### Modeling Topological Polymers

Clayton Shonkwiler

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 $$\mathbb{R}^\mathcal{E}$$.

By analogy with vector calculus:

Definition. The gradient of a function $$f$$ is the vector field

(\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 $$(\text{gradient fields})^d\subset \left(\mathbb{R}^\mathcal{E}\right)^d$$.

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]

Topological polymers

Size exclusion chromatograph

$$\lim_{n \to \infty}\frac{1}{\mathcal{V}(\mathcal{G}_n)} \mathbb{E}[R_g^2(\mathcal{G}_n)]$$

$$\frac{17}{162}\approx 0.105$$

$$\frac{107}{810}\approx 0.132$$

$$\frac{109}{810}\approx 0.135$$

$$\frac{31}{162}\approx 0.191$$

$$\frac{43}{162}\approx 0.265$$

$$\frac{49}{162}\approx 0.302$$

“an extremely compact 3D conformation, achieving exceptionally thermostable bioactivities”

### Open questions

• Topological type of graph embedding?
• What if the graph is a random graph?

# Thank you!

#### Modeling Topological Polymers

By Clayton Shonkwiler

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