1,367 research outputs found
Partial differential equations for self-organization in cellular and developmental biology
Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field
A hyperbolic model of chemotaxis on a network: a numerical study
In this paper we deal with a semilinear hyperbolic chemotaxis model in one
space dimension evolving on a network, with suitable transmission conditions at
nodes. This framework is motivated by tissue-engineering scaffolds used for
improving wound healing. We introduce a numerical scheme, which guarantees
global mass densities conservation. Moreover our scheme is able to yield a
correct approximation of the effects of the source term at equilibrium. Several
numerical tests are presented to show the behavior of solutions and to discuss
the stability and the accuracy of our approximation
Dynamical strategies for obstacle avoidance during Dictyostelium discoideum aggregation: a Multi-agent system model
Chemotaxis, the movement of an organism in response to chemical stimuli, is a
typical feature of many microbiological systems. In particular, the social
amoeba \textit{Disctyostelium discoideum} is widely used as a model organism,
but it is not still clear how it behaves in heterogeneous environments. A few
models focusing on mechanical features have already addressed the question;
however, we suggest that phenomenological models focusing on the population
dynamics may provide new meaningful data. Consequently, by means of a specific
Multi-agent system model, we study the dynamical features emerging from complex
social interactions among individuals belonging to amoeba colonies.\\ After
defining an appropriate metric to quantitatively estimate the gathering
process, we find that: a) obstacles play the role of local topological
perturbation, as they alter the flux of chemical signals; b) physical obstacles
(blocking the cellular motion and the chemical flux) and purely chemical
obstacles (only interfering with chemical flux) elicit similar dynamical
behaviors; c) a minimal program for robustly gathering simulated cells does not
involve mechanisms for obstacle sensing and avoidance; d) fluctuations of the
dynamics concur in preventing multiple stable clusters. Comparing those
findings with previous results, we speculate about the fact that chemotactic
cells can avoid obstacles by simply following the altered chemical gradient.
Social interactions are sufficient to guarantee the aggregation of the whole
colony past numerous obstacles
Reactions, Diffusion and Volume Exclusion in a Heterogeneous System of Interacting Particles
Complex biological and physical transport processes are often described
through systems of interacting particles. Excluded-volume effects on these
transport processes are well studied, however the interplay between volume
exclusion and reactions between heterogenous particles is less well known. In
this paper we develop a novel framework for modeling reaction-diffusion
processes which directly incorporates volume exclusion. From an off-lattice
microscopic individual based model we use the Fokker--Planck equation and the
method of matched asymptotic expansions to derive a low-dimensional macroscopic
system of nonlinear partial differential equations describing the evolution of
the particles. A biologically motivated, hybrid model of chemotaxis with volume
exclusion is explored, where reactions occur at rates dependent upon the
chemotactic environment. Further, we show that for reactions due to contact
interactions the appropriate reaction term in the macroscopic model is of lower
order in the asymptotic expansion than the nonlinear diffusion term. However,
we find that the next reaction term in the expansion is needed to ensure good
agreement with simulations of the microscopic model. Our macroscopic model
allows for more direct parameterization to experimental data than the models
available to date.Comment: 13 pages, 4 figure
Mesoscopic and continuum modelling of angiogenesis
Angiogenesis is the formation of new blood vessels from pre-existing ones in
response to chemical signals secreted by, for example, a wound or a tumour. In
this paper, we propose a mesoscopic lattice-based model of angiogenesis, in
which processes that include proliferation and cell movement are considered as
stochastic events. By studying the dependence of the model on the lattice
spacing and the number of cells involved, we are able to derive the
deterministic continuum limit of our equations and compare it to similar
existing models of angiogenesis. We further identify conditions under which the
use of continuum models is justified, and others for which stochastic or
discrete effects dominate. We also compare different stochastic models for the
movement of endothelial tip cells which have the same macroscopic,
deterministic behaviour, but lead to markedly different behaviour in terms of
production of new vessel cells.Comment: 48 pages, 13 figure
Taxis Equations for Amoeboid Cells
The classical macroscopic chemotaxis equations have previously been derived
from an individual-based description of the tactic response of cells that use a
"run-and-tumble" strategy in response to environmental cues. Here we derive
macroscopic equations for the more complex type of behavioral response
characteristic of crawling cells, which detect a signal, extract directional
information from a scalar concentration field, and change their motile behavior
accordingly. We present several models of increasing complexity for which the
derivation of population-level equations is possible, and we show how
experimentally-measured statistics can be obtained from the transport equation
formalism. We also show that amoeboid cells that do not adapt to constant
signals can still aggregate in steady gradients, but not in response to
periodic waves. This is in contrast to the case of cells that use a
"run-and-tumble" strategy, where adaptation is essential.Comment: 35 pages, submitted to the Journal of Mathematical Biolog
A Unified Term for Directed and Undirected Motility in Collective Cell Invasion
In this paper we develop mathematical models for collective cell motility.
Initially we develop a model using a linear diffusion-advection type equation
and fit the parameters to data from cell motility assays. This approach is
helpful in classifying the results of cell motility assay experiments. In
particular, this model can determine degrees of directed versus undirected
collective cell motility. Next we develop a model using a nonlinear diffusion
term that is able capture in a unified way directed and undirected collective
cell motility. Finally we apply the nonlinear diffusion approach to a problem
in tumor cell invasion, noting that neither chemotaxis or haptotaxis are
present in the system under consideration in this article
A Review of Mathematical Models for the Formation of\ud Vascular Networks
Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud
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A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features
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