/scmb

this talk!

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

Polyethylene

Nicole Gordine [CC BY 3.0] from Wikimedia Commons

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

**Simple model:** The freely jointed chain/random walk model dates back to Kuhn in the 1930s.

**Rayleigh (1919):** The probability density function of the end-to-end vector \(\vec{\ell}\) of a 3D equilateral random walk is

\Phi(\vec{\ell}) = \frac{1}{2\pi^2 \|\vec{\ell}\|} \int_0^\infty y \sin (\|\vec{\ell}\| y) \operatorname{sinc}^N y \operatorname{d}\!y

**Corollary** (see e.g. de Gennes or Khokhlov–Grosberg–Pande): The expected end-to-end distance and radius of gyration of a random walk of \(N\) segments scale like \(\sqrt{N}\).

*Modern polymer physics is based on the analogy between a polymer chain and a random walk.*

Generate \(N\) independent uniform random points in \(\mathbb{R}^3\) according to your favorite probability distribution and treat them as an ordered list of edge vectors.

**Self-avoiding walks on the lattice** (see, Madras–Slade): Scaling exponent agrees with polymers in dilute solution with good solvent.

**Thick off-lattice random walks** (see Plunkett–Chapman): Markov chain simulation algorithm and experimental evidence of the same scaling exponent as lattice SAWs.

Knot complexity in DNA from P4 tailless mutants

**Is this surprising?**

**Lattice Polygons:** Reduces the problem to combinatorics

(see, e.g., Sumners–Whittington or many works from the Arsuaga–Vazquez Lab)

cf. Diao

**Off-Lattice Markov Chains:** Randomized simulation

(see, e.g., Alvarado–Calvo–Millett)

*Topologically constrained random walk* \(\Leftrightarrow\) *point in some (nice!) configuration space*

Knowledge of the (differential, symplectic, algebraic) geometry of these conformation spaces leads to both

**theorems and fast numerical algorithms** for studying and TCRWs in \(\mathbb{R}^3\).

The space of *framed polygons* of fixed total length but variable edge lengths lifts to the complex Grassmannian \(\operatorname{Gr}_2(\mathbb{C}^N)\) (see Hausmann–Knutson, Howard–Manon–Millson, and Cantarella–Deguchi–Shonkwiler).

**Algorithm** (w/ Cantarella & Deguchi)

Simulation of framed \(N\)-gons in \(\Theta(n)\) time.

**Theorem** (w/ Cantarella, Grosberg, & Kusner)

The expected total curvature of framed \(N\)-gons is \(\frac{\pi}{2}N + \frac{\pi}{4}\frac{2N}{2N-3}\).

**Corollary**

At least \(\frac{1}{3}\) of hexagons and \(\frac{1}{11}\) of heptagons are knotted.

**Theorem **(w/ Cantarella, Needham, Stewart)

The probability that a random triangle is obtuse is

\(\frac{3}{2}-\frac{3\ln 2}{\pi}\approx0.838\)

The space of equilateral \(N\)-gons can be constructed as a *symplectic reduction* of the Grassmannian (see Kapovich–Millson and Hausmann–Knutson):

\operatorname{ePol}(N)=\operatorname{Gr}_2(\mathbb{C}^N)/\!/\!_{\vec{1}}U(1)^{N-1}

Continuous symmetry \(\Rightarrow\) conserved quantity

Rotations around \(N-3\) chords \(d_i\) by \(N-3\) angles \(\theta_i\) commute.

**Theorem **(w/ Cantarella)

The joint distribution of \(d_1,\ldots , d_{N-3}\) and \(\theta_1, \ldots , \theta_{N-3}\) are all uniform on their domains.

Therefore, simulating equilateral \(N\)-gons is equivalent to sampling random points in the convex polytope of \(d_i\)’s and random angles \(\theta_i\).

The \((N-3)\)-dimensional* moment polytope* \(\mathcal{P}_N \subset \mathbb{R}^{N-3}\) is defined by the triangle inequalities

0 \leq d_i \leq 2

1 \leq d_i + d_{i-1}

|d_i - d_{i-1}| \leq 1

0 \leq d_{N-3} \leq 2

**Algorithm** (w/ Cantarella)

A uniformly ergodic Markov chain for simulating random equilateral \(n\)-gons.

More generally, a uniformly ergodic Markov chain for simulating random points from *any* toric symplectic manifold.

The same algorithm works *even better* for tightly confined polygons.

**Theorem** (Hake)

The fraction of random equilateral 6-gons which are knotted is no bigger than \(\frac{14-3\pi}{192} < \frac{1}{42}\).

**Algorithm** (w/ Cantarella, Duplantier, & Uehara)

Direct sampling of equilateral \(N\)-gons in \(\Theta(N^{5/2})\) time.

**Theorem** (Eddy; see poster session)

The stick number of the \(9_{43}\) and \(9_{48}\) knots is 9.

**Theorem **(w/ Cantarella, Chapman, and Reiter)

The closest equilateral polygon to an open chain is given by *geometric median closure*.

1QMG – From KnotProt

3L05

Wood-based nanofibrillated cellulose

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

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

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

If \(\lambda_i\) are the eigenvalues of \(L\), the expected squared radius of gyration of a Gaussian random embedding of \(\mathfrak{G}\) in \(\mathbb{R}^3\) is

\frac{3}{\mathfrak{V}} \sum \frac{1}{\lambda_i}

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

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

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]

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

ladder graph (subdivided)

Topological polymers

Size exclusion chromatograph

larger molecule

smaller molecule

**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(\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 \(K_{3,3}\)!

The symplectic geometry of closed equilateral random walks in 3-space

J. Cantarella & C. Shonkwiler

*Annals of Applied Probability ***26 **(2016), no. 1, 549–596

A fast direct sampling algorithm for equilateral closed polygons

J. Cantarella, B. Duplantier, C. Shonkwiler, & E. Uehara

*Journal of Physics A* **49** (2016), no. 27, 275202

Funding: Simons Foundation

Probability theory of random polygons from the quaternionic viewpoint

J. Cantarella, T. Deguchi, & C. Shonkwiler

*Communications on Pure and Applied Mathematics* **67** (2014), no. 10, 1658–1699

The expected total curvature of random polygons

J. Cantarella, A.Y. Grosberg, R. Kusner, & C. Shonkwiler

*American Journal of Mathematics* **137** (2015), no. 2, 411–438

Spherical geometry and the least symmetric triangle

L. Bowden, A. Haynes, C. Shonkwiler, & A. Shukert

*Geometriae Dedicata* (2018), https://doi.org/10.1007/s10711-018-0327-4

Random triangles and polygons in the plane

J. Cantarella, T. Needham, C. Shonkwiler, & G. Stewart

*The American Mathematical Monthly*, to appear, arXiv:1702.01027 [math.MG]