### Topological Polymers and Random Embeddings of Graphs

Clayton Shonkwiler

https://shonkwiler.org

September, 2022

/erice22

This talk!

### Collaborators

Jason Cantarella

U. of Georgia

Tetsuo Deguchi

Ochanomizu U.

Erica Uehara

Ochanomizu U.

Funding: Simons Foundation (#524120, J.C.; #709150, C.S.), Japan Science and Technology Agency (CREST JPMJCR19T4, Deguchi Lab), Japan Society for the Promotion of Science (KAKENHI JP17H06463)

### Equilibrium Distributions for Large Molecules

Want to define a probability distribution on the positions of $$\mathcal{V}$$ points (i.e., monomers) in $$\mathbb{R}^3$$.

If positions are coupled (by a bond, steric effect, etc.), add an edge between points, forming a graph $$\mathcal{G}$$.

### Edge Distributions

Interactions are symmetric probability distributions on edge vectors.

Problem.

Edge vectors are not independent when there are loops.

### Topological Polymers

A topological polymer joins monomers in any (multi)graph type.

Elasticity theory (1940s–1980s, James, Guth, Flory, Eichinger, etc.).

Structure graph $$\mathcal{G}$$

Edges i.i.d. Gaussian conditioned on $$\mathcal{G}$$

The expected variation size $$\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+$$.

The expected variation size $$\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{974299}{765600}$$.

The expected variation size $$\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+$$.

Graph Laplacian of $$\mathcal{G}$$

The expected variation size $$\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+$$.

### Our Approach

Problem.

Classical elasticity assumes mean-zero Gaussians. This assumption is built into the theory.

Solution.

New formalism which handles arbitrary distributions in a clean, provable way.

### Classical Materials

Natural elastic materials tend to have extremely complicated, random graph types

Wood-based nanofibrillated cellulose

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

### New Materials

Synthetic chemists can now produce simple topological polymers in usable quantities.

$$\theta$$-curves in solution at the Tezuka lab

### Ansatz

Linear polymers

Topological polymers

Independently

Conditioned on graph type

Edges chosen from some $$O(d)$$-invariant distribution $$\mu$$.

What does this mean?

### Chain Groups

Let $$\mathcal{G}$$ be a (directed) graph with $$\mathcal{E}$$ edges and $$\mathcal{V}$$ vertices.

Definition.

The vector space $$\operatorname{VC}$$ of vertex chains is the vector space of (formal) linear combinations of vertices:

$$x = x_1 v_1 + \dots + x_{\mathcal{V}}v_{\mathcal{V}}$$.

Definition.

The vector space $$\operatorname{EC}$$ of edge chains is the vector space of (formal) linear combinations of edges:

$$w = w_1 e_1 + \dots + w_{\mathcal{E}}e_{\mathcal{E}}$$.

### Boundaries

Definition.

The boundary map $$\partial : \operatorname{EG} \to \operatorname{VC}$$ is defined by

$$\partial(e_i) = \operatorname{head}(e_i) - \operatorname{tail}(e_i)$$.

\partial(e_4)=v_4-v_2

Definition.

$$\operatorname{ker} \partial \subset \operatorname{EC}$$ is the loop space of $$\mathcal{G}$$.

### Boundaries

Definition.

Every $$w \in \ker \partial \subset \operatorname{EC}$$ is a linear combination of closed loops. $$\dim\ker \partial$$ is the cycle rank $$\xi(\mathcal{G}) = \mathcal{E} - \mathcal{V}+1$$; i.e., the first Betti number of $$\mathcal{G}$$.

$$\xi(\mathcal{G}) = \frac{1}{2} \sum_{i=1}^{\mathcal{V}} \left( \deg(v_i)-2\right) + 1$$.

$$-$$

$$=$$

### Embedding Spaces

The chain spaces encode the topology of the graph. The embedding into $$\mathbb{R}^d$$ is determined by:

Definition.

The space of vertex positions $$\operatorname{VP} := \operatorname{Hom}(\operatorname{VC},\mathbb{R}^d)$$.

Definition.

The space of edge displacements $$\operatorname{ED} = \operatorname{Hom}(\operatorname{EC},\mathbb{R}^d)$$.

Definition.

The displacement map $$\operatorname{disp}: \operatorname{VP} \to \operatorname{ED}$$ is given by

$$\operatorname{disp}(X)(e_i) = X(\operatorname{head}(e_i)) - X(\operatorname{tail}(e_i))$$.

