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Timetable (CGPW04)

New mathematical and computational problems involved in cell motility, morphogenesis and pattern formation

Monday 7th December 2015 to Friday 11th December 2015

Monday 7th December 2015
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director) INI 1
10:00 to 11:00 Mark Chaplain
Mathematical modelling of angiogenesis in wounds, tumours and retinae: The good, the bad and the beautiful
Angiogenesis is the growth of a new network of blood vessels from a pre-existing vasculature. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration and proliferation; degradation of tissue; new capillary vessel (sprout) formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is blood flow associated with the nascent network, the subsequent growth of the network evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix and perfusion-related haemodynamic forces in a manner that may be described as both adaptive and dynamic. Angiogenesis is a vital component of both normal and pathological processes such as wound healing, solid tumour growth and retinal development.  

In this talk we will present a basic mathematical model for the development of a vascular network which simultaneously couples vessel growth with blood flow through the vessels - a dynamic adaptive vasculature model. We will then apply the model to three different biological scenarios: (i) tumour-induced angiogenesis; (ii) wound healing and (iii) the developing retina. The computational simulation results will be compared with experimental data and the predictions of the model discussed with regard to scheduling of the delivery of chemotherapy drugs to solid tumours.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Daphne Weihs
Making Holes: Identifying How Metastatic Cancer Cells Apply Force to Invade Their Microenvironment
The process of invasion is of special importance in cancer metastasis, the main cause of death in cancer patients. Cells typically penetrate a matrix by degrading it or by squeezing through pores. However, cell mechanics and forces applied by cells especially during the initial stages of metastatic penetration, as metastatic cells indent a substrate, are still unknown. We measure the forces that cells apply to an impenetrable, synthetic 2-dimensional gel-matrix, effectively limiting cells to rely only on mechanical-interactions; gels are non-degradable polyacrylamide with sub-micron pores. We show that single metastatic breast-cancer cells will apply force to an impenetrable gel, and indent it in attempted invasion, when the gel is in the appropriate stiffness range; benign cells do not indent the gels. The metastatic cells require gel-substrates to be soft enough to indent, yet stiff enough to grip and generate force on. Cells develop grip handles and pull the underlying gel s inwards and upwards bringing the nucleus into the indentation concavity. We reveal a special coordinated role for the nucleus and the cytoskeleton when a single cell attempts to invade the impenetrable barrier. The actin, nucleus, and microtubules reorganize in sequence, with the actin at the leading edge of the cell. Cells repeatedly attempt penetration over several hours and then relocate, indicating an advanced mechano-transduction feedback loop. We use finite element analysis to identify force application patterns to maximize indentations, by varying cell size, shape and the locations and magnitudes of the mechanical loads applied by cells. We demonstrate that cells must combine lateral forces and significant normal forces to achieve the large, experimentally observed gel-indentations. The systems and analysis approaches shown here reveal cell adaptability and force application mechanisms.

