Past OCCAM Wednesday morning Events

The good use of condiments is one of the secrets of a tasty quiche. If you want to delight your guests, add a pinch of ground pepper or cinnamon to the yellow liquid formed by the mix of the eggs and the crème fraiche. Here, is a surprise : even if the liquid is at rest, the pinch of milled pepper spreads by itself at the surface of the mixture. It expands in a circular way, and within a few seconds, it covers an area equal to several times its initial one. Why does it spread like that ? What factors influence this dispersion ? I will present some experiments and mathematical models of this process.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY     *****

Reduced ODE models describing coarsening dynamics of droplets in nanometric polymer film interacting on solid substrate in the presence of large slippage at the liquid/solid interface are derived from one-dimensional lubrication equations. In the limiting case of the infinite slip length corresponding to the free suspended films a collision/absorption model then arises and is solved explicitly. The exact collision law is derived. Existence of a threshold at which the collision rates switch from algebraic to exponential ones is shown.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY     *****

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON MONDAY     *****

Ultralow interfacial tension mixtures have interfacial tensions that are 1,000 times, or more, lower than typical oil-water systems. Despite the recent utility of ultralow interfacial tension mixtures in industry and research, quantifying the interfacial tension remains challenging. Here I describe a technique that measures ultralow interfacial tensions by magnetically deflecting paramagnetic spheres in a co-flow microfluidic device. This method involves the tuning of the distance between the co-flowing interface and the magnetic field source, and observing the behavior of the magnetic particles as they approach the liquid-liquid interface--the particles either pass through or are trapped. I demonstrate the effectiveness of this technique for measuring very low interfacial tensions by testing solutions of different surfactant concentrations, and show that the results are comparable with measurements made using a spinning drop tensiometer.

*****     PLEASE NOTE THIS SEMINAR TAKES PLACE ON MONDAY     *****

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 24TH JUNE 2013     *****

Energy equations describing magnetic and inertial confinement functions (ICF) are strongly coupled, time dependent non-linear PDEs. The huge disparity of the coefficients in the coupled non-linear equations brings tremendous numerical difficulties to get high resolution solutions. It results in highly ill-conditioned linear systems in each non-linear iteration. Solving the resulted non-linear systems is time-consuming which takes up to 90% in the total simulation time. Many customized numerical techniques have to be employed to get a robust and accurate solution.This talk will present an inexact Newton-Krylov-Schwarz framework to solve the problem, demonstrating how to integrate preconditioning, partial Jacobian matrix forming techniques, parallel computing techniques with the Newton-Krylov solvers to solve the challenging problem. The numerical results will be shown and other numerical problems will be mentioned.

*****     If anyone is planning to take the 11.36 train after the seminar to the NA conference in Glasgow a taxi from the Gibson building is being arranged. Please contact Jude, @email, to book a place in the taxi.     *****

*****     PLEASE NOTE THIS SEMINAR WILL COMMENCE AT 12.00     *****

I will present experimental work on collective dynamics in two different systems: (i) a collection of self propelled droplets and (ii) coupled mechanical oscillators.  

In the first part, I will talk about microswimmers made from water-in-oil emulsion droplets. Following a brief description of the swimming mechanism, I will discuss some of the collective effects that emerge in quasi 1 and 2 dimensional confinements of swimming droplets. Specifically, I dwell on hydrodynamic and volume exclusion interactions, only through which these droplets can couple their motions. 

In the second part, I will present recent results about an intriguing dynamic known as a chimera state. In the world of coupled oscillators, a chimera state is the co-existence of synchrony and asynchrony in a population of identical oscillators, which are coupled nonlocally. Following nearly 10 years of intense theoretical research, it has been an imminent question whether these chimera states exist in real systems. Recently, we built an experiment with of springs, swings and metronomes and realised, for the first time, these symmetry breaking states in a purely physical system.

*****     PLEASE NOTE THIS SEMINAR WILL COMMENCE AT 12.00     *****

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 17TH JUNE 2013     *****

Computing is an exercise of discretization of the real world into space, time, and value. While discretization in time and space is well understood in the sciences, discretization of value is a scientific domain full of opportunity. Maxeler's Multiscale Dataflow Computing allows the programmer to finely trade off discretization of value with real performance measured in wallclock time.

In this talk I will show the connection between discretization of value and Kolmogorov Complexity on one hand and approximation theory on the other. Utilizing the above concepts together with building general purpose computing systems based on dataflow concepts, has enabled us to deliver production systems for Oil & Gas imaging (modelling, multiple elimination, RTM, Geomechanics), Finance Risk (derivatives modelling and scenario analysis), as well as many scientific application such as computing weather models, Astrochemistry, and brain simulations. Algorithms range from 3D Finite Difference, Finite Elements (sparse matrix solvers), pattern matching, conjugate gradient optimization, to communication protocols and bitcoin calculations. Published results of users of our machines show a 20-50x total advantage in computations per unit space (1U) and computations per Watt.

*****     PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON MONDAY 17TH JUNE 2013     *****

***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****

Signal transduction pathways are sophisticated information processing machinery in the cell that is arguably taking advantage of highly non-idealistic natures of intracellular environments for its optimum operations. In this study, we focused on effects of intracellular macromolecular crowding on signal transduction pathways using single-particle simulations. We have previously shown that rebinding of kinases to substrates can remarkably increase processivity of dual-phosphorylation reactions and change both steady-state and transient responses of the reaction network. We found that molecular crowding drastically enhances the rebinding effect, and it shows nonlinear time dependency although kinetics at the macroscopic level still follows the conventional model in dilute media. We applied the rate law revised on the basis of these calculations to MEK-ERK system and compared it with experimental measurements.

***** PLEASE NOTE THAT THIS WILL TAKE PLACE ON TUESDAY 11TH JUNE ****

*****     PLEASE NOTE THAT THIS WILL TAKE PLACE ON FRIDAY 7TH JUNE     *****

To follow

********** PLEASE NOTE THE SPECIAL TIME **********

Total generalised variation (TGV) was introduced by Bredies et al. as a high quality regulariser for variational problems arising in mathematical image processing like denoising and deblurring. The main advantage over the classical total variation regularisation is the elimination of the undesirable stairscasing effect. In this talk we will give a small introduction to TGV and provide some properties of the exact solutions to the L^{2}-TGV model in the one dimensional case.

In this talk aimed at a general audience I will discuss the ways in which we have used mathematical models of the regulation of haematopoiesis (blood cell production) to understand haematological diseases, and suggest successful treatment strategies for these diseases. At the end I will talk about our current work on tailoring chemotherapy so that it has less damaging effects on the haematopoietic system and, consequently, improve the quality of life for patients being treated for a variety of tumours.

******************** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY ********************

Well-known iterative schemes for the solution of ill-posed linear equations are the Landweber iteration, the cg-iteration and semi-iterative algorithms like the $\nu$-methods. After introducing these methods, we show that for ill-posed problems a slight modification of the underlying three-term recurrence relation of the $\nu$-methods provides accelerated Landweber algorithms with better performance properties than the $\nu$-methods. The new semi-iterative methods are based on the family of co-dilated ultraspherical polynomials. Compared to the standard $\nu$-methods, the residual polynomials of the modified methods have a faster decay at the origin. This results in an earlier termination of the iteration if the spectrum of the involved operator is clustered around the origin. The convergence order of the modified methods turns out to be the same as for the original $\nu$-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is determined. At the end, the new semi-iterative methods are used to solve a parameter identification problem obtained from a model in elastography.

Cell polarity in the rod-shaped bacterium Myxococcus xanthus is crucial for the direction of movement of individual cells. Polarity is governed by a regulatory system characterized by a dynamic spatiotemporal oscillation of proteins between the opposite cell poles. A mathematical framework for a minimal macroscopic model is presented which produces self-sustained regular oscillations of the protein concentrations. The mathematical model is based on a reaction-diffusion PDE system and is independent of external triggers. Necessary conditions on the reaction terms leading to oscillating solutions are derived theoretically. Possible scenarios for protein interaction are numerically tested for robustness against parameter variation. Finally, possible extensions of the model will be addressed.

