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.

Knots are ubiquitous in nature and their analysis has important implications in a wide variety of fields including fluid dynamics, material science and molecular and structural biology. In many systems particles are found in crowded environments hence it is natural to rigorously characterize the properties of knots in confined volumes. In this work we combine analytical and numerical work on the simple cubic lattice to determine the minimal number of lattice steps, minimum step number, needed to make a knot inside a tubular region. Our complementary approaches help us establish a detailed enumeration of minimal knot lengths and/or conformations of knots in tubular regions. Analytical results characterize the types of knots and links that can be embedded in a tubular regions and determines the minimum number of steps required to construct all 2-bridge knots and links up to ten crossings in the (2 × 1)-tube. Simulation results, on the other hand, estimate the minimum step number and provide exact trajectories of all knot types up to eight crossings for wider tubular regions. These findings not only determine what knots and links can be built in a highly confined volume but also provide 7 To whom questions concerning the numerical studies should be addressed. 8 To whom questions concerning the analytical studies should be addressed.