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12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:15 Stéphanou Angélique
An integrated computational approach for the design of patient-specific virtual tumours
The design of a patient-specific virtual tumour is an important step towards personalized medicine since the virtual tumour can be used to define the most adapted and efficient treatment protocol. However this requires to capture the description of many key events of tumour development, including angiogenesis, matrix remodelling, hypoxia, cell heterogeneity that will all influence the tumour growth kinetics and degree of tumour invasiveness. To that end, an integrated hybrid and multiscale approach has been developed based on data acquired on a preclinical mouse model as a proof of concept. Fluorescence imaging is exploited to build case-specific virtual tumours and to validate their spatiotemporal evolution. The validity of the model will be discussed as well as its potential to identify the best therapeutic strategy for each individual tumour case.
14:15 to 15:00 Amit Gefen
The structural integrity of cells under sustained mechanical deformations: The key for understanding pressure ulcers
Sustained internal mechanical loads in tissues which develop during immobile weight-bearing postures such as while in bed or in a chair were identified as a fundamental cause for the onset and progression of pressure ulcers (PUs), particularly of the deep tissue injury (DTI) type. The sustained loading may compromise tissue viability either directly, by distorting cell shapes, or indirectly, by distorting the vasculature or lymphatic networks or, at the micro-scale, by distorting cellular organelles involved in regulating transport, e.g. the plasma membrane (PM). This talk will review our record of published research concerning the effects of sustained deformations across the different hierarchical scales: tissue-scale [cm], meso-scale [mm] and cell-scale [μm], with a focus on how sustained bodyweight loads eventually compromise homeostasis and cell viability. The evolution of our work to test our central hypothesis will be shown, specifically, that macroscopic tissue defo rmations translate to cell-level deformations and in particular, to localized tensile strains in the plasma membrane (PM) of cells. These localized PM stretches increase the permeability of the PM which, over time, could disrupt vital transport processes such as the function of ion channels, endocytosis and exocytosis. Viability of tissues exposed to sustained loading in the context of pressure ulcer development should therefore be investigated in all dimensional scales, from the macro to micro, and in particular at a cell scale, in order to provide complete understanding of the aetiology of PUs and DTIs and for identifying individuals for whom and conditions at which the susceptibility to these injuries might be greater. Emerging relevant methods of cell permeability quantification and modeling such as multiscale and multiphysics modeling will be highlighted, as they contribute substantially to the aetiological research in this field.
15:00 to 15:30 Afternoon Tea
15:30 to 16:15 Dumitru Trucu
Structured Models of Cell Migration Incorporating Membrane Reactions
The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes enables a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this work we establish a general spatio-temporal-structural framework that enables the description of surface- bound reaction processes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with two concrete examples arising from cancer invasion.
16:15 to 17:00 Dagmar Iber
From Networks to Function – Computational Models of Organogenesis
One of the major challenges in biology concerns the integration of data across length and time scales into a consistent framework: how do macroscopic properties and functionalities arise from the molecular regulatory networks ­and how do they evolve? Morphogenesis provides an excellent model system to study how simple molecular networks robustly control complex pattern forming processes on the macroscopic scale in spite of molecular noise, and how important functional variants can evolve from small genetic changes. Recent advances in 3D imaging technologies, computer algorithms, and computer power now allow us to develop and analyse increasingly realistic models of biological control. In my talk, I will show how data-based modelling can be used to define mechanisms for fundamental developmental processes and I will discuss the computational challenges that arise.
17:00 to 18:00 Welcome Wine Reception & Poster Session
Tuesday 8th December 2015
09:00 to 10:00 Alan Champneys
Turing bifurcation, wave-pinning or localised patterns for cell polarity formation; three sides of the same coin?
In this talk I shall present recent work in collaboration with students Nicolas Verschuren and with Victor Brena motivated by problems of cellular level polarity formation motivated by a range of problems in plant biology. After reviewing some existing theories based on reaction-diffusion modelling, I will present some work on plant root hair formation in Arabidopsis in collaboration with Claire Grierson. By modelling the kinetics of the plant rho proteins, or ROPs, it will be argued that the key mechanism can be explained by the formation of a localised patch, which arises due to the presence of a subcritical Turing bifurcation and the recent theory of so-called homoclinic snaking. To see what happens in wild type, one needs to include spatial gradients, such that the dynamics of the patch can be explained asymptotically with the help of Michael Ward's semi-strong analysis technique. The mechanism is contrasted with that of the recent theory of wave pinning in mass-conservative reaction-diffusion equations. It is argued that small source and loss terms are biologically motivated by actions of the nucleus controlling the process and by proteins being recycled as symmetry-breaking takes hold. A new study is then undertaken of what happens under introduction of small source and loss terms to a canonical wave-pinning model. It is shown that localised patterns develop into snakes in one limit and in other limit develop into pinned fronts. A new asymptotic analysis shows how front selection occurs in the limit that the source and loss terms tend to zero.
10:00 to 11:00 Chandrasekhar Venkataraman
Free Boundary Problems from a Model for Receptor-Ligand Dynamics
Co-authors: Charles Elliott (Warwick), Thomas Ranner (Leeds)

We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling that models receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the ECM.

