Equilateral Polygons, Shapes of Ring Polymers, and an Application to Frame Theory

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

03.17.18

/osu18

This talk!

Statistical physics viewpoint

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

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

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Key Idea

Loop \(\Leftrightarrow\) point in some (nice!) conformation space

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

theorems and fast numerical algorithms.

Grassmannians...

As in Erik’s talk, points in the Grassmannian \(\mathrm{Gr}_2(\mathbb{C}^n)\) are conformations of closed, framed polygons in \(\mathbb{R}^3\).

Independently spinning the frames corresponds to the action by the torus of diagonal elements of \(U(n)\).

Think of this as a discrete, 3-dimensional version of Younes–Michor–Shah–Mumford’s shape space. Tom has developed the smooth version.

Torus Action

Scalar matrices act trivially on \(\mathrm{Gr}_2(\mathbb{C}^n)\), so there is an effective \(U(1)^{n-1}\) action, and \(\mathrm{Gr}_2(\mathbb{C}^n)/U(1)^{n-1}\) is the space of closed polygons in \(\mathbb{R}^3\).

This is not a symplectic or algebraic operation:

\mathrm{dim}\,\mathrm{Gr}_2(\mathbb{C}^n)/U(1)^{n-1}=4n-8-(n-1)=3n-7
dimGr2(Cn)/U(1)n1=4n8(n1)=3n7\mathrm{dim}\,\mathrm{Gr}_2(\mathbb{C}^n)/U(1)^{n-1}=4n-8-(n-1)=3n-7

To get symplectic/algebraic geometry, need to take the symplectic reduction/GIT quotient

\mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_r U(1)^{n-1} \quad \text{or} \quad \mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_L(\mathbb{C}^*)^{n-1}
Gr2(Cn)//rU(1)n1orGr2(Cn)//L(C)n1\mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_r U(1)^{n-1} \quad \text{or} \quad \mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_L(\mathbb{C}^*)^{n-1}

which depends on a choice of moment map fiber/line bundle, but is a symplectic manifold/projective variety.

Fixing Edgelengths

Geometrically, this choice is equivalent to choosing the lengths \(r_1,\ldots , r_n\) of the edges in the polygon.

Definition. \(\mathrm{Pol}(n,r) := \mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_r U(1)^{n-1}\) is the conformation space of closed polygons in \(\mathbb{R}^3\) with edgelength vector \(r=(r_1,\ldots , r_n)\).

In polymer physics, primarily interested in equilateral polygons, i.e., the space \(\mathrm{ePol}(n) := \mathrm{Pol}(n,(1,\ldots , 1))\).

Equilateral Polygons

This is the space of arclength-parametrized discrete closed curves in \(\mathbb{R}^3\). See Millson–Zombro for the generalization to smooth or Lipschitz closed curves.

Symmetries

The Gelfand–Tsetlin integrable system on \(\mathrm{Gr}_2(\mathbb{C}^n)\) descends to \(n-3\) commuting symmetries on \(\mathrm{Pol}(n,r)\).

The corresponding conserved quantities are the distances \(d_i\) from the first vertex to the \((i+2)\)nd vertex.

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

Sampling

Theorem (with Cantarella, 2016)

The joint distribution of \(d_1,\ldots , d_{n-3}\) is uniform on \(\mathcal{P}_n\) and \(\theta_1, \ldots , \theta_{n-3}\) are each uniform on \([0,2\pi]\).

Corollary

Any algorithm for sampling the convex polytope \(\mathcal{P}_n\subseteq \mathbb{R}^{n-3}\) gives an algorithm for sampling \(\mathrm{ePol}(n)\).

A Sampling Algorithm

Introduce fake chordlengths \(d_0=1=d_{n-2}\) and make the linear change of variables

\(s_i = d_i - d_{i-1} \text{ for } 1 \leq i \leq n-2\).

