/ag17

This talk!

A ring polymer in solution takes on an ensemble of random shapes, with topology (knot type!) as the unique conserved quantity.

Knotted DNA

Wassermann et al.

*Science* 229, 171–174

Knot complexity in DNA from P4 tailless mutants

Arsuaga et al., *PNAS* 99 (2002), 5373–5377

**Is this surprising?**

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

*Modern polymer physics is based on the analogy between a polymer chain and a random walk.*

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Generate \(n\) independent uniform random points on \(S^2\) and treat them as an ordered list of edge vectors.

Alvarado, Calvo, Millett, *J. Stat. Phys.* 143 (2011), 102–138

*Random Polygon* \(\Leftrightarrow\) *point in some (nice!) configuration space*

Knowledge of the (differential, symplectic, algebraic) geometry of these conformation spaces leads to both

**theorems and fast numerical algorithms** for studying and comparing polygons in \(\mathbb{R}^3\).

A random closed \(n\)-edge polygon is a \(k\)-edge random walk and an \((n-k)\)-edge random walk, conditioned on having the same end-to-end distance.

**Classical Fact:** The density of the end-to-end vector of an \(n\)-step random walk is

\phi_n(\ell) = \frac{2\ell}{\pi} \int_0^\infty x \sin \ell x \,\mathrm{sinc}^n x \,\mathrm{d}x

$\phi_n(\ell) = \frac{2\ell}{\pi} \int_0^\infty x \sin \ell x \,\mathrm{sinc}^n x \,\mathrm{d}x$

= \frac{2^{-n-1}}{\pi
\ell(n-2)!} \sum_{k=0}^{n-1} (-1)^k \binom{n-1}{k} \left([-2k+\ell +n-2]_+^
{n-2}\right.

$= \frac{2^{-n-1}}{\pi
\ell(n-2)!} \sum_{k=0}^{n-1} (-1)^k \binom{n-1}{k} \left([-2k+\ell +n-2]_+^
{n-2}\right.$

\left.-
[-2 k+\ell +n]_+^{n-2}\right)

$\left.-
[-2 k+\ell +n]_+^{n-2}\right)$

**Proof:** Fourier transform (since \(\mathrm{sinc}\) is the transform of the boxcar function).

This is piecewise-polynomial in \(\ell\) of degree \(n-3\)

**Proposition (with Cantarella):** The pdf of the chord connecting \(v_1\) with \(v_{k+1}\) in an \(n\)-gon is

\frac{4 \pi \ell^2}{C_n} \phi_k(\ell) \phi_{n-k}(\ell)

$\frac{4 \pi \ell^2}{C_n} \phi_k(\ell) \phi_{n-k}(\ell)$

where \(C_n = 2^{n-5}\pi^{n-4} \int_{-\infty}^{\infty} x^2 \,\mathrm{sinc}^n x \,\mathrm{d}x\).

**Fact:** This is piecewise-polynomial in \(\ell\) of degree \(n-4\).

Why are the expectations rational?

Why degree \(n-4\)?

*Where the heck are the polyhedra?*

Continuous symmetry \(\Rightarrow\) conserved quantity

**Duistermaat–Heckman Theorem (stated informally):** *On a \(2m\)-dimensional symplectic manifold, \(d\) commuting Hamiltonian symmetries (a Hamiltonian \(T^d\)-action) induce \(d\) conserved quantities (momenta).*

*The joint distribution of the momenta is continuous, piecewise polynomial, degree \(\leq m-d\).*

\(n\)-gons up to rotation are \(2m=(2n-6)\)-dimensional, so chord length is piecewise polynomial of degree \(\leq\)

\(m-1=(n-3)-1=n-4\)

**We actually have more symmetries!**

Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.

**Theorem (with Cantarella, ’16):** The joint distribution of \(d_1,\ldots , d_{n-3}\) and \(\theta_1, \ldots , \theta_{n-3}\) are all uniform on their domains.

**Proof:** Check that D–H applies (this is the hard part, since the torus action is not defined everywhere).

Then count: \(m=n-3\) and we have \(d=n-3\) symmetries, so the pdf of the \(d_i\) is piecewise polynomial of degree \(\leq\)

\(m-d = (n-3)-(n-3)=0\).

