Using Differential Geometry to Model Complex Biopolymers

Clayton Shonkwiler

Colorado State University

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
Φ()=12π20ysin(y)sincNyd​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?

Sampling random rings is hard

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

(see, e.g., Alvarado–Calvo–Millett)

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}
ePol(N)=Gr2(CN)/​/​1U(1)N1\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
0di20 \leq d_i \leq 2
1 \leq d_i + d_{i-1}
1di+di11 \leq d_i + d_{i-1}
|d_i - d_{i-1}| \leq 1
didi11|d_i - d_{i-1}| \leq 1
0 \leq d_{N-3} \leq 2
0dN320 \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

Small knots in proteins persist

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

Expected radius of gyration

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}
3V1λi\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=[3011010000030110100010301001001103000010011030000110000310010100013100001000131000010001310000110013]L=\left[ \begin{array}{cccccccccc} 3 &amp; 0 &amp; -1 &amp; -1 &amp; 0 &amp; -1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 3 &amp; 0 &amp; -1 &amp; -1 &amp; 0 &amp; -1 &amp; 0 &amp; 0 &amp; 0 \\ -1 &amp; 0 &amp; 3 &amp; 0 &amp; -1 &amp; 0 &amp; 0 &amp; -1 &amp; 0 &amp; 0 \\ -1 &amp; -1 &amp; 0 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; -1 &amp; 0 \\ 0 &amp; -1 &amp; -1 &amp; 0 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; -1 \\ -1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 3 &amp; -1 &amp; 0 &amp; 0 &amp; -1 \\ 0 &amp; -1 &amp; 0 &amp; 0 &amp; 0 &amp; -1 &amp; 3 &amp; -1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1 &amp; 0 &amp; 0 &amp; 0 &amp; -1 &amp; 3 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; -1 &amp; 0 &amp; 0 &amp; 0 &amp; -1 &amp; 3 &amp; -1 \\ 0 &amp; 0 &amp; 0 &amp; 0 &amp; -1 &amp; -1 &amp; 0 &amp; 0 &amp; -1 &amp; 3 \\ \end{array} \right]

Two Synthetic Topological Polymers

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

ladder graph (subdivided)

Topological polymers

Size exclusion chromatograph

Distinguishing graph types in the lab

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[Rg2(K3,3)]=108261V+60V2+17V3486V20.12+0.035VE[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[Rg2(ladder)]=5401305V+372V2+109V32430V20.15+0.045VE[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

Using Differential Geometry to Model Complex Biopolymers

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