107 research outputs found
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
Mechanochemical models for calcium waves in embryonic epithelia
In embryogenesis, epithelial cells, acting as individual entities or as
coordinated aggregates in a tissue, exhibit strong coupling between
chemical signalling and mechanical responses to internally or externally
applied stresses. Intercellular communication in combination with such
coordination of morphogenetic movements can lead to drastic modifications in the calcium distribution in the cells. In this paper we extend
the recent mechanochemical model in [K. Kaouri, P.K. Maini, P.A.
Skourides, N. Christodoulou, S.J. Chapman. J. Math. Biol., 78 (2019)
2059–2092], for an epithelial continuum in one dimension, to a more
realistic multi-dimensional case. The resulting parametrised governing
equations consist of an advection-diffusion-reaction system for calcium
signalling coupled with active-stress linear viscoelasticity and equipped
with pure Neumann boundary conditions. We implement a mixed finite
element method for the simulation of this complex multiphysics problem. Special care is taken in the treatment of the stress-free boundary
conditions for the viscoelasticity in order to eliminate rigid motions
from the space of admissible displacements. The stability and solvability of the continuous weak formulation is shown using fixed-point
theory. We investigate numerically the solutions of this system and
show that solitary waves and periodic wavetrains of calcium propagate
through the embryonic epithelial sheet. We analyse the bifurcations of
the system guided by the bifurcation analysis of the one-dimensional model. We also demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with contraction. This coupled model
can be employed to gain insights into recent experimental observations
in the context of embryogenesis, but also in other biological systems
such as cancer cells, wound healing, keratinocytes, or white blood cells
Mechanochemical models for calcium waves in embryonic epithelia
In embryogenesis, epithelial cells, acting as individual entities or as
coordinated aggregates in a tissue, exhibit strong coupling between
chemical signalling and mechanical responses to internally or externally
applied stresses. Intercellular communication in combination with such
coordination of morphogenetic movements can lead to drastic modifications in the calcium distribution in the cells. In this paper we extend
the recent mechanochemical model in [K. Kaouri, P.K. Maini, P.A.
Skourides, N. Christodoulou, S.J. Chapman. J. Math. Biol., 78 (2019)
2059–2092], for an epithelial continuum in one dimension, to a more
realistic multi-dimensional case. The resulting parametrised governing
equations consist of an advection-diffusion-reaction system for calcium
signalling coupled with active-stress linear viscoelasticity and equipped
with pure Neumann boundary conditions. We implement a mixed finite
element method for the simulation of this complex multiphysics problem. Special care is taken in the treatment of the stress-free boundary
conditions for the viscoelasticity in order to eliminate rigid motions
from the space of admissible displacements. The stability and solvability of the continuous weak formulation is shown using fixed-point
theory. We investigate numerically the solutions of this system and
show that solitary waves and periodic wavetrains of calcium propagate
through the embryonic epithelial sheet. We analyse the bifurcations of
the system guided by the bifurcation analysis of the one-dimensional model. We also demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with contraction. This coupled model
can be employed to gain insights into recent experimental observations
in the context of embryogenesis, but also in other biological systems
such as cancer cells, wound healing, keratinocytes, or white blood cells
Turing Patterning in Stratified Domains
Reaction-diffusion processes across layered media arise in several scientific
domains such as pattern-forming E. coli on agar substrates,
epidermal-mesenchymal coupling in development, and symmetry-breaking in cell
polarisation. We develop a modelling framework for bi-layer reaction-diffusion
systems and relate it to a range of existing models. We derive conditions for
diffusion-driven instability of a spatially homogeneous equilibrium analogous
to the classical conditions for a Turing instability in the simplest nontrivial
setting where one domain has a standard reaction-diffusion system, and the
other permits only diffusion. Due to the transverse coupling between these two
regions, standard techniques for computing eigenfunctions of the Laplacian
cannot be applied, and so we propose an alternative method to compute the
dispersion relation directly. We compare instability conditions with full
numerical simulations to demonstrate impacts of the geometry and coupling
parameters on patterning, and explore various experimentally-relevant
asymptotic regimes. In the regime where the first domain is suitably thin, we
recover a simple modulation of the standard Turing conditions, and find that
often the broad impact of the diffusion-only domain is to reduce the ability of
the system to form patterns. We also demonstrate complex impacts of this
coupling on pattern formation. For instance, we exhibit non-monotonicity of
pattern-forming instabilities with respect to geometric and coupling
parameters, and highlight an instability from a nontrivial interaction between
kinetics in one domain and diffusion in the other. These results are valuable
for informing design choices in applications such as synthetic engineering of
Turing patterns, but also for understanding the role of stratified media in
modulating pattern-forming processes in developmental biology and beyond.Comment: 25 pages, 7 figure
Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry
Conditions for self-organisation via Turing’s mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further
A Review on Mechanics and Mechanical Properties of 2D Materials - Graphene and Beyond
Since the first successful synthesis of graphene just over a decade ago, a
variety of two-dimensional (2D) materials (e.g., transition
metal-dichalcogenides, hexagonal boron-nitride, etc.) have been discovered.
Among the many unique and attractive properties of 2D materials, mechanical
properties play important roles in manufacturing, integration and performance
for their potential applications. Mechanics is indispensable in the study of
mechanical properties, both experimentally and theoretically. The coupling
between the mechanical and other physical properties (thermal, electronic,
optical) is also of great interest in exploring novel applications, where
mechanics has to be combined with condensed matter physics to establish a
scalable theoretical framework. Moreover, mechanical interactions between 2D
materials and various substrate materials are essential for integrated device
applications of 2D materials, for which the mechanics of interfaces (adhesion
and friction) has to be developed for the 2D materials. Here we review recent
theoretical and experimental works related to mechanics and mechanical
properties of 2D materials. While graphene is the most studied 2D material to
date, we expect continual growth of interest in the mechanics of other 2D
materials beyond graphene
Towards an integrated experimental-theoretical approach for assessing the mechanistic basis of hair and feather morphogenesis
In his seminal 1952 paper, ‘The Chemical Basis of Morphogenesis’, Alan Turing lays down a milestone in the application of theoretical approaches to understand complex biological processes. His deceptively simple demonstration that a system of reacting and diffusing chemicals could, under certain conditions, generate spatial patterning out of homogeneity provided an elegant solution to the problem of how one of nature's most intricate events occurs: the emergence of structure and form in the developing embryo. The molecular revolution that has taken place during the six decades following this landmark publication has now placed this generation of theoreticians and biologists in an excellent position to rigorously test the theory and, encouragingly, a number of systems have emerged that appear to conform to some of Turing's fundamental ideas. In this paper, we describe the history and more recent integration between experiment and theory in one of the key models for understanding pattern formation: the emergence of feathers and hair in the skins of birds and mammals
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