Writhe

knot-crossing-plus

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knot-crossing-minus

Reidemeister move 1

File:Simulation of an elastic rod relieving torsional stress by forming coils.ogv Writhe is a concept in mathematics and topology that quantifies the amount of twisting or coiling of a curve, such as a DNA molecule or a knot. It is particularly significant in the study of DNA supercoiling, where it helps in understanding the structural properties of DNA and its biological functions. The writhe of a closed curve in three-dimensional space is a measure of its departure from being planar. It is a scalar quantity that can be positive or negative, depending on the direction of the twists.

Definition

The writhe of a curve can be mathematically defined using the Gauss linking integral, which is a measure of the linking number between two curves. For a single closed curve, the writhe is calculated by considering the curve as being split into two closely spaced, parallel curves, and then computing the linking number of these two curves. This approach allows the writhe to capture the global topology of the curve, including its self-intersections and overall shape.

Calculation

To calculate the writhe, one typically projects the three-dimensional curve onto a plane and counts the number of crossings, taking into account their signs (positive or negative) based on the orientation of the crossing. The calculation involves integrating over all possible directions of projection, which makes the writhe a non-local geometric property of the curve. This integral calculation reflects the complexity and the global nature of the writhe as a topological invariant.

Applications

In biology, the concept of writhe is crucial for understanding the behavior of DNA molecules. DNA supercoiling, which is influenced by the writhe, affects the accessibility of the genetic code for transcription and replication. Enzymes such as topoisomerases can change the writhe of DNA, thereby regulating its supercoiling and influencing biological processes.

In knot theory, a branch of topology, the writhe is used to distinguish between different knots and links. It is part of the Jones polynomial, a knot invariant that is used for this purpose. The writhe contributes to the understanding of the spatial configuration of knots and their mathematical properties.

See Also

• Knot theory
• DNA supercoiling
• Topoisomerase
• Gauss linking integral