Then \(\sum s_i = d_{n-2} - d_0 = 0\), so \(s_{n-2}\) is determined by \(s_1, \ldots , s_{n-3}\)

and the 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

become

\(-1 \leq s_i \leq 1, -1 \leq \sum_{i=1}^{n-3} s_i \leq 1\)

\(\sum_{j=1}^i s_j + \sum_{j=1}^{i+1}s_j \geq -1\)

The polytope is surprisingly large!

Let \(\mathcal{C}_n \subset \mathbb{R}^{n-3}\) be determined by

\(-1 \leq s_i \leq 1, -1 \leq \sum_{i=1}^{n-3} s_i \leq 1\)

\(\sum_{j=1}^i s_j + \sum_{j=1}^{i+1}s_j \geq -1\)

\(\mathcal{C}_5\)

\(\mathcal{C}_6\)

Proposition (with Cantarella, Duplantier, Uehara, 2016)

The probability that a point in the hypercube lies in \(\mathcal{C}_n\) is asymptotic to

\(6 \sqrt{\frac{6}{\pi}}\frac{1}{n^{3/2}}\)

Direct sampling

Action-Angle Method

Theorem  (with Cantarella, Duplantier, Uehara, 2016)

The action-angle method directly samples polygon space in expected time \(\Theta(n^{5/2})\).

  • Generate \((s_1,\ldots , s_{n-3})\) uniformly on \([-1,1]^{n-3}\)
  • Test whether \((s_1,\ldots , s_{n-3})\in \mathcal{C}_n\)
  • Change to \((d_1, \ldots , d_{n-3})\) coordinates
  • Generate dihedral angles from \(T^{n-3}\)
  • Build corresponding polygon

\(O(n)\) time

acceptance probability \(\sim \frac{1}{n^{3/2}}\)

RandomDiagonals[n_] := 
  Accumulate[
   Join[{1}, RandomVariate[UniformDistribution[{-1, 1}], n]]];

InMomentPolytopeQ[d_] := 
  And[Last[d] >= 0, Last[d] <= 2, 
   And @@ (Total[#] >= 1 & /@ Partition[d, 2, 1])];

DiagonalSample[n_] := Module[{d},
   For[d = RandomDiagonals[n], ! InMomentPolytopeQ[d], , 
    d = RandomDiagonals[n]];
   d[[2 ;;]]
   ];

Diagonal sampling in 3 lines of code

A random 10,000-gon

Frames

Definition. A frame in a Hilbert space \(\mathcal{H}\) is a collection \(\{f_i\}\) of elements of \(\mathcal{H}\) so that

A\|x \|^2 \leq \sum |\langle f_i, x\rangle|^2 \leq B\|x\|^2
Ax2fi,x2Bx2A\|x \|^2 \leq \sum |\langle f_i, x\rangle|^2 \leq B\|x\|^2

for all \(x \in \mathcal{H}\). If \(A=B\), the frame is tight. If \(\mathcal{H} = \mathbb{R}^d\) or \(\mathbb{C}^d\), the frame is finite. If \(\|f_i\|=1\) for all \(i\), the frame is unit norm. Finite unit norm tight frames are called FUNTFs.

Lemma. A frame \(F\) in \(\mathbb{C}^d\) is tight \(\Leftrightarrow\) \(T_F^*T_F = \lambda \mathbb{I}_d\).

A finite frame \(F\) in \(\mathbb{C}^d\) has corresponding analysis operator \(T_F: \mathbb{C}^d \to \mathbb{C}^n\) given by \(T_F(v) = (\langle v, f_1\rangle, \ldots , \langle x, f_n\rangle)\), synthesis operator \(T_F^*\), and frame operator \(T_F^*T_F\).

Example. The Mercedes–Benz frame in \(\mathbb{R}^2\) is a FUNTF.

Lemma. A FUNTF in \(\mathbb{C}^d\) has \(T_F^*T_F=\frac{n}{d}\mathbb{I}_d\).

FUNTFs

\(\mathcal{F}^n_{\frac{n}{d} \mathbb{I}_d} =\{F \subseteq \mathbb{C}^d : T_F^*T_F=\frac{n}{d}\mathbb{I}_d\} \simeq \mathrm{St}_d(\mathbb{C}^n) \), the Stiefel manifold of orthonormal \(d\)-frames in \(\mathbb{C}^n\), and hence \(U(d)\backslash \mathcal{F}^n_{\frac{n}{d}\mathbb{I}_d} \simeq \mathrm{Gr}_d(\mathbb{C}^n)\).

If \(S\) is an invertible, positive definite, Hermitian \(d \times d\) matrix, let \(\mathcal{F}^n_S := \{F = \{f_i\}_{i=1}^n \subseteq \mathbb{C}^d : T_F^*T_F=S\}\)

Let \(\mathcal{F}^n_S(r)\subseteq\mathcal{F}^n_S\) be the space of frames with frame operator \(S\) and \(\|f_i\| = r_i\) for all \(i\).

Proposition. \(U(d)\backslash\mathcal{F}^n_{\frac{n}{d}\mathbb{I}_d}(r)/U(1)^{n-1} \simeq \mathrm{Gr}_d(\mathbb{C}^n)/\!/\!_r U(1)^{n-1}=\mathrm{Pol}(n,r)\)

FUNTFs in \(\mathbb{C}^2\)

Theorem (2018)

A modification of the action-angle algorithm directly samples FUNTFs in \(\mathbb{C}^2\) in \(\Theta(n^{5/2})\) time.

Equilateral polygons in \(\mathbb{R}^3\) lift to FUNTFs in \(\mathbb{C}^2\)!

Histogram of coherences of length-6 FUNTFs in \(\mathbb{C}^2\)

Histogram of coherences of length-4 FUNTFs in \(\mathbb{C}^2\)

Coherences of lifts of Sloane's optimal point packings \(S^2\), compared to the Toth bound

More generally...

Similar symplectic reasoning leads to:

Theorem (with Needham, 2018)

If \(S\) is an invertible, positive-definite, \(d \times d\) matrix, the space \(\mathcal{F}^n_S(r)\) (of length-\(n\) frames in \(\mathbb{C}^d\) with frame operator \(S\) and \(\|f_i\|=r_i\)) is path-connected.

This generalizes Cahill, Mixon, and Strawn’s (2017) resolution of the frame homotopy conjecture, which showed that the space of FUNTFs in \(\mathbb{C}^d\) is path-connected.

Problems

  • Does distance in the Grassmannian serve as a reasonable proxy for structural similarity physical loops (e.g., ring biopolymers)?
  • What is going on with coherences of length-4 FUNTFs in \(\mathbb{C}^2\)?
  • Is there a (fast) sampling algorithm for FUNTFs in \(\mathbb{C}^d\)?
  • What is the corresponding 2D/real story?

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

Symplectic geometry of spaces of frames

T. Needham & C. Shonkwiler

In preparation

Equilateral Polygons, Shapes of Ring Polymers, and an Application to Frame Theory

By Clayton Shonkwiler

Equilateral Polygons, Shapes of Ring Polymers, and an Application to Frame Theory

Random flights in 3-space forming a closed loop, or random polygons, are a standard simplified model of so-called ring polymers like bacterial DNA. Equilateral random polygons, where all steps are the same length, are particularly interesting (and challenging) in this context. In this talk I will describe an (almost) toric Kähler structure on the moduli space of equilateral polygons and show how to exploit this structure to get a fast algorithm to directly sample the space. Using work of Hausmann–Knutson, the Kähler structure on the space of equilateral polygons can be realized as the Kähler reduction of the standard Kähler structure on the Grassmannian of 2-planes in complex n-space. This means that equilateral polygons in 3-space can be lifted to Finite Unit-Norm Tight Frames (FUNTFs) in complex 2-space. I will describe how to modify the polygon sampler to produce a FUNTF sampler and show that optimal packings in the 2-sphere lift to FUNTFs with low coherence.

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