Any microscopic particle placed in a
fluid will move about randomly. This effect, known as Brownian
motion, is caused by collisions between the particle and the
surrounding molecules. Two hallmarks of Brownian motion are that
particles spread out linearly over time, and the probability of
finding a particle at a certain position at a given time is
mathematically described by a Gaussian function (a bell curve).
However, in certain situations, such as individual nematodes (a type
of roundworm) or microscopic beads on lipid tubes, this probability
behaves quite differently—sometimes as an exponential function. At
first, this phenomenon, now observed in a large range of systems,
seems to violate a universal mathematical law known as the central
limit theorem, which predicts that this probability should converge
to a Gaussian function. Here, we establish a physical minimal model
for such “Brownian yet non-Gaussian” diffusion.

Using analytical calculations and
simulations, we show that both the linear spread of particles and an
exponential probability distribution can be reconciled when the
intensity of the random jiggling of the particles itself becomes a
random function of time. We augment the standard Langevin equation—a
differential equation that describes the Brownian motion of a
particle—with a random noise strength. This “diffusing diffusivity”
has an inherent correlation time that defines a crossover from the
non-Gaussian probability seen on short time scales to a long-time
Gaussian.

Our minimal model for the
diffusing-diffusivity approach to Brownian yet non-Gaussian
diffusion is very versatile, and we believe that it will help
establish this approach as a new paradigm in the physics of
stochastic processes.

Bacterial communities undergo collective
behavioural switches upon producing and sensing diffusible signal
molecules; a mechanism referred to as Quorum Sensing (QS).
Exemplarily, biofilm organic matrices are built concertedly by
bacteria in several environments. QS scope in bacterial ecology has
been debated for over 20 years. Different perspectives counterpose the
role of density reporter for populations to that of local environment
diffusivity probe for individual cells. Here we devise a model system
where tubes of different heights contain matrix-embedded producers and
sensors. These tubes allow non-limiting signal diffusion from one open
end, thereby showing that population spatial extension away from an
open boundary can be a main critical factor in QS. Experimental data,
successfully recapitulated by a comprehensive mathematical model,
demonstrate how tube height can overtake the role of producer density
in triggering sensor activation. The biotic degradation of the signal
is found to play a major role and to be species-specific and entirely
feedback-independent.

The presence of knots has been observed
in a small fraction of single-domain proteins and related to their
thermodynamic and kinetic properties. The exchanging of identical
structural elements, typical of domain-swapped proteins, makes such
dimers suitable candidates to validate the possibility that mutual
entanglement between chains may play a similar role for protein
complexes. We suggest that such entanglement is captured by the
linking number. This represents, for two closed curves, the number of
times that each curve winds around the other. We show that closing the
curves is not necessary, as a novel parameter *G*′, termed
Gaussian entanglement, is strongly correlated with the linking number.
Based on 110 non redundant domain-swapped dimers, our analysis
evidences a high fraction of chains with a significant intertwining,
that is with |*G′*| > 1. We report that Nature promotes
configurations with negative mutual entanglement and surprisingly, it
seems to suppress intertwining in long protein dimers. Supported by
numerical simulations of dimer dissociation, our results provide a
novel topology-based classification of protein-swapped dimers together
with some preliminary evidence of its impact on their physical and
biological properties.