuses that appear in the Yoshimura model. Hence, the TM α-helical barrel and the Yoshimura origami model differ
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in important shape characteristics.
The second origami model we compared the β-barrel/α-helical barrel to is the Kresling origami model (Figure 2D). The "chain saw" crease pattern (Figure 3D) of the Kresling model appears in the post-buckling mode when compressing a thin cylindrical shell while applying a twist load. The modified Kresling folding pattern was applied to the design of a stent, with a cylinder shape. The stent is inserted into the body folded, and when it reaches a desired location it unfolds to assure that the passage is kept open (i.e. veins) – similar to the way the β-barrel supports the cell membrane [40]. We applied a FEA to simulate the post-buckling structure of a cylinder, which is subject to twist forces that are associated with the Kresling pattern and to compare the buckled shape with the β-barrel/α-helical barrel shapes. The Azulay
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Journal Pre-proof cylinders, with a 1: 1×106 ratio scaled-up models of the OmpA β-barrel and the TM α-helical barrel of the MscL, are subjected to a combination of axial and twist forces. The dimensions of the phenomenological models in the FEA are specified in Table 1. Interestingly, in the FEA strain rate map result of the twist buckled cylinders, the number of folds in the twist postbuckling mode of a scaled up model of the β-barrel is identical to the number of β-strands in the β-barrel (Figure 6B). Similarly, the number of folds in the twist post-buckling mode of a scaled up FEA cylinder model of the α-helical barrel is
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Figure 6. FEA results of the buckling mode of thin-walled cylindrical shells :(A) β-barrel cylinder model subjected to axial load that are indicated by horizontal arrows, α is the angle between the β-strand and the cylinder axis. (B) β-barrel cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (C) The MscL TM cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (D) The MscL TM cylinder model subjected to a combination of axial forces that are indicated by horizontal arrows and twist moment indicated by curvature arrow loads. (E) The top view of the MscL TM cylinder model after buckling and the top view of the α-helical TM barrel of the MscL.
The buckling mode-shape is a repeatable rotated pattern that looks like mountains and valleys [41], and the membrane compression is maximized at a specific angle to the axial direction which is determined by the geometry [33]. Hence, the valley/mountain lines in the buckled pattern take the same specific angle which appears to represent a minimum on the load-deflection curve. This means that the structure folds easily taking extra load at a reduced stiffness. The tilt angle of the folds with the cylinder axis in the twist post-buckling mode of the β-barrel model (Figure 6B) is ~30O, similar to angle between the β-strand axes and the OmpA barrel axis, which is ~36O-40O [38-39]. Furthermore, Azulay p. 12
Journal Pre-proof when folding the Kresling pattern from paper, additional fold lines appear between the creases. These new "undesired" creases follow a pattern, which is similar to the pattern of hydrogen bonds between the tilted strands in the β-barrels. For the FEA α-helical barrel (cylinder) model the tilt angle of the creases is ~30º and the tilt angle of the α-helices in the MscL is ~30º-40º. Hence, the tilt angle in the two cases is similar. In addition, the number of mountain creases in the outer circle and the valley creases in the inner circle are identical, and in correlation with the number of α-helices in TM2 and TM1 (Figure 6E).
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A summary of the comparison of the shape and structural features and characteristics of the OmpA β-barrel and MscL α-
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helical TM barrel protein structures as well as the Yosimura and the Kresling origami models is presented in Table 2.
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OmpA (β-barrel)
Yoshimura origami model *
Kresling origami model *
TM domain of MscL (α-helical barrel)
Yoshimura origami model **
Kresling origami model **
Geometrical classification of the structure
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Cylinder shape
Number of domains
One domain
A single pattern model
A single pattern model
One domain
A single pattern model
A single pattern model
Number of chains in each domain
Eight β-strands
12 tilted crease (FEA axial buckling analysis)
Eight tilted creases (FEA axial-twist buckling analysis)
10 α-helices arranged in two circles – five in TM1 and five in TM2
10 tilted creases (FEA axial buckling analysis)
10 tilted crease (FEA axial-twist buckling). Five mountains in the outer circle and five valleys in the inner circle
Shape feature
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Axisymmetric
Tilted β-strands
Tiled rhombuses crease pattern with tilted creases
Tilted mountain/valley creases
Tilted α-helices
Tiled rhombuses crease pattern with tilted creases
Tilted mountain/valley creases
Tilt angle of the βstrands and β-barrel axis is ~36O-40O
* Tilt angle of the diamond crease with the cylinder axis is ~10O
* Tilt angle of the crease is ~30O
Tilt angle of the αhelices and the αhelical barrel axis is ~30 O-40O
** Tilt angle of the diamond crease with the cylinder axis is ~20O
** Tilt angle of the creases in the FEA model of the α-helical cylinder is ~30O
Angle between the tilted creases and the hydrogen bonds is ~90O
Angle (δ) between the creases of the rhombuses is 45O
Angle (δ) between the tilted crease and undesired creases in the folded model are ~90O
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* The result is for a cylinder that is modeled as a thin cylinder with diameter/height that is proportional to the OmpA β-barrel ** The result is for a cylinder that is modeled as a thin cylinder with diameter/height that is proportional to the TM α-helical barrel domain of the MscL Azulay
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Journal Pre-proof In the last step, we analyze the protein structure based on the structural characteristics of the analog origami model the forces that may be applied on the protein and the function it fulfils in its environment. As a result of osmotic change the α-helical barrel adjusts by opening acting as a gate, while the β-barrel supports the membrane without changing its conformation. When the membrane lipid gets thicker, the α-helices angle with the α-helical barrel axis decreases stretching the barrel height and decreasing its radius (Figure 7B). When the membrane lipid gets thinner the α-helices angle with the α-helical barrel axis increases, hence, the α-helical barrel shortens and widens (Figure
