or
/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
1QMG – Acetohydroxyacid isomeroreductase
Suppose \(e_1,\ldots , e_n\) are the edges of a random walk in \(\mathbb{R}^d\).
By Chernoff’s inequality,
end-to-end distance: 16.99
distance to closed: 5.64
end-to-end distance: 17.76
distance to closed: 0.68
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.
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:
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.
\(\text{gmc}(X)_i = \frac{x_i-\text{gm}(X)}{\|x_i - \text{gm}(X)\|}\)
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
\(\nabla \mathrm{Ad}_X(y) = \frac{1}{n}\sum_i \frac{x_i-y}{\|x_i-y\|}\).
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 \(X\).
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
3L05A
2HOCA
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
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 \(r<\frac{5}{1000}\),
Corollary
For any \(\alpha < \frac{5}{1000} \sqrt{\frac{n}{2}}\),
Similar results hold in any dimension.
Provable: For \(n>\)1,280,000, \(\mathbb{P}(d(X,\mathrm{Pol}(n,3))<4)\geq 0.999\).
Actual: For \(n\geq10\), \(\mathbb{P}(d(X,\mathrm{Pol}(n,3))<3)\geq 0.999\).
Recall that \(\mathrm{gm}(X)\) is the unique minimizer of the convex function \(\mathrm{Ad}_X(y)\).
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!).
Open and closed random walks with fixed edgelengths in \(\mathbb{R}^d\)
Jason Cantarella, Kyle Chapman, Philipp Reiter, & Clayton Shonkwiler
Funding: Simons Foundation