Spontaneous dimensional reduction and ground state degeneracy in a simple chain model

Letter

Spontaneous dimensional reduction and ground state degeneracy in a simple chain model

Tatjana Škrbić, Trinh Xuan Hoang, Achille Giacometti, Amos Maritan, and Jayanth R. Banavar

Phys. Rev. E 104, L012101 – Published 6 July 2021

Abstract

Chain molecules play a key role in the polymer field and in living cells. Our focus is on a new homopolymer model of a linear chain molecule subject to an attractive self-interaction promoting compactness. We analyze the model using simple analytic arguments complemented by extensive computer simulations. We find several striking results: there is a first-order transition from a high-temperature random coil phase to a highly unusual low-temperature phase; the modular ground states exhibit significant degeneracy; the ground state structures exhibit spontaneous dimensional reduction and have a two-layer structure; and the ground states are assembled from secondary motifs of helices and strands connected by tight loops. We discuss the similarities and notable differences between the ground state structures [we call these PoSSuM (Planar Structures with Secondary Motifs)] in the phase and protein native state structures.

Received 2 February 2021
Revised 13 March 2021
Accepted 8 April 2021

DOI:https://doi.org/10.1103/PhysRevE.104.L012101

Physics Subject Headings (PhySH)

Phase transitions

Statistical Physics & Thermodynamics

Authors & Affiliations

Tatjana Škrbić ^*

Department of Physics and Institute for Fundamental Science, University of Oregon, Eugene, Oregon 97403, USA and Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari di Venezia, Campus Scientifico, Edificio Alfa, via Torino 155, 30170 Venezia Mestre, Italy

Trinh Xuan Hoang ^†

Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi 11108, Vietnam

Achille Giacometti ^‡

Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari di Venezia, Campus Scientifico, Edificio Alfa, via Torino 155, 30170 Venezia Mestre, Italy and European Center for Living Technologies (ECLT), Ca' Bottacin, Dorsoduro 3911 Calle Crosera, 30123 Venezia, Italy

Amos Maritan^§

Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, 35131 Padova, Italy

Jayanth R. Banavar ^∥

Department of Physics and Institute for Fundamental Science, University of Oregon, Eugene, Oregon 97403, USA

^*tskrbic@uoregon.edu
^†hoang@iop.vast.ac.vn
^‡achille.giacometti@unive.it
^§amos.maritan@unipd.it
^∥Corresponding author: banavar@uoregon.edu

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Vol. 104, Iss. 1 — July 2021

