18,917 research outputs found
The fractional Keller-Segel model
The Keller-Segel model is a system of partial differential equations
modelling chemotactic aggregation in cellular systems. This model has blowing
up solutions for large enough initial conditions in dimensions d >= 2, but all
the solutions are regular in one dimension; a mathematical fact that crucially
affects the patterns that can form in the biological system. One of the
strongest assumptions of the Keller-Segel model is the diffusive character of
the cellular motion, known to be false in many situations. We extend this model
to such situations in which the cellular dispersal is better modelled by a
fractional operator. We analyze this fractional Keller-Segel model and find
that all solutions are again globally bounded in time in one dimension. This
fact shows the robustness of the main biological conclusions obtained from the
Keller-Segel model
Self Trapping of a Single Bacterium in its Own Chemoattractant
Bacteria (e.g. E. Coli) are very sensitive to certain chemoattractants (e.g.
asparate) which they themselves produce. This leads to chemical instabilities
in a uniform population. We discuss here the different case of a single
bacterium, following the general scheme of Brenner, Levitov and Budrene. We
show that in one and two dimensions (in a capillary or in a thin film) the
bacterium can become self-trapped in its cloud of attractant. This should occur
if a certain coupling constant is larger than unity. We then estimate the
reduced diffusion D_eff of the bacterium in the strong coupling limit, and find
D_eff ~ 1/g.Comment: 4 pages, absolutely no figure
From cellular properties to population asymptotics in the Population Balance Equation
Proliferating cell populations at steady state growth often exhibit broad
protein distributions with exponential tails. The sources of this variation and
its universality are of much theoretical interest. Here we address the problem
by asymptotic analysis of the Population Balance Equation. We show that the
steady state distribution tail is determined by a combination of protein
production and cell division and is insensitive to other model details. Under
general conditions this tail is exponential with a dependence on parameters
consistent with experiment. We discuss the conditions for this effect to be
dominant over other sources of variation and the relation to experiments.Comment: Exact solution of Eq. 9 is adde
Skating on a Film of Air: Drops Impacting on a Surface
Drops impacting on a surface are ubiquitous in our everyday experience. This
impact is understood within a commonly accepted hydrodynamic picture: it is
initiated by a rapid shock and a subsequent ejection of a sheet leading to
beautiful splashing patterns. However, this picture ignores the essential role
of the air that is trapped between the impacting drop and the surface. Here we
describe a new imaging modality that is sensitive to the behavior right at the
surface. We show that a very thin film of air, only a few tens of nanometers
thick, remains trapped between the falling drop and the surface as the drop
spreads. The thin film of air serves to lubricate the drop enabling the fluid
to skate on the air film laterally outward at surprisingly high velocities,
consistent with theoretical predictions. Eventually this thin film of air must
break down as the fluid wets the surface. We suggest that this occurs in a
spinodal-like fashion, and causes a very rapid spreading of a wetting front
outwards; simultaneously the wetting fluid spreads inward much more slowly,
trapping a bubble of air within the drop. Our results show that the dynamics of
impacting drops are much more complex than previously thought and exhibit a
rich array of unexpected phenomena that require rethinking classical paradigms.Comment: 4 pages, 4 figure
Sonoluminescing air bubbles rectify argon
The dynamics of single bubble sonoluminescence (SBSL) strongly depends on the
percentage of inert gas within the bubble. We propose a theory for this
dependence, based on a combination of principles from sonochemistry and
hydrodynamic stability. The nitrogen and oxygen dissociation and subsequent
reaction to water soluble gases implies that strongly forced air bubbles
eventually consist of pure argon. Thus it is the partial argon (or any other
inert gas) pressure which is relevant for stability. The theory provides
quantitative explanations for many aspects of SBSL.Comment: 4 page
The Two Fluid Drop Snap-off Problem: Experiments and Theory
We address the dynamics of a drop with viscosity breaking up
inside another fluid of viscosity . For , a scaling theory
predicts the time evolution of the drop shape near the point of snap-off which
is in excellent agreement with experiment and previous simulations of Lister
and Stone. We also investigate the dependence of the shape and
breaking rate.Comment: 4 pages, 3 figure
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