Abstracts of Invited Presentations
Title: Three-Dimensional Models of Myxobacteria Aggregation and Morphogenesis
Abstract:
Myxobacteria are a model system for studying cell-cell interaction and
organization preceeding differentiation. When starved, tens of thousands of
Myxobacteria cells change their motility, align, stream and form aggregates,
which then develop into fruiting bodies. While cell aggregation has
canonically been modeled as the result of chemotaxis, growing evidence shows
that Myxobacteria organization depends on a contact-mediated cell signaling
mechanism. In this talk we will describe the first three-dimensional model
of cell aggregation in Myxobacteria based on a short-range cell-cell
communication. We will demonstrate that the same 3D discrete stochastic
model simulates for different values of parameters, all stages of
myxobacteria aggregation: the formation of a traffic jam, which then
triggers formation of an aggregation center culminating in a fruiting body
formation.
In the second half of the talk we will present a computational model
of the developmental phenomena.To illustrate general approach, we will
describe a simulation of the skeletal pattern in a growing embryonic
vertebrate
limb.
Title: The Short and Long of Complexity in Soft Electronic Matter
Abstract:
A substantial change is presently taking place in experimental and
theoretical approaches to large classes of "strongly correlated"
materials, reflecting growing evidence that multiscale complexity (in
space and time) is frequently both intrinsic and functional, and
further that intimate relationships between hierarchies of functional
scales constitute essential "Systems" or "Networks". This complexity
provides qualitatively new avenues for predictive design of
technological materials, including intrinsically nanoscale structures.
In a search for underpinning concepts and principles, the prevalence
of coexisting short- and long-range forces has become apparent as the
key in some major class of materials with emergent "landscapes"
of
spatio-temporal patterns and associated glassy dynamics. We review
our recent work in these directions in the context of
elastically-driven textures in (a) directionally-bonded electronic
materials (particularly doped transition metal oxides)(1), and (b)
bubble opening patterns in DNA and their role in transcription
initiation.(2) We emphasize the essential role of local
constraints,requiring faithful representations of lattice-scale
effects.
(1) K. Ahn et al, Nature 428, 401 (2004) and refs therein ; J-X Zhu
et al, Phys Rev Lett 91 ,057004 (2003).
(2 )C. Choi et al, Nucleic Acids Research 32, 1589 (2004) ; Euro Phys
Lett 68, 127 (2004).
Presentation: Slides (PDF)
Title: Sparks, spikes and bumps in biological networks: from discrete models
to localised solutions
Abstract:
Many different types of cell, and indeed networks of cells, are capable
of composing global signals (travelling waves) from elementary events.
At the sub-cellular level these elementary events may take the form of
sparks of elevated calcium, arising via calcium release from internal
stores. If whole cell models are assumed to be of reaction-diffusion
type then powerful techniques from continuum mechanics can be brought
to bear in studying nonlinear calcium waves. However, in models that
respect the discrete nature of internal stores translation symmetry is
broken and one cannot use such techniques. Importantly, this loss of
translation symmetry is a prerequisite for the existence of a saltatory
wave. There is a similar loss of translation symmetry in the
dendritic tree of a neural cell studded with so-called spines. In this
case the elementary events underlying a global signal are spikes of
voltage. I will discuss exactly soluble models that give a feel for
wave speed and stability for both these physiologically significant
saltatory waves. Moving away from the sub-cellular scale it is
interesting to raise the issue of whether discrete or particle like
properties can emerge in tissue level models. To give an affirmative
example I will present a continuum model of neural tissue that supports
localised bumps of activity. Instabilities of these bumps are further
shown to lead to the emergence of both localised breathers and
travelling waves. Travelling pulses in this model truly have a
discrete character in the sense that they scatter as auto-solitons.
Presentation: Slides (PDF), and
movies -
AECR.mpg,
Animation.mpg,
BreathingPulse.mpg,
BumpToBreather.mpg,
BumpToPulse.mpg,
Collide_Sym.mpg,
MovieJapan3.mpg,
Periodic_d085.mpg,
Pulse_d085.mpg,
Spiral.mpg,
Spiral2.mpg,
Splitter.mpg,
WaveProf.mpg
Title: Atomistic potential energy surfaces and solid mechanics
Abstract:
How does collective "solid mechanics" behaviour
emerge from (accurate quantum-mechanical or approximate)
many-atom potential energy surfaces?