### A Wild Functor Appears!

Proposition.

The map $$\operatorname{disp}:\operatorname{VP} \to \operatorname{ED}$$ is equal to the map $$\partial^\ast$$ induced by the contravariant functor $$\operatorname{Hom}(-,\mathbb{R}^d)$$.

Proposition.

If $$\mathcal{G}$$ is connected,

$$\operatorname{im}\partial^\ast = \{W \in \operatorname{ED} : W(u) = 0$$ for all $$u \in \ker \partial\}$$.

### Random Graph Embeddings

Theorem.

The space of assignments of edge displacements compatible with the graph type $$\mathcal{G}$$ is the linear subspace

$$\operatorname{im}\operatorname{disp} = \operatorname{im}\partial^\ast$$.

Definition.

A probability measure $$\mu$$ on $$\operatorname{ED}$$ is admissible if it has finite first moment and is invariant under the diagonal action of $$O(d)$$.

“A probability distribution on embeddings of $$\mathcal{G}$$ is the restriction of an admissible probability measure $$\mu$$ to the $$d(\mathcal{V}-1)$$-dimensional subspace $$\operatorname{im}\partial^\ast \subset \operatorname{ED}$$.”

### Gaussian Embeddings – Phantom Network Theory

Definition.

The distribution $$\mu$$ on ED is the standard Gaussian; restriction to $$\operatorname{im}\partial^\ast$$ is standard Gaussian on that subspace.

$$\operatorname{ED} \simeq \mathbb{R}^{3\mathcal{E}}$$

$$\operatorname{VP} \simeq \mathbb{R}^{3\mathcal{V}}$$

$$\partial^\ast$$

$$\operatorname{im}\partial^\ast$$

?

$$\partial^{\ast +}$$

$$(\operatorname{ker}\partial^\ast)^\bot$$

### The Correct Inner Product on VP

$$(\operatorname{ED}, \langle\, , \,\rangle)$$

$$(\operatorname{VP}, \langle \, , \, \rangle_{\widetilde{L}^\ast})$$

$$\partial^\ast$$

$$\operatorname{im}\partial^\ast$$

$$\partial^{\ast +}$$

$$(\operatorname{ker}\partial^\ast)^\bot$$

Definition.

The graph Laplacian $$L: \operatorname{VC} \to \operatorname{VC}$$ is $$L:=\partial \partial^T$$.

Proposition.

With the inner product $$\langle X, Y \rangle_{\widetilde{L}^\ast} = \langle X, \widetilde{L}^\ast Y \rangle$$ on VP, $$\partial^\ast$$ and $$\partial^{\ast +}$$ are partial isometries.

### Sampling

• Compute pseudoinverse $$\partial^{\ast +}: \operatorname{ED} \to \operatorname{VP}$$.
• Sample $$W$$ from conditional distribution on $$\operatorname{im}\partial^\ast \subset \operatorname{ED}$$.
• Construct vertex positions $$X = \partial^{\ast +} W$$.

### Realistic Graphs aren’t Arbitrary

Definition.

For a multigraph $$\mathcal{G}$$, let $$\mathcal{G}_n$$ be the graph created by subdividing each edge of $$\mathcal{G}$$ into $$n$$ edges.

Observation.

In synthetic polymers, $$n \sim$$ # of persistence lengths along each edge of the structure graph.

### Chain Maps and Structure Graphs

Idea.

The junction positions in a random embedding of a subdivided graph ought to be some random embedding of the structure graph.

Definition.

Given $$\mathcal{G}$$ and $$\mathcal{G}'$$ and $$f_0: \operatorname{VC}' \to \operatorname{VC}$$, $$f_1: \operatorname{EC}' \to \operatorname{EC}$$, $$f_0$$ and $$f_1$$ are chain maps if $$\partial f_1 = f_0 \partial'$$.

### Structure Graph Distribution

Suppose $$f_0,f_1$$ are injective chain maps between $$\mathcal{G}'$$ and $$\mathcal{G}$$ with the same cycle rank, $$\mu$$ an admissible measure on $$\operatorname{ED}$$ compatible with $$\mathcal{G}$$, and $$\mu' = (f_1^\ast)_\sharp$$ the pushforward on $$\operatorname{ED}'$$.

The probability measure $$\nu_{\mathcal{G}'}'$$ on $$\operatorname{VP}'$$ induced by $$\mu'$$ exists and is the pushforward under $$\operatorname{proj}\operatorname{im}(\partial')^{\ast+}$$ of $$\nu_{\mathcal{G}}$$ on $$\operatorname{VP}$$ induced by $$\mu$$.

### Example: $$(m,n)$$ $$\theta$$-Graph

Corollary.

The expected (squared) distance between junctions in an $$m$$-arc $$\theta$$-graph with $$n$$ edges along each arc in a Gaussian random embedding in $$\mathbb{R}^d$$ is $$d \frac{n}{m}$$.

$$f_0,f_1$$

$$\mu' = \mathcal{N}(\vec{0},n)$$ on $$(\mathbb{R}^d)^m$$

$$\operatorname{im}(\partial')^\ast \subset \operatorname{ED}'$$ is $$\operatorname{diag} \mathbb{R}^d \subset (\mathbb{R}^d)^m$$

$$\mu_{\mathcal{G}'}' = \mathcal{N}(\vec{0},n)$$ on $$\operatorname{im} (\partial ')^\ast$$.

$$W(w) \sim\mathcal{N}(0,\frac{n}{m})$$ on coord. $$\mathbb{R}^d \subset (\mathbb{R}^d)^m$$.

### Freely-Jointed Networks

Definition.

If the measure $$\mu$$ on $$\operatorname{ED}$$ is the submanifold measure on the product of unit spheres $$(S^2)^{\mathcal{E}} \subset \operatorname{ED} = (\mathbb{R}^3)^{\mathcal{E}}$$, call the resulting model a freely jointed network.

### Junction–Junction Distance

With the obvious chain maps

$$f_0,f_1$$

can compute $$\mu'$$ explicitly. Junction–junction distances are explicit 6D numerical integrals.

Comparison with Markov chain experiments

### What happens as $$n \to \infty$$?

Definition.

The normalized graph Laplacian $$\mathcal{L}(\mathcal{G})$$ is given by

\mathcal{L}(\mathcal{G}) = \begin{cases}1-\frac{2\times\text{\# loop edges}}{\operatorname{deg}(v_i)} & i = j \\ -\frac{k}{\sqrt{\operatorname{deg}(v_i) \operatorname{deg}(v_j)}} & \text{if }v_i,v_j \text{ joined by } k \text{ edges} \\ 0 & \text{else}\end{cases}

$$\lim_{n \to \infty}\frac{1}{\mathcal{V}(\mathcal{G}_n)}\mathbb{E}[R_g^2(\mathcal{G}_n)] = \frac{1}{\mathcal{E}(\mathcal{G})^2}\left(\operatorname{tr}\mathcal{L}^+(\mathcal{G})+\frac{1}{3}\operatorname{Loops}(\mathcal{G})-\frac{1}{6}\right)$$.

### Experimental Measurements of Relative Size

Size Exclusion Chromatography apparatus

Honda Lab

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

$$\frac{17}{49}\approx 0.347$$

$$\frac{107}{245}\approx 0.437$$

$$\frac{109}{245}\approx 0.445$$

$$\frac{31}{49}\approx 0.633$$

$$\frac{43}{49}\approx 0.878$$

$$1$$

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

### Comparing Theory and Simulation

We performed molecular dynamics simulations using LAMMPS on the TSUBAME supercomputer at Tokyo Tech. These included self-avoidance, so radii of gyration fit to

\mathbb{E}[R_g^2; \mathcal{G}_n] = C_{\mathcal{G}}\mathcal{V}(\mathcal{G}_n)^{1.176}+\Delta_{\mathcal{G}}

and we could estimate $$g(\mathcal{G}_\infty,\mathcal{G}_\infty^{\text{tree}}) = \frac{C_{\mathcal{G}}}{C_{\text{tree}}}$$.

# Thank you!

### References

J. Cantarella, T. Deguchi, C. Shonkwiler, E. Uehara

Radius of gyration, contraction factors, and subdivisions of topological polymers

preprint, 2020, arXiv:2004.06199

J. Cantarella, T. Deguchi, C. Shonkwiler, E. Uehara

Random graph embeddings with general edge potentials

preprint, 2022, arXiv:2205.09049

#### Topological Polymers and Random Embeddings of Graphs

By Clayton Shonkwiler

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