for knot diagrams

hchapman.org/talks/birs_steadystate

The Topology of Nucleic Acids:

Research at the Interface
of Low-Dimensional Topology,

Polymer Physics and Molecular Biology

Banff International Research Station

Banff, Alberta

March 24^{th} 2019

**Crossings** are the primary components of knot diagram models

Crossings may be viewed as **self-contacts** where enzymes may act to change DNA topology

Strand passage action, *e.g.* topo-IV

Coherent smoothing action, *e.g.* XerCD-*dif*-FtsK complex

For each prime knot type \(K\), sample diagrams representing \(K\) from a **weighted** distribution:

- Sample SAPs representing \(K\) uniformly of desired sizes, using BFACF
- Randomly project to 50–100 diagrams
- Result is a list of diagrams for \(K\), weighted by occurrence: sample from this list uniformly

Aim to answer this by applying the same techniques, now
using **uniform distribution on knotted diagrams**

One step of a **diagram Markov chain** takes as input a knot diagram, performs with some probability a **(Reidemeister) transition**, and returns the resulting knot diagram

Knot diagrams for \(K\) are sampled

**uniformly for any given size**, and- approximately
**each size is equally likely**

- Maximum diagram size considered: \(70\) crossings
- Enumeration step ran with flatness parameter \(\Delta = 0.99\) until update parameter \(f \le 10^{-6}\)
- Enumeration step took approximately 1d to tune all 36 knot types
- Samples drawn every \(1.5 \times 10^{5}\) attempted transitions
- \(100,000\) diagrams sampled for each knot type (approx. \(1,500\) samples per diagram size)
- Data for each knot type gathered in approximately 25m, totalling 15h

\(n\) | Rooted | Rooted/\(4n\) | Exact (CCM '16) |
---|---|---|---|

6 | 12 | .5 | 1 |

7 | 911.68 | 32.56 | 34 |

8 | 40 207.17 | 1 256.47 | 1 267 |

9 | 1 370 185.44 | 38 060.71 | 38 199 |

10 | 40 659 767.57 | 1 016 494.19 | 1 015 996 |

11 | 1 103 328 823.05 | 25 075 655.07 | N/A |

Exact enumeration data from Cantarella-C-Mastin '16

Now we just:

- For each prime knot type \(K\), sample diagrams representing \(K\) from the uniform distribution of a desired size range, using fixed-knot type diagram MCMC

Then, we apply the same analysis as in Hua et al.

In either study, after sampling a diagram of type \(K\):

- Reduce the diagram fully by \(\mathrm{RI}^{-}\) and \(\mathrm{RII}^-\) moves

(except in the case of the unknot)

In either study, after sampling a diagram of type \(K\):

- Reduce the diagram fully by \(\mathrm{RI}^{-}\) and \(\mathrm{RII}^-\) moves

(except in the case of the unknot) - Pick a crossing uniformly at random and make a crossing change
- If the resulting knot type is sufficiently small (\(cr(L) \le 8\)) and prime, record it

Gathering data for each prime knot type \(K\) of up to 8 crossings yields a square stochastic matrix \(M\)

The expected average crossing number over all projections of a \(M\)-edge Gaussian random polygon is \(O(M \ln M)\)

Can rescale \(x\)-axis by \(e^{W_0(n)}\) where \(W_0\) is the main branch of the Lambert \(W\) function (satisfying \(z = W(ze^z)\))

- Do these data show a key difference between diagrams and space polygons?
- Uniform diagram sampler allows new tests for null hypotheses in many other previous experiments (e.g. Stolz et al. '17 work on XerCD-dif-FtsK in E. coli)
- Wang-Landau approximate enumeration provides new enumeration of numbers of diagrams for given knot types