Complex systems emerging from many biochemical applications often exhibit multiscale and multiphysics (MSMP) features: The systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical MSMP system may come across scales with macroscopic, mesoscopic and microscopic kinetics,
deterministic and stochastic dynamics, continuous and discrete state space, fastscale and slow-scale reactions, and species of both large and small populations. These complex features present great challenges in the modeling and simulation practice. The goal of our research is to develop innovative computational methods and rigorous fundamental theories to answer these challenges. In this talk we will start with introduction of basic stochastic simulation algorithms for biochemical systems and multiscale
features in the stochastic cell cycle model of budding yeast. With detailed analysis of these multiscale features, we will introduce recent progress on simulation algorithms and computational theories for multiscale stochastic systems, including tau-leaping methods, slow-scale SSA, and the hybrid method. 

***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY 19TH FEBRUARY *****

With "fully anisotropic" I describe diffusion models of the form u_t=\nabla \nabla (D(x) u), where the diffusion tensor appears inside both derivatives. This model arises naturally in the modeling of brain tumor spread and wolf movement and other applications. Since this model does not satisfy a maximum principle, it can lead to interesting spatial pattern formation, even in the linear case. I will present a detailed derivation of this model and discuss its application to brain tumors and wolf movement. Furthermore, I will present an example where, in the linear case, the solution blows-up in infinite time; a quite surprising result for a linear parabolic equation (joint work with K.J. Painter and M. Winkler).

I propose to present modeling and experimental data about the organization of telomeres (ends of the chromosomes): the 32 telomeres in Yeast form few local aggregates. We built a model of diffusion-aggregation-dissociation for a finite number of particles to estimate the number of cluster and the number of telomere/cluster and other quantities. We will further present based on eingenvalue expansion for the Fokker-Planck operator, asymptotic estimation for the mean time a piece of a polymer (DNA) finds a small target in the nucleus.

When two drops come into contact they will rapidly merge and form a single drop. Here we address the coalescence of drops on a substrate, focussing on the initial dynamics just after contact. For very viscous drops we present similarity solutions for the bridge that connects the two drops, the size of which grows linearly with time. Both the dynamics and the self-similar bridge profiles are verified quantitatively by experiments. We then consider the coalescence of water drops, for which viscosity can be neglected and liquid inertia takes over. Once again, we find that experiments display a self-similar dynamics, but now the bridge size grows with a power-law $t^{2/3}$. We provide a scaling theory for this behavior, based on geometric arguments. The main result for both viscous and inertial drops is that the contact angle is important as it determines the geometry of coalescence -- yet, the contact line dynamics appears irrelevant for the early stages of coalescence.

***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON THURSDAY 31ST JANUARY *****

The contact angle of a liquid droplet on a surface can be controlled by making the droplet part of a capacitive structure where the droplet contact area forms one electrode to create an electrowetting-on-dielectric (EWOD) configuration [1]. EWOD introduces a capacitive energy associated with the charging of the solid-liquid interface, in addition to the surface free energy, to allow the contact angle, and hence effective hydrophilicity of a surface, to be controlled using a voltage. However, the substrate must include an electrode coated with a thin, and typically hydrophobic, solid insulating layer and the liquid must be conducting, typically a salt solution, and have a direct electrical contact. In this seminar I show that reversible voltage programmed control of droplet wetting of a surface can be achieved using non-conducting dielectric liquids and without direct electrical contact. The approach is based on non-uniform electric fields generated via interdigitated electrodes and liquid dielectrophoresis to alter the energy balance of a droplet on a solid surface (Fig. 1a,b). Data is shown for thick droplets demonstrating the change in the cosine of the contact angle is proportional to the square of the applied voltage and it is shown theoretically why this equation, similar to that found for EWOD can be expected [2]. I also show that as the droplet spreads and becomes a film, the dominant change in surface free energy to be expected occurs by a wrinkling/undulation of the liquid-vapor interface (Fig. 1c) [3,4]. This type of wrinkle is shown to be a method to create a voltage programmable phase grating [5]. Finally, I argue that dielectrowetting can be used to modify the dynamic contact angle observed during droplet spreading and that this is described by a modified form of the Hoffman-de Gennes law for the relationship between edge speed and contact angle. In this dynamic situation, three distinct regimes can be predicted theoretically and are observed experimentally. These correspond to an exponential approach to equilibrium, a pure Tanner’s law type power law and a voltage determined superspreading power law behavior [6]. 