We prove the existence and uniqueness of a solution to the model and we also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to the limiting problems, which take the form of free boundary problems posed on the cell surface. We also present numerical simulations in a realistic geometry.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Fred Vermolen
Cell-Based Modelling of Wound Contraction, the Immune System and Angiogenesis
Wound contraction and angiogenesis are biological processes that often take place during healing of wounds and in tumor development. To model these processes, one distinguishes between different types of models, which are descriptive at several scales, ranging from cellular scale (micro-scale) to the tissue scale (macro-scale). The models are on the macro-scale are based on continuum hypotheses, which means that one sets up and solves partial differential equations with the associated boundary and initial conditions. On the smallest scale one models all kinds of cell phenomena on a molecular level. In this talk, we will consider colonies of cells, which are treated as discrete entities, as well as chemical and mechanical signals that are modelled as sets of partial differential equations. Hence, the current approach is a hybride one.

The process of angiogenesis, which is the formation of a vascular network in tissues, is often modeled by using principles based on cell densities in a continuum approach or on hybride cellular-continuum level where one uses cellular automata (in particular cellular Potts) models. In this study, we abandon the lattice needed to model the cell positions in cellular automata modelling and instead, we apply a continuous cell-based approach to simulate three-dimensional angiogenesis. Next to the application of this modelling strategy to angiogenesis, we discuss the application of the formalism to wound contraction.
Next to angiogenesis, a cell deformation and migration model will be presented, where the cell boundary, as well as the boundary of the nucleus is divided into a set of discrete points. Cell migration is modelled in terms of random walk and chemotaxis. The deformation of the nucleus is a novel step in literature.

The talk will describe some of the mathematical issues encountered in these models and further some animations will be shown to illustrate the potential merits of our approaches.
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:15 Alexander Hunt
DTI-Based Multiscale Modelling of Glioma
Co-Authors: C. Surulescu (TU Kaiserslautern), C. Engwer (WWU Munster)

Glioma invasion is a multiscale process ranging from subcellular biolo- gical events to the growth of a solid tumour mass. Key features involved in this process are migration and proliferation. The former is modelled using kinetic transport equations including medical data, which reveal the brain structure in detail. For characterising proliferation we use two alternative approaches: one relying on the go-or-growth dichotomy and involving two tumour cell populations, and the other paying attention to cell-tissue interactions seen as the onset of the biological processes leading to cell division.

For the biomedical application it is interesting to also include some therapy approach. We propose a new model combining receptor binding inhibition with radiation therapy and perform numerical simulations for the di
erent settings.
14:15 to 15:00 Nikolaos Sfakianakis
Filament Based Lamellipodium Model (FBLM) modeling and numerical simulations
The cytoskeleton is a cellular skeleton inside the cytoplasm of living cells. The front of the cytoskeleton, also known as lamellipodium and is the driving mechanism of cell motility and is comprised by long double helix polymers of actin protein termed actin-filaments. The actin-filaments polymerize/depolymerize and exhibit a series of physical properties like elasticity, friction with the substrate, crosslink binding, repulsion, myosin-drive contractility, nucleation, fragmentation, capping and more.

In this talk we address the FBLM that describes the above (microspcopic) dynamics of the actin-filaments and results to the (macroscopic) movement of the cell. We introduce the Finite Element Method (FEM) used to simulate this system and present numerical experiments exhibiting the motility of the cells in a series biological scenaria (including chemotactic and haptotactic influence) and compare our results with on-vitro experiments.

