Modeling Topological Polymers

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

09.28.19

/siam19

This talk!

Collaborators

Jason Cantarella

U. of Georgia

Tetsuo Deguchi

Ochanomizu U.

Erica Uehara

Ochanomizu U.

Funding: Simons Foundation

Linear polymers

A linear polymer is a chain of molecular units with free ends.

Polyethylene

Nicole Gordine [CC BY 3.0] from Wikimedia Commons

Shape of linear polymers

In solution, linear polymers become crumpled:

Protonated P2VP

Roiter–Minko, J. Am. Chem. Soc. 127 (2005), 15688-15689

[CC BY-SA 3.0], from Wikimedia Commons

Ring polymers

Octamethylcyclotetrasiloxane

(Common in cosmetics, bad for fish)

Ring biopolymers

Most known cyclic polymers are biological

Material properties

Ring polymers have weird properties; e.g.,

Thermus aquaticus

Uses cyclic archaeol, a heat-resistant lipid

Grand Prismatic Spring

Home of t. aquaticus; 170ºF

Topological polymers

A topological polymer joins monomers in some graph type:

Petersen graph

In biology

Topological biopolymers have graph types that are extremely complicated (and thought to be random):

Wood-based nanofibrillated cellulose

Qspheroid4 [CC BY-SA 4.0], from Wikimedia Commons

Synthetic topological polymers

The Tezuka lab in Tokyo can now synthesize many topological polymers in usable quantities

Y. Tezuka, Acc. Chem. Res. 50 (2017), 2661–2672

Main Question

What is the probability distribution on the shapes of topological polymers in solution?

Ansatz

Linear polymers

Ring polymers

Topological polymers

Random walks with independent steps

Random walks with steps conditioned on closure

Random walks with steps conditioned on ???

Functions and vector fields

Suppose \(\mathcal{G}\) is a directed graph with \(\mathcal{V}\) vertices and \(\mathcal{E}\) edges.

Definition. A function on \(\mathcal{G}\) is a map \(f:\{v_1,\dots , v_\mathcal{V}\} \to \mathbb{R}\). Functions are vectors in \(\mathbb{R}^\mathcal{V}\).

Definition. A vector field on \(\mathcal{G}\) is a map \(w:\{e_1,\dots , e_\mathcal{E}\} \to \mathbb{R}\). Vector fields are vectors in \(\mathbb{R}^\mathcal{E}\).

Gradient and divergence

By analogy with vector calculus:

Definition. The gradient of a function \(f\) is the vector field

(\operatorname{grad} f)(e_i) = f(\operatorname{head} e_i) - f(\operatorname{tail} e_i).

(\operatorname{div} w)(v_i) = \sum_{j=1}^\mathfrak{E} \begin{cases} +w(e_j) & v_i = \operatorname{head} e_j \\ -w(e_j) & v_i = \operatorname{tail} e_j \\ 0 & \text{else} \end{cases}

Definition. The divergence of a vector field \(w\) is the function

Gradient and divergence as matrices

\operatorname{grad}_{ij} = \begin{cases} +1 & v_j = \operatorname{head} e_i \\ -1 & v_j = \operatorname{tail} e_i \\ 0 & \text{else} \end{cases}

\operatorname{div}_{ij} = \begin{cases} +1 & v_i = \operatorname{head} e_j \\ -1 & v_i = \operatorname{tail} e_j \\ 0 & \text{else} \end{cases}

So if \(B = \operatorname{div}\), which is \(\mathcal{V} \times \mathcal{E}\), then \(\operatorname{grad} = B^T\).

\(-B B^T = L\), the graph Laplacian.

Helmholtz’s Theorem

Fact.

The space \(\mathbb{R}^\mathcal{E}\) of vector fields on \(\mathcal{G}\) has an orthogonal decomposition

\mathbb{R}^\mathcal{E} = (\text{gradient fields}) \oplus (\text{divergence-free fields})

Corollary.

A vector field \(w\) is a gradient (conservative field) if and only if the (signed) sum of \(w\) around every loop in \(\mathcal{G}\) vanishes.

Gaussian embeddings

Definition.

A function \(f:\{v_i\} \to \mathbb{R}^d\) determines an embedding of \(\mathfrak{G}\) into \(\mathbb{R}^d\). The displacement vectors between adjacent vertices are given by \(\operatorname{grad}f\).

A Gaussian random embedding of \(\mathcal{G}\) has displacements sampled from a standard multivariate Gaussian on \((\text{gradient fields})^d\subset \left(\mathbb{R}^\mathcal{E}\right)^d\).

Expected radius of gyration

Theorem (w/ Cantarella, Deguchi, & Uehara; also Estrada–Hatano)

If \(\lambda_i\) are the nonzero eigenvalues of \(L\), the expected squared radius of gyration of a Gaussian random embedding of \(\mathcal{G}\) in \(\mathbb{R}^d\) is

\frac{d}{\mathcal{V}} \sum \frac{1}{\lambda_i}

This quantity is called the Kirchhoff index of \(\mathcal{G}\).

Suppose \(\mathcal{G}\) is a graph with \(\mathcal{V}\) vertices. Let \(L\) be the graph Laplacian of \(\mathcal{G}\).

L=\left[ \begin{array}{cccccccccc} 3 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 3 & 0 & -1 & 0 & 0 & -1 & 0 & 0 \\ -1 & -1 & 0 & 3 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & -1 & 0 & 3 & 0 & 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 & 0 & 3 & -1 & 0 & 0 & -1 \\ 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 \\ 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 3 \\ \end{array} \right]

Subdivisions

Current topological polymers are subdivisions of relatively simple graphs.

Notation. If \(\mathcal{G}\) is a (multi-)graph, \(\mathcal{G}_n\) is its \(n\)th subdivision.

Subdivision Theorem

Theorem (w/ Cantarella, Deguchi, & Uehara)

\lim_{n \to \infty} \frac{1}{\mathcal{V}(\mathcal{G}_n)} \mathbb{E}[R_g^2(\mathcal{G}_n)] = \frac{2 \operatorname{Loops}(\mathcal{G})-1}{4 \mathcal{E}(\mathcal{G})^2} + \frac{3}{2\mathcal{E}(\mathcal{G})^2} \sum \frac{1}{\lambda_k'}

where the \(\lambda_k'\) are the nonzero eigenvalues of the normalized graph Laplacian \(\mathcal{L}\) of \(\mathcal{G}\).

\mathcal{L} = T^{-1/2}LT^{-1/2}

\(T\) is the diagonal matrix of degrees of vertices of \(\mathcal{G}\).

\mathcal{L} = \left[ \begin{array}{cccccc} 1 & -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} & 0 & 0 \\ -\frac{1}{3} & 1 & 0 & 0 & -\frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{3} & 0 & 1 & 0 & -\frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{3} & 0 & 0 & 1 & -\frac{1}{3} & -\frac{1}{3} \\ 0 & -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} & 1 & 0 \\ 0 & -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} & 0 & 1 \\ \end{array} \right]