I will discuss basic open problems and recent mathematical results,
including
(1) identification of the 3D fcc (face-centered cubic)
lattice as a local minimizer of the celebrated Lennard-Jones pair
interaction model
(2) an analysis of the Cauchy-Born rule for elasticity tensor
determination
in a simple 2D atomistic model. We find that the CB rule
-- is actually a theorem when short-range interactions are dominant
-- but fails when long-range interactions are dominant or when
sufficiently strong pressure is applied. In the failure regime,
atomic-scale fluctuations of the atomic position field become favourable.
Joint work with Florian Theil (Warwick).
References: G.Friesecke, F.Theil, Validity and failure of the Cauchy-Born
Hypothesis in a two-dimensional mass-spring lattice, J.Nonl.Sci 12 445-478
2002;
G.Friesecke, F.Theil, Periodic crystals as local minimizers of pair
potential energies, preprint 2005
Title: Fluid-dynamic models of discrete traffic and product flows
Abstract:
We present approaches to deriving macroscopic, fluid-dynamic equations for
the flow of individual vehicles in street networks and the flow of
packages in production processes. Depending on the degree of
simplification, we obtain equations capable of describing emergent
stop-and-go waves or equations for queuing networks describing the
spreading of shock waves. The models are suitable for the analytical
understanding of instabilities such as phantom traffic jams or bullwhip
effects. They also help to derive the essential dynamical features
required to derive suitable strategies of on-line control. This is
relevant for future approaches towards more flexible traffic flow control
and production scheduling under varying or perturbed conditions.
Presentation: Video, Slides (PDF)
Title: The Role of Fluctuations in Theories for Granular Materials
Abstract:
Granular materials exhibit behavior characteristic of gases, liquids, or
solids, depending on the conditions to which they are subjected. The
formulation of continuum theories appropriate for each of these regimes
involves the incorporation of fluctuations of velocity or position at the
particle scale and suitable determination of the strength of the
fluctuations in terms of the average fields of interest. We'll review such
theories, discuss their implementation, and highlight their limitations.
Presentation: Slides (PDF)
Title: The Onset of Calcium Oscillations
Abstract:
Calcium oscillations are an important means of signalling in many cell
types. These oscillations have been studied extensively using ordinary
differential equation (ode) models, some with a high degree of complexity.
These models generally show that the onset of oscillations is via a
subcritical Hopf bifurcation.
The experimental data, however, show that calcium release is not
nearly as regular as the ode models suggest, but is highly stochastic
in nature. In fact, these data show that oscillations are initially
very irregular with large average interspike interval and large
variance, eventually settling into a regular oscillation as the
bifurcation parameter (in this case [IP_3]) is increased.
The reason for the discrepancy between data and models is
that calcium release is via events that are fundamentally stochastic
in time and discrete in space. Furthermore, the assumptions that
permit a whole cell ode model, namely uniformly distributed calcium and
the law of large numbers, do not apply under many physiological
conditions.
In this talk, I will describe a modeling approach that takes the
stochastic and spatially discrete nature of calcium release into
account, with the result that the onset of calcium oscillations
agrees with experimental observations.
Presentation: Video, Slides (PDF)
Title: What can we learn from discrete models in photorefractive materials
and Bose-Einstein condensates ? Some old results, some new results and
some future challenges
Abstract:
In this talk, we will give an, admittedly subjective, view of
some recent developments in nonlinear optics and soft condensed-matter
physics, emphasizing the role of discrete systems in the relevant
findings. We will explain how discreteness (or rather "effective
discreteness") arises in these contexts and will discuss some notes
of caution as to what not to expect from such approaches. The binding
theme of the presentation will be the, so called, discrete nonlinear
Schrödinger model and its variants. We will use that model to present
some successful connections of theory with experiments in both
photorefractive media and Bose-Einstein condensates in optical lattices,
will then discuss some recent results and close with a personal view
of some interesting directions of current investigation and future study.
Presentation: Video, Slides (PDF)
Title: From discrete system simulations to constitutive continuum theories
Abstract:
One essential question in material science and mechanics is, how
to bridge the gap between the microscopic picture (molecular dynamics
or discrete element simulations of particulate flows) and a
macroscopic description on the level of a continuum theory.