Joint work(-s) with Chr. Schmeiser, D. Oelz, A. Manhart, V. Small

15:00 to 15:30 Afternoon Tea
15:30 to 16:15 Natalie Emken
A continuous reaction-diffusion-advection model for the establishment of actin-mediated polarity in yeast
Co-author: Prof. Dr. Christian Engwer (Institute for Computational and Applied Mathematics, University of Muenster)
Cell polarity plays a crucial role for many different cell types. In the yeast cell, the model system to study the underlying mechanisms of polarization, the GTPase Cdc42 is a key regulator of this process. Its clustering relies on multiple parallel acting mechanisms. A common model explains polarity by a Turing-type mechanism. Based on reaction-diffusion equations it simulates a Bem1-mediated Cdc42 recruitment. Since cell polarity occurs even in the absence of Bem1, recent papers emphasize the exchange between the cytosol and the plasma membrane. However, studies combining biological experiments and mathematical simulations also suggest an actin-mediated feedback of Cdc42. Stochastic vesicle trafficking models demonstrate that transport of Cdc42 via actin cables can either reinforce or perturb polarization . We present a minimal mathematical model, based on reaction-diffusion-advection equations, that is able to reproduce the experimentally observed phenomena, in particular those of knock-down experiments. Contrary to former approaches which only incorporate the diffusive transport, our system explicitly simulates exocytosis and endocytosis of Cdc42. Vesicles move along actin cables, thus we further consider actin polymerization and depolymerization. Since we consider five substances, either cytosolic or membrane-bound, and model the full geometry we have a coupled bulk-surface problem. We present numerical results in 3D and compare those to experimental data. This way, we show that the model is able to reproduce experimentally observed pathological cases and demonstrate how vesicle transport could reinforce polarity. Based on this specific model, we develop a general system of three membrane reaction-diffusion equations coupled to two diffusion equations inside the cell. We perform a linearized stability analysis and derive conditions for a transport-mediated instability. We complete our theoretical analysis by numerical simulations for different geometries.
16:15 to 17:00 Rachel Bearon
Continuum models for motile cells in shear flow
Many micro-organisms such as bacteria and algae swim in fluid environments. This swimming behaviour can interact with fluid motions to generate transport which differs both from that experienced by passive tracers in flow and micro-swimmers in the absence of flow. I will give examples of how population-level models can be derived to describe the spatio-temporal distribution of such swimmers, including a model for slender bacteria which undergo run-and-tumble chemotaxis in a channel (Bearon et al J. Fluid Mech. 2015). The continuum model developed can describe an experimentally observed phenomenon of trapping in high shear which existing drift-diffusion models are unable to capture.
Wednesday 9th December 2015
09:00 to 10:00 Veronica Grieneisen
Multilevel approach to cell and tissue polarity and traffic jams
In this talk I wish to compare and contrast cell and tissue polarity between very diverse organisms. Computational approaches combined with molecular studies and in vivo microscopy were necessary to reveal how polarity is coordinated and linked on three different levels: on the scale of the tissue, the cellular and subcellular tissue level. At the single cell level, a spatially uniform activation and patterning of GTPases can cause polarity to emerge spontaneously, independent of spatial pre-patterns or localized polarizing signals. We argue that plants and animals have inherited this same “unicellular mode” of establishing cell polarity, and that multicellular coordination has thereafter diverged using this underlying mechanism as a building block: Being capable of intracellular partitioning, neighbouring plant cells that are separated by cell wall then coordinate their polarities - through indirect cell-cell coupling. This is resultant from changes in concentration level of a phytohormone, auxin, inbetween and along cells.

In the specific case of pavement cells of leaves (jigsaw-shaped cells with interlocking lobes and indentations), this phenomenon comes about as interdigitation, and requires the opposite response of identical neighbouring cells to the same local auxin signal in the cell wall, between the cells. Our theoretical work identifies key requirements for such indirect cell-cell signalling that that gives rise to correct interdigitation. These requirements, based on known molecular interactions, can then be extrapolated to other multi-cellular tissues, to understand the interdependency between cell and tissue polarity.

Extrapolating these findings we further show how animal cells, capable of direct cell-cell coupling, can establish, through similar principles, robust tissue coordination. In the end of our talk, I will also show how established tissue polarity in plants requires extra conditions of regulation, to avoid issues of traffic jam in relation to nutrient uptake.

10:00 to 11:00 Yasin Dagdas
Membrane remodeling and cellular morphogenesis during plant tissue colonization
One of the biggest challenges of 21st century is feeding the growing human population. Plant pathogens cause devastating yield losses in staple crops and pose a serious threat to global food security. According to recent reports, plant pathogens cause crop losses that if alleviated would feed at least 700 million people. To prevent crop loses due to pathogens, we have to understand plant-microbe interactions at molecular and systems level. To facilitate plant colonization these deadly microbes evolved unique infection strategies. They form specialized infection cells that involve tightly controlled spatiotemporal repolarization events. Additionally, pathogens also induce extensive membrane remodeling within plant tissues. They initiate plant infection via invagination of plant plasma membrane and reorient cellular resources to these infection cells. They can occupy up to 80% of plant cell volume without inducing any immune response. We have limited knowledge on the molecular details of infection cell morphogenesis and cellular reprogramming during plant infection. I will present our recent results on rice blast and Irish potato famine pathogen and propose research questions that can be answered using mathematical modeling.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Angela Stevens
Stochastic particle models and chemotactic/haptotactic motion of cells
Co-authors: Stefan Grosskinsky (University of Warwick), Daniel Marahrens (formerly MPI MIS Leipzig), Juan Velazquez (University of Bonn)

In this talk the first equation within a class of well known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites. They interact directly among themselves only on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically the limiting procedure and its proofs are based on results by Koukkous and Kipnis/Landim. Further PDE-ODE based haptotaxis models are discussed and their relation to attractive reinforced random walks.
12:30 to 13:30 Lunch at INI
13:30 to 13:45 Welcome & Introduction - John Toland (Isaac Newton Institute) & Jane Leeks (Turing Gateway to Mathematics) INI 1
13:45 to 14:30 Mark Chaplain
Case Studies in Cancer Modelling
14:30 to 15:00 Neil Dalchau
Systems Biology Research at Microsoft
15:00 to 15:20 Afternoon Tea
15:20 to 15:50 Jamie Meredith
Grand Challenge and Other Funding Opportunities at Cancer Research UK
15:50 to 16:20 Jonathan Stott
Quantitative Biology Research at Astra Zeneca
16:20 to 17:00 Open Discussion INI 1
17:00 to 18:00 Wine Reception/Networking
19:30 to 22:00 Conference Dinner at Christ's College
Thursday 10th December 2015
09:00 to 10:00 Katarina Wolf
Principles of single and collective cancer cell migration: Impact by environmental substrate guidance
The migration of single cells or cell collectives in the multicellular organism is a complex process that takes place during a wide range of physiological functions in the organism, as well as during disease, such as cancer. Cell migration takes mainly place within complex extracellular matrix (ECM) environments of different dimensionality, structure, and spacing. Hence basic concepts on cell migration have been extended from 2D into the 3D context. Examples will be shown on how mesenchymal, amoeboid or collective migration modes are maintained or modulated by the tissue architecture in vitro as well as in vivo.
10:00 to 11:00 Luigi Preziosi
Multiscale modelling for cell motility
Several problems regarding cell motility and morphogenesis are characterized by the contemporary presence of cells that behave as single entities and cells that follow and cluster around them. From the mathematical point of view, describing such phenomena requires the development of mathematical models in which virtual cells can switch from a discrete to a continuous description. Keeping this in mind, the talk aims at presenting some ideas on how to do that, making for instance use of measure theory or introducing the concept of bubble functions.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Pierre Degond
Modelling of cross-linked fiber networks and tissue self-organization
In this talk, we will derive a continuum model for the dynamics of a network of cross-linked fibers. We will outline how this model can be applied to the modelling of tissue self-organization.
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:15 Niklas Kolbe
Mathematical modelling and simulation of an Epithelial-Mesenchymal-like transition in cancer cells
Recent biological work has revealed the existence of cells within the body of a tumour that differ in their level of differentiation from the bulk of the cancer cells. Compared to the more usual differentiated cancer cells, these cancer stem cells exhibit higher motility, they are more resilient to therapy, and are able to metastasise to secondary locations within the organism. They seem to transition from the differentiated cancer cells via a (de-)differentiation program, termed Epithelial-Mesenchymal Transition, which can also be found in normal tissue. The compound of the tumour as well as its internal dynamics affect the extracellular environment, in particular the invasion of the Extracellular Matrix.  

