10,555 research outputs found
Cellular pattern formation during Dictyostelium aggregation
The development of multicellularity in the life cycle of Dictyostelium discoideum provides a paradigm model system for biological pattern formation. Previously, mathematical models have shown how a collective pattern of cell communication by waves of the messenger molecule cyclic adenosine 3′5′-monophosphate (cAMP) arises from excitable local cAMP kinetics and cAMP diffusion. Here we derive a model of the actual cell aggregation process by considering the chemotactic cell response to cAMP and its interplay with the cAMP dynamics. Cell density, which previously has been treated as a spatially homogeneous parameter, is a crucial variable of the aggregation model. We find that the coupled dynamics of cell chemotaxis and cAMP reaction-diffusion lead to the break-up of the initially uniform cell layer and to the formation of the striking cell stream morphology which characterizes the aggregation process in situ. By a combination of stability analysis and two-dimensional simulations of the model equations, we show cell streaming to be the consequence of the growth of a small-amplitude pattern in cell density forced by the large-amplitude cAMP waves, thus representing a novel scenario of spatial patterning in a cell chemotaxis system. The instability mechanism is further analysed by means of an analytic caricature of the model, and the condition for chemotaxis-driven instability is found to be very similar to the one obtained for the standard (non-oscillatory) Keller-Segel system. The growing cell stream pattern feeds back into the cAMP dynamics, which can explain in some detail experimental observations on the time evolution of the cAMP wave pattern, and suggests the characterization of the Dictyostelium aggregation field as a self-organized excitable medium
Mathematical models for chemotaxis and their applications in self-organisation phenomena
Chemotaxis is a fundamental guidance mechanism of cells and organisms,
responsible for attracting microbes to food, embryonic cells into developing
tissues, immune cells to infection sites, animals towards potential mates, and
mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of
the bedrock of mathematical biology, a go-to-choice for modellers and analysts
alike. For the former it is simple yet recapitulates numerous phenomena; the
latter are attracted to these rich dynamics. Here I review the adoption of PKS
systems when explaining self-organisation processes. I consider their
foundation, returning to the initial efforts of Patlak and Keller and Segel,
and briefly describe their patterning properties. Applications of PKS systems
are considered in their diverse areas, including microbiology, development,
immunology, cancer, ecology and crime. In each case a historical perspective is
provided on the evidence for chemotactic behaviour, followed by a review of
modelling efforts; a compendium of the models is included as an Appendix.
Finally, a half-serious/half-tongue-in-cheek model is developed to explain how
cliques form in academia. Assumptions in which scholars alter their research
line according to available problems leads to clustering of academics and the
formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog
Mathematical description of bacterial traveling pulses
The Keller-Segel system has been widely proposed as a model for bacterial
waves driven by chemotactic processes. Current experiments on {\em E. coli}
have shown precise structure of traveling pulses. We present here an
alternative mathematical description of traveling pulses at a macroscopic
scale. This modeling task is complemented with numerical simulations in
accordance with the experimental observations. Our model is derived from an
accurate kinetic description of the mesoscopic run-and-tumble process performed
by bacteria. This model can account for recent experimental observations with
{\em E. coli}. Qualitative agreements include the asymmetry of the pulse and
transition in the collective behaviour (clustered motion versus dispersion). In
addition we can capture quantitatively the main characteristics of the pulse
such as the speed and the relative size of tails. This work opens several
experimental and theoretical perspectives. Coefficients at the macroscopic
level are derived from considerations at the cellular scale. For instance the
stiffness of the signal integration process turns out to have a strong effect
on collective motion. Furthermore the bottom-up scaling allows to perform
preliminary mathematical analysis and write efficient numerical schemes. This
model is intended as a predictive tool for the investigation of bacterial
collective motion
Arrested phase separation in reproducing bacteria: a generic route to pattern formation?