The former involves impulses/contact-forces and
collisions/deformations, whereas the latter concerns tensorial
quantities like the stress or the velocity gradient.
Applying a consistent averaging formalism, one can obtain scalar-
and vector-fields as well as classical tensorial macroscopic
quantities like fabric, stress or velocity gradient and, in addition,
micropolar quantities like curvature or couple-stress.
Generic features in such systems involve enormous inhomogeneities,
an-isotropic material behavior, and memory (i.e. history dependence).
The stress, strain rate, and structure tensors are, in general,
not co-linear, after a small shear deformation. The material acts
against shear by gaining stiffness (contacts) as opposed to
the compressive shear direction.
Examples for the micro-macro transition for powders will be given
in the framework of unconsolidated granular materials.
Presentation: Slides (PDF)
Title: The Transition to Suspension in
Granular Flows
Abstract:
Naturally occuring suspension currents, such as powder
snow avalanches and turbidity currents, are formed by a dense granular
flow interacting with the ambient fluid. We explain how this process
occurs and to what extent it can be described by continuum theories. We
compare the theory with small scale experiments and full-scale
observations of natural avalanches.
Presentation: Slides (PDF)
Alexander Mielke (joint work with Johannes Giannoulis)
Title: Dynamics of modulated pulses in discrete lattices via
the nonlinear Schrödinger equation
Abstract:
The propagation of acoustic or optical pulses in crystals can be described
via the Hamiltonian dynamics of a discrete periodic lattice of particles
interacting via suitable interaction potentials. We consider pulses which
are weak modulations of a basic harmonic pattern which is characterized
by a wave vector. In suitable scaling regimes such a pulse moves with
the associated group velocity and is deformed by dispertion and nonlinear
interaction, which can be described on the macroscopic, continuum level
via the nonlinear Schrödinger equation. We discuss the formal derivation
as well as the mathematical justification of the multiscale problem.
Presentation: Video, Slides (PDF)
Title: Modelling volcanic processes and hazards
Abstract:
Volcanoes are dynamic non-linear systems with resulting complex
behaviours. Due to the hazards they pose a variety of modeling
approaches can be used to understand the dynamics of the eruptions,
help to forecast activity and develop risk assessments of the hazardous
phenomena. Volacnoes provide a good way of illustrating the strengths
and limitations of different modeling strategies applied to natural
systems.
Title: Bistable Waves in Discrete Media
Abstract:
We consider reaction-diffusion equations with a bistable reaction term.
The equations considered include Nagumo equations, FitzHugh-Nagumo
equations
and a class of coupled Nagumo equations. Models that are discrete in space
are often appropriate when there is an inherent length scale such as the
distance between nodes of Ranvier in myelinated nerve fibers or the
interatomic
distance in materials. We focus on traveling wave solutions for some "simple"
models. Phenomena considered include propagation failure due to
the inherent length scale, lattice induced anisotropy in higher space
dimensions,
synchronization of waves in coupled systems, wave speedup due to temporal
discretization, existence of pulse and front solutions and attractors for
some FitzHugh-Nagumo equations, and time permitting some recent work on
waves in
inhomogeneous discrete media.
Presentation: Slides (PDF)
Title: Macroscopic traffic flow models: glamour and misery
Abstract:
Macroscopic (flow) models have a long tradition in the modelling
of traffic flow. Starting from the Lighthill Whitham (Richards) (LWR) approach
of a first order equation for the density, a couple of different models have
been suggested over time. This includes first order, second order and
even third order models invented to catch more complex phenomena. One of
the most fundamental critiques is the one put forward by Daganzo (A requiem
on second-order theories), who pointed out a number of serious short-comings
of higher-order (especially second order) models. However, in reacting to this,
a very clever new type of models has been invented to circumvent the
short-comings Daganzo has pointed out, which unfortunately are still
rarely used in applications.
A very old critique of this models, even the first order ones, is that they
are unsuitable to describe traffic flow because of the small numbers of
particles involved. (In a 100 m box there are at most 14 vehicles.) This
goes hand-in-hand with their inability to model fluctuations. These models,
by their very construction are mean values of an underlying stochastic theory,
which is given by a suitable Boltzmann equation.
