Clayton Shonkwiler PRO
Mathematician and artist
/erice22
This talk!
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)
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 \(\mathcal{G}\).
Interactions are symmetric probability distributions on edge vectors.
Problem.
Edge vectors are not independent when there are loops.
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 \(\mathcal{G}\)
Theorem [Estrada–Hatano, James–Guth]
The expected variation size \(\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+\).
Theorem [Estrada–Hatano, James–Guth]
The expected variation size \(\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{974299}{765600}\).
Theorem [Estrada–Hatano, James–Guth]
The expected variation size \(\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+\).
Graph Laplacian of \(\mathcal{G}\)
Theorem [Estrada–Hatano, James–Guth]
The expected variation size \(\mathbb{E}[\sum \|\delta p_i\|^2] = \frac{d}{\mathfrak{V}} \operatorname{tr}L^+\).
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.
Natural elastic materials tend to have extremely complicated, random graph types
Wood-based nanofibrillated cellulose
Qspheroid4 [CC BY-SA 4.0], from Wikimedia Commons
Synthetic chemists can now produce simple topological polymers in usable quantities.
\(\theta\)-curves in solution at the Tezuka lab
Linear polymers
Topological polymers
Independently
Conditioned on graph type
Edges chosen from some \(O(d)\)-invariant distribution \(\mu\).
What does this mean?
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:
\(w = w_1 e_1 + \dots + w_{\mathcal{E}}e_{\mathcal{E}}\).
Definition.
The boundary map \(\partial : \operatorname{EG} \to \operatorname{VC}\) is defined by
\(\partial(e_i) = \operatorname{head}(e_i) - \operatorname{tail}(e_i)\).
Definition.
\(\operatorname{ker} \partial \subset \operatorname{EC}\) is the loop space of \(\mathcal{G}\).
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 \(\mathcal{G}\).
\(\xi(\mathcal{G}) = \frac{1}{2} \sum_{i=1}^{\mathcal{V}} \left( \deg(v_i)-2\right) + 1\).
\(-\)
\(=\)
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
\(\operatorname{disp}(X)(e_i) = X(\operatorname{head}(e_i)) - X(\operatorname{tail}(e_i))\).
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 \(u \in \ker \partial\}\).
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 \(\operatorname{im}\partial^\ast \subset \operatorname{ED}\).”
Definition.
The distribution \(\mu\) on ED is the standard Gaussian; restriction to \(\operatorname{im}\partial^\ast\) is standard Gaussian on that subspace.
\(\operatorname{ED} \simeq \mathbb{R}^{3\mathcal{E}}\)
\(\operatorname{VP} \simeq \mathbb{R}^{3\mathcal{V}}\)
\(\partial^\ast\)
\(\operatorname{im}\partial^\ast\)
?
\(\partial^{\ast +}\)
\((\operatorname{ker}\partial^\ast)^\bot\)
\((\operatorname{ED}, \langle\, , \,\rangle)\)
\((\operatorname{VP}, \langle \, , \, \rangle_{\widetilde{L}^\ast})\)
\(\partial^\ast\)
\(\operatorname{im}\partial^\ast\)
\(\partial^{\ast +}\)
\((\operatorname{ker}\partial^\ast)^\bot\)
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.
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.
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 \(\partial f_1 = f_0 \partial'\).
Theorem [with Cantarella, Deguchi, Uehara (2022)]
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 \(\mu\).
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. \(\mathbb{R}^d \subset (\mathbb{R}^d)^m\).
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.
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
Definition.
The normalized graph Laplacian \(\mathcal{L}(\mathcal{G})\) is given by
Theorem [with Cantarella, Deguchi, Uehara (2020)]
\(\lim_{n \to \infty}\frac{1}{\mathcal{V}(\mathcal{G}_n)}\mathbb{E}[R_g^2(\mathcal{G}_n)] = \frac{1}{\mathcal{E}(\mathcal{G})^2}\left(\operatorname{tr}\mathcal{L}^+(\mathcal{G})+\frac{1}{3}\operatorname{Loops}(\mathcal{G})-\frac{1}{6}\right)\).
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”
We performed molecular dynamics simulations using LAMMPS on the TSUBAME supercomputer at Tokyo Tech. These included self-avoidance, so radii of gyration fit to
and we could estimate \(g(\mathcal{G}_\infty,\mathcal{G}_\infty^{\text{tree}}) = \frac{C_{\mathcal{G}}}{C_{\text{tree}}}\).
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
By Clayton Shonkwiler