Writhe

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 (or any closed, simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.^[1]

Writhe of link diagrams

In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings.

A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. If as you travel along a link component and cross over a crossing, the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule.


Positive crossing		Negative crossing

For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.

A Type I Reidemeister move changes the writhe by 1

The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is not an isotopy invariant of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.

Writhe of a closed curve

Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, R³. By viewing the curve from different vantage points, one can obtain different projections and draw the corresponding knot diagrams. Its Wr (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.^[2] Hence, writhe in this situation can take on any real number as a possible value.^[1]

We can calculate Wr with an integral. Let $C$ be a smooth, simple, closed curve and let $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ be points on $C$ . Then the writhe is equal to the Gauss integral

Wr=\frac{1}{4\pi}\int_{C}\int_{C}d\mathbf{r}_{1}\times d\mathbf{r}_{2}\cdot\frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}

Numerically approximating the Gauss integral for writhe of a curve in space

Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of $N$ line segments. A procedure that has first been derived by Levitt ^[3] for the description of protein folding and later been used for supercoiled DNA by Klenin and Langowski ^[4] is to compute

Wr=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j<i}\frac{\Omega_{ij}}{4\pi}

where $\Omega_{ij}/{4\pi}$ is the exact evaluation of the double integral over line segments $i$ and $j$ ; note that $\Omega_{ij}=\Omega_{ji}$ and $\Omega_{i,i+1}=\Omega_{ii}=0$ .^[4]

To evaluate $\Omega_{ij}/{4\pi}$ for given segments numbered $i$ and $j$ , number the endpoints of the two segments 1, 2, 3, and 4. Let $r_{pq}$ be the vector that begins at endpoint $p$ and ends at endpoint $q$ . Define the following quantities:^[4]

n_{1}=\frac{r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|},\; n_{2}=\frac{r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|},\; n_{3}=\frac{r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|},\; n_{4}=\frac{r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}

Then we calculate^[4]

\Omega^{*}=\arcsin\left(n_{1}\cdot n_{2}\right)+\arcsin\left(n_{2}\cdot n_{3}\right)+\arcsin\left(n_{3}\cdot n_{4}\right)+\arcsin\left(n_{4}\cdot n_{1}\right).

Finally, we compensate for the possible sign difference and divide by $4\pi$ to obtain^[4]

\frac{\Omega}{4\pi}=\frac{\Omega^{*}}{4\pi}\text{sign}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).

In addition, other methods to calculate writhe are fully described mathematically and algorithmically in.^[4]

Applications in DNA topology

A simulation of an elastic rod relieving torsional stress by forming coils

DNA will coil if you twist it, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.^[1]

Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically in ^[5] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number” .

References

1 2 3 Bates, Andrew (2005). DNA Topology. Oxford University Press. pp. 36–37. ISBN 0198506554.
↑ Cimasoni, David (2001). "Computing the Writhe of a Knot". Journal of Knot Theory and Its Ramifications 10 (387). doi:10.1142/S0218216501000913.
↑ Levitt, M (1986). "Protein Folding by Restrained Energy Minimization and Molecular Dynamics". J. Mol. Biol. 170: 723–764. doi:10.1016/s0022-2836(83)80129-6.
1 2 3 4 5 6 Klenin, K; Langowski, J (2000). "Computation of writhe in modeling of supercoiled DNA". Biopolymers 54: 307–317. doi:10.1002/1097-0282(20001015)54:5<307::aid-bip20>3.0.co;2-y.
↑ Fuller, F B (1971). "The writhing number of a space curve". Proceedings of the National Academy of Sciences 68: 815–819. doi:10.1073/pnas.68.4.815. PMC 389050. PMID 5279522.