or

### Loop Closure is Surprisingly Non-Destructive

Clayton Shonkwiler

http://shonkwiler.org

07.19.18

/cta18

This talk!

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

### Is it almost closed?

1QMG – Acetohydroxyacid isomeroreductase

### Most random walks are almost closed(?)

Suppose $$e_1,\ldots , e_n$$ are the edges of a random walk in $$\mathbb{R}^d$$.

\mathbb{P}\left(\left\|\frac{1}{n}\sum_i e_i \right\| < r\right) \geq 1-2d e^{-\frac{nr^2}{2d}}
$\mathbb{P}\left(\left\|\frac{1}{n}\sum_i e_i \right\| < r\right) \geq 1-2d e^{-\frac{nr^2}{2d}}$

By Chernoff’s inequality,

### Fixing edgelengths is trickier

end-to-end distance: 16.99

distance to closed: 5.64

end-to-end distance: 17.76

distance to closed: 0.68

### Distance to a closed equilateral polygon

Proposition. As $$n \to \infty$$, distance to the closest polygon to a random walk in $$\mathbb{R}^d$$ converges in distribution to a Nakagami$$\left(\frac{d}{2},\frac{d}{d-1}\right)$$ distribution.

### The geometric median

Definition

A geometric median (or Fermat-Weber point) of a collection $$X=\{x_1,\ldots , x_n\}$$ of points in $$\mathbb{R}^d$$ is any point closest to the $$x_i$$:

$$\text{gm}(X)=\text{argmin}_y \sum \|x_i-y\|$$

Definition

A point cloud has a nice geometric median if:

• $$\text{gm}(X)$$ is unique ($$\Leftarrow X$$ is not linear)
• $$\text{gm}(X)$$ is not one of the $$x_i$$

### Geometric median closure

Definition

If the edge cloud $$X$$ of an equilateral arm has a nice geometric median, the geometric median closure $$\text{gmc}(X)$$ recenters the edge cloud at the geometric median.

$$\text{gmc}(X)_i = \frac{x_i-\text{gm}(X)}{\|x_i - \text{gm}(X)\|}$$

### The geometric median closure is closed

Proposition (with Cantarella, Chapman, and Reiter)

If it exists, the geometric median closure of an arm is a closed polygon.

Proof

$$\text{gm}(X)$$ minimizes the average distance function

$$\mathrm{Ad}_X(y) = \frac{1}{n}\sum_i \|x_i - y\|$$,

which is convex everywhere and smooth away from the $$x_i$$, and

$$\nabla \mathrm{Ad}_X(y) = \frac{1}{n}\sum_i \frac{x_i-y}{\|x_i-y\|}$$.

### The geometric median closure is optimal

Definition

An arm or polygon $$X$$ is given by $$n$$ edge vectors $$x_i \in \mathbb{R}^d$$, or a single point in $$\mathbb{R}^{dn}$$. The distance between $$X$$ and $$Y$$ is the Euclidean distance between these points in $$\mathbb{R}^{dn}$$.

Theorem (with Cantarella, Chapman, and Reiter)

If $$X$$ is an equilateral arm in $$\mathbb{R}^d$$ with a geometric median closure, then $$\text{gmc}(X)$$ is the closest equilateral polygon to $$X$$.

### Some neat bounds

Suppose $$X=(x_1,\ldots , x_n)$$ consists of the edges of an $$n$$-step random walk in $$\mathbb{R}^d$$. Let $$\mu=\|\mathrm{gm}(X)\|$$.

Lemma. $$d(X,\mathrm{Pol}(n,d))<\mu\sqrt{2}\sqrt{n}$$

In fact, $$d(X,\mathrm{Pol}(n,d)) \sim \mu\sqrt{\frac{d-1}{d}}\sqrt{n}$$.

Lemma. If $$d_\mathrm{max-angular}(X,Y):=\max_i \angle(x_i,y_i)$$, then

d_\mathrm{max-angular}(X,\mathrm{Pol}(n,d)) < \arcsin\mu
$d_\mathrm{max-angular}(X,\mathrm{Pol}(n,d)) < \arcsin\mu$

3L05A

2HOCA

### Closure mostly preserves small knots

Knotted core size vs. Knotted closure probability

Knotted core size likelihood

~70% of closures knotted

~15% of closures knotted

~15% of proteins

~70% of proteins

For all proteins in KnotProt as of July 2, 2018

### Main Theorem

Theorem (with Cantarella, Chapman, and Reiter)

If $$X$$ consists of the edges of a random walk in $$\mathbb{R}^3$$ and $$\mu=\|\mathrm{gm}(X)\|$$, then for any $$r<\frac{5}{1000}$$,

\mathbb{P}(\mu < r) \geq 1-6e^{-n\frac{r^2}{9}}
$\mathbb{P}(\mu < r) \geq 1-6e^{-n\frac{r^2}{9}}$

Corollary

For any $$\alpha < \frac{5}{1000} \sqrt{\frac{n}{2}}$$,

\mathbb{P}(d(X,\mathrm{Pol}(n,3)<\alpha)\geq 1-6 e^{-\frac{\alpha^2}{4}}
$\mathbb{P}(d(X,\mathrm{Pol}(n,3)<\alpha)\geq 1-6 e^{-\frac{\alpha^2}{4}}$

Similar results hold in any dimension.

### Provable vs. actual distance to closure

Provable: For $$n>$$1,280,000, $$\mathbb{P}(d(X,\mathrm{Pol}(n,3))<4)\geq 0.999$$.

Actual: For $$n\geq10$$, $$\mathbb{P}(d(X,\mathrm{Pol}(n,3))<3)\geq 0.999$$.

### Flow of the proof

Recall that $$\mathrm{gm}(X)$$ is the unique minimizer of the convex function $$\mathrm{Ad}_X(y)$$.

1. The minimum eigenvalue of the Hessian of $$\mathrm{Ad}_X$$ is very likely to be bounded below near the origin.
2. $$\|\nabla \mathrm{Ad}_X(0)\|$$ is very likely to be small.
3. By Taylor’s theorem, the radial directional derivative must be positive outside some small ball, so the point where $$\nabla \mathrm{Ad}(y) = 0$$—namely, $$\mathrm{gm}(X)$$—must be inside this ball, and hence close to the origin.

### Moral of the story

Closing a random walk is very unlikely to mess up the local structure of the walk.

Random walks are surprisingly close to closed polygons, for any $$n$$, in any dimension, and for any fixed choice of edgelengths (not just equilateral!).

# Thank you!

### Reference

Open and closed random walks with fixed edgelengths in $$\mathbb{R}^d$$

Jason Cantarella, Kyle Chapman, Philipp Reiter, & Clayton Shonkwiler

arXiv: 1806.00079

Funding: Simons Foundation