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Images

Figure 1
Coordinate system used in our study. Here we show four consecutive main chain spheres along the chain. The bond length $b = 1$ sets the length scale of the model. The main chain spheres (shown in gray with centers marked by little red spheres numbered 1 through 4) have a diameter $σ$ larger than $b$ leading to an overlap of successive spheres. The yellow side spheres of diameter $σ_{s}$ are attached to the main chain spheres tangentially in the negative normal direction. The angles between successive bonds are held constant at a value $θ_{0}$ . The range of attractive interactions is denoted by $R$ . $μ_{3}$ is the dihedral angle between the planes ( $π_{1}$ and $π_{2}$ ) defined by sphere centers (1,2,3) and (2,3,4), respectively. It is also the angle between successive binormal vectors in a Frenet coordinate system at sphere centers 2 and 3.
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Figure 2
Sketches of ideal assembly of secondary motifs. Panels I and II show snapshots of an individual helix and strand with perfect commensurability attributes. The model has $θ = θ^{*} = 98 . 213^{\circ}$ and the optimal values of $μ = μ^{*} = 41 . 410^{\circ}$ and $180^{\circ}$ for the helix and strand, respectively. There is commensurability both in the number of main chain spheres per repeat unit (four for the helix and two for the strand) as well as equality of the lengths of the repeat units [helix ( $i, i + 4$ ) distance = strand ( $i, i + 2$ ) distance = 1.512 in units of the bond length]. The small spheres outline the backbone of the chain, the large gray spheres are the main chain spheres (diameter = $4 / 3$ ), the little yellow (for helix) (orange, for strand) spheres are the side spheres (diameter = $2 / 3$ ) pointed in the negative normal direction. A solitary strand has no local contacts, whereas the local energy score per interior helix main chain sphere is $- 3$ arising from energies of $- 1, - 1 / 2, - 1 / 2$ , and $- 1$ for four contacts of sphere $i$ with spheres $i - 4, i - 3, i + 3, i + 4$ , respectively. Panels III, IV, and V show the perfect fit of a helix with a strand; two helices of opposite chirality; and three strands. The three subpanels show three views (front, top, and side) of each of these assemblies, in particular, revealing the two layer structure. Panels III(a), IV(a), and V(a) show a trace of the backbone of the secondary motifs (without main chain or side spheres) and the three nonlocal contacts made by each internal sphere of the secondary motif. For a strand, the total number of nonlocal contacts is six (three from one side and three from the other), whereas for a sphere in a helix, there are four local contacts (yielding a favorable energy of $- 3$ –recall $x = - 1 / 2$ ) and three nonlocal contacts resulting in a net favorable energy of $- 6$ . Indeed, each of the interior spheres of any of these assemblies has a net energy of $- 6$ yielding the highly degenerate ground state. Our analysis of the PoSSuM phase in this paper is carried out with $θ = 97^{\circ}$ in the vicinity of $θ^{*}$ . The flexibility in the $θ$ and $μ$ angles allows an excellent match of the ground state PoSSuM structures with the perfect commensurability structures shown here.
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Figure 3
Gallery of conformations. All the panels depict conformations of chain length $N = 80$ except for VII, which shows domain formation for a $N = 160$ chain. Panel IX is a globular conformation for the case of a chain with no side spheres. The presence of side spheres results in nearly degenerate PoSSuM conformations of two helices (I), three helices (II), a sheet assembled from strands (III), a curved sheet (IV), conformations which include a switch between a helix and strand (V and VI), and a pivotal helix that links two sheets at right angles to each other (VIII). For several panels, two or three perspectives are shown to illustrate the tight packing and the layered structure. The gray circles depict the main chain spheres. The side spheres are not shown. The little circles lie at the centers of the main chain spheres. As a guide to the eye, we color a helix sphere center blue, a strand sphere center red, a turn sphere center yellow, and a hybrid sphere center in the vicinity of a helix-strand switch green. These assignments are made based on $μ$ values, the number of main chain spheres per turn, and the number of local and nonlocal contacts.
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Figure 4
Temperature dependence of the PoSSuM model. (a) Plot of the specific heat as a function temperature for three chain lengths. The peak sharpens and increases approximately linearly with $N$ suggesting that the transition is first order. The inset is a plot of the distribution of the canonical energy for $N = 80$ for three temperatures in the vicinity of the transition. (b) The distribution of $μ$ at three temperatures. At a temperature of 1.0, there is already a signature of incipient secondary motifs. The three peaked structure at the temperature of 0.4 underscores the presence of secondary motifs. (c) The two layered structure of the PoSSuM ground states is shown by the green histogram fitted as the sum of two Gaussians. The red histograms indicate the spatial spread of the side spheres. In order to determine the direction perpendicular to the layers, we used the eigenvector corresponding to the smallest eigenvalue of the moment of inertia matrix. The graph is an average over 40 distinct ground state PoSSuM conformations of $N = 80$ . (d) is the corresponding plot for the globular phase that is obtained when the side chains are removed from the model. Note the lack of layering and the higher spread of the main chain spheres.
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Figure 5
A single constant temperature Monte Carlo run for $N = 80$ at a reduced temperature of 0.595, around 2% below the transition temperature. The energy is plotted as a function of Monte-Carlo time and shows many switches between the unfolded state and the PoSSuM state. Twenty well-formed nearly degenerate PoSSuM configurations with distinct topologies are shown at the bottom of the figure in the same order as their appearance in the Monte Carlo run.
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