Ignoring all these difficulties, our own work is even more basic: we are simply
asking the question: how well can these models describe the phenomena observed.
This is being done by directly comparing them with reality; so far it seems
that Daganzo is right who has pointed out that higher-order theories are not
only erroneous, by also superfluous: it is very hard to find phenomena that
cannot be modelled by LWR-theory. (It is a completely different story whether
they can be understood by LWR.)
Abstracts of Contributed Presentations
Title: Master equation approach to the study of phase change processes in data storage media
Abstract:
The dynamics of crystallization in phase-change materials is investigated
using a master equation approach. We develop a novel model using the
thermodynamics of the processes involved. Some partial analytical results are
obtained for the isothermal case and for large cluster sizes, but principally
numerical simulations are used to investigate the model.
Title: Oblique shock waves of granular flows
Abstract:
Oblique shocks can be generated when granular flows are deflected inwardly by
wedges. We assume mass and momentum to be conserved across shocks, forming the
granular jump condition. An oblique shock relation is derived according to
this condition. It predicts two solutions for oblique shocks, which are
observed as well from experiments and numerics. Based on the shock relation,
the intersection of oblique shocks of opposite families is analyzed. When the
refracted shocks are attached, they are still theoretically predictable. If
the oblique shocks are of unequal-strengths, a slip-line occurs downstream of
the intersection, and is in the direction of the intersected flows.
Title: Discrete state models in chaotic population dynamics
Abstract:
Both time and population size can be treated either as discrete or continuous
in population dynamics. Traditionally, time was chosed according to the
character of the population, however, the size was always considered as a real
number. Recently, it became obvious that if the dynamics is chaotic,
integer-based discrete models give radically different predictions. Real
populations obviously consist of integer numbers of individuals, however, it
is equally obvious that random noise has to be added to the deterministic
model. Such noisy, discrete models display ambivalent behaviour. In case of
low noise they resemble the discrete deterministic models, in case of high
noise they predict statistics similar to the continuous model. In real
populations the noise is intermediate, and models in this intermediate zone
show complex behaviour.
Using recent results from the theory dynamicals
systems as well as experimental data from the Beetle Team we propose some
qualitative explanations and guidelines for numerical simulation.
Title: Granular avalanches, particle-size segregation, shock-waves and pattern formation
Abstract:
Granular avalanches are one of the most abundant grain transport mechanisms in
both the natural environment and industry, with applications ranging from snow
avalanches, debris-flows, lahars and pyroclastic flows to processes in the
bulk chemical, pharmaceutical, agricultural, food-processing and mining
industries. Even in simple flows, such as pouring one's cornflakes into a bowl
at breakfast, it is avalanches that transport the grains! This talk gives an
overview of some of the simple continuum theories that are used to model these
flows, pointing out their weaknesses, their successes and some of their
applications. During the avalanches, particles of different sizes often
segregate out into inversely graded layers with the large grains on top of the
fines. Depending on whether these layers are brought to rest by deposition at
the base, or through the propagation of shock waves that bring the grains to
rest en masse, strikingly different patterns are formed in the resulting
deposit. Experiments will be demonstrated to show the development of the
patterns and some first attempts to model these systems with continuum and
discrete models will be discussed.
Title: Discrete statistical stability, scale-free distributions and the evolution of the WWW
Abstract:
Consideration is given to the convergence properties of sums of identical,
independently distributed random variables drawn from a class of discrete
distributions with power-law tails. Different limiting distributions, and
rates of convergence to these limits, are identified and depend on the index
ν of the tail. For indices in the range -2< ν <-1, the limiting
distributions evolve towards the discrete analogue of the L vy-stable
distributions, for which the mean and higher order moments do not exist. For
indices ν<-2, the limiting distribution evolves to a Poisson
distribution, but the rate of convergence to the Poisson at the marginal value
ν = -2 is extraordinarily slow. These results are applied to the
evolution of scale-free random networks. It is shown that any change in the
evolution of the network topology of the WWW, for example, is unlikely to be
yet evident. Moreover, it is shown that treating discrete scale-free behaviour
with continuum or mean-field models or approximations can lead to incorrect
results since the discrete distributions do not have a dense limit. The
modifications to these results caused by the existence of an outer scale to
the distributions are discussed and quantified.
