We introduce a method to estimate continuum percolation thresholds and illustrate its usefulness by investigating geometric percolation of noninteracting line segments and disks in two spatial dimensions. These examples serve as models for electrical percolation of elongated and flat nanofillers in thin film composites. While the standard contact volume argument and extensions thereof in connectedness percolation theory yield accurate predictions for slender nanofillers in three dimensions, they fail to do so in two dimensions, making our test a stringent one. In fact, neither a systematic order-by-order correction to the standard argument nor invoking the connectedness version of the Percus-Yevick approximation yield significant improvements for either type of particle. Making use of simple geometric considerations, our new method predicts a percolation threshold of ρ c l 2 ≈ 5.83 for segments of length l, which is close to the ρ c l 2 ≈ 5.64 found in Monte Carlo simulations. For disks of area a we find ρ c a ≈ 1.00, close to the Monte Carlo result of ρ c a ≈ 1.13. We discuss the shortcomings of the conventional approaches and explain how usage of the nearest-neighbor distribution in our method bypasses those complications.

Inspired by recent experiments on the spontaneous assembly of virus-like particles from a solution containing a synthetic coat protein and double-stranded DNA [Carlos Calcines-Cruz, Ilya J. Finkelstein, and Armando Hernandez-Garcia, Nano Lett. 21 (2021), 2752-2757], we put forward a kinetic model that has as main ingredients a stochastic nucleation and a deterministic growth process. The efficiency and rate of the packaging of the DNA turn out to strongly increase by introducing proteins onto the DNA template that are modified using CRISPR-Cas techniques to bind specifically at predesignated locations, mimicking assembly signals in viruses. Our model shows that treating these proteins as nucleation-inducing diffusion barriers is sufficient to explain experimentally observed increase in encapsulation efficiency, but only if the nucleation rate is sufficiently high. We find an optimum in the encapsulation kinetics for conditions where the number of packaging signals is equal to the number of nucleation events that can occur during time required to fully encapsulate the DNA template, presuming that the nucleation events can only take place adjacent to a packaging signal. Our theory is in satisfactory agreement with the available experimental data.

The binding of ligands to distinct sites at proteins or at protein clusters is often cooperative or anti-cooperative due to allosteric signalling between those sites. The allostery is usually attributed to a configurational change of the proteins from a relaxed to a configurationally different tense state. Alternatively, as originally proposed by Cooper and Dryden, a tense state may be achieved by merely restricting the thermal vibrations of the protein around its mean configuration. In this work, we provide theoretical tools to investigate fluctuation allostery using cooling and titration experiments in which ligands regulate dimerisation, or ring or chain formation. We discuss in detail how ligands may regulate the supramolecular (co)polymerisation of liganded and unliganded proteins.

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We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal...