ИСТИНА

In recent years there have seen plethora of new theoretical and experimental studies of chromosome structure and dynamics, both in eukaryotic and prokaryotic cells. Combination of theoretical considerations (e.g., necessity that chromosome parts disentangle easily during the transcription) and experimental observations (presence of distinct chromosome territories, locus-locus contact maps obtained by the Hi-C method) make us believe that at least in many cases spatial organization of chromosome is self-similar on different lengthscales, and resembles the so-called fractal globule. Study of the chromosome matter dynamics provides important information about DNA organization, and is crucial in order to understand the functioning of the cell machinery. Contemporary single-particle tracking techniques have allowed to study the statistics of loci displacements in both eukaryotic and prokaryotic cells, in both cases the statistics of these displacement seems to be well described by fractal Brownian motion with Hurst exponent close to 0.2, which is significantly different from the exponent 0.25 predicted by the classical Rouse model of polymer dynamics. In literature, there exist at least two competing explanations for this slower-than-Rouse subdiffusion: either it is caused by the viscoelastic properties of the surrounding intra-cellular medium [1], or by the packing of the chromosome into a fractal globule [2]. Here we present a theoretical framework which allows us to unify these two possible explanations and calculate the locus-locus correlation functions for a Gaussian polymer chain (chromosome) packed with arbitrary fractal dimension and surrounded by viscoelastic medium whose dynamics is described by the fractal Langevin equation. Our results are a direct generalization of those obtained in [3] for an ideal chain (ν=1/2) in viscoelastic media, and we use the beta-model suggested in [4] to model polymer conformations with arbitrary fractal dimensions. We show that the presence of viscoelastic media does not change the fractal dimension of the equilibrium polymer conformation, but simply slows down the relaxation in the system. The characteristic relaxation times of the chain are calculated and concrete forms of correlation decay functions obtained. Moreover, by direct calculation of the generalized partition function for a Gaussian chain with an arbitrary fractal dimension, in particular, we conclude that the probability of contact between distinct monomers has the exponent 1 for the compact state, that is pretty close to the HI-C results with human [5] and mice [6] chromosomes. As a result, we suggest a theoretical framework allowing to distinguish between the effects of topological interactions (fractal dimension) and influence of media viscoelasticity, and provide a way to recover both fractal dimension of packing and the characteristics of the viscoelastic media from the results of experiments on one-point and two-point DNA locus tracking. [1] S. C. Weber, J. A. Theriot, A. J. Spakowitz, Physical Review E, 82, 011913 (2010). [2] M.V. Tamm , L.I. Nazarov, A.A. Gavrilov, A.V. Chertovich, Phys. Rev. Letters, 114, 178102 (2015). [3] T. J. Lampo, A. S. Kennard, A. J. Spakowitz, Biophysical Journal, 110, 338 (2016). [4] A. Amitai , D. Holcman, Physical Review E, 88, 052604 (2013). [5] E. Lieberman-Aiden et al, Science, 326, 289 (2009). [6] Y. Zhang et al., Cell, 148, 908 (2012).