Based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the Anderson model at the transition point. A simple theoretical model where such statistics can be obtained analytically is briefly discussed. A few models with similar behavior are considered. In particular, we analytically found the eigenvalue statistics of a Poisson-distributed matrix perturbed by a rank one matrix, which is a good model for spectral statistics of a singular billiard.

We study theoretical “design principles” for transcription factor (TF)–DNA interaction in bacteria, focusing particularly on the statistical interaction of the TFs with the genomic background (i.e., the genome without the target sites). We introduce and motivate the concept of programmability, i.e., the ability to set the threshold concentration for TF binding over a wide range merely by mutating the binding sequence of a target site. This functional demand, together with physical constraints arising from the thermodynamics and kinetics of TF–DNA interaction, leads us to a narrow range of “optimal” interaction parameters. We find that this parameter set agrees well with experimental data for the interaction parameters of a few exemplary prokaryotic TFs, which indicates that TF–DNA interaction is indeed programmable. We suggest further experiments to test whether this is a general feature for a large class of TFs.

For the emergence of early life, the formation of biopolymers such as RNA is essential. However, the addition of nucleotide monomers to existing oligonucleotides requires millimolar concentrations. Even in such optimistic settings, no polymerization of RNA longer than about 20 bases could be demonstrated. How then could self-replicating ribozymes appear, for which recent experiments suggest a minimal length of 200 nt? Here, we demonstrate a mechanism to bridge this gap: the escalated polymerization of nucleotides by a spatially confined thermal gradient. The gradient accumulates monomers by thermophoresis and convection while retaining longer polymers exponentially better. Polymerization and accumulation become mutually self-enhancing and result in a hyperexponential escalation of polymer length. We describe this escalation theoretically under the conservative assumption of reversible polymerization. Taking into account the separately measured thermophoretic properties of RNA, we extrapolate the results for primordial RNA polymerization inside a temperature gradient in pores or fissures of rocks. With a dilute, nanomolar concentration of monomers the model predicts that a pore length of 5 cm and a temperature difference of 10 K suffice to polymerize 200-mers of RNA in micromolar concentrations. The probability to generate these long RNAs is raised by a factor of >10 ⁶⁰⁰ compared with polymerization in a physical equilibrium. We experimentally validate the theory with the reversible polymerization of DNA blocks in a laser-driven thermal trap. The results confirm that a thermal gradient can significantly enlarge the available sequence space for the emergence of catalytically active polymers.