Acknowledgements

GM acknowledges the contributions of colleagues Professor Carl Brown, Dr. Mike Newton, Dr. Gary Wells and Mr Naresh Sampara at Nottingham Trent University who were central to the development of this work. EPSRC funding under grant EP/E063489/1 is also gratefully acknowledged.

References

[1]   F. Mugele and J.C. Baret, “Electrowetting: From basics to applications”, J. Phys.: Condens. Matt., 2005, 17, R705-R774.

[2]  G. McHale, C.V. Brown, M.I. Newton, G.G. Wells and N. Sampara, “Dielectrowetting driven spreading of droplets”, Phys. Rev. Lett., 2011, 107, art. 186101.

[3]  C.V. Brown, W. Al-Shabib, G.G. Wells, G. McHale and M.I. Newton, “Amplitude scaling of a static wrinkle at an oil-air interface created by dielectrophoresis forces”, Appl. Phys. Lett., 2010,  97, art. 242904.

[4]  C.V. Brown, G. McHale and N.J. Mottram, “Analysis of a static wrinkle on the surface of a thin dielectric liquid layer formed by dielectrophoresis forces”, J. Appl. Phys. 2011, 110 art. 024107.

[5]  C.V. Brown, G. G. Wells, M.I. Newton and G. McHale, “Voltage-programmable liquid optical interface”, Nature Photonics, 2009, 3, 403-405.

[6]  C.V. Brown, G. McHale and N. Sampara, “Voltage induced superspreading of droplets”, submitted (2012)

One of the main problems occurring in the aorta is the development of aneurysms, in which case the artery wall thickens and its diameter increases. Suffice to say that many other factors may be involved in this process. These include, amongst others, geometry, non-homogeneous material, anisotropy, growth, remodeling, age, etc. In this talk, we examine the bifurcation of inflated thick-walled cylindrical shells under axial loading and its interpretation in terms of the mechanical response of arterial tissue and the formation and propagation of aneurysms. We will show that this mechanical approach is able to capture features of the mechanisms involved during the formation and propagation of aneurysms.

Recent advances in engineered muscle tissue attached to a synthetic substrate motivates the development of appropriate constitutive and numerical models. Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta. In this talk we present a   continuum model that accounts for the stimulation of muscle fibers by introducing multiple stress-free reference configurations and for the hysteretic response by specifying a pseudo-elastic energy function. A simple example representing uniaxial loading-unloading is used to validate and verify the characteristics of the model. Then, based on experimental data of muscular thin films, a more complex case shows the qualitative potential of Manduca muscle tissue in active biohybrid constructs.

Although the importance of hydrologic uncertainty is widely recognized it is rarely considered in control problems, especially real-time control. One of the reasons is that stochastic control is computationally expensive, especially when control decisions are derived from spatially distributed models. This talk reviews relevant control concepts and describes how reduced order models can make stochastic control feasible for computationally demanding applications. The ideas are illustrated with a classic problem -- hydraulic control of a moving contaminant plume.

Russian mathematician V.I.Arnold conjectured that convex, homogeneous bodies with less than four equilibria (also called mono-monostatic) may exist. Not only did his conjecture turn out to be true, the newly discovered objects show various interesting features. Our goal is to give an overview of these findings as well as to present some new results. We will point out that mono-monostatic bodies are neither flat, nor thin, they are not similar to typical objects with more equilibria and they are hard to approximate by polyhedra. Despite these "negative" traits, there seems to be strong indication that these forms appear in Nature due to their special mechanical properties.

Telomeres, non-coding terminal structures of DNA strands, consist of repetitive long tandem repeats of a specific length. An absence of an enzyme, telomerase, in certain cellular structures requires an alternative telomerase-independent pathway for telomeric sequence length regulation. Besides linear telomeres other configurations such as telomeric circles and telomeric loops were experimentally observed. They are suspected to play an important role in a universal mechanism for stabilization of the ends of linear DNA that possibly dates back to pre-telomerase ages. We propose a mathematical model that captures biophysical interactions of various telomeric structures on a short time scale and that is able to reproduce experimental measurements in mtDNA of yeast. Moreover, the model opens up a couple of interesting mathematical problems such as validity of a quasi-steady state approximation and dynamic properties of discrete coagulation-fragmentation systems. We also identify and estimate key factors influencing the length distribution of telomeric circles, loops and strand invasions using numerical simulations.