In this talk we introduce a model that combines the transition between the afore-mentioned types of cancer cells based on the (microscopic) dynamics of the Epidermal Growth Factors, with the (macroscopic) invasion of the Extracellular Matrix by the cancer cell ensemble. Moreover, we present numerical experiments exhibiting the dynamics of both types of cancer cells and elaborate on the numerical methods that we use.

[1] N. Hellmann, N. Kolbe, N. Sfakianakis: A mathematical insight in the epithelial–mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix, Bull Braz Math Soc (2016).  

[2] N. Kolbe, J. Katuchova, N. Sfakianakis, N. Hellmann, M. Lukacova: A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model, Appl Math Comp (2015).
14:15 to 15:00 Pasquale Ciarletta
Chemo-mechanical modeling of morphogenesis in living matter
Life phenomena result from the mutual equilibrium between the living matter and the surrounding media. A network of servo-mechanisms physiologically restores the stable equilibrium between the interior matter of a living entity in the face of external perturbative agents. In particular, living cells can balance exogenous and endogenous forces using an iterative process, also known as mechano-reciprocity. Hence, not only living matter can adapt through epigenetic remodelling to the external physical cues, but it can also respond by activating gene regulatory processes, which may also drive the onset of pathologies, e.g. solid tumours. Moreover, living materials have the striking ability to change actively their micro-structural organization in order to adjust their functions to the surrounding media, developing a state of internal tension, which even persists after the removal of any external loading. This complex mechanical and biochemical interaction can finally control morp hogenesis during growth and remodelling, leading to shape instabilities characterized by a complex morphological phase diagram. In this lecture, I will introduce few mathematical s of mechanobiology and morphogenesis in living materials [1,2], with several applications concerning solid tumours [3], gastro-intestinal organogenesis [4], bacterial colonies [5] and nerve fibers [6]. [1] Ciarletta P, Ambrosi D, Maugin G A, Preziosi L. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER, 2013, 36, 23-28. [2] Ciarletta P, Preziosi L, Maugin GA. JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, 2013, 61, 852-872; [3] Ciarletta P. Buckling instability in growing tumour spheroids. PHYSICAL REVIEW LETTERS, 2013, 110. [4] Ciarletta P., Balbi V., Kuhl, E. Pattern selection in growing tubular tissues. PHYSICAL REVIEW LETTERS, 2014, 113, 248101. [5] Giverso, C., Verani M., Ciarletta P. JOURNAL OF THE ROYAL SOCIETY INTERFACE, 2015, 12 [6] Taffetani M., Ciarletta P, PHYSICAL REVIEW E, 2015,91
15:00 to 15:30 Afternoon Tea
15:30 to 16:15 Pia Domschke
Mathematical modelling of cancer invasion: The role of cell adhesion variability
Co-authors: Dumitru Trucu (University of Dundee), Alf Gerisch (TU Darmstadt), Mark Chaplain (University of St Andrews)

Cancer invasion is a complex process occurring across several spatial and temporal scales, perhaps the three most important being the intracellular, cellular and tissue scales. Key biological processes occurring during invasion are the secretion of matrix degrading enzymes, cell proliferation, the loss of cell-cell adhesion on one hand and enhanced cell-matrix adhesion on the other hand, as well as active migration. The ability of cancer cells to alter or degrade the surrounding tissue enables the cancer cells to locally invade the neighbouring region. The movement of cancer cells occurs through chemotaxis as well as haptotaxis and is supported by the binding and unbinding of molecules on the cell surface to other cells and/or the extracellular matrix (ECM). The number and strength of these binding proteins define the magnitude of cell-cell and cell-matrix adhesion and are modified by the cell’s microenvironment. Hence, the movement of the cells is not only determined l ocally but depends on the neighbourhood of the cell.

