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## Abstract

Various modeling and simulations have been done to study the organization of chromosomes under different circumstance and for different purposes. In this thesis, we would first study the influence of bending rigidity and spatial confinement on the organization of the chromatin. More concretely, the effect of heterogeneity and the definition of contacts will be addressed. We find that the definition of a contact does not change the asymptotic behavior of the contact probability. The heterogeneity of bending rigidity is shown to render the chain more flexible by comparing the persistent length and the contact probability of homogeneous and heterogeneous chains. In addition, we simulate semiflexible chains in rectangular confinements with different aspect ratios. An oscillation in the contact probability and the orientational correlation function is found due to the spiraling of polymer when the box size is small enough.

The entanglement of chains is another important aspect when studying chromatins. The processes of disentanglement of two flexible chains are studied using the Monte Carlo simulation. Specifically, several measurements such as the inter-contact of chains, dynamic structure factor are analyzed in the process. When only the excluded volume interaction exists in the system, the average time required for segregation is barely influenced by the initial configurations of the two chains according to our results. However, the intertwinement of chains indeed could impede the segregation at a small time scale. The number of contacts inside a self-avoiding chain is also analyzed. It is found that the total number $N_c$ grows linearly with the length of a free chain, while in cubic confinement it grows quadratically. The distribution function of contacts number between two halves $N_c(AB)$ shows a power-law decay behavior and then an exponential decay for a free chain. In confinement, the function has a maximum. As the chain becomes longer, the percentage of inter-half contacts among the total contacts has a power-law decay behavior with an exponent close to -1, which supports that the number of contacts between two halves is finite even when the chain is infinitely long.

Finally, we studied the fractality and the topology in the self-avoiding walks. Specifically, we calculate the fractal dimension and growth rates of the Betti numbers of the system. These growth rates can be viewed as a topological signature for different systems. The intra-contacts of the self-avoiding walk is a subset of the original walk, and we find that this subset may have a slight multifractal property. In addition, the topological exponents are also different from the self-avoiding walk. Further, each contact gives rise to the formation of a loop. To elucidate how these loops influence the structure of the self-avoiding walk, we delete the loops in a similar way to the loop-erased random walk, thus producing a new walk: loop-deleted self-avoiding walk (LDSAW). The critical exponent of LDSAW is approximated by studying the scaling behavior of mean end-to-end distance, and the dependence of the mean length of LDSAW on the length of the original self-avoiding walk. Afterward, the fractal dimension and growth rates of Betti numbers of this LDSAW are calculated. The same calculations are also performed on the projection and random subsets of self-avoiding walks.

Item Type: | Dissertation |
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Supervisor: | Heermann, Prof. Dr. Dieter W. |

Place of Publication: | Heidelberg |

Date of thesis defense: | 13 November 2019 |

Date Deposited: | 25 Nov 2019 10:46 |

Date: | 2019 |

Faculties / Institutes: | The Faculty of Physics and Astronomy > Institute for Theoretical Physics |

Subjects: | 500 Natural sciences and mathematics |

Controlled Keywords: | chromosomes organization, Monte Carlo simulation, Contacts, loop-deleted self-avoiding walk, Fractal |