In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove replication links by local reconnection. We previously showed mathematically that there is a unique minimal pathway of unlinking replication links by reconnection while stepwise reducing the topological complexity. However, the possibility that reconnection preserves or increases topological complexity is biologically plausible. In this case, are there other unlinking pathways? Which is the most probable? We consider these questions in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 different substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total of 881 different topologies. We conclude that the minimal pathway of unlinking replication links that was found under more stringent assumptions is the most probable. We also present exact results on unlinking a 6-crossing replication link. These results point to a general process of topology simplification by local reconnection, with applications going beyond DNA.

It is widely recognized that the three-dimensional (3D) architecture of eukaryotic chromatin plays an important role in processes such as gene regulation and cancer-driving gene fusions. Observing or inferring this 3D structure at even modest resolutions had been problematic, since genomes are highly condensed and traditional assays are coarse. However, recently devised high-throughput molecular techniques have changed this situation. Notably, the development of a suite of chromatin conformation capture (CCC) assays has enabled elicitation of contacts—spatially close chromosomal loci—which have provided insights into chromatin architecture. Most analysis of CCC data has focused on the contact level, with less effort directed toward obtaining 3D reconstructions and evaluating the accuracy and reproducibility thereof. While questions of accuracy must be addressed experimentally, questions of reproducibility can be addressed statistically—the purpose of this paper. We use a constrained optimization technique to reconstruct chromatin configurations for a number of closely related yeast datasets and assess reproducibility using four metrics that measure the distance between 3D configurations. The first of these, Procrustes fitting, measures configuration closeness after applying reflection, rotation, translation, and scaling-based alignment of the structures. The others base comparisons on the within-configuration inter-point distance matrix. Inferential results for these metrics rely on suitable permutation approaches. Results indicate that distance matrix-based approaches are preferable to Procrustes analysis, not because of the metrics per se but rather on account of the ability to customize permutation schemes to handle within-chromosome contiguity. It has recently been emphasized that the use of constrained optimization approaches to 3D architecture reconstruction are prone to being trapped in local minima. Our methods of reproducibility assessment provide a means for comparing 3D reconstruction solutions so that we can discern between local and global optima by contrasting solutions under perturbed inputs.

We demonstrate that AlphaFold and AlphaFold Multimer, implemented within the ColabFold suite, can accurately predict the structures of the furin enzyme with known six residue inhibitory peptides. Noting the similarity of the peptide inhibitors to polybasic furin cleavage domain insertion region of the SARS-CoV-2, which begins at P681, we implement this approach to study the wild type furin cleavage domain for the virus and several mutants. We introduce mutations in silico for alpha, omicron, and delta variants, for several sequences which have been rarely observed, for sequences which have not yet been observed, for other coronaviruses (NL63, OC43, HUK1a, HUK1b, MERS, and 229E), and for the H5N1 flu. We show that interfacial hydrogen bonds between the furin cleavage domain and furin are a good measure of binding strength that correlate well with endpoint binding free energy estimates, and conclude that among all candidate viral sequences studied, delta is near the very top binding strength within statistical accuracy. However, the binding strength of several rare sequences match delta within statistical accuracy. We find that the furin S1 pocket is optimized for binding arginine as opposed to lysine. This residue, typically at sequence position five, contains the most hydrogen bonds to the furin, and hydrogen bond count for just this residue shows a strong positive correlation with the overall hydrogen bond count. We demonstrate that the root mean square backbone C-alpha fluctuation of the first residue in the furin cleavage domain has a strong negative correlation with the interfacial hydrogen bond count. We show by considering the variation with the number of basic residues that the maximum mean number of interfacial hydrogen bonds expected is 15.7 at 4 basic residues.

Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the ∞ × 2 × 1 lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved.

Motivation: The position of chromosomes in the interphase nucleus is believed to be associated with a number of biological processes. Here, we present a web-based application that helps analyze the relative position of chromosomes during interphase in human cells, based on observed radiogenic chromosome aberrations. The inputs of the program are a table of yields of pairwise chromosome interchanges and a proposed chromosome geometric cluster. Each can either be uploaded or selected from provided datasets. The main outputs are P-values for the proposed chromosome clusters. SCHIP is designed to be used by a number of scientific communities interested in nuclear architecture, including cancer and cell biologists, radiation biologists and mathematical/computational biologists. Availability:http://cramer.stat.ub.es/schip Contact:jarsuaga@cc.ucsf.edu Supplementary information:http://cramer.stat.ub.es/schip/help.htm