A **superhelix** is a molecular structure in which a helix is itself coiled into a helix. This is significant to both proteins and genetic material, such as overwound circular DNA.

The earliest significant reference in molecular biology is from 1971, by F. B. Fuller:

A geometric invariant of a space curve, the writhing number, is defined and studied. For the central curve of a twisted cord the writhing number measures the extent to which coiling of the central curve has relieved local twisting of the cord. This study originated in response to questions that arise in the study of supercoiled double-stranded DNA rings.

^{ [1] }

About the writhing number, mathematician W. F. Pohl says:

It is well known that the writhing number is a standard measure of the global geometry of a closed space curve.

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Contrary to intuition, a topological property, the linking number, arises from the geometric properties twist and writhe according to the following relationship:

*L*_{k}=*T*+*W*,

where *L*_{k} is the linking number, *W* is the writhe and *T* is the twist of the coil.

The linking number refers to the number of times that one strand wraps around the other. In DNA this property does not change and can only be modified by specialized enzymes called topoisomerases.

- DNA supercoil (superhelical DNA)
- Knot theory

The **alpha helix** (**α-helix**) is a common motif in the secondary structure of proteins and is a right hand-helix conformation in which every backbone N−H group hydrogen bonds to the backbone C=O group of the amino acid located four residues earlier along the protein sequence.

A **helix**, plural **helixes** or **helices**, is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word *helix* comes from the Greek word *ἕλιξ*, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid.

In mathematics and physics, the **right-hand rule** is a common mnemonic for understanding orientation of axes in three-dimensional space.

In geometry, a **secant** is a line that intersects a curve at a minimum of two distinct points. The word *secant* comes from the Latin word *secare*, meaning *to cut*. In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points.

In mathematics, a **modular equation** is an algebraic equation satisfied by *moduli*, in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.

The **nucleoid** is an irregularly shaped region within the prokaryotic cell that contains all or most of the genetic material. The chromosome of a prokaryote is circular, and its length is very large compared to the cell dimensions needing it to be compacted in order to fit. In contrast to the nucleus of a eukaryotic cell, it is not surrounded by a nuclear membrane. Instead, the nucleoid forms by condensation and functional arrangement with the help of chromosomal architectural proteins and RNA molecules as well as DNA supercoiling. The length of a genome widely varies and a cell may contain multiple copies of it.

In knot theory, there are several competing notions of the quantity **writhe**, or *Wr*. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve in such a way that does not change its topology, one may still change its writhe.

In knot theory, a knot or link diagram is **alternating** if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is **alternating** if it has an alternating diagram.

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 term entered popular culture with the publication in 1968 of *The Double Helix: A Personal Account of the Discovery of the Structure of DNA* by James Watson.

The **Tait conjectures** are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

In the elementary differential geometry of curves in three dimensions, the **torsion** of a curve measures how sharply it is twisting out of the plane of curvature. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

**DNA supercoiling** refers to the over- or under-winding of a DNA strand, and is an expression of the strain on that strand. Supercoiling is important in a number of biological processes, such as compacting DNA, and by regulating access to the genetic code, DNA supercoiling strongly affects DNA metabolism and possibly gene expression. Additionally, certain enzymes such as topoisomerases are able to change DNA topology to facilitate functions such as DNA replication or transcription. Mathematical expressions are used to describe supercoiling by comparing different coiled states to relaxed B-form DNA.

In geometry, a **vertex**, often denoted by letters such as , , , , is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

In geometry, an **edge** is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a **side**. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.

**Molecular models of DNA structures** are representations of the molecular geometry and topology of deoxyribonucleic acid (DNA) molecules using one of several means, with the aim of simplifying and presenting the essential, physical and chemical, properties of DNA molecular structures either *in vivo* or *in vitro*. These representations include closely packed spheres made of plastic, metal wires for *skeletal models*, graphic computations and animations by computers, artistic rendering. Computer molecular models also allow animations and molecular dynamics simulations that are very important for understanding how DNA functions *in vivo*.

**Nucleic acid structure** refers to the structure of nucleic acids such as DNA and RNA. Chemically speaking, DNA and RNA are very similar. Nucleic acid structure is often divided into four different levels: primary, secondary, tertiary, and quaternary.

**Ribbon theory** is a strand of mathematics within topology that has seen particular application as regards DNA.

In addition to the variety of verified DNA structures, there have been a range of proposed DNA models that have either been disproven, or lack evidence.

In mathematics by a **ribbon** is meant a smooth space curve given by a three-dimensional vector , depending continuously on the curve arc-length , together with a smoothly varying unit vector perpendicular to at each point.

In mathematics, **twist** is the rate of rotation of a smooth ribbon around the space curve , where is the arc length of and a unit vector perpendicular at each point to . Since the ribbon has edges and the twist measures the average winding of the curve around and along the curve . According to Love (1944) twist is defined by

- ↑ Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF).
*Proceedings of the National Academy of Sciences*.**68**(4): 815–819. Bibcode:1971PNAS...68..815B. doi: 10.1073/pnas.68.4.815 . MR 0278197. PMC 389050 . PMID 5279522. - ↑ Pohl, William F. (1968). "The self-linking number of a closed space curve".
*Journal of Mathematics and Mechanics*.**17**(10): 975–985. doi: 10.1512/iumj.1968.17.17060 . MR 0222777.

- Weisstein, Eric W. "Călugăreanu Theorem".
*MathWorld*. - Weisstein, Eric W. "Writhe".
*MathWorld*. - Weisstein, Eric W. "Twist".
*MathWorld*. - DNA Structure and Topology at Molecular Biochemistry II: The Bello Lectures.

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