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7C) [42]. Also, α-helical barrel should function under axial hydro pressure, which may result in bucking [43], and it is
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similar to pressure vessels that twist-buckle due to inside/outside pressure [44]. Considering the structures of the α-
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helical barrel and the Kresling model, axial, radial and tangential forces will cause the α-helices to tilt and change the
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Figure 7. Conformations of the α-helical barrel in membrane lipid: (A) Membrane and α-helical barrel have the same thickness; (B) Membrane thickens, and α-helical barrel adjusts by decreasing the tile angle of the α-helices with the barrel axis, and decreasing the diameter of the barrel. The red arrows indicate lateral pressure by membrane curvature strain. (C) Membrane becomes thinners, and the α-helical barrel adjusts to a decrease in external pressure by increasing the tilt angle of the α-helices and increasing the diameter of the barrel allowing molecules to enter. The red error indicate forces from external pressure, or from a decrease in the tension to the membrane cell, [42] [2]. Following the above comparison, initially the two origami models, the Yoshimura and the Kresling models, look similar to the β-barrel and α-helical barrel protein structures. However, further analysis demonstrated that the β-barrel/αhelical barrel and the Yoshimura model that appears in axial buckling differ in structural characteristics, and that βbarrel/α-helical barrel proteins structure and Kresling origami model that has a twist-buckling pattern are in agreement. Azulay p. 15
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CONCLUSIONS A new approach for the comparing of protein structures with origami models is presented. In the process, a protein of interest is selected, and analog origami model/s are identified. Based on the features of the analog origami models and the function that the protein performs, the structural characteristics of the protein are analyzed.
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As a proof of concept, a comparison and analysis of OmpA β-barrel and α-helical TM barrel of the MscL with the Yoshimura and Kresling origami models is performed. The function that the proteins fulfills as well as the unique
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mechanical characteristics of the origami models, shed light on the way that these protein structures are designed to
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fulfill their function. The Yoshimura model appears in the post-buckling mode of a thin wall cylinder that is subjected to
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axial force. The Kresling model appears in the post-buckling mode of a thin wall cylinder, that is subjected to a
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combination of axial and twist forces. It is shown that OmpA β-barrel and TM α-helical barrel of the MscL differ from the shape of the Yoshimura model and are analogue with the Kresling model.
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Based on the analogy and the characteristics of the Kresling model, it is suggested that these protein structures are specifically designed to adjust diameter and height by changing the β-strands/α-helices tilt in response to axial, radial or
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twist loads from changes in the lipid of the membrane. However, due to hydrogen bonds that link the β-strands, the
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OmpA β-barrel is stiff and does not change its diameter [45], and at this stage we can only explain how the structural characteristics of the Kresling model contribute to the function of the TM α-helical barrel of the MscL. Hence, given the function that the protein fulfils and the applied forces, the analogue origami models provide possible insight into why, through the course of evolution, the twisted structure of the α-helical barrel that is similar to Kresling origami model was selected. Following the above, the proposed approach should be applied to further analysis of proteins that can contribute to the understanding of protein structures and function. The mechanical-structural characteristics that will be identified in the
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Journal Pre-proof comparison of proteins and origami models can be applied in the design of de-novo proteins and in the design of nanomechanisms in non-biological applications.
AUTHOR CONTRIBUTIONS
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H.A. conceived the research. H.A. and A.L. performed analysis. N.Q. supervised the project. All authors provided critical
CONFLICT OF INTEREST
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The authors declare that they have no conflict of interest.
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feedback, helped shape the research, the analysis and wrote the manuscript.
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Graphical abstract
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Highlights Following are the highlights of the proposed paper: 1) A general approach, that combines knowledge from origami research, for comparing origami models and protein structures for studying the structural characteristics and function of proteins, is proposed. 2) A test case for the approach is given: β-barrel and α-helical barrel structures are compared with the Kresling and Yoshimura origami models. 3) It is shown that the structures of the β-barrel and α-helical barrel are in agreement with the shape and structural characteristics of the Kresling model. The resemblance of the protein structures to the Kresling model and the model characteristics indicate that the circular twist of the β-strands and α-helices in the barrels create a pattern that is analog with the post-buckling mode shape of a cylindrical shell. 4) Through the analogy it is explained how the structural characteristic can help the α-helical barrel to adjust length and diameter in response to changes in the membrane structure. However, such conformation only apply to the α-helical barrel, since the β-barrel, in spite of resembles to the Kresling model, remains stiff due to hydrogen bonds between the β-strands.
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