Topological correlations in dense lattice trivial knots

1. Topological correlations in dense lattice trivial knots Sergei Nechaev LPTMS (Orsay, France)

2. Structure of the talk • Biophysical motivations for the conside-ration of topological correlations in lattice knots; • “Statistical topology” of disordered systems: topology as a “quenched disorder”; • Conditional distributions and expectations of highest degree of polynomial invariants; • “Brownian bridges” in hyperbolic spaces.

3. 1. Biophysical motivations 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

4. 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?

5. Theoretical prediction of a “crumpled” structure:A.Grosberg, E.Schakhnovich, S.N. (J. Physique, 1988) We considered a dense state of an unknotted polymer ring after a temperature jump from q–temp. to T (T<q) This structure resembles self-similarPeano curve This structure is thermodynamically favorable!

6. How to prove the existence of a crumpled structure? 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). (a) (b)

7. We consider a quasi-two-dimensional system corresponding to a polymer globule in a thin slit

8. 2. Statistical topology of disordered systems 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”. To the vertex kwe assign the value of a “disorder” depending on the crossing type:

9. 3. Conditional distributions of knot invariants 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.

10. 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. 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.

11. Polynomial invariants and “Potts spin glass” Write as a partition function of Potts spin system with disordered interactions on the dual lattice where

12. Results Typical unconditional knot complexity nof a random knot has asymptotic behavior: Typical conditionalknot complexity n*of a daughter (quasi)knot,which is a part of a parent trivial knot, has asymptotic behavior: Relative complexityn*/Nof a daughter knot tends to 0:

13. “Brownian bridges” in hyperbolic spaces 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. Example: Classical Fuerstenberg theorem, 1963

14. Random walks in Lobachevsky plane Elongation of a single random walk in hyperbolic geometry Radial distribution function is

15. 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

16. Toy model of hierarchically overlapping intervals

17. For the overlapping probability we have generated an ensemble of contact maps. The typical plots and the output are as follows (d = 3): We see the block-hierarchical structure (size 64x64)

18. 2. Random block-hierarchical adjacency matrices of random graphs (contact maps) What is the density of eigenvalues for such a matrix for typical distributions of matrix elements?

19. Matrix elements are defined as follows where , and is the level of the hierarchy Example: 1 2 3 4 5 6 7 8 0 1 1 1 1 2 1 2 0 0 3 8 3 0 4 1 1 1 5 7 4 6 1 6 5 1 1 7 1 8

20. 3. Scale-free spectral density (μ = 0.2) • Semi–log plot of the distribution of eigenvalues for N = 256 (solid line) and N = 2048(dashed line); • The left– and right–hand tails of for N = 256 in log–log coordinates.

21. Comparison of spectral densities of random hierarchical and Erdös-Rényi randomgraphs • The semi–log plot of thespectral densities for: random hierarchical graphs for N = 256 and μ = 0.2 (solidline), random Erdös-Rényi graphs for N = 256 and p = 0.2 (dotted line), for N = 256 and p = 0.02 (dashedline); • The central part of the figure (a) in the linear scale.

22. Numerical verification of the power-law behavior for Gaussian distribution of hierarchical adjacency matrix

23. 4. Distribution of “motifs” in hierarchical networks Consider subgraphs-triads Define statistical significanceZk 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”).

24. m m Distribution of motifs in the superfamily II (in the classification of U. Alon et al) The found distribution of motifs in hierarchical networks is very similar to the distribution of motifs in the superfamily II (networks of neurons)