# A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Our Protein Dataset

#### 2.2. Numerical Simulations of a Chain of Tethered Tangent Spheres

_{att}= 1.6σ between all pairs of spheres and magnitude ε, which sets the characteristic energy scale. The attractive interaction causes the chain to become compact at low temperatures.

_{i}, i = 1, 2, … M. Each simulation represents a replica or a system copy in thermal equilibrium. The key advantage is the possibility of swapping replicas at different temperatures without affecting the equilibrium condition at each temperature. This permits rapid equilibration even when there is a rugged free energy landscape. In each Monte Carlo (MC) simulation of one replica, new moves are accepted with the standard Metropolis acceptance probabilities [67]. We ensure that the number of swaps that entail the exchange of replicas is large enough to ensure the fidelity of the statistics. The efficiency of the RE scheme depends on the number of replicas, the selected set of temperatures as well as of the swap moves frequency. For best performance, the acceptance rate of swaps is tuned to be around 20% [68]. The RE simulation results are conveniently analyzed using the weighted histogram analysis method [69]. We employed 30 replicas with a finer temperature mesh at lower values of the reduced temperatures (k

_{B}T/ε) in the range k

_{B}T/ε = 0.3–0.5 with a separation of neighboring temperatures of 0.02. In the k

_{B}T/ε = 0.5–1 interval, the separation of neighboring temperatures was 0.05, and for k

_{B}T/ε = 1–4, the separation interval was 0.2. We allowed for the RE swaps only between neighboring temperatures. The exchange moves were attempted every 100 MC steps per monomer. The length of the simulations was 10

^{9}MC steps per monomer and per replica.

_{B}T/ε ~ 3 (ε denotes the energy scale of attraction), there is a coil-to-globule transition (signaled by a kink in the specific heat per bead C

_{V}/Nk

_{B}). This is a finite-size counterpart of the θ-point for this system. At low temperatures k

_{B}T/ε ~ 0.4, there is a second ‘transition’ into a compact globule phase, signaled by the maxima of the specific heat per bead C

_{V}/Nk

_{B}. The results we will present for the tangent sphere model are at infinite temperature and k

_{B}T/ε = 0.3.

## 3. General Considerations

#### Three-Body Interactions and the Self-Avoidance of a Continuum Tube

## 4. Results

#### 4.1. Power Law Scaling

_{α}atoms along the backbones of globular proteins when the Ramachandran ω angles characterizing a consecutive triplet have canonical values of |ω| ~ 180°. Case (a) occurs in around 99.7% of the cases in globular proteins yielding the trans isomeric conformation of a peptide backbone, where the two neighboring C

_{α}atoms are on opposite sides of the peptide bond, with a bond length approximately equal to 3.81 Å [63]. (b) Consecutive triplets (quartets) of C

_{α}atoms in globular proteins in which at least one of the two Ramachandran ω angles has a rare non-canonical value of |ω|≈ 0° that occurs in ~0.3% of cases. This happens when two neighboring C

_{α}atoms are on the same side of the peptide bond, resulting in a shorter bond length of around ~2.95 Å [63]. This corresponds to the so-called cis-conformation of a protein backbone [80]. Cases (a) and (b) are combined for the quartets because they show very similar behavior. (c) Three (four) points selected from a two-step (three-step) self-avoiding random walk of a tangent sphere model (hard spheres of diameter 3.81 Å and bond length 3.81 Å with no other interaction besides the non-overlapping of hard spheres (polymer at infinite temperature). (d) Three (four) points selected from a two-step (three-step) self-avoiding random walk of a tangent sphere model (spheres of diameter 3.81 Å and bond length 3.81 Å) subject to an attractive square-well interaction of range R

_{att}= 1.6σ ≈ 6Å (polymer at low temperature). (e) Three (four) points chosen randomly within a three-dimensional sphere of unit radius. (f) Three (four) points selected as points on a two-step (three-step) random walk (no self-avoidance or steric constraints) in three dimensions with a fixed bond length of 3.81 Å (corresponding to the distance between consecutive C

_{α}atoms in proteins [63]).

**Figure 2.**(

**a**) Cumulative probability distributions of the inverse radii X = 1/R of circles drawn through three consecutive points along different classes of chains: (blue) 965,122 triplets of the backbones of globular proteins in our data set (defined by C

_{α}atoms) when both the Ramachandran ω angles characterizing a consecutive triplet have canonical values of |ω| ≈ 180°; (red) 5774 consecutive triplets of C

_{α}atoms in globular proteins in which at least one of the two Ramachandran ω angles is|ω| ≈ 0°; (green) 16,391,622 triplets taken from ≈200,000 low temperature (k

_{B}T/ε = 0.3) configurations, obtained using replica-exchange (RE) simulations, of a chain of 80 tangent spheres of diameter σ with an attractive square well potential of range R

