Supramolecular fibers are prominent structures in biology and chemistry. A quantitative understanding of molecular exchange pathways in these one-dimensional aggregates was obtained by a combination of super-resolution stochastic optical reconstruction microscopy and stochastic simulation. The potential of this methodology is demonstrated with a set of well-defined synthetic building blocks that self-assemble into supramolecular fibrils. Previous ensemble measurements hid all molecular phenomena underpinning monomer exchange, but the molecular pathway determined from single-aggregate studies revealed unexpected homogeneous exchange along the polymer backbone. These results pave the way for experimental investigation of the structure and exchange pathways of synthetic and natural supramolecular fibers.

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ (1, 2). Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed.The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with N nodes is investigated when N → ∞. The paper is part of a sequel of three papers. The other two papers study the case where τ ∈ (2, 3), and τ ∈ (3, ∞), respectively.The main result of this paper is that the graph distance converges for τ ∈ (1, 2) to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most N α for some α > 0. For τ −1 < α < (τ − 1) −1 , the hopcount converges to 3 in probability, while for α > (τ − 1) −1 , the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.

Random fluctuations are inherent to all complex molecular systems. Although nature has evolved mechanisms to control stochastic events to achieve the desired biological output, reproducing this in synthetic systems represents a significant challenge. Here we present an artificial platform that enables us to exploit stochasticity to direct motile behavior. We found that enzymes, when confined to the fluidic polymer membrane of a core-shell coacervate, were distributed stochastically in time and space. This resulted in a transient, asymmetric configuration of propulsive units, which imparted motility to such coacervates in presence of substrate. This mechanism was confirmed by stochastic modelling and simulations in silico. Furthermore, we showed that a deeper understanding of the mechanism of stochasticity could be utilized to modulate the motion output. Conceptually, this work represents a leap in design philosophy in the construction of synthetic systems with life-like behaviors.

Personalized PageRank is an algorithm to classify the improtance of web pages on a user-dependent basis. We introduce two generalizations of Personalized PageRank with nodedependent restart. The first generalization is based on the proportion of visits to nodes before the restart, whereas the second generalization is based on the probability of visited node just before the restart. In the original case of constant restart probability, the two measures coincide. We discuss interesting particular cases of restart probabilities and restart distributions. We show that the both generalizations of Personalized PageRank have an elegant expression connecting the so-called direct and reverse Personalized PageRanks that yield a symmetry property of these Personalized PageRanks.

Orthogonality between the subcarriers of an orthogonal frequency division multiplexing (OFDM) system is affected by phase noise, which causes inter-carrier interference (ICI). The distribution of this interference term is studied in this paper. The distribution of the ICI for large number of carriers is derived and it is shown that the complex Gaussian approximation, generally applied in previous literature, is not valid and that the ICI term exhibits thicker tails. An analysis of the tail probabilities confirms these finding and shows that bit-error probabilities are severely underestimated when the Gaussian approximation for the ICI term is used, leading to too optimistic design criteria. Results from a simulation study confirm the analytical findings and show the validity of the limit distribution, obtained under the assumption of a large number of subcarriers, already for a modest number of subcarriers.Index Terms-Inter-carrier interference, limit distribution, orthogonal frequency division multiplexing, phase noise, stochastic integral, tail probabilities.