Circuit topology

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Circuit topology relations in a chain with two binary contacts. Schematic description of circuit topology.jpg
Circuit topology relations in a chain with two binary contacts.

The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Proteins fold via the formation of contacts of various natures, including hydrogen bonds, disulfide bonds, and beta-beta interactions. [1] RNA molecules fold by forming hydrogen bonds between nucleotides, forming nested or non-nested structures. Contacts in the genome are established via protein bridges including CTCF and cohesins and are measured by technologies including Hi-C. [2] Circuit topology categorises the topological arrangement of these physical contacts, that are referred to as hard contacts (or h-contacts). Furthermore, chains can fold via knotting (or the formation of "soft" contacts (s-contacts)). Circuit topology uses a similar language to categorise both "soft" and "hard" contacts, and provides a full description of a folded linear chain. In this framework, a "circuit" refers to a segment of the chain where each contact site within the segment forms connections with other contact sites within the same segment, and thus is not left unpaired. A folded chain can thus be studied based on its constituting circuits.

Contents

A simple example of a folded chain is a chain with two hard contacts. For a chain with two binary contacts, three arrangements are available: parallel (P), series (S), and crossed (X). For a chain with n contacts, the topology can be described by an n by n matrix in which each element illustrates the relation between a pair of contacts and may take one of the three states, P, S and X. Multivalent contacts can also be categorised in full or via decomposition into several binary contacts. Similarly, circuit topology allows for the classification of the pairwise arrangements of chain crossings and tangles, thus providing a complete 3D description of folded chains. Furthermore, one can apply circuit topology operations to soft and hard contacts to generate complex folds, using a bottom-up engineering approach.

Both knot theory and circuit topology aim to describe chain entanglement, making it important to understand their relationship. Knot theory considers any entangled chain as a connected sum of prime knots, which are themselves undecomposable. Circuit topology splits any entangled chains (including prime knots) into basic structural units called soft contacts, and lists simple rules on how soft contacts can be put together. [3] [4] An advantage of circuit topology is that it can be applied to open linear chains with intra-chain interactions, so-called hard contacts. [5] This enabled topological analysis of proteins and genomes, which are often described as "unknot" in knot theory. [6] [7] Finally, circuit topology enables studying interactions between hard contacts and entanglements and can identify slip knots, while knot theory typically overlooks hard contacts and split knots. Thus, circuit topology serves as a complementary approach to knot theory.

Circuit topology has implications for folding kinetics and molecular evolution and has been applied to engineer polymers including molecular origami. [8] Circuit topology along with contact order and size are determinants of the folding rate of linear polymers. [9] The approach can also be used for medical applications including the prediction of pathogenicity of mutations.

Further reading

Related Research Articles

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Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, responding to stimuli, providing structure to cells and organisms, and transporting molecules from one location to another. Proteins differ from one another primarily in their sequence of amino acids, which is dictated by the nucleotide sequence of their genes, and which usually results in protein folding into a specific 3D structure that determines its activity.

<span class="mw-page-title-main">Topology</span> Branch of mathematics

Topology is the part of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

<span class="mw-page-title-main">Protein folding</span> Change of a linear protein chain to a 3D structure

Protein folding is the physical process by which a protein, after synthesis by a ribosome as a linear chain of amino acids, changes from an unstable random coil into a more ordered three-dimensional structure. This structure permits the protein to become biologically functional.

<span class="mw-page-title-main">Knot theory</span> Study of mathematical knots

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

<span class="mw-page-title-main">Soft matter</span> Subfield of condensed matter physics

Soft matter or soft condensed matter is a subfield of condensed matter comprising a variety of physical systems that are deformed or structurally altered by thermal or mechanical stress of the magnitude of thermal fluctuations. These materials share an important common feature in that predominant physical behaviors occur at an energy scale comparable with room temperature thermal energy, and that entropy is considered the dominant factor. At these temperatures, quantum aspects are generally unimportant. Soft materials include liquids, colloids, polymers, foams, gels, granular materials, liquid crystals, flesh, and a number of biomaterials. When soft materials interact favorably with surfaces, they become squashed without an external compressive force. Pierre-Gilles de Gennes, who has been called the "founding father of soft matter," received the Nobel Prize in Physics in 1991 for discovering that methods developed for studying order phenomena in simple systems can be generalized to the more complex cases found in soft matter, in particular, to the behaviors of liquid crystals and polymers.

<span class="mw-page-title-main">Protein structure</span> Three-dimensional arrangement of atoms in an amino acid-chain molecule

Protein structure is the three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers – specifically polypeptides – formed from sequences of amino acids, which are the monomers of the polymer. A single amino acid monomer may also be called a residue, which indicates a repeating unit of a polymer. Proteins form by amino acids undergoing condensation reactions, in which the amino acids lose one water molecule per reaction in order to attach to one another with a peptide bond. By convention, a chain under 30 amino acids is often identified as a peptide, rather than a protein. To be able to perform their biological function, proteins fold into one or more specific spatial conformations driven by a number of non-covalent interactions, such as hydrogen bonding, ionic interactions, Van der Waals forces, and hydrophobic packing. To understand the functions of proteins at a molecular level, it is often necessary to determine their three-dimensional structure. This is the topic of the scientific field of structural biology, which employs techniques such as X-ray crystallography, NMR spectroscopy, cryo-electron microscopy (cryo-EM) and dual polarisation interferometry, to determine the structure of proteins.

