# Steady states for crossing changes in knot diagrams

Harrison Chapman

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 24th 2019

## Knot Diagrams

Crossings are the primary components of knot diagram models

## Crossings as Self-contacts

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

## Hua, Nguyen, Raghavan, Arsuaga, Vasquez (2007)

Calculated transition probabilities between knot types and steady-state probabilities of DNA molecules under the action of topo-IV.

## Issue

Action of topo-IV is easily expressed diagrammatically, but complex to represent with self-avoiding polygons (SAPs) alone

## Strategy

Sample SAPs, then transfer to diagrams to study action of topo-IV

## Strategy of Hua et al.

For each prime knot type $$K$$, sample diagrams representing $$K$$ from a weighted distribution:

1. Sample SAPs representing $$K$$ uniformly of desired sizes, using BFACF
2. Randomly project to 50–100 diagrams
3. Result is a list of diagrams for $$K$$, weighted by occurrence: sample from this list uniformly

## Idea

Crossings in diagrams drawn under weighted distribution likely represent crossings close in original SAPs

## Question

How tied are the results of Hua et al. to the projection-weighted distribution on diagrams?

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

## Diagram MCMC

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

Explore all knot shadows by ignoring crossing signs; get all knot diagrams by adding crossing information

## Fixed Knot Type Markov Chain

Explore all diagrams of fixed knot type by respecting crossing signs (c.f. BFACF and lattice polygons)

## C–Rechnitzer

The Wang-Landau Markov chain on diagrams of knot type $$K$$ has stationary distribution where the probability that an $$n$$-crossing diagram $$D$$ representing $$K$$ is sampled is, $\pi(D) \propto \frac{1}{g_n(K)} \approx \frac{1}{k_n(K)}.$

Knot diagrams for $$K$$ are sampled

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

## Under the (Sampling) Hood

• 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

## Approximate Enumeration: $$6_1$$

$$n$$ 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

## Strategy for Uniformly Sampled Diagrams

Now we just:

1. 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.

## Steady States for Topo-IV Action

In either study, after sampling a diagram of type $$K$$:

1. Reduce the diagram fully by $$\mathrm{RI}^{-}$$ and $$\mathrm{RII}^-$$ moves
(except in the case of the unknot)

## Steady States for Topo-IV Action

In either study, after sampling a diagram of type $$K$$:

1. Reduce the diagram fully by $$\mathrm{RI}^{-}$$ and $$\mathrm{RII}^-$$ moves
(except in the case of the unknot)
2. Pick a crossing uniformly at random and make a crossing change
3. If the resulting knot type is sufficiently small ($$cr(L) \le 8$$) and prime, record it

After normalizing, result is a vector of transition probabilities for knot type $$K$$

## Steady States for Topo-IV Action

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

The eigenvector associated to the largest eigenvalue of $$M$$ is the steady state of the system.

## Rescaling the $$x$$-axis?

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 $$z = W(ze^z)$$)