We present a generic mechanism by which reproducing microorganisms, with a
diffusivity that depends on the local population density, can form stable
patterns. It is known that a decrease of swimming speed with density can
promote separation into bulk phases of two coexisting densities; this is
opposed by the logistic law for birth and death which allows only a single
uniform density to be stable. The result of this contest is an arrested
nonequilibrium phase separation in which dense droplets or rings become
separated by less dense regions, with a characteristic steady-state length
scale. Cell division mainly occurs in the dilute regions and cell death in the
dense ones, with a continuous flux between these sustained by the diffusivity
gradient. We formulate a mathematical model of this in a case involving
run-and-tumble bacteria, and make connections with a wider class of mechanisms
for density-dependent motility. No chemotaxis is assumed in the model, yet it
predicts the formation of patterns strikingly similar to those believed to
result from chemotactic behavior
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
On the stability of homogeneous solutions to some aggregation models
Vasculogenesis, i.e. self-assembly of endothelial cells leading to capillary network formation, has been the object of many experimental investigations in recent years, due to its relevance both in physiological and in pathological conditions. We performed a detailed linear stability analysis of two models of in vitro vasculogenesis, with the aim of checking their potential for structure formation starting from initial data representing a continuum cell monolayer. The first model turns out to be unstable at low cell densities, while pressure stabilizes it at high densities. The second model is instead stable at low cell densities. Detailed information about the instability regions and the structure of the critical wave numbers are obtained in several interesting limiting cases. We expect that altogether, this information will be useful for further comparisons of the two models with experiments
Studies of Bacterial Branching Growth using Reaction-Diffusion Models for Colonial Development
Various bacterial strains exhibit colonial branching patterns during growth
on poor substrates. These patterns reflect bacterial cooperative
self-organization and cybernetic processes of communication, regulation and
control employed during colonial development. One method of modeling is the
continuous, or coupled reaction-diffusion approach, in which continuous time
evolution equations describe the bacterial density and the concentration of the
relevant chemical fields. In the context of branching growth, this idea has
been pursued by a number of groups. We present an additional model which
includes a lubrication fluid excreted by the bacteria. We also add fields of
chemotactic agents to the other models. We then present a critique of this
whole enterprise with focus on the models' potential for revealing new
biological features.Comment: 1 latex file, 40 gif/jpeg files (compressed into tar-gzip). Physica
A, in pres
On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis
We consider the phenomenon of collapse in the critical Keller-Segel equation
(KS) which models chemotactic aggregation of micro-organisms underlying many
social activities, e.g. fruiting body development and biofilm formation. Also
KS describes the collapse of a gas of self-gravitating Brownian particles. We
find the fluctuation spectrum around the collapsing family of steady states for
these equations, which is instrumental in derivation of the critical collapse
law. To this end we develop a rigorous version of the method of matched
asymptotics for the spectral analysis of a class of second order differential
operators containing the linearized Keller-Segel operators (and as we argue
linearized operators appearing in nonlinear evolution problems). We explain how
the results we obtain are used to derive the critical collapse law, as well as
for proving its stability.Comment: 22 pages, 1 figur
Development and applications of a model for cellular response to multiple chemotactic cues
The chemotactic response of a cell population to a single chemical species has been characterized experimentally for many cell types and has been extensively studied from a theoretical standpoint. However, cells frequently have multiple receptor types and can detect and respond chemotactically to more than one chemical. How these signals are integrated within the cell is not known, and we therefore adopt a macroscopic phenomenological approach to this problem. In this paper we derive and analyze chemotactic models based on partial differential (chemotaxis) equations for cell movement in response to multiple chemotactic cues. Our derivation generalizes the approach of Othmer and Stevens [29], who have recently developed a modeling framework for studying different chemotactic responses to a single chemical species. The importance of such a generalization is illustrated by the effect of multiple chemical cues on the chemotactic sensitivity and the spatial pattern of cell densities in several examples. We demonstrate that the model can generate the complex patterns observed on the skin of certain animal species and we indicate how the chemotactic response can be viewed as a form of positional indicator
Synthetic Chemotaxis and Collective Behavior in Active Matter
Conspectus: The ability to navigate in chemical gradients, called chemotaxis,
is crucial for the survival of microorganisms. It allows them to find food and
to escape from toxins. Many microorganisms can produce the chemicals to which
they respond themselves and use chemotaxis for signalling which can be seen as
a basic form of communication. Remarkably, the past decade has let to the
development of synthetic microswimmers like e.g. autophoretic Janus colloids,
which can self-propel through a solvent, analogously to bacteria and other
microorganims. The mechanism underlying their self-propulsion involves the
production of certain chemicals. The same chemicals involved in the
self-propulsion mechanism also act on other microswimmers and bias their
swimming direction towards (or away from) the producing microswimmer. Synthetic
microswimmers therefore provide a synthetic analogue to chemotactic motile
microorganisms. When these interactions are attractive, they commonly lead to
clusters, even at low particle density. These clusters may either proceed
towards macrophase separation, resembling Dictyostelium aggregation, or, as
shown very recently, lead to dynamic clusters of self-limited size (dynamic
clustering) as seen in experiments in autophoretic Janus colloids. Besides the
classical case where chemical interactions are attractive, this Account
discusses, as its main focus, repulsive chemical interactions, which can create
a new and less known avenue to pattern formation in active systems leading to a
variety of pattern, including clusters which are surrounded by shells of
chemicals, travelling waves and more complex continously reshaping patterns. In
all these cases `synthetic signalling' can crucially determine the collective
behavior of synthetic microswimmer ensembles and can be used as a design
principle to create patterns in motile active particles
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