We explore the spatio-temporal dynamics of a mathematical model of cancer invasion, where cell-cell and cell-matrix adhesion are accounted for through non-local interaction terms. A non-local model of cancer invasion for a single cancer cell population is extended to a structured-population model with n cancer cell sub-populations, which may mutate into each other. The change of adhesion properties during the growth of the cancer is investigated through time-dependent adhesion parameters within the cancer cell sub-populations as well as those between the cancer cells and the components of the extracellular matrix. We focus on one and two cancer cell sub-population models in two spatial dimensions, which show heterogeneous dynamics in our computational simulation results.
16:15 to 17:00 Poster Session
Friday 11th December 2015
09:00 to 10:00 Jose A. Carrillo
Swarming Models with Repulsive-Attractive Effects: Pattern Stability
I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
10:00 to 11:00 Nicholas Hill
Multiscale modelling of pressure and flow in the pulmonary circulation
A multiscale computational model has been developed to predict flow and pressure in the pulmonary circulation, in which the flow and pressure in the smaller blood vessels are described using linearised equations in pairs of asymmetric structured trees joined at the roots. The geometric and elastic properties of all the blood vessels are described by physiological parameters. Magnetic resonance imaging (MRI) is used to determine the geometry of the large pulmonary arteries and veins, and to measure the cardiac output from the right ventricle. The flow in the large blood vessels is solved using a Lax--Wendroff scheme, and the admittances of the structured trees provide the boundary conditions linking each large artery to its respective large vein. The model predicts flow and pressure in both the large and small vessels down to 50 microns in radius, providing important data about the local physiological environment experienced by tissues and cells.

The results of simulating various pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment of circulatory diseases within the lung. We use wave intensity analysis to study the propagation of forward and backward, compression and decompression waves in our model. The approximations for the pulse wave velocity used in experiments on wave intensity analysis are assessed, and reflected waves lower the peak pressure in the right ventricle.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Alf Gerisch
Nonlocal models for interaction driven cell movement
Authors: A. Gerisch, K.J. Painter, J.M. Bloomfield, J.A. Sherratt    

Instructing others to move is fundamental for many populations, whether animal or cellular. Often such commands are transmitted by contact, such that an instruction is relayed directly from signaller to receiver: for cells, this can occur via receptor–ligand mediated interactions at their membranes, potentially at a distance if a cell extends long filopodia. Given that commands ranging from attractive to repelling can be transmitted over variable distances and between cells of the same (homotypic) or different (heterotypic) type, these mechanisms can clearly have a significant impact on the organisation of a tissue.   In this talk we consider nonlocal models based on integro-PDEs to model contact based cell movement. We describe some specific applications and highlight the mathematical and numerical challenges that the models present.
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:15 Arik Yochelis
Pattern formation by molecular motors in cellular protrusions
Co-authors: S. Ebrahim (NIH, US), B. Millis (NIH, US), R. Cui (NIH, US), B. Kachar (NIH, US), M. Naoz (Weizmann Institute of Science, Israel), N. S. Gov (Weizmann Institute of Science, Israel)

Actin-based cellular protrusions are an ubiquitous feature of cells, performing a variety of critical functions ranging from cell-cell communication to cell motility. The formation and maintenance of these protrusions relies on the transport of proteins via myosin motors, to the protrusion tip. While tip-directed motion leads to accumulation of motors (and their molecular cargo) at the protrusion tip, it is observed that motors also form rearward moving, periodic and isolated aggregates. Not only that these aggregates are apparently important to the recycling of the motors but also their origins and mechanisms are open puzzles. Motivated by novel experiments, a mass conserving nonlinear reaction-diffusion-advection model is proposed. Analysis of the model provides insights into pattern selection mechanisms, i.e., how local and global bifurcations, and boundary conditions lead to emergence of pulses and traveling waves. These pattern selection mechanisms are found not only to qualitatively agree with empirical observations but open new vistas to the transport phenomena by molecular motors in general.

