### Using Differential Geometry to Model Complex Biopolymers

Clayton Shonkwiler

http://shonkwiler.org

01.28.19

/scmb

this talk!

### 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

### Random Walks

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

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.

### Simulating random walks in $$\mathbb{R}^3$$ is easy

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.

### Alternatives & Refinements

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.

### Ring Polymers

Knot complexity in DNA from P4 tailless mutants

Is this surprising?

### Two Standard Strategies

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

### Ansatz

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 $$\mathbb{R}^3$$.

### Grassmannians and framed polygons

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.

### Planar polygons

Theorem (w/ Cantarella, Needham, Stewart)

The probability that a random triangle is obtuse is

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

### Equilateral Polygons

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

Continuous symmetry $$\Rightarrow$$ conserved quantity

### $$N-3$$ commuting symmetries

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

### Chord distributions

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 $$\theta_i$$.

### A polytope

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
$0 \leq d_i \leq 2$
1 \leq d_i + d_{i-1}
$1 \leq d_i + d_{i-1}$
|d_i - d_{i-1}| \leq 1
$|d_i - d_{i-1}| \leq 1$
0 \leq d_{N-3} \leq 2
$0 \leq d_{N-3} \leq 2$

### Simulations

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.

### Viral DNA and Confined Polygons

The same algorithm works even better for tightly confined polygons.

### Consequences & Refinements

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.

### Open Knots

Theorem (w/ Cantarella, Chapman, and Reiter)

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

1QMG – From KnotProt

3L05

### More Complicated Topological Biopolymers

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

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

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]
$L=\left[ \begin{array}{cccccccccc} 3 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 3 & 0 & -1 & 0 & 0 & -1 & 0 & 0 \\ -1 & -1 & 0 & 3 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & -1 & 0 & 3 & 0 & 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 & 0 & 3 & -1 & 0 & 0 & -1 \\ 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 3 & -1 \\ 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 3 \\ \end{array} \right]$

### Two Synthetic Topological Polymers

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

Topological polymers

Size exclusion chromatograph

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 $$K_{3,3}$$!

# Thank you!

### References

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

J. Phys. A Highlight of 2016

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]

#### Using Differential Geometry to Model Complex Biopolymers

By Clayton Shonkwiler

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