### Modeling Topological Polymers

Clayton Shonkwiler

http://shonkwiler.org

11.08.18

### 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 $$\mathfrak{G}$$ is a directed graph with $$\mathfrak{V}$$ vertices and $$\mathfrak{E}$$ edges.

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

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

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

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

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

### Helmholtz’s Theorem

Fact.

The space $$\mathbb{R}^\mathfrak{E}$$ of vector fields on $$\mathfrak{G}$$ has an orthogonal decomposition

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

Corollary.

A vector field $$w$$ is a gradient (conservative field) if and only if the (signed) sum of $$w$$ around every loop in $$\mathfrak{G}$$ 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 $$\mathfrak{G}$$ has displacements sampled from a standard multivariate Gaussian on $$(\text{gradient fields})^d\subset \left(\mathbb{R}^\mathfrak{E}\right)^d$$.

### Useful lemma

Lemma. The projections of a Gaussian random embedding of $$\mathfrak{G}$$ in $$\mathbb{R}^d$$ onto each coordinate axis are independent Gaussian random embeddings of $$\mathfrak{G}$$ into $$\mathbb{R}$$.

So we can restrict to Gaussian embeddings of $$\mathfrak{G}$$ in $$\mathbb{R}$$.

### Main Theorem

Since $$\operatorname{grad}f=0 \,\Longleftrightarrow f$$ is a constant function, $$\operatorname{grad}f$$ only determines $$f$$ up to a constant. So assume our random embeddings are centered; i.e., $$\sum f(v_i) = 0$$.

Theorem [w/ CDU; cf. James, 1947]

The distribution of vertex positions on the $$(\mathfrak{V}-1)$$-dimensional subspace of centered embeddings is

\mathcal{N}(0,(BB^T)^+) = \mathcal{N}(0,L^+)
$\mathcal{N}(0,(BB^T)^+) = \mathcal{N}(0,L^+)$

$$BB^T = \operatorname{div}\operatorname{grad} = L$$ is the graph Laplacian

L_{ij} = \begin{cases} \operatorname{deg}(v_i) & i=j \\ -1 & v_i,v_j \text{ joined by an edge} \\ 0 & \text{else} \end{cases}
$L_{ij} = \begin{cases} \operatorname{deg}(v_i) & i=j \\ -1 & v_i,v_j \text{ joined by an edge} \\ 0 & \text{else} \end{cases}$

### Pseudoinverse

For symmetric matrices like $$L$$, with eigenvalues $$\lambda_i$$ and eigenvectors $$v_i$$, the pseudoinverse $$L^+$$ is the symmetric matrix defined by:

• The eigenvalues $$\lambda_i'$$ of $$L^+$$ are
\lambda_i' = \begin{cases} \frac{1}{\lambda_i} & \lambda_i \neq 0 \\ 0 & \lambda_i = 0 \end{cases}
$\lambda_i' = \begin{cases} \frac{1}{\lambda_i} & \lambda_i \neq 0 \\ 0 & \lambda_i = 0 \end{cases}$
• The eigenvectors $$v_i'$$ of $$L^+$$ are $$v_i' = v_i$$.

### Sampling algorithm

Algorithm.

• Generate $$w$$ from $$\mathcal{N}(0,I_\mathfrak{E})$$
• Let $$f = {B^T}^+ w$$, which is $$\mathcal{N}(0,L^+)$$

### Expected distances

Theorem [w/ CDU; cf. James, 1947]

The distribution of vertex positions on the $$(\mathfrak{V}-1)$$-dimensional subspace of centered embeddings is

\mathcal{N}(0,(BB^T)^+) = \mathcal{N}(0,L^+)
$\mathcal{N}(0,(BB^T)^+) = \mathcal{N}(0,L^+)$

Corollary.

The expected squared distance between vertex $$i$$ and vertex $$j$$ is

E[(f_i-f_j)^2] = E[f_i^2+f_j^2-2f_if_j] = L_{ii}^+ + L_{jj}^+ - L_{ij}^+-L_{ji}^+
$E[(f_i-f_j)^2] = E[f_i^2+f_j^2-2f_if_j] = L_{ii}^+ + L_{jj}^+ - L_{ij}^+-L_{ji}^+$

### Example: multitheta graphs

Definition.

An $$(m,n)$$-theta graph consists of $$m$$ arcs of $$n$$ edges connecting two junctions.

$$(5,20)$$-theta graph

Theorem [Deguchi–Uehara, 2017]

The expected squared distance between junctions is $$\frac{dn}{m}$$.

### Amazing fact about $$L^+$$

Proposition [Nash–Williams (resistors, 1960s), James (springs, 1947)]

The expression $$L_{ii}^++L_{jj}^+-L_{ij}^+-L_{ji}^+$$ is:

• the resistance between $$v_i$$ and $$v_j$$ if all edges of $$\mathfrak{G}$$ are unit resistors;
• the reciprocal of the force between $$v_i$$ and $$v_j$$ if all edges are unit springs and $$v_i$$ and $$v_j$$ are one unit apart.

Randall Munro, [CC BY-NC 2.5], from xkcd

### Deguchi–Uehara result, redux

R=R_1+R_2+\dots +R_n
$R=R_1+R_2+\dots +R_n$
\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}
$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

Theorem [w/ CDU; cf. Estrada–Hatano, 2010]

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

\frac{d}{\mathfrak{V}} \sum \frac{1}{\lambda_i} = \frac{d}{\mathfrak{V}} \operatorname{tr} L^+
$\frac{d}{\mathfrak{V}} \sum \frac{1}{\lambda_i} = \frac{d}{\mathfrak{V}} \operatorname{tr} L^+$

This quantity is called the Kirchhoff index of $$\mathfrak{G}$$.

### Distinguishing graph types

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

### The suspects

T. Suzuki et al., J. Am. Chem. Soc. 136 (2014), 10148–10155

$$K_{3,3}$$ (subdivided)

Topological polymers

Size exclusion chromatograph

### Distinguishing graph types in the lab

T. Suzuki et al., J. Am. Chem. Soc. 136 (2014), 10148–10155

larger molecule

smaller molecule

### Different sizes

Proposition [with Cantarella, Deguchi, & Uehara]

If each edge is subdivided equally to make $$\mathfrak{V}$$ vertices total:

E[R_g^2(K_{3,3})] = \frac{108 - 261 \mathfrak{V} + 60 \mathfrak{V}^2 + 17 \mathfrak{V}^3}{486 \mathfrak{V}^2} \sim 0.12 + 0.035 \mathfrak{V}
$E[R_g^2(K_{3,3})] = \frac{108 - 261 \mathfrak{V} + 60 \mathfrak{V}^2 + 17 \mathfrak{V}^3}{486 \mathfrak{V}^2} \sim 0.12 + 0.035 \mathfrak{V}$
E[R_g^2(\text{ladder})] = \frac{540 - 1305 \mathfrak{V} + 372 \mathfrak{V}^2 + 109 \mathfrak{V}^3}{2430 \mathfrak{V}^2} \sim 0.15 + 0.045 \mathfrak{V}
$E[R_g^2(\text{ladder})] = \frac{540 - 1305 \mathfrak{V} + 372 \mathfrak{V}^2 + 109 \mathfrak{V}^3}{2430 \mathfrak{V}^2} \sim 0.15 + 0.045 \mathfrak{V}$

So the smaller molecule is predicted to be $$K_{3,3}$$!

### Open questions

• Expectations are computable by computer algebra; are there analytic formulae for graph subdivisions?
• 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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