Past OCCAM Wednesday morning Events

Successful navigation through a complicated and evolving environment is a fundamental task carried out by an enormous range of organisms, with migration paths staggering in their length and intricacy. Selecting a path requires the detection, processing and integration of a myriad of cues drawn from the surrounding environment and in many instances it is the intrinsic orientation of the environment that provides a valuable navigational aid.

In this talk I will describe the use of transport models to describe migration in oriented environments, and demonstrate the scaling approaches that allow us to derive macroscopic models for movement.

I will illustrate the methods through a number of apposite examples, including the migration of cells in the extracellular matrix, the macroscopic growth of brain tumours and the movement of wolves in boreal forest.

Cell locomotion is essential for early development, angiogenesis, tissue regeneration, the immune response, and wound healing in multicellular organisms, and plays a very deleterious role in cancer metastasis in humans. Locomotion involves the detection and transduction of extracellular chemical and mechanical signals, integration of the signals into an intracellular signal, and the spatio-temporal control of the intracellular biochemical and mechanical responses that lead to force generation, morphological changes and directed movement. While many single-celled organisms use flagella or cilia to swim, there are two basic modes of movement used by eukaryotic cells that lack such structures -- mesenchymal and amoeboid. The former, which can be characterized as `crawling' in fibroblasts or `gliding' in keratocytes, involves the extension of finger-like filopodia or pseudopodia and/or broad flat lamellipodia, whose protrusion is driven by actin polymerization at the leading edge. This mode dominates in cells such as fibroblasts when moving on a 2D substrate. In the amoeboid mode, which does not rely on strong adhesion, cells are more rounded and employ shape changes to move -- in effect 'jostling through the crowd' or `swimming'. Here force generation relies more heavily on actin bundles and on the control of myosin contractility. Leukocytes use this mode for movement through the extracellular matrix in the absence of adhesion sites, as does Dictyostelium discoideum when cells sort in the slug. However, recent experiments have shown that numerous cell types display enormous plasticity in locomotion in that they sense the mechanical properties of their environment and adjust the balance between the modes accordingly by altering the balance between parallel signal transduction pathways. Thus pure crawling and pure swimming are the extremes on a continuum of locomotion strategies, but many cells can sense their environment and use the most efficient strategy in a given context. We will discuss some of the mathematical and computational challenges that this diversity poses.

A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed.

This modelling approach incorporates recent experimental findings on behaviour of locusts. It exhibits nontrivial dynamics with a "phase change" behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations with nonlocal and nonlinear right hand side is derived and analyzed. The existence of its solutions is proven and the systemʼs long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model when the number of individuals grows. This is a joint work with Radek Erban.

Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change.

In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions and for accelerating invasion waves.

Would you like to solve a partial differential equation efficiently with a relative error of 10% or would you prefer to wait a bit longer and solve it with an error of only 1% ? Is it sufficient to know that the error is about 1% (having no idea what the `about' means) or would you prefer to have reliable information that the error is guaranteed to be below the required tolerance?

Answering these questions is necessary for the efficient and reliable numerical solution of practically any mathematical problem. In the context of numerical solution of partial differential equations, the crucial tool is the adaptive algorithm with suitable error indicators and estimators. I will overview the adaptive algorithm and its variants. I will concentrate on the a posteriori error estimators with the emphasis on the guaranteed ones.

We define a set of nonlocal operators and develop a nonlocal vector calculus that mimics the classical differential vector calculus. Included are the definitions of nonlocal divergence, gradient, and curl operators and the derivation of nonlocal integral theorems and identities. We indicate how, through certain limiting processes, the nonlocal operators are connected to their differential counterparts. The nonlocal operators are shown to appear in nonlocal models for diffusion and in the nonlocal, spatial derivative free, peridynamics continuum model for solid mechanics. We show, for example, that unlike elliptic partial differential equations, steady state versions of the nonlocal models do not necessary result in the smoothing of data. We also briefly consider finite element methods for nonlocal problems, focusing on solutions containing jump discontinuities; in this setting, nonlocal models can lead to optimally accurate approximations.

I describe recent work on the synchronization of IP3R calcium channels in the interior of cells. Hybrid  models of calcium release couple deterministic equations for diffusion and reactions of calcium ions to stochastic gating transitions of channels. I discuss the validity of such models as well as numerical methods.Hybrid models were used to simulate cooperative release events for clusters of channels. I show that for these so-called puffs the mixing assumption for reactants does not hold. Consequently, useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels.

Based on the spatial approach, a Markovian model can be inferred, representing well calcium puffs in neuronal cells. I then describe further reduction of the stochastic model and the synchronization arising for small channel numbers. Finally, the effects of calcium binding proteins on duration of release is discussed.

Colonies of motile microorganisms, the cytoskeleton and its components, cells and tissues have much in common with soft condensed matter systems (i.e. liquid crystals, amphiphiles, colloids etc.), but also exhibit behaviors that do not appear in inanimate matter and that are crucial for biological functions.

These unique properties arise when the constituent particles are active: they consume energy from internal and external sources and dissipate it by moving through the medium they inhabit. In this talk I will give a brief introduction to the notion of "active matter" and present some recent results on the hydrodynamics of active nematics suspensions in two dimensions.

As many of you are no doubt aware, a Study Group of particular significance will take place in Thuwal, Saudi Arabia, from 23rd January - 26th January 2011.

The problem statements, in their preliminary form can be found at:

http://people.maths.ox.ac.uk/bentahar/KSG/

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes.

In this talk, I will present a new theory to study the phenomenon of cavitation in soft solids that, contrary to existing approaches,

simultaneously: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii)  applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and

(iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as the homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. Then, by means of a novel iterated homogenization procedure, exact solutions are constructed for such a problem. These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation - corresponding to the event when the initially infinitesimal cavities suddenly grow into finite sizes - can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the cavities plays the role of "time" and the applied load plays the role of "space".

An application of the theory to the case of Ne-Hookean solids containing a random isotropic distribution of vacuous defects will be presented.

Modern differential geometry is the art of the abstract that can be pictured. Continuum mechanics is the abstract description of concrete material phenomena. Their encounter, therefore, is as inevitable and as beautiful as the proverbial chance meeting of an umbrella and a sewing machine on a dissecting table. In this rather non-technical and lighthearted talk, some of the surprising connections between the two disciplines will be explored with a view at stimulating the interest of applied mathematicians.

Cryopreservation (using temperatures down to that of liquid nitrogen at

–196 °C) is the only way to preserve viability and function of mammalian cells for research and transplantation and is integral to the quickly evolving field of regenerative medicine. To cryopreserve tissues, cryoprotective agents (CPAs) must be loaded into the tissue. The loading is critical because of the high concentrations required and the toxicity of the CPAs. Our mathematical model of CPA transport in cartilage describes multi-component, multi-directional, non-dilute transport coupled to mechanics of elastic porous media in a shrinking and swelling domain.

Parameters are obtained by fitting experimental data. We show that predictions agree with independent spatially and temporally resolved MRI experimental measurements. This research has contributed significantly to our interdisciplinary group’s ability to cryopreserve human articular cartilage.

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium and find an instability for a threshold value of the twist.

We shall discuss a simple low order numerical integration scheme for ODEs and DAEs. The scheme has a parameter that allows for regularization of Jacobian of stiff problems and for numerically elucidating multi-scale response, if any, in some problems.