Title: Collapse of Granular Columns
Abstract:
To follow.
Title: Dynamics and pattern formation in invasive tumor growth
Abstract:
In this work, we study the in-vitro dynamics of the most malignant form of the
primary brain tumor: Glioblastoma Multiforme. Typically, the growing tumor
consists of the inner dense proliferating zone and the outer less dense
invasive region. Experiments with different type of cells show qualitatively
different behavior. Wild type cells invade a spherically symmetric manner, but
mutant cells are organized in tenuous branches. We formulate a model for this
sort of growth using two coupled reaction-diffusion equations for the cell and
nutrient concentrations. When the ratio of the nutrient and cell diffusion
coefficients exceeds some critical value, the plane propagating front becomes
unstable with respect to transversal perturbations. The instability threshold
and the full phase-plane diagram in the parameter space are determined. We
also analyze the role of cell-cell adhesion, both in continuous and discrete
(lattice) models. The results are in a good agreement with experimental
findings for the two types of cells.
Title: Micromechanical modelling of granular media with evaluation of microstructures
Abstract:
Non-linear deformation behaviors (macro-scale phenomena) of granular materials
are controlled by constituent grain properties (micro-scale element) and
microstructure formed (meso-scale unit). Grain shape, grain strength and
stiffness, and resistant friction angle between grains assign grain
properties. Microstructures in granular material are formed such as the
connection paths of contact points between grains. In this study, the
deformation-failure behaviors of granular materials with different grain
shapes and friction properties and reinforced by mixing with different
material are simulated by DEM in two-dimension. On the basis of analysis
results, changes in microstructure and localization with macro deformation are
observed. In addition, the micromechanical model is proposed by using
micro-polar theory and homogenization scheme with account for particle
rotation.
Title: Granular Segregation in a Rotating Polygonal Drum
Abstract:
The mixing in a rotating drum of granular materials is a widely used
industrial process. The chaotic motion of the individual particles has
been well demonstrated. When the particles are of varying sizes however
segregation takes place adding strong ordering to the system. A simple
method will be introduced and demonstrated, providing a way of predicting
the patterns fromed by this segregation in arbitrary convex drums.
Title: Orientation-induced instabilities in the collective motion of self-propelled particles: discrete and continuous models
Abstract:
Collective motion is a challenging example of self-organization with
widespread biological applications ranging from swarms of bacteria to
mammalian herds. Despite the fact that in each case the interactions between
individuals are of a different nature, it is possible to determine common
requirements for self-organization.
Here, we focus on the implications of alignment and analyze 'ferromagnetic'
and 'liquid crystal'-like alignment mechanisms. While the former only admits
parallel alignment, the latter allows parallel and anti-parallel alignment.
For both alignment mechanisms we have obtained numerical evidence of a
phase-transition related to the noise introduced into the system through an
individual-based model. In the case of parallel and anti-parallel alignment,
we observe spontaneous symmetry breaking of the rotation symmetry leading to
orientational order of the particles.
We also present a simple continuum model that captures the essential features
of the orientation dynamics of the system. This model consists of an
integrodifferential equation where spatial and angular dependences are
included and the kernels are physically motivated. The linear stability
analysis of the model reveals that there are orientation instabilities
triggered by different modes in the ferromagnetic and liquid-crystal-like
interactions. Possible interpretations will be discussed.
Title: A multiscale model for damage using gradient flows of Young measures
Abstract:
We study local minimisers of Young measures describing the deformation
gradient of a body. The Young measures are defined as quasi-static limit of a
gradient flow with respect to a modified Wasserstein metric [1]. This idea is
applied to a model for damage/fracture. We prove existence for the time
discretized model and study the asymptotic behaviour. The resulting model
describes the stability of a body under small loads and damage under large
loads using only the elastic energy density as material parameter [2]. Finally
we prove the convergence of the time discretized model to a gradient flow
[3].
An asymptotic analysis of recent variational fracture models [4,5] shows an
interesting connection between these seemingly very different approaches to
our approach [6].
This is joint work with Isaac Chenchiah (Max-Planck-Institute Leipzig) and
Johannes Zimmer (University of Bath).
