HPC-Europa2 TAM 2009 Abstracts
Bruno Koobus, Bijan Mohammadi, Frank Nicoud
Institut de Mathematiques et de Modellisation de Montpellier (I3M)
We describe a few examples of how HPC can impact applied mathematics research for engineering, medical and environmental applications. Examples which will be presented range from biomedical fluid dynamics to littoral evolution analysis and aeronautics. Algorithmic and implementation issues will be discussed and some theoretical sub-optimal choices will be justified from the point of view of HPC. The talk ends with some remarks on current researches at the institute on future algorithms for HPC on GPUs.
Gabriel Staffelbach (CERFACS)
Combustion process is at the root of most energy production systems. The giant leaps performed in computer science have allowed to use simulation to better our understanding of combustion even in complex cases such as piston engines, or rocket engines. This presentation focuses on the use of the AVBP code applied to the simulation of industrial cases. It discusses and illustrates the application of high performance computing for Computational Fluid Dynamics (CFD) ifor aeronautical turbines f. Special attention is given to unsteady dangerous behaviours of flames which these combustors often exhibit ( failed ignition, high-level vibrations, wall burn out).
Spezia Riccardo (CNRS - Université d'Evry)
Light-harvesting (LH) complexes are used by photosynthetic organisms to increase the overall efficiency of photosynthesis. This is accomplished by harvesting light energy and funneling it to the reaction center, where it is converted into electrochemical potential. Dinoflagellates, unicellular algae constituting one of the most important classes of phytoplankton, use a water-soluble LH complex called peridinin-chlorophyll-a-protein (PCP) with a 4:1 peridinin/chlorophyll ratio. The presence of peridinin molecules in the PCPs enables the organism to collect light in the visible spectral region where chlorophyll poorly absorbs. The peridinins of PCPs are also able to play a photoprotective role by efficient quenching of the chlorophyll triplet states, which may occasionally be populated, thus preventing the formation of the highly toxic singlet oxygen. Infrared spectroscopy, and in particular time-resolved IR difference spectroscopy is a well-established technique which has been successfully used to investigate photophysical phenomena and photochemical reactions taking place in photosynthetic reaction center and LH complexes. This technique allows reaction-induced changes in both the protein and the cofactors to be monitored. In both static and time-resolved IR band assignment remains a difficult task. Vibrational frequencies can be obtained from theoretical calculations performed at molecular level such that the band assignment can be subsequently performed. In studying IR signal a key signature is provided by the carbonyl function of the lactone ring. In particular the effect of different protein environments of the four peridinins present in PCP complex, was evocated to understand IR spectroscopy . Based on this motivation, a set of IR and Resonance Raman experiments is going to be performed by Dr. A.Mezzetti of the University of Lille (France) to understand environmental effects on vibrational properties of peridinin. At this end three prototypical solvents were used in simulations: 1) an apolar/aprotic solvent, like cyclohexane; 2) a polar/aprotic solvent, like deuterated acetonitrile; 3) a polar/protic solvent like ethanol. Ab-initio molecular dynamics should help for the band assignment that is a particularly important task to rationalize all those experiments. At this end we set-up a mixed QM/MM approach where the peridinin is treated at DFT level while the different solvents at a classical level. This is achieved by using the CPMD package coupled with Gromos96. The active collaboration with prof. L. Guidoni of University of Lâ€™Aquila (former at University of Rome) with the HPC fellowship was fundamental for the realization of the project. We have performed simulations in different solvents and vibrational spectra of key functions show the effect of different solute-solvent interactions at a microscopic level.
 Mezzetti A. and Spezia, R. 2008. Spectroscopy: Int. J., 22, 235-250.
Postek Eligiusz (University of Sheffield)
The cytoskeleton (CSK) can be modelled as a tensegrity structure . The role of the CSK is continuously discovered . The developed computer code can serve as a physical module of an agent based model of the epithelial tissue . In other words, it should solve the mechanical problem for an assumed number and configuration of the cells. The mechanical problem is formulated in the Updated Lagrangian formulation and valid for large deformations. The tensegrity structures cannot exist without prestressing. They are mechanisms without prestressing forces. This implies nonlinearity under applied external load. Our structural problem is solved using the incremental solution with Newton-Raphson technique. The single cell will be modelled by an elementary tensegrity structure derived from icosahedron. It consists of 24 tendons and 6 struts. When testing, the structure performs stiffening effect in tension. Having the elementary structure of a cell we can model different cell assemblies. The cell assemblies can stand for a piece of tissue. The design sensitivity analysis is used for calculation of the design sensitivity derivatives, ie, derivatives of the state variables with respect to the design parameters. In our case, the design parameters are the cross-sectional areas and Youngâ€™s moduli in particular bars. The special case of the design parameters is, for example, the cross-sectional areas of the bars in entire single cell. The latter can tell us how the properties of entire cell influence the constrained displacements and stresses. We will employ direct differentiation method to calculate the DS gradients. It requires solving at each incremental step as many additional right hand sides as the design variables. The original feature of the program is the design sensitivity analysis of nonlinear systems with the local derivatives calculated analytically. The derivatives are calculated on two scales, firstly on the element level (tendons and struts) and secondly they are assembled to get the fields (displacements, stresses) derivatives with respect to the properties of entire cell. Further analysis, having already tested program as described above will serve for the analysis of the epitheliome tissue. The other problems in the scope of interest are the influence of the different imperfections, like, for example, additional stresses imposed on the tissue when cells divide. It will be possible to include the program into the framework of the agent module as well.
 Stamenovic D., 2005, Effects of cytoskeletal prestress on cell rheological behaviour, 1, Acta Biomaterialia, pp. 255-262.
 Ainsworth C., 2008, â€œStretching the imaginationâ€, Nature, 456, 1 Dec, pp. 696-699.
 Pogson, M.; Smallwood, R.; Qwarnstrom, E.; Holcombe, M., 2006, â€œFormal agent-based modelling of intracellular chemical interactionsâ€, 85, Biosystems, pp. 37-45.
Alvioli Massimiliano (University of Perugia)
