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Edited by: Guowei Wei, Michigan State University, USA

Reviewed by: Guillaume Witz, Harvard University, USA; Graziano Vernizzi, Siena College, USA

*Correspondence: Mariel Vazquez, Department of Mathematics, University of California, Davis, One Shields Ave, Davis, CA 95616, USA

This article was submitted to Mathematics of Biomolecules, a section of the journal Frontiers in Molecular Biosciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Understanding the folding of the human genome is a key challenge of modern structural biology. The emergence of chromatin conformation capture assays (e.g., Hi-C) has revolutionized chromosome biology and provided new insights into the three dimensional structure of the genome. The experimental data are highly complex and need to be analyzed with quantitative tools. It has been argued that the data obtained from Hi-C assays are consistent with a fractal organization of the genome. A key characteristic of the fractal globule is the lack of topological complexity (knotting or inter-linking). However, the absence of topological complexity contradicts results from polymer physics showing that the entanglement of long linear polymers in a confined volume increases rapidly with the length and with decreasing volume.

The 3-dimensional (3D) organization of the genome is a key functional component of the cell, and errors in this organization are associated with a wide range of diseases (Mitelman et al.,

In the context of Hi-C analyses of human cell lines (described in Section 2.1), Lieberman-Aiden et al. (

In Hi-C experiments, genomic DNA is cross-linked and linearized into fragments using restriction enzymes. The ends of these crosslinked fragments are biotinilated and ligated. Biotinilated junctions, termed

Our approach is based on, and extends, the BFACF algorithm. BFACF is a dynamic Monte Carlo method acting on the space of self-avoiding polygons in the simple cubic lattice (^{3}) by performing one of the three local moves described in Figure _{0}. Within this range, the choice of

_{1} smoothly embedded in ^{3}, a minimal step lattice realization of 3_{1}, and the resulting BFACF globule. This BFACF globule is a 4000-step embedding of the knot within a sphere of radius 10.5 obtained using the modified BFACF algorithm described in Section 2. _{1}. The slope of the linear fit is in excellent agreement with the experimental data of Lieberman-Aiden et al. (_{31)n} for

Inspired by the decondensation process that occurs at the end of metaphase and by the work of Rosa and Everaers (^{7} BFACF steps (with

_{contact} |
_{R(s)} |
|||
---|---|---|---|---|

Fractal globule | −0.993 | 0.2763 | ||

Equilibrium globule | −1.508 | 0.1753 | ||

Experimental data | −1.08 | |||

Unknot 0_{1} |
0.25 0.25 |
−1.0688±0.003 | [−1.075, −1.063 ] | 0.3559±0.002 |

0.25 0.25 |
−1.0670±0.004 | [−1.075, −1.059 ] | 0.3555±0.002 | |

0.25 0.25 |
−1.0672±0.003 | [−1.073, −1.061 ] | 0.3553±0.002 | |

0.25 0.25 |
−1.0673±0.003 | [−1.073, −1.061 ] | 0.3553±0.002 | |

0.25 |
−1.0712±0.003 | [−1.077, −1.065 ] | 0.3558±0.002 | |

0.25 |
−1.0690±0.003 | [−1.075, −1.063 ] | 0.3554±0.002 | |

0.25 |
−1.0615±0.003 | [−1.067, −1.055 ] | 0.3546±0.002 | |

0.25 |
−0.9742±0.004 | [−0.983, −0.966 ] | 0.3439±0.003 | |

−1.0843±0.003 | [−1.091, −1.078 ] | 0.3565±0.002 | ||

−1.0764±0.003 | [−1.083, −1.070 ] | 0.3558±0.002 | ||

−1.0491±0.003 | [−1.055, −1.043 ] | 0.3531±0.002 | ||

−1.1545±0.007 | [−1.168, −1.141 ] | 0.4105±0.003 | ||

Trefoil 3_{1} |
0.10 0.25 0.25 | −1.0848±0.003 | [−1.091, −1.078 ] | 0.3584±0.001 |

5-torus knot 5_{1} |
0.10 0.25 0.25 | −1.0862±0.003 | [−1.093, −1.080 ] | 0.3593±0.001 |