[1] M.O.Rieger: A model for hysteresis in mechanics using local minimizers of
Young measures, in: Progress in Nonlinear Differential Equations and Their
Applications, Volume 63.
[2] M.O.Rieger and J.Zimmer: Young measure flow as a model for damage, in
preparation.
[3] I.Chenchiah, M.O.Rieger and J.Zimmer: Gradient flows in asymmetric metric
spaces, in preparation.
[4] G.Dal Maso and R.Toader: A model for the quasi-static growth of brittle
fractures based on local minimization, Math. Models Methods Appl. Sci., 2002.
[5] G.A.Francfort and J.-J.Marigo: Revisiting brittle fracture as an energy
minimization problem, J. Mech. Phys. Solids, 1998.
[6] M.O.Rieger and P.Tilli: On the Gamma-limit of the Mumford-Shah Functional,
Calc.Var., (electr.) 2004.
Title: The Tissue Mechanics of Tumour Growth
Abstract:
The unchecked growth of a solid tumor produces solid stress, causing
deformation of the surrounding tissue. This stress can result in
clinical complications, especially in confined environments such as the
brain, and may also be responsible for pathophysiological anomalies
such as the collapse of blood and lymphatic vessels. High stress levels may
also inhibit further cell division within tumors. Unfortunately,
little is known about the dynamics of stress accumulation in tumors or its
effects on cell biology. We present a mathematical model for tumor
growth in a confined, elastic environment such as living tissue. The model,
developed from theories of thermal expansion using the current
configuration of the material element, allows the stresses within the
growing tumor and the surrounding medium to be calculated. The
experimental observation that confining environments limit the growth of
tumor spheroids to less than the limit imposed by nutrient diffusion
is incorporated into the model using a stress dependent rate for tumor
growth. The model is validated against experiments for MU89 tumor
spheroid growth in Type VII agarose gel. Using the mathematical model and
the experimental evidence we show that the tumor cell size
is reduced by solid stress inside the tumor spheroid. This leads to the
interesting possibility that cell size could be a direct indicator of solid
stress level inside the tumors in clinical setting.
Roose et al. (2004) Microvascular Research 66:204-212
Title: Dispersive Continuum Models and Peierls Dislocation
Abstract:
Within the Peierls-Nabarro model of a dislocation, we present certain higher
gradient continuum models involving length scales that modify the simple
dispersion relation of classical continuum. We construct the displacement
field for such elastic models and obtain kinetic relations that resemble the
kinetic relations obtained from the discrete model of a crystal lattice with
only nearest neighbour interactions and various atomisitic simulations. The
emission of phonons has been assumed as the only mechanism for dissipation of
energy and we find that certain features of the dispersion relation play
dominant role in dislocation dynamics. The important role played by small
length scales in the phenomena involving small waves has been highlighted
while staying within the realm of continuum mechanics. In these
microscopically conservative models the dissipation at a macroscale occurs due
to generation of waves of small wavelengths.
Title: Elastoplastic waves in antiplane shear
Abstract:
The antiplane shear model is a simplified form of the
continuum mechanics description of granular flow.
Both of these models are dynamically
ill-posed, exemplifying a failure of the continuum
description of a discrete system. Further simplification of the antiplane
shear model
and discretization in space results in a well-posed system
that has solutions reminiscent of shear bands.
By re-introducing discreteness to the
continuum model, a phenomenon like that in the discrete physical system
appears.
In some regions of parameter space, the solution exhibits
elastic-plastic transitions which propagate with fixed speed, and interactions
between these and conventional elastic and plastic waves produce periodic
solutions, in which the size of the "shear band" oscillates.
I will present numerical experiments demonstrating the bifurcations to and
within these periodic solutions, which include period doubling as well as
changes in the form of solution which are less straightforward to characterize.
Title: Connections between wave structures in micro- and macroscopic highway traffic models
Abstract:
We analyse the wave-like solutions of a class of microscopic models
of highway traffic via their homogenized PDE limits. Different families
of waves are classified via bifurcations of an associated planar ODE
problem, and we display novel types of solution which cannot be produced
by the classical Lighthill-Whitham-Richards (LWR) framework. The new types
of solution correspond to so-called `under-compressive' waves.
Finally, we show how the theory works in the presence of spatial
inhomogeneity (e.g. a bottleneck or merge).
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