<span class="mw-page-title-main">Molecular knot</span> Molecule whose structure resembles a knot

In chemistry, a molecular knot is a mechanically interlocked molecular architecture that is analogous to a macroscopic knot. Naturally-forming molecular knots are found in organic molecules like DNA, RNA, and proteins. It is not certain that naturally occurring knots are evolutionarily advantageous to nucleic acids or proteins, though knotting is thought to play a role in the structure, stability, and function of knotted biological molecules. The mechanism by which knots naturally form in molecules, and the mechanism by which a molecule is stabilized or improved by knotting, is ambiguous. The study of molecular knots involves the formation and applications of both naturally occurring and chemically synthesized molecular knots. Applying chemical topology and knot theory to molecular knots allows biologists to better understand the structures and synthesis of knotted organic molecules.

<span class="mw-page-title-main">Nucleic acid double helix</span> Structure formed by double-stranded molecules

In molecular biology, the term double helix refers to the structure formed by double-stranded molecules of nucleic acids such as DNA. The double helical structure of a nucleic acid complex arises as a consequence of its secondary structure, and is a fundamental component in determining its tertiary structure. The structure was discovered by Rosalind Franklin, her student Raymond Gosling, James Watson, and Francis Crick, while the term "double helix" entered popular culture with the 1968 publication of Watson's The Double Helix: A Personal Account of the Discovery of the Structure of DNA.

<span class="mw-page-title-main">Intrinsically disordered proteins</span> Protein without a fixed 3D structure

In molecular biology, an intrinsically disordered protein (IDP) is a protein that lacks a fixed or ordered three-dimensional structure, typically in the absence of its macromolecular interaction partners, such as other proteins or RNA. IDPs range from fully unstructured to partially structured and include random coil, molten globule-like aggregates, or flexible linkers in large multi-domain proteins. They are sometimes considered as a separate class of proteins along with globular, fibrous and membrane proteins.

The contact order of a protein is a measure of the locality of the inter-amino acid contacts in the protein's native state tertiary structure. It is calculated as the average sequence distance between residues that form native contacts in the folded protein divided by the total length of the protein. Higher contact orders indicate longer folding times, and low contact order has been suggested as a predictor of potential downhill folding, or protein folding that occurs without a free energy barrier. This effect is thought to be due to the lower loss of conformational entropy associated with the formation of local as opposed to nonlocal contacts.

A peculiarity of thermal motion of very long linear macromolecules in entangled polymer melts or concentrated polymer solutions is reptation. Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another. Pierre-Gilles de Gennes introduced the concept of reptation into polymer physics in 1971 to explain the dependence of the mobility of a macromolecule on its length. Reptation is used as a mechanism to explain viscous flow in an amorphous polymer. Sir Sam Edwards and Masao Doi later refined reptation theory. Similar phenomena also occur in proteins.

<span class="mw-page-title-main">History of knot theory</span>

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In chemistry, topology provides a way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms ethereal wave functions must fit together. Molecular topology is a part of mathematical chemistry dealing with the algebraic description of chemical compounds so allowing a unique and easy characterization of them.

<span class="mw-page-title-main">Knotted protein</span> Proteins with backbone entangled in a knot

Knotted proteins are proteins whose backbones entangle themselves in a knot. One can imagine pulling a protein chain from both termini, as though pulling a string from both ends. When a knotted protein is “pulled” from both termini, it does not get disentangled. Knotted proteins are very rare, making up only about one percent of the proteins in the Protein Data Bank, and their folding mechanisms and function are not well understood. Although there are experimental and theoretical studies that hint to some answers, systematic answers to these questions have not yet been found.

<span class="mw-page-title-main">Polymer architecture</span>

Polymer architecture in polymer science relates to the way branching leads to a deviation from a strictly linear polymer chain. Branching may occur randomly or reactions may be designed so that specific architectures are targeted. It is an important microstructural feature. A polymer's architecture affects many of its physical properties including solution viscosity, melt viscosity, solubility in various solvents, glass transition temperature and the size of individual polymer coils in solution.

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<span class="mw-page-title-main">Arc diagram</span> Graph drawing with vertices on a line

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<span class="mw-page-title-main">Knotted polymers</span>

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An intramolecular cyclization reaction is where the growing polymer chain reacts with a vinyl functional group on its own chain, rather than with another growing chain in the reaction system. In this way the growing polymer chain covalently links to itself in a fashion similar to that of a knot in a piece of string. As such, single chain cyclized/knotted polymers consist of many of these links, as opposed to other polymer architectures including branched and crosslinked polymers that are formed by two or more polymer chains in combination.

<span class="mw-page-title-main">Alireza Mashaghi</span> Physician-scientist and biophysicist at Leiden University

Alireza Mashaghi is a physician-scientist and biophysicist at Leiden University. He is known for his contributions to single-molecule analysis of chaperone assisted protein folding, molecular topology and medical systems biophysics and bioengineering. He is a leading advocate for interdisciplinary research and education in medicine and pharmaceutical sciences.

Topological polymers may refer to a polymeric molecule that possesses unique spatial features, such as linear, branched, or cyclic architectures. It could also refer to polymer networks that exhibit distinct topologies owing to special crosslinkers. When self-assembling or crosslinking in a certain way, polymeric species with simple topological identity could also demonstrate complicated topological structures in a larger spatial scale. Topological structures, along with the chemical composition, determine the macroscopic physical properties of polymeric materials.

References

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