_{att}= 1.6σ ≈ 6 Å; (orange) 25,598,976 triplets obtained from MC simulations at T = ∞; (purple) 143,557,206 triplets of points chosen uniformly from within a unit sphere in three dimensions; and (black) 100,000,000 two-step random walks in three dimensions with fixed bond length of 3.81 Å. The gray dashed line has a slope of 2 and is a guide to the eye. (

**b**) Cumulative probability distributions of the inverse radii X = 1/R of spheres, whose surface passes through four consecutive points in different classes of chains: (blue) 957,723 local quartets along the backbones of globular proteins (employing C

_{α}atoms); (green) 16,181,473 local quartets selected from ≈200,000 chain configurations, obtained from RE simulations, of a chain of 80 tangent spheres of diameter σ subject to an attractive square-well potential of range R

_{att}= 1.6σ ≈ 6 Å in the low-temperature phase (k

_{B}T/ε = 0.3); (orange) 25,270,784 local quartets obtained from MC simulations at T = ∞; (purple) 112,754,340 quartets of points chosen uniformly within a unit sphere in three dimensions; and (black) 100,000,000 three-step random walks in three dimensions with a fixed bond length of 3.81 Å. The gray dashed line is a guide to the eye and has a slope of 1. In all simulations, the bond lengths have been chosen to be 3.81 Å, equal to the mean value of the distance between the two consecutive C

_{α}atoms along the protein chain. The distinctive behaviors of the purple curves (corresponding to the random points cases) occur because one can obtain circles of arbitrarily small radii, a situation precluded in the other cases due to steric considerations. The behaviors of the tangent polymer model at high and low temperatures are essentially the same. The local behavior is governed by the same steric constraints in both cases, and the CDF does not change. In contrast, for real polymers, recent experimental studies [81,82] have shown the importance of mechanical properties in determining the local curvature in the context of super-lubricity at the single-molecule level.

_{α}atoms along the protein chain are much closer to one another, and the backbone in many of these cases has PRO residues. This stiffens up the protein backbone with respect to the canonical case, permitting large bond bending angles, θ, that can almost reach ≈180°.

#### 4.2. Rationalization of the Power-Law Exponent

^{d−2}

^{d−2}[1 − (Xb/2)

^{2}]

^{(d−3)/2}

^{2}is much smaller than 1 and yields good power law behavior, as observed in Figure 2a and Figure 3a. The pivotal quantity that determines the asymptotic exponent is the behavior of p(θ) when θ approaches 180° and the three points become co-linear. The difference in behavior in 2 and 3 dimensions is shown in Figure 3b. The numerical simulations are in good accord with the prediction of Equation (1).

#### 4.3. Chain Geometries

_{α}≈ 92°, the β-residues for θ

_{β}≈ 120°, and the loop-residues have a pronounced maximum close to the helical value and a less prominent maximum around 111°, see Figure 6a. These peaks are also reflected in the high-density regions in the (θ,μ) cross plot of protein native state structures, shown in Figure 4c. Figure 6b shows the histograms of the relevant non-local length scales for all five cases. All five curves exhibit a single peak denoting a relevant non-local length scale. Table 1 is a compilation of these characteristic local and non-local length scales.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Sketch of the axis of a self-avoiding continuum tube (depicted in blue). The points A, B, and C lie alongside each other on the tube axis whereas point D is a nearby point from another part of the tube. The three-body prescription is to draw circles through all triplets of points on the tube axis and ensure that none of the radii is smaller than the tube radius. For a local triplet of points, one obtains the local radius of curvature whereas the non-local radius is a measure of the distance of closest approach of two parts of the tube.

**Figure 3.**(

**a**) Cumulative probability distributions of the inverse radii X = 1/R of the circles drawn through three points of a two-step random walk with a fixed bond length of 3.81 Å, when the walk is performed in two dimensions (red) and in three dimensions (blue). (

**b**) The distribution of the bond bending angle θ in the two-step random walk in two dimensions is uniform p(θ) = const. (red histogram), while, in three dimensions, p(θ) = sin θ (blue histogram and the green line). (

**c**) Cumulative probability distributions of the inverse radii X = 1/R of the spheres drawn through four points of a three-step random walk with a fixed bond length of 3.81 Å, when the walk is performed in three dimensions (red) and for a constrained random walk. The constraint arises because the first three points (first two steps of the random walk) are sampled from a plane (sampling in two dimensions—after all, any three points do lie in a plane), while the final step (fourth point) is sampled in full three-dimensional space (blue). The constrained random walk corresponds to a fractal dimension 2 < d < 3 for the effective sampling of the variables controlling 1/R and reduces the steepness of the power law.

