Effective Hamiltonian of topologically stabilized polymer states

Polovnikov, Kirill; Nechaev, Sergei; Tamm, M. V.

doi:10.1039/c8sm00785c

Cited by 34 publications

(51 citation statements)

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“…It is exactly what happens in the case of the Hamiltonian (5) with power-law decaying coefficients (6) (see [8]). This allows one to use (3) and (4) for the mean-square distance between the monomers, and distance-distance correlations, respectively, resulting in the following expression for the correlator r(k, n, m), which, as it turns out, depends in this case on a single variable χ = s 1 /s 2 = (n − k)/(m − n)…”

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confidence: 56%

“…Apart from the fundamental interest of our results for the study of general properties of fractional Brownian motion and for polymer physics, the probabilities of multi-body interactions are of significant importance for biophysical applications. Recently in [8] we proposed to use the fractional Brownian motion trajectories as an analytically simple semiquantitative model of collapsed polymer conformations with fractal dimension d f > 2, including polymer conformations in topologically stabilized states, such as melts of nonconcatenated rings and chromosomes in living nuclei. Technically, our computation of the multiloop probabilities is based on a specific way of constructing ensembles of discretized fractional Brownian trajectories based on a Gibbs sampling with the quadratic Hamiltonian suggested in [8].…”

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confidence: 99%

“…It has been shown recently in [8] that in the limit of large N one can approximate P (X) for the subdiffusive (d f > 2) fractal Brownian motion (fBm) by a Gibbs measure with a simple quadratic Hamiltonian:…”

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confidence: 99%

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Many-body contacts in fractal polymer chains and fractional Brownian trajectories

2019

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We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension d f visits the same spot n ≥ 3 times, at given moments t1, ..., tn, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with d f = 2, the resulting many-body contact probabilities cannot be factorized into a product of single loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently, the probabilities we calculate here can be interpreted as probabilities of multi-body contacts in a fractal polymer conformation with the same fractal dimension d f . This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically-stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.

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confidence: 56%

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confidence: 99%

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Many-body contacts in fractal polymer chains and fractional Brownian trajectories

2019

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“…Such a memory can arise due to some equilibrium topological state of chromatin, or could be a result of partial relaxation of mitotic chromosomes 36 . Notable examples of , typically appearing in the chromatin context for chain embedding in a three-dimensional space, are for ideal chain and for the crumpled globule 12 , 37 – 39 .…”

Section: Stochastic Block Model With Polymer Contact Probabilitymentioning

confidence: 99%

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Polovnikov

Gorsky

Nechaev

et al. 2020

Sci Rep

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Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the polymer block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices. Many real-world stochastic networks split into self-organized communities. Social networks feature circles of friends 1-3 , colleagues 2 , members of a karate club 1 , communities of dolphins 4 etc. Cellular networks demonstrate modular organization, which optimizes crucial biological processes and relationships, such as synchronization of neurons in the connectome 5, 6 , efficiency of metabolic pathways 7, 8 ], genes specialization 9 or interaction between enhancers and promoters 10. Interest to polymer modular networks has appeared recently in the context of genome spatial folding. Proximity of chromatin loci in space is believed to be deeply connected with gene regulation and function. Hi-C experiments 11-13 provide the genome-wide colocalization data of chromatin loci. As the main outcome of the experiment, large genome-wide matrices of contacts from each individual cell or from the population are produced. Analyses of these matrices has revealed that the eukaryotic genome is organized in various and biologically relevant communities, whose main function is to insulate some regions of DNA and to provide easy access to the others. In particular, the data collected from a population of cells suggest that transcribed ("active") chromatin segregates from the, "inactive" one, forming two compartments in the bulk of the nucleus 12, 14. Within compartments chromatin is organized further as a set of topologically-associated domains (TADs) 15-17 that regulate chromatin folding at finer scales. However, interpretation and validation of communities in individual cells remains vaguely defined due to sparsity of respective data. The broad field of applications of ...

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“…To test this hypothesis we turned to budding yeast. The three dimensional organization of the budding yeast genome is much simpler than that of the human genome, its chromosome territory structure is much less pronounced, and its overall organization is believed to be closer to equilibrium than that of the mammalian genome 56,57 . On the other hand, the budding yeast genome preserves some of the basic packaging features of higher organisms 21,53,58 and presents very strong evidence for the Rabl configuration 49,50,59,60 .…”

Section: Introductionmentioning

confidence: 99%

The Rabl configuration limits topological entanglement of chromosomes in budding yeast

Pouokam

Cruz

Burgess

et al. 2019

Sci Rep

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The three dimensional organization of genomes remains mostly unknown due to their high degree of condensation. Biophysical studies predict that condensation promotes the topological entanglement of chromatin fibers and the inhibition of function. How organisms balance between functionally active genomes and a high degree of condensation remains to be determined. Here we hypothesize that the Rabl configuration, characterized by the attachment of centromeres and telomeres to the nuclear envelope, helps to reduce the topological entanglement of chromosomes. To test this hypothesis we developed a novel method to quantify chromosome entanglement complexity in 3D reconstructions obtained from Chromosome Conformation Capture (CCC) data. Applying this method to published data of the yeast genome, we show that computational models implementing the attachment of telomeres or centromeres alone are not sufficient to obtain the reduced entanglement complexity observed in 3D reconstructions. It is only when the centromeres and telomeres are attached to the nuclear envelope (i.e. the Rabl configuration) that the complexity of entanglement of the genome is comparable to that of the 3D reconstructions. We therefore suggest that the Rabl configuration is an essential player in the simplification of the entanglement of chromatin fibers.

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Effective Hamiltonian of topologically stabilized polymer states

Cited by 34 publications

References 50 publications

Many-body contacts in fractal polymer chains and fractional Brownian trajectories

Many-body contacts in fractal polymer chains and fractional Brownian trajectories

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

The Rabl configuration limits topological entanglement of chromosomes in budding yeast

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