5-twist knot 5_{2} |
0.10 0.25 0.25 | −1.0842±0.003 | [−1.091, −1.078 ] | 0.3596±0.001 |

9-torus knot 9_{1} |
0.10 0.25 0.25 | −1.0860±0.003 | [−1.093, −1.079 ] | 0.3606±0.001 |

20 trefoils (_{31)20} |
0.10 0.25 0.25 | −1.0792±0.003 | [−1.086, −1.073 ] | 0.3514±0.002 |

40 trefoils (_{31)40} |
0.10 0.25 0.25 | −0.9190±0.011 | [−0.941, −0.897 ] | 0.3045±0.005 |

60 trefoils (_{31)60} |
0.10 0.25 0.25 | −0.6556±0.013 | [−0.682, −0.629 ] | 0.2590±0.005 |

100 trefoils (_{31)100} |
0.10 0.25 0.25 | −0.5584±0.035 | [−0.628, −0.489 ] | 0.1952±0.002 |

Simulation results are shown in Table _{contact}); 95% confidence intervals for the contact probability data; and slope of the end-to-end distance _{R(s)}. The first three rows show the experimental and simulated data in Lieberman-Aiden et al. (

To determine the combination of parameters [

The parameters that better minimized the difference between the experimentally observed slope and the simulated slope for the unknotted conformation were (0.10, 0.25, 0.25). These parameters were then used to generate BFACF globules with knot types: torus knots (3_{1}, 5_{1}, and 9_{1}), twist knot (5_{2}), and connected sums of _{31)n} = 3_{1}#3_{1}#…#3_{1}. We refer the reader to Murasugi (

Next we considered the knotted portion of each BFACF globule. Our preliminary data suggest that the knots are not localized. We analyzed two sets of 10^{4} conformations for knots 3_{1} and 9_{1} obtained with the optimal parameter combination (0.10, 0.25, 0.25). We cut each knot at a pair of points at distance 1 from each other and excised as much of the conformation as possible while still retaining the initial topology. The procedure was repeated multiple times on each conformation. The smallest knotted arc had 1012 edges for 3_{1} (i.e., 25.3% of the total length) and 1452 edges for the 9_{1} (i.e., 36.3% of the total length). Note that a minimal 3_{1} knot in ^{3} has 24 edges (Diao, _{1} within the BFACF globule would occupy 0.6% of the total length. In a connected sum, a minimal (_{31)n} can be tied with 20

The widespread and growing interest in the experimental characterization of 3D chromatin structure is driven by the underlying hypothesis that structure is tightly related to function. In particular, gene regulation and cancer-driving gene fusions are believed to be strongly influenced by the 3D organization of the genome (Mitelman et al.,

Lieberman-Aiden et al. (_{1} and 9_{1} suggest that the knotted portion of the BFACF globules is not localized. If this is a general trend then it implies that large-scale topological complexity is compatible with Hi-C data. We will explore this question, as well as the comparison to other models (Yokota et al.,

The topology of genome is a problem that we are just beginning to understand. Studies on lower organisms such as viruses and trypanosomes have revealed high levels of topological knotting and interlinking. In fact, theoretical studies of different polymer models have widely shown that the knotting and linking probability grows rapidly upon confinement (Arsuaga et al.,

Motivated by the results and conclusions in Lieberman-Aiden et al. (

Computer programs used in this work are available through the Knotplot software. Data generated in this study are available upon request from the authors of the paper.

JA, MS, and MV designed research; JA, RJ, RGS, RHS, MS, and MV performed research; RGS, RHS, and RJ contributed new analytic tools; JA, RJ, RHS, MS, and MV analyzed data; and JA, MS, and MV wrote the paper.

We thank the referees for their careful review of the manuscript. This research was primarily supported by NIH-R01GM109457 (JA, MV, and MS). Other support came from DMS1057284 (MV and RHS), DMS0920887 (RJ and RGS), CMB training grant GM 007067 (RJ). The authors thank B. McAuley, M. Puoukham, D. Koenig, and W. Wright for helpful discussions, M. S. Flanner for contributing scripting expertise in matlab, python, and c++, and Barbara Ustanko, ELS, for editorial assistance with this manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{4}| theory in four dimensions