Since the pdf is continuous and the domain is connected, it must be constant.

The \((n-3)\)-dimensional* moment polytope* \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities

0 \leq d_i \leq 2

$0 \leq d_i \leq 2$

1 \leq d_i + d_{i-1}

$1 \leq d_i + d_{i-1}$

|d_i - d_{i-1}| \leq 1

$|d_i - d_{i-1}| \leq 1$

0 \leq d_{n-3} \leq 2

$0 \leq d_{n-3} \leq 2$

**Theorem (with Cantarella, Duplantier, Uehara, ’16):** A direct sampling algorithm for equilateral \(n\)-gons with expected performance \(O(n^{5/2})\).

If we let \(s_i = d_i - d_{i-1}\) for \(i=1,\ldots , n-2\) and \(s_i \in [-1,1]\), then we have \(|d_i - d_{i-1}|\leq 1\).

**Proposition (with Cantarella, Duplantier, Uehara):** If we build \(d_i\) from \(s_i\) sampled uniformly from the hypercube \([-1,1]^{n-3}\), the \(d_i\) obey the triangle inequalities with probability asymptotic to

6 \sqrt{\frac{6}{\pi}} n^{-3/2}.

$6 \sqrt{\frac{6}{\pi}} n^{-3/2}.$

```
RandomDiagonals[n_] :=
Accumulate[
Join[{1}, RandomVariate[UniformDistribution[{-1, 1}], n]]];
InMomentPolytopeQ[d_] :=
And[Last[d] >= 0, Last[d] <= 2,
And @@ (Total[#] >= 1 & /@ Partition[d, 2, 1])];
DiagonalSample[n_] := Module[{d},
For[d = RandomDiagonals[n], ! InMomentPolytopeQ[d], ,
d = RandomDiagonals[n]];
d[[2 ;;]]
];
```

Not the first direct sampler (Grosberg–Moore, Diao–Ernst–Montemayor–Ziegler), but simple and fast.

If we round every point \((d_1),\ldots , d_{n-3})\) in the moment polytope to the nearest point with half-integer coordinates \(\frac{1}{2}(x_1, \ldots , x_{n-3})\) …

… and subdivide the regions of attraction by

ordering and sign of \(d_i - \frac{1}{2} x_i\), we get a collection of identical, orthogonal simplices of equal area (this works for any \(n\)):

**Proposition*:** The number of half-integer points in \(\mathcal{P}_n\) is the Motzkin number

M_{n-3} = \sum_{k=0}^{\lfloor \frac{n-2}{2}\rfloor} \binom{n-2}{2k} C_k

$M_{n-3} = \sum_{k=0}^{\lfloor \frac{n-2}{2}\rfloor} \binom{n-2}{2k} C_k$

\sim \sqrt{3} \frac{1 + \frac{1}{16(n-3)}}{(2n-3)\sqrt{(n-1)\pi}} 3^{n-2}

$\sim \sqrt{3} \frac{1 + \frac{1}{16(n-3)}}{(2n-3)\sqrt{(n-1)\pi}} 3^{n-2}$

Therefore, there are exponentially many simplices in this decomposition. ☹️

**Theorem (K. Chapman, in progress):** With \(O(n^3)\) preprocessing of the transition probabilities, a direct sampling algorithm for equilateral \(n\)-gons in \(O(n^2 (\log n)^2)\) time.

The simplices are indexed by certain permutations.

We can recursively construct Lehmer codes of valid permutations using a Markov chain.

- What’s the corresponding story for planar \(n\)-gons?
- Polygons form a Kähler manifold, so there’s a well-defined distance between \(n\)-gons. How should we optimally register polygons? Can we find explicit geodesics?
- What about more complicated polymer topologies?

The symplectic geometry of closed equilateral random walks in 3-space

J. Cantarella & C. Shonkwiler

*Annals of Applied Probability* **26** (2016), no. 1, 549–596

A fast direct sampling algorithm for equilateral closed polygons

J. Cantarella, B. Duplantier, C. Shonkwiler, & E. Uehara

*Journal of Physics A* **49** (2016), no. 27, 275202

Sampling knots using orthoschemes

K. Chapman

In preparation