**Figure 4.**(θ,μ) cross plots of the bond bending angle θ versus dihedral angle μ for 966,505 randomly chosen monomers belonging to two different polymer classes and for the same number of residues in protein native state structures. (

**a**) (light green) Polymer chain consisting of 80 tangent spheres of diameter σ subject to an attractive square well potential of range R

_{att}= 1.6σ ≈ 6 Å at low temperature (k

_{B}T/ε = 0.3) studied using RE simulations. (

**b**) (orange) Polymer chain consisting of 80 tangent spheres of diameter σ at infinite temperature accessed by means of MC simulations with the only interaction being steric avoidance of all pairs of spheres. (

**c**) (blue) For 966,505 residues of the 4391 globular proteins in our data set.

**Figure 5.**Frequency distribution of the sequence separation |i − j| along the chain of the nearest non-local contact of bead i found at location j. In panel (

**a**), the blue points indicate 313,574 α-residues out of a total of 975,287 residues in our data set from 4391 globular proteins; the red points are 214,501 β-beads; and the purple points denote an analysis of 442,821 loop-residues. (

**b**) the green and yellow points show the distinct smooth behaviors of a tangent polymer model at low (k

_{B}T/ε = 0.3) and infinite temperatures. The behavior is monotonic, with the closest non-local contacts always being close along the sequence. The inset shows the specific heat per bead C

_{V}/Nk

_{B}for an 80-bead long tangent sphere polymer as a function of the reduced temperature k

_{B}T/ε. There are two continuous (second order) ‘transitions’: at a high temperature ~3, there is a coil-to-globule transition and at ~0.4, there is a transition into a compact globule.

**Figure 6.**(

**a**) Frequency distribution of the local radius of curvature R for five different classes of consecutive triplets. We consider only pure protein triplets in which all three residues are in the same structural class. There are 256,154 α triplets (blue) comprising ~26% of all 970,896 triplets. There are 134,643 β triplets (red) and 313,923 loop triplets (purple). There are 16,391,622 low-temperature polymer triplets (green) and 25,270,784 infinite-temperature triplets (orange). (

**b**) Frequency distributions of the distances to the closest non-local contact, defined as |j − i| ≥ 3, of monomers belonging to different classes with the same color code as in panel (

**a**).

**Figure 7.**Representative conformations of chains of length 80: (

**A**) a low-temperature globule and (

**B**) an infinite-temperature coil of a generic tangent polymer chain; (

**C**) all-α protein [PDB code: 3bqp, chain B]; (

**D**) all-β protein [PDB code: 1bdo, chain A]; and (

**E**) α/β protein [PDB code: 3l9, chain X]. The color coding of the conformation does not represent secondary motifs but rather depicts how far a monomer is along the sequence. The chain beginning is depicted in red color that morphs towards blue at the chain end. The contact maps shown in the other panels are those indicating the closest non-local contact j of a given monomer. The black points indicate those that are found within 6 Å, and the red points are those that are found further away than 6 Å of these configurations. The choice of 6 Å is dictated by the fact that the radial distribution function of proteins exhibits a pronounced minimum at this value [61].

**Figure 8.**Scaled values of the local radius and minimal non-local distance for the five conformations shown in Figure 7. For each residue of the three proteins, depending on its type (‘α’, ‘β’, or ‘loop’), these quantities are appropriately scaled with the value of the corresponding characteristic length scale presented in Table 1. (

**a**) All-α protein 80 residues long [PDB code: 3bqp, chain B]. (

**b**) All-β protein 80 residues long [PDB code: 1bdo, chain A]. (

**c**) α/β protein 80 residues long [PDB code: 3l9, chain X]. In the top three panels, the blue ribbons indicate α-helical parts of the sequence, the red ribbons indicate β-sheets, and the purple ribbons indicate loop regions. (

**d**) A tangent polymer conformation at k

_{B}T/ε = 0.3. (

**e**) A tangent polymer conformation at T = ∞.

**Table 1.**Characteristic local and non-local length scales for three residue types in proteins and for monomers of a tangent polymer chain in the low and high-temperature phases.

Monomers | Most Frequent Value of Local Radius [Å] | Relevant Non-Local Length Scale [Å] |
---|---|---|

α—residues in proteins | 2.73 | 5.06 |

β—residues in proteins | 3.6 | 4.67 |

loop—residues in proteins | 2.75 | 5.26 |

Spheres in tangent polymers at low T | 2.21 | 3.81 |

Spheres in tangent polymers at T = ∞ | 2.47 | 7.72 |

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## Share and Cite

**MDPI and ACS Style**

Škrbić, T.; Giacometti, A.; Hoang, T.X.; Maritan, A.; Banavar, J.R.
A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures. *Polymers* **2024**, *16*, 502.
https://doi.org/10.3390/polym16040502

**AMA Style**

Škrbić T, Giacometti A, Hoang TX, Maritan A, Banavar JR.
A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures. *Polymers*. 2024; 16(4):502.
https://doi.org/10.3390/polym16040502

**Chicago/Turabian Style**

Škrbić, Tatjana, Achille Giacometti, Trinh X. Hoang, Amos Maritan, and Jayanth R. Banavar.
2024. "A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures" *Polymers* 16, no. 4: 502.
https://doi.org/10.3390/polym16040502