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14:15 to 15:00 Christian Mazza
Self-organization and pattern formation in auxin flux
The plant hormone auxin is instrumental for plant growth and morphogenesis. In the shoot apical meristem , the auxin flux is polarized through its interplay with PIN proteins. Concentration based mathematical models of the auxin flux permit to explain some aspects of phyllotaxis , where auxin accumulation points act as auxin sinks and correspond to primordia. Simulations show that these models can reproduce geometrically regular patterns like spirals in sunflowers or Fibonacci numbers. We propose a mathematical study of a related non-linear o.d.e. using Markov chain theory. We will next consider a concurrent model which is based on the so-called flux hypothesis, and show that it can explain the self-organization of plant vascular systems.

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15:00 to 15:30 Afternoon Tea
15:30 to 16:15 Jocelyn Etienne
Cells and embryos as flowing shells: analytical and numerical approaches for viscoelastic liquid shells
Actomyosin networks are known to be much denser at external surfaces of cells and early embryos than in their bulk. They are also known to be the major mechanical element allowing the cell to maintain its shape and governing its dynamics: myosin molecular motors convert biochemical energy into mechanical action, which can resolve in increased tension or deformation depending on boundary conditions [1].

Similarly, during early morphogenesis of embryos, actomyosin forms a surfacic continuum, seamed at cell-cell boundaries by so called adherens junctions, over a thikness of less than a micron at the outer surface of the 50 -micron ellipsoidal embryo. Gene expression is known to lead to successive patternings of myosin density within this actomyosin continuum, which in turn is necessary for the large morphogenetic movements of early embryogenesis to occur. However, while we know that such a myosin patternings are causal, the mechanism by which they govern the correct morphogenetic flows remains unclear. Decyphering it necessitates to resolve the mechanical balance of the embryo with the myosin force-production as a source term.

After presenting the general problem in a closed form suitable for mathematical analysis, I will present three particular cases:
- Single cells in a liquid bridge-like geometry, allowing a partial analytical resolution of the viscoelastic mechanical problem.
- Ventral furrow formation of the Drosophila embryo, for which elasticity approaches are possible at short times.
- The surface flow during germ-band extension of Drosophila, for which we have developped a new surface finite element technique allowing us to solve compressible Stokes-like problems in which velocities are tangential to a curved surface.

[1] J. Étienne, J. Fouchard, D. Mitrossilis, N. Bufi, P. Durand-Smet and A. Asnacios, 2015. Cells as liquid motors: Mechanosensitivity emerges from collective dynamics of actomyosin cortex. Proc. Natl. Acad. Sci. USA 112(9):2740–2745.

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16:15 to 17:00 Stephen Watson
Emergent Parabolic Scaling of Nano-Faceting Crystal Growth
Nano-faceted crystals answer the call for self-assembled, physico-chemically tailored materials, with those arising from a kinetically mediated response to free-energy dis-equilibria (thermokinetics) holding the greatest promise. The dynamics of slightly undercooled crystal-melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment-detachment limited kinetics offer a model system for the study of thermokinetic effects. The fundamental non-equilibrium feature of this dynamics is explicated through our discovery of 1D convex- and concave- translating fronts ( solitons) whose constant asymptotic angles provably deviate from the thermodynamically expected Wulff angles in direct proportion to the degree of undercooling. These thermokinetic solitons induce a novel emergent facet dynamics, which is exactly characterised via an original geometric matched-asymptotic analysis. We thereby discover an emergent parabolic symmetry of its coarsening facet ensembles, which naturally implies the universal scaling law L ~ t^{1/2} for the growth in time t of the characteristic length L .

Related Links
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons