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Circuit Topology Analysis of Polymer Folding Reactions - PubMed

  • ️Wed Jan 01 2020

Review

Circuit Topology Analysis of Polymer Folding Reactions

Maziar Heidari et al. ACS Cent Sci. 2020.

Abstract

Circuit topology is emerging as a versatile measure to classify the internal structures of folded linear polymers such as proteins and nucleic acids. The topology framework can be applied to a wide range of problems, most notably molecular folding reactions that are central to biology and molecular engineering. In this Outlook, we discuss the state-of-the art of the technology and elaborate on the opportunities and challenges that lie ahead.

Copyright © 2020 American Chemical Society.

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Conflict of interest statement

The authors declare no competing financial interest.

Figures

Figure 1
Figure 1

Configuration of a folded chain with 6 contact pairs is depicted in the left panel. Each contact along the chain is marked with different colors. The corresponding circuit topology of the folded chain is shown in the middle panel. The loop pairs are wired in series (S), parallel (P), or cross (X) topology. The inverse parallel topology (P–1) is assumed to be the same as parallel topology (P). Since the arrangement of the contacts is analogous to the arrangement of elements in an electrical circuit, the mathematical framework is dubbed “circuit topology”. The circuit topology entropy of a folded chain configuration with 9-contact pairs is shown in right panel as a function of the series (ns) and parallel (np) topological fractions. Reproduced from ref (22) 2015 with permission from the Royal Society of Chemistry.

Figure 2
Figure 2

(a) Configuration of a folded chain with 11 contact pairs is illustrated. The contacts can exemplify local loops in the RNA molecules or β–β interactions in proteins. The circuit topology matrix built on the high resolution information, that is, the contact pairs, is shown in the top middle. The contact pairs are categorized in series (S), parallel (P), cross (X), and inverted parallel (P–1). If the local contacts are grouped into single coarse-grained (CG) contact, the number of contacts reduce to four CG contacts, and the new low-resolution circuit topology is shown in the top right matrix. (b) Two molecular chains with contact pairs formed in the concerted series (CS) and concerted parallel (CP) arrangements are illustrated. In these cases, a single binding site on the chain forms two contacts and its valency rises up to two. By slight change of the definition, one can regard CP as P and CS as S, although a full consideration is also an option of course. (c) Snapshot and circuit topology of the folding trajectory of the N-terminal domain of ribosomal protein L9 (2HFV). First, the N-terminal hairpin forms. Next, the α helix forms a contact in concerted parallel with the hairpin, contacting the first strand. This second contact helps initiate the hydrogen bonding of the third β strand to the first, yielding three contacts in concerted parallel relation. Contact marked with * is not numbered in the molecular structure because the final α helix was excluded in folding simulations. The dashed and solid lines in the circuit topology matrix indicate the two folding steps.

Figure 3
Figure 3

(a) Schematic representation of the three mechanical unfolding methods: pulling, threading, and pulling by threading. The number of pathways and efficiency of unfolding are listed for a 3-contact chain with a specific topology, L1PL2, L1SL3, L2SL3 (see eq 1). (b) Number of pathways for unfolding a 5-contact chain using pulling, pulling by threading, and threading methods. The total number of contact pairs in each topology is Ns + Np + Nx = 10, where Ns, Np, and Nx are the numbers of contact pairs in series, parallel and cross topologies, respectively. The plots in the figure show the number pathways as a function of Ns and Np (Nx follows from these numbers). The color codes the number of pathways. Reproduced from ref (41). Copyright 2018 American Chemical Society.

Figure 4
Figure 4

Folding time maps of a chain under different types of internal restraints. The folding time of a chain with no external perturbation (control case) is shown on the top. The folding time map in the presence of a chaperone is shown in the lower panels, where the chaperone binds to a native contact of the chain (lower left) and where it binds to the chain and forms an external contact (lower right). In both cases, the diffusion of the contact pairs of the chaperone is three times smaller than the diffusion of the chain’s native binding sites. The axes of the ternary plots are topological fractions in series (S), parallel (P), and cross (X). Reproduced from ref (42) 2017 with permission from the PCCP Owner Societies.

Figure 5
Figure 5

Fraction of topological circuits and corresponding averaged contact order (CO) as a function of the confining radius, Rc for a linear polymer chain (see eq 1). The dashed line indicates the transition confinement radius, Rct, at which all topological fractions are equal. Topological circuits of two intramolecular loops under spherical confinement with radius Rc are illustrated on the right side. Reproduced from ref (45) 2017 with permission from the Royal Society of Chemistry.

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