We present here a nonbiased probabilistic method that allows us to consistently analyze knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists of analyzing the spectrum of knots formed by multiple closures of the same open walk through random points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border zone of the configuration space of two or more knot types. We also characterize the scaling behavior of linear random knots.

Using numerical simulations we investigate how overall dimensions of random knots scale with their length. We demonstrate that when closed non-self-avoiding random trajectories are divided into groups consisting of individual knot types, then each such group shows the scaling exponent of Ϸ0.588 that is typical for self-avoiding walks. However, when all generated knots are grouped together, their scaling exponent becomes equal to 0.5 (as in non-self-avoiding random walks). We explain here this apparent paradox. We introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots. We also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots.

The production of beneficial public goods is common in the microbial world, and so is cheating -the exploitation of public goods by nonproducing mutants. Here, we examine co-evolutionary dynamics between cooperators and cheats and ask whether cooperators can evolve strategies to reduce the burden of exploitation, and whether cheats in turn can improve their exploitation abilities. We evolved cooperators of the bacterium Pseudomonas aeruginosa, producing the shareable iron-scavenging siderophore pyoverdine, together with cheats, defective in pyoverdine production but proficient in uptake. We found that cooperators managed to co-exist with cheats in 56% of all replicates over approximately 150 generations of experimental evolution. Growth and competition assays revealed that co-existence was fostered by a combination of general adaptions to the media and specific adaptions to the co-evolving opponent. Phenotypic screening and whole-genome resequencing of evolved clones confirmed this pattern, and suggest that cooperators became less exploitable by cheats because they significantly reduced their pyoverdine investment. Cheats, meanwhile, improved exploitation efficiency through mutations blocking the costly pyoverdine-signalling pathway. Moreover, cooperators and cheats evolved reduced motility, a pattern that likely represents adaptation to laboratory conditions, but at the same time also affects social interactions by reducing strain mixing and pyoverdine sharing. Overall, we observed parallel evolution, where co-existence of cooperators and cheats was enabled by a combination of adaptations to the abiotic and social environment and their interactions.