# Séminaire du LPTMS : Kirill Polovnikov ( Institut Curie and Skoltech)

## From topologically-stabilized polymer states to KPZ universalities in polymers

### Kirill Polovnikov (Institut Curie and Skoltech)

Hybrid: onsite seminar + zoom.

https://cnrs.zoom.us/j/98891142520?pwd=bVVJUGQrcHdLL2xaU2Z2ckYyU0lzdz09

Meeting ID: 988 9114 2520

Passcode: c8qvg3

Two non-concatenated rings cannot become catenated in the course of their dynamics, which leads to an effective topological repulsion between them. Computer simulations and some phenomenological models have shown that in the melt state these long-range interactions induce space-filling conformations for each ring with the fractal dimension $df$ equal to the dimension of the embedding space D. One of the main difficulties towards analytical description of such polymer states (e.g. a system of many unknotted and mutually non-concatenated rings) is the lack of a tractable Hamiltonian fixing the appropriate topological invariants. Thus, structural and dynamic correlations in these peculiar states remain poorly understood, being one of the central problems in Polymer Physics. Intriguingly, the conformations of chromosomes in the nucleus follow precisely the statistics of topologically-stabilized polymers further folded into a sequence of short-scale loops [1]. Topology of the genome and how it evolves during the cell cycle [2] is becoming an increasingly important question in Chromosomes Biophysics.

In my talk I will first present some of our results on the construction of an effective Hamiltonian of fractal polymer states with arbitrary fractal dimension df>=2 [3-5]. Such a Hamiltonian represents a fully-connected weighted network with the power-law decaying weights away from the main diagonal in the corresponding Laplacian matrix. This approach maps fractal polymer conformations onto the trajectories of subdiffusive fractional Brownian motion (fBm) and connects their power spectrum with the eigenvalue spectrum of the Laplacian. I will then apply the developed fBm framework to the particular case of topologically-stabilized polymers in D=3 and show that it (i) yields an analytically tractable model of fractal polymer dynamics in the form of a fractional Langevin equation [3,5] and (ii) can be used to build a minimal model of chromosomes as fractal polymer chains with quenched disorder of short-scale loops [1]. The loops disorder can be treated using a diagrammatic technique with the ultimate result explaining a beautiful universality of the contact probability laws observed in Hi-C experiments for various cell types. Furthermore, the model allows to infer the typical sizes of chromosomal loops from the data.

In the second part, I will use the de-Gennes scaling approach to show that the tail of the end-to-end probability density in a topologically-stabilized polymer is, in fact, non-Gaussian and corresponds to the right tail of the Tracy-Widom distribution. Interestingly, the critical exponent of the space-filling folding in three dimensions D=3 also corresponds to the growth exponent of the 1D KPZ, 1/D=1/3=beta. I will discuss two other polymer physics settings with long-ranged interactions exhibiting the full set of 1D KPZ exponents:

(b) a classical model of polymer dynamics with hydrodynamic interactions (Zimm) and

(c) a novel model of a polymer stretched over a cylindrical surface (« curved Pincus ») [6].

Equilbirium KPZ-like behavior in both models can be derived using scaling or the Lifshitz optimal fluctuation arguments. I thus suggest that, despite different underlying physics, long-range interactions in polymers (topology, hydrodynamics or space curvature) at equilibrium might create a similar law of fluctuations as a local but non-linear one-dimensional dynamics in the steady-state.

References:

[1] Polovnikov, K., Belan, S., Imakaev, M., Brandão, H. B., & Mirny, L. A. (2022). Fractal polymer with loops recapitulates key features of chromosome organization. bioRxiv, 2022-02.

[2] Hildebrand, E. M., Polovnikov, K., Dekker, B., Liu, Y., Lafontaine, D. L., Fox, A. N., … & Dekker, J. (2022). Chromosome decompaction and cohesin direct Topoisomerase II activity to establish and maintain an unentangled interphase genome. bioRxiv, 2022-10.

[3] Polovnikov, K., Nechaev, S., & Tamm, M. V. (2018). Effective Hamiltonian of topologically stabilized polymer states. Soft matter, 14(31), 6561-6570.

[4] Polovnikov, K. E., Nechaev, S., & Tamm, M. V. (2019). Many-body contacts in fractal polymer chains and fractional Brownian trajectories. Physical Review E, 99(3), 032501.

[5] Polovnikov, K. E., Gherardi, M., Cosentino-Lagomarsino, M., & Tamm, M. V. (2018). Fractal folding and medium viscoelasticity contribute jointly to chromosome dynamics. Physical review letters, 120(8), 088101.

[6] Polovnikov, K. E., Nechaev, S. K., & Grosberg, A. Y. (2022). Stretching of a Fractal Polymer around a Disc Reveals Kardar-Parisi-Zhang Scaling. Physical Review Letters, 129(9), 097801.

**Date/Time : ** 14/03/2023 - *11:00 - 12:00*

**Location : ** Salle des séminaires du FAST et du LPTMS, bâtiment Pascal n°530

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