Topological correlations in dense lattice trivial knots

Sergei Nechaev

LPTMS (Orsay, France)

Structure of the talk

1.Biophysical motivations for the conside-ration of topological correlations in lattice knots;

2.“Statistical topology” of disordered systems: topology as a “quenched disorder”;

3.Conditional distributions and expectations of highest degree of polynomial invariants;

4.“Brownian bridges” in hyperbolic spaces.

Double helix of DNA could reach length ~ 2 m, consists of ~ 3 billions base pairs and is packed in a cell nucleus of size of ~ 20 micrometers.

During transcription the DNA fragment should “disentangle” form densely packed state and should “fold back” after.

How to do that fast and reversibly???

The possible answer is contained in an experimental work on human genome (E. Lieberman-Aiden, et al, Science, 2009):

DNA forms a “crumpled” compact nontrivial fractal structure without knots

1. Biophysical motivations

The classical theory of coil-to-globule phase transitions (Lifshitz, Grosberg, Khokhlov, 1968-1980) states:

At low temperatures the macromolecule (with “open” ends) forms a compact weakly fluctuating drop-like “globular” structure.

What is the geometry of an unknotted fluctuating polymer ring in a compact (“globular”) phase?

We considered a dense state of an unknotted polymer ring after a temperature jump from –temp. to T (T< )

This structure resembles self-similar Peano curve

This structure is thermodynamically favorable!

Theoretical prediction of a “crumpled” structure: A.Grosberg, E.Schakhnovich, S.N. (J. Physique, 1988)

In a compact state two scenarios of a microstructure formation could be realized. Either folds deeply penetrate each other as shown in (a), or folds of all scales stay segregated in the space as shown in (b).

How to prove the existence of a crumpled structure?

(a) (b)

We consider a quasi-two-dimensional system corresponding to a polymer globule

in a thin slit

In a globular phase one can separate topological and spatial fluctuations. We model the globule by a dense knot diagram completely filling the rectangular lattice . Thus, we keep only the “topological disorder”.

h wL L

To the vertex k we assign the value of a “disorder” depending on the crossing type:

1kb 1kb

1kb

2. Statistical topology of disordered systems

What is a typical topological state of a “daughter” (quasi)knot under the condition that the “parent” knot is trivial?

Quasiknot – a part of a knot equipped with boundary conditions.

tL hL

wL

3. Conditional distributions of knot invariants

How to characterize topological states?

It is sufficient for statistical purposes to discriminate knots by degrees of polynomial (Kauffman) invariants:

For trivial knots n ~ 1, for very complex knots n ~ N.

| |

ln ( )lim

ln K

A

f An

A

Remark 1. Degree n reflects nonabelian character of topological interactions.

Remark 2. Degree n defines the metric space and allows to range the knots by their complexities.

Remark 3. Since is a partition function, the degree n has a sense of a free energy.

( )Kf A

Write as a partition function of Potts spin system with disordered interactions on the dual lattice

where

{ } { , }

( ,{ }) ,{ } exp ( , ) ( , )K ij ij ij ij i ji j

f A C A J A

24 2 2( , ) ln ; ij

ij ijJ A A q A A

Polynomial invariants and “Potts spin glass”

( )Kf A

Results

Typical conditional knot complexity n* of a daughter (quasi)knot, which is a part of a parent trivial knot, has asymptotic behavior:

t wn N L L

tL

wL

hL

* h wn N L L

* const lim lim 0N N

n N

N N

Relative complexity n*/N of a daughter knot tends to 0:

Typical unconditional knot complexity n of a random knot has asymptotic behavior:

The space of all topological states in our model is non-Euclidean and is the space of constant negative curvature (Lobachevsky space).

The random walk in Lobachevsky geometry can be modeled by multiplication of random noncommutative unomodular matrices.

Brownian bridge = conditional random walk in Lobachevsky space.

1 11 1 2 2

1 11 1 2 2

... ...n n N N N N

n n N N N N

a b a b a ba b a b

c d c d c dc d c d

1 0

0 1

maxNe *

maxne

Example:

Classical Fuerstenberg theorem, 1963

“Brownian bridges” in hyperbolic spaces

Elongation of a single random walk in hyperbolic geometry

Radial distribution function is

( , ) ( , )

( , 0) ( )

W t D W tt

W t

x x

x x

Random walks in Lobachevsky plane

11ik

i k

g gx xg

2 2 2 22

1 0sinh ;

0 sinhikds d d g

2

2

/ 4 /(4 )

3

/ 4/(4 )

( , )cosh cosh4 2 ( )

4 sinh

tD tD

tDtD

e eW t d

tD

ee

tD

Bunch of M random walks in Lobachevsky plane

Probability to create a watermelon of two random walks in Lobachevsky plane

where

In general one has:

Thus,

For M ≥ 3 the trajectories are elongated in hyperbolic geometry

( , | 2) ( , ) ( , ) ( )W t W t W t P

( , ) 2 sinhP t

1 1( , | ) ( , ) ( ) ( , )sinhM M M MW t M W t P W t

2/ 4 / 2

/(4 )1 / 2 1

( , )2 ( ) sinh

MtD MM tD

M M M

eW t e

tD

Toy model of hierarchically overlapping intervals

For the overlapping probability

we have generated an ensemble of contact maps.

The typical plots and the output are as follows (= 3):

( ) ~ 1 1P x x

We see the block-hierarchical structure (size 64x64)

0

0

0

0

0

0

0

0

t1

(2)

t1(3)

t1(4)

t1

(1)

t1(1)

t2(1)

t2

(1) t2(1)

t2

(1)

t2(1)

t2

(1) t2(1)

t2(1)

t2(2)

t2

(2)

t2

(2)

t2(2)

t2(2)

t2

(2)

t2(2)

t2(2)

t3(1)

t3(1)

t3

(1)

t3(1)

t3

(1)

t3

(1)

t3

(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3

(1)

t3(1)

t3

(1)

t3

(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3(1)

t3(1)t1

(2)

t1

(3)

t1(4)

T =

What is the density of eigenvalues for such a matrix for typical distributions of matrix elements?

( )

2. Random block-hierarchical adjacency matrices of random graphs (contact maps)

0

0

0

0

0

0

0

0

t1

(2)

t1(3)

t1(4)

t1

(1)

t1(1)

t2(1)

t2

(1) t2(1)

t2

(1)

t2(1)

t2

(1) t2(1)

t2(1)

t2(2)

t2

(2)

t2

(2)

t2(2)

t2(2)

t2

(2)

t2(2)

t2(2)

t3(1)

t3(1)

t3

(1)

t3(1)

t3

(1)

t3

(1)

t3

(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3

(1)

t3(1)

t3

(1)

t3

(1)

t3

(1)

t3(1)

t3(1)

t3(1)

t3

(1)

t3(1)

t3(1)t1

(2)

t1

(3)

t1(4)

T =

Matrix elements are defined as follows

where , and is the level of the hierarchy

( )1 with probability

0 with probability 1n

qt

q

q e

0

1

1

0

1

1

1

0

1

1

0

1

1

1

1 2

47

6 5

38

2 3 4 5 6 7 81

2

3

6

7

8

1

4

5

Example:

3. Scale-free spectral density (μ = 0.2) (a) Semi–log plot of the distribution of eigenvalues for

N = 256 (solid line) and N = 2048 (dashed line);

(b) The left– and right–hand tails of for N = 256 in log–log coordinates.

( )

( )

Comparison of spectral densities of random hierarchical and Erdös-Rényi random graphs

(a) The semi–log plot of the spectral densities for: random hierarchical graphs for N = 256 and μ = 0.2 (solid line), random Erdös-Rényi graphs for N = 256 and p = 0.2 (dotted line), for N = 256 and p = 0.02 (dashed line);

(b) The central part of the figure (a) in the linear scale.

Numerical verification of the power-law behavior for Gaussian distribution of hierarchical

adjacency matrix

8 82 2

4. Distribution of “motifs” in hierarchical networks

Consider subgraphs-triads

Define statistical significance Zk with respect to randomized networks with the same connectivity

and

Consider a vector p = { p1,…, p13 }.

According to U. Alon et al (Science, 2004) all networks fall into 4 “superfamilies” with respect to distribution of components of the vector p (“motifs”).

randk kk

k

N NZ

13

2

1k k k

k

p Z Z

The found distribution of motifs in hierarchical networks is very similar to the distribution of motifs in the superfamily II

(networks of neurons)

Distribution of motifs in the superfamily II (in the classification of U. Alon et al )