For a given positive measure on a fixed domain, Gaussian quadrature routines can be defined via their property of integrating an arbitrary polynomial of degree $2n+1$ exactly using only $n+1$ quadrature nodes. In the special case of Gauss--Jacobi quadrature, this means that $$\int_{-1}^1 (1+x)^\alpha(1-x)^\beta f(x) dx = \sum_{j=0}^{n} w_j f(x_j), \quad \alpha, \beta > -1, $$ whenever $f(x)$ is a polynomial of degree at most $2n+1$. When $f$ is not a polynomial, but a function analytic in a neighbourhood of $[-1,1]$, the above is not an equality but an approximation that converges exponentially fast as $n$ is increased.

An undergraduate mathematician taking a numerical analysis course could tell you that the nodes $x_j$ are roots of the Jacobi polynomial $P^{\alpha,\beta}_{n+1}(x)$, the degree $n+1$ polynomial orthogonal with respect to the given weight, and that the quadrature weight at each node is related to the derivative $P'^{\alpha,\beta}_{n+1}(x_j)$. However, these values are not generally known in closed form, and we must compute them numerically... but how?

Traditional approaches involve applying the recurrence relation satisfied by the orthogonal polynomials, or solving the Jacobi matrix eigenvalue problem in the algorithm of Golub and Welsch, but these methods are inherently limited by a minimal complexity of $O(n^2)$. The current state-of-the-art is the $O(n)$ algorithm developed by Glasier, Liu, and Rokhlin, which hops from root to root using a Newton iteration evaluated with a Taylor series defined by the ODE satisfied by $P^{\alpha,\beta}_{n+1}$.

We propose an alternative approach, whereby the function and derivative evaluations required in the Newton iteration are computed independently in $O(1)$ operations per point using certain well-known asymptotic expansions. We shall review these expansions, outline the new algorithm, and demonstrate improvements in both accuracy and efficiency. 

We present some recent results concerning domain wall motion in one-dimensional nanowires, including the existence, velocity and stability of travelling-wave and precessing solutions.  We consider the case of unixial anisotropy, characteristic of wires with symmetrical (e.g., circular) cross-section, as opposed to strongly anisotropic geometries (films and strips) that have received greater attention.  This is joint work with Arseni Goussev and Valeriy Slastikov.

Actin polymerization in vivo is regulated spatially and temporally by a web of signalling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodelling actin meshes and bundles. In a recent work we studied how molecular motors regulate growth dynamics of elongated organelles of living cells. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers towards the tip, even the concentration bump of G-actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G-actin along the filopodium is rather non-trivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this non-monotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.

The reaction-diffusion master equation (RDME) is a popular model in systems biology. In the RDME, diffusion is modeled as discrete jumps between voxels in the computational domain. However, it has been demonstrated that a more fine-grained model is required to resolve all the dynamics of some highly diffusion-limited systems.

In Greenʼs Function Reaction Dynamics (GFRD), a method based on the Smoluchowski model, diffusion is modeled continuously in space.

This will be more accurate at fine scales, but also less efficient than a discrete-space model.

We have developed a hybrid method, combining the RDME and the GFRD method, making it possible to do the more expensive fine-grained simulations only for the species and in the parts of space where it is required in order to resolve all the dynamics, and more coarse-grained simulations where possible. We have applied this method to a model of a MAPK-pathway, and managed to reduce the number of molecules simulated with GFRD by orders of magnitude and without an appreciable loss of accuracy.

In this talk, I will introduce stochastic models to describe the state of the chemical networks using continuous-time Markov chains.
First, I will talk about the multiscale approximation method developed by Ball, Kurtz, Popovic, and Rempala (2006). Extending their method, we construct a general multiscale approximation in chemical reaction networks. We embed a stochastic model for a chemical reaction network into a family of models parameterized by a large parameter N. If reaction rate constants and species numbers vary over a wide range, we scale these numbers by powers of the parameter N. We develop a systematic approach to choose an appropriate set of scaling exponents. When the scaling suggests subnetworks have di erent time-scales, the subnetwork in each time scale is approximated by a limiting model involving a subset of reactions and species.

After that, I will briefly introduce Gaussian approximation using a central limit theorem, which gives a model with more detailed uctuations which may be not captured by the limiting models in multiscale approximations.

Next, we consider modeling of a chemical network with both reaction and diffusion.
We discretize the spatial domain into several computational cells and model diffusion as a reaction where the molecule of species in one computational cell moves to the neighboring one. In this case, the important question is how many computational cells we need to use for discretization to get balance between e ective diffusion rates and reaction rates both of which depend on the computational cell size. We derive a condition under which concentration of species converges to its uniform solution exponentially. Replacing a system domain size in this condition by computational cell size in our stochastic model, we derive an upper bound
for the computational cell size.

Finally, I will talk about stochastic reaction-diffusion models of pattern formation. Spatially distributed signals called morphogens influence gene expression that determines phenotype identity of cells. Generally, different cell types are segregated by boundary between
them determined by a threshold value of some state variables. Our question is how sensitive the location of the boundary to variation in parameters. We suggest a stochastic model for boundary determination using signaling schemes for patterning and investigate their effects on the variability of the boundary determination between cells.

Atherosclerosis is the leading cause of death, both above and below age 65, in the United States and all Western countries. Its earliest prelesion events appear to be the transmural (across the wall)-pressure (DP)-driven advection of large molecules such as low-density lipoprotein (LDL) cholesterol from the blood into the inner wall layers across the monolayer of endothelial cells that tile the blood-wall interface. This transport occurs through the junctions around rare (~one cell every few thousand) endothelial cells whose junctions are wide enough to allow large molecules to pass. These LDL molecules can bind to extracellular matrix (ECM) in the wall’s thin subendothelial intima (SI) layer and accumulate there. On the other hand, the overall transmural water flow can dilute the local intima LDL concentration, thereby slowing its kinetics of binding to ECM, and flushes unbound lipid from the wall. An understanding of the nature of this water flow is clearly critical.

            We have found that rat aortic endothelial cells express the ubiquitous membrane water-channel protein aquaporin-1 (AQP), and that blocking its water channel or knocking down its expression significantly reduces the apparent hydraulic conductivity Lp of the endothelium and, consequently of the entire wall. This decrease has an unexpected and strong DP -dependence. We present a fluid mechanics theory based on the premise that DP compacts the SI, which, as we show, lowers its Lp. The theory shows that blocking or knocking down AQP flow changes the critical DP at which this compaction occurs and explains our observed dependence of Lp on DP. Such compaction may affect lipid transport and accumulation in vivo. However, AQP’s sharp water selectivity gives rise to an oncotic paradox: the SI should quickly become hypotonic and shut down this AQP flow. The mass transfer problem resolve this paradox. The importance of aquaporin-based, rather than simply junctional water transport is that transport via protein channels allows for the possibility of active control of vessel Lp by up- or down-regulation of protein expression. We show that rat aortic endothelial cells significantly change their AQP numbers in response to chronic hypertension (high blood pressure), which may help explain the as yet poorly-understood fact that hypertension correlates with atherosclerosis. We also consider lowering AQP numbers as a strategy to affect disease progression.

Kernel functions are suitable tools for multivariate scattered data approximation. In this talk, we discuss the conditioning and stability of optimal reconstruction schemes from multivariate scattered data by using

(conditionally) positive definite kernel functions. Our discussion first provides basic Riesz-type stability estimates for the utilized reconstruction method, before particular focus is placed on upper and lower bounds of the Lebesgue constants.

If time allows, we will finally draw our attention to relevant aspects concerning the stability of penalized least squares approximation.

In many fields of science and engineering, such as fluid or structural mechanics and electric circuit design, large scale dynamical systems need to be simulated, optimized or controlled. They are often described by discretizations of systems of nonlinear partial differential equations yielding high-dimensional discrete phase spaces. For this reason, in recent decades, research has mainly focused on the development of sophisticated analytical and numerical tools to help understand the overall system behavior. During this time meshless methods have enjoyed significant interest in the research community and in some commercial simulators (e.g., LS-DYNA). In this talk I will describe a normalized-corrected meshless method which ensures linear completeness and improved accuracy. The resulting scheme not only provides first order consistency O(h) but also alleviates the particle deficiency (kernel support incompleteness) problem at the boundary. Furthermore, a number of improvements to the kernel derivative approximation are proposed.

To illustrate the performance of the meshless method, we present results for different problems from various fields of science and engineering (i.e. nano-tubes modelling, solid mechanics, damage mechanics, fluid mechanics, coupled interactions of solids and fluids). Special attention is paid to fluid flow in porous media. The primary attraction of the present approach is that it provides a weak formulation for Darcy's law which can be used in further development of meshless methods.

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. We introduce the ASAP algorithm, for the graph realization problem in R^d, given a sparse and noisy set of distance measurements associated to the edges of a globally rigid graph. ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The reflections and rotations are estimated using a recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. Furthermore, the algorithm successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose SP-ASAP, a faster version of ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that ASAP and SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms. Time permitting, we briefly discuss the analogy between the graph realization and the low-rank matrix completion problems, as well as an application of synchronization over Z_2 and its variations to bipartite multislice networks.

Mathematical models of chemotactic movement of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular signaling chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities that ar biologically unrealistic. Here we present a microscopic model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We show that this model permits travelling wave solutions and predicts the formation of other bacterial patterns such as radial and spiral streams. We also present connections of this microscopic model with macroscopic models of bacterial chemotaxis. This is joint work with Radek Erban, Benjamin Franz, Hyung Ju Hwang, and Kevin J.

Painter.

Many swimming microorganisms inhabit heterogeneous environments consisting of solid particles immersed in viscous fluid. Such environments require the organisms attempting to move through them to negotiate both hydrodynamic forces and geometric constraints. Here, we study this kind of locomotion by first observing the kinematics of the small nematode and model organism Caenorhabditis elegans in fluid-filled, micro-pillar arrays. We then compare its dynamics with those given by numerical simulations of a purely mechanical worm model that accounts only for the hydrodynamic and contact interactions with the obstacles. We demonstrate that these interactions allow simple undulators to achieve speeds as much as an order of magnitude greater than their free-swimming values. More generally, what appears as behavior and sensing can sometimes be explained through simple mechanics.

We will consider a simplified model for on-chip power distribution networks of array bonded integrated circuits. In this model the voltage is the solution of a Poisson equation in an infinite planar domain whose boundary is an array of circular or square pads of size $\epsilon$. We deal with the singular limit as $\epsilon\to 0$ and we are interested in deriving an explicit formula for the maximum voltage drop in the domain in terms of a power series in $\epsilon$. A procedure based on the method of matched asymptotic expansions will be presented to compute all the successive terms in the approximation, which can be interpreted as using multipole solutions of equations involving spatial derivatives of $\delta$-functions.

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this talk we discuss efficient, scalable techniques for solving fractional-in-space reaction diffusion equations combining the finite element method with robust techniques for computing the fractional power of a matrix times a vector. We shall demonstrate the methods on a number examples which show the qualitative difference in solution profiles between standard and fractional diffusion models.

Diffusive process with discontinuous coefficients provide significant computational challenges. We consider the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. Rather than seeking to construct discretizations that match the interface, we consider the use of regularly-shaped meshes so that the interface "cuts'' through the cells (elements or volumes). Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. We develop an adjoint based a posteriori error analysis technique to estimate the error in a given quantity of interest (functional of the solution). In order to employ this method, we first construct a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.

In biological cells, molecules are transported actively or by diffusion and react with each other when they are close.

The reactions occur with certain probability and there are few molecules of some chemical species. Therefore, a stochastic model is more accurate compared to a deterministic, macroscopic model for the concentrations based on partial differential equations.

At the mesoscopic level, the domain is partitioned into voxels or compartments. The molecules may react with other molecules in the same voxel and move between voxels by diffusion or active transport. At a finer, microscopic level, each individual molecule is tracked, it moves by Brownian motion and reacts with other molecules according to the Smoluchowski equation. The accuracy and efficiency of the simulations are improved by coupling the two levels and only using the micro model when it is necessary for the accuracy or when a meso description is unknown.

Algorithms for simulations with the mesoscopic, microscopic and meso-micro models will be described and applied to systems in molecular biology in three space dimensions.