•  Both methods have better performance on the 3.1 knot, a much simpler knot than the 9.21.

•  When only considering the most prevalent knot type, the two methods are practically the same.

•  For both methods, the dashed lines lie slightly below the solid, indicating that when a method’s closures return the correct knot type most often, it is usually found by at least 50% of the closures.

Introduction

Comparing the extent to which closure methods recover the knot type of a closed knot from open subarcs

Emily Vecchia and Dr. Eric J. Rawdon ! University of St. Thomas

•  An open chain knot is a single curve in three-dimensional space with two endpoints.

•  Numerous physical systems such as polymers, DNA, and proteins contain open chain knots.

•  Understanding the connection between structure and function in these physical systems requires a definition of open chain knotting.

Thanks to Dr. Eric Rawdon, Erin Brine-Doyle, Madeline Shogren, Dr. Rob Scharein (KnotPlot), and Dr. Jason Canterella (plCurve).

Methodology

Motivation

•  Each tested open chain is taken as a subchain from a closed knot and classified using each closure method. These classifications are then used to see how well each closure method predicts the knot type of the closed knot from which the open chain came.

•  The process begins with the generation of 100 random closed 100-edge polygons in three-space for 23 different knot types.

•  The knotting in each open subchain (10,000 for each polygon) is classified using the different closure methods.

•  The knot type of the closed polygon is compared to the knot types of its subchains when they have been closed using each of the closure methods.

•  Classical knot theory focuses on knotted loops with no endpoints. Open chain knotting is not yet defined mathematically.

•  The goal is to analyze and compare different methods researchers have used for classifying knots in open chains by determining how well each closure method predicts traditional knot theory.

•  In practice, one may not know the amount of edges “missing” for a subchain. As is, the random arc method cannot be used in this situation. We would like to test ways of estimating the edges missing.

•  We will test a variation of the infinite method in which the rays are extended in different directions.

Results, statistics, and analysis

Protein folding: To knot or not to knot?, Eugene Shakhnovich, January 2011.

Protein structure

Open chain

Closed knot

Predicting knot type

Distribution difference

Closure methods

Infinite rays

closure method1

Random equilateral arc

closure method

•  The philosophy behind this method is that there is a portion of the knot that is “missing.”

•  This method extends parallel rays from the endpoints of the given subchain.

Problem and goal

•  Given a subchain from a closed knot of known type, the graph below shows the percentage of the time each closure method correctly identifies the original knot’s type.

•  Solid lines: the closure method’s guess is the most prevalent knot type returned out of the 100 times the subchain was closed.

•  Dashed lines: the closure method’s guess is the most prevalent knot type and it appears over 50% of the time.

Future work

Acknowledgements

0

0.05

0.1

0.15

0.2

0.25

0.3

1 11 21 31 41 51 61 71 81 91

3.1

9.21

Average distance between subchain classification

Aver

age

leas

t squ

ares

dist

ance

Number of edges

•  The least squares difference between the distributions of closed knot types for each subchain was computed and averaged over number of edges and original knot type.

•  For both types, the methods are closest near 90 edges.

Average of returning the knot type of the original knot

Number of edges

Aver

age

of fi

ndin

g or

igin

al k

not

Type

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

1 Sułkowska, J. I., E. J. Rawdon, K.C. Millett, J. N. Onuchic, and A. Stasiak, Proc Natl Acad Sci USA 109(2012): E1715-23. 4.

A closed knot and two of its subchains

•  The knot is closed by connecting these rays at infinity. In practice, it is sufficient to connect via a line segment when the rays are “outside” the knot.

•  We close in 100 directions per subchain, and these 100 closures are used to estimate the probability distribution of knot types in all possible directions.

•  To close the knot, a random equilateral arc is created and concatenated with the subchain. The arc has the same number of edges as the portion removed from the closed knot from which the subchain came.

•  The knot is closed 100 times to estimate the distribution of knot types from all possible random closures with this number of edges.

•  At low numbers of edges, the distance is due to the fact that the infinite method will always return the unknot, which is not the case for the random arc method.

•  At high numbers of edges, the random arc is almost always correct, so the change in difference comes from the infinite method.

Infinite 3.1

Infinite 3.1 50%

Random arc 3.1

Random arc 3.1 50%

Infinite 9.21

Infinite 9.21 50%

Random arc 9.21

Random arc 9.21 50%