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Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-typeKlattice 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 ofK. 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$\mathrm{\infty }\times 2\times 1$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.

 

Establishing the host range for novel viruses remains a challenge. Here, we address the challenge of identifying non-human animal coronaviruses that may infect humans by creating an artificial neural network model that learns from spike protein sequences of alpha and beta coronaviruses and their binding annotation to their host receptor. The proposed method produces a human-Binding Potential (h-BiP) score that distinguishes, with high accuracy, the binding potential among coronaviruses. Three viruses, previously unknown to bind human receptors, were identified: Bat coronavirus BtCoV/133/2005 and Pipistrellus abramus bat coronavirus HKU5-related (both MERS related viruses), andRhinolophus affiniscoronavirus isolate LYRa3 (a SARS related virus). We further analyze the binding properties of BtCoV/133/2005 and LYRa3 using molecular dynamics. To test whether this model can be used for surveillance of novel coronaviruses, we re-trained the model on a set that excludes SARS-CoV-2 and all viral sequences released after the SARS-CoV-2 was published. The results predict the binding of SARS-CoV-2 with a human receptor, indicating that machine learning methods are an excellent tool for the prediction of host expansion events.

 

We study equilibrium configurations of double-stranded DNA in a cylindrical viral capsid. The state of the encapsidated DNA consists of a disordered inner core enclosed by an ordered outer region, next to the capsid wall. The DNA configuration is described by a unit helical vector field, tangent to an associated centre curve, passing through properly selected locations. We postulate an expression for the energy of the encapsulated DNA based on that of columnar chromonic liquid crystals. A thorough analysis of the Euler–Lagrange equations yields multiple solutions. We demonstrate that there is a trivial, non-helical solution, together with two solutions with non-zero helicity of opposite sign. Using bifurcation analysis, we derive the conditions for local stability and determine when the preferred coiling state is helical. The bifurcation parameters are the ratio of the twist versus the bend moduli of DNA and the ratio between the sizes of the ordered and the disordered regions. 

Site-specific recombination is an enzymatic process where two sites of precise sequence and orientation along a circle come together, are cleaved, and the ends are recombined. Site-specific recombination on a knotted substrate produces another knot or a two-component link depending on the relative orientation of the sites prior to recombination. Mathematically, site-specific recombination is modeled as coherent (knot to link) or non-coherent (knot to knot) banding. We here survey recent developments in the study of non-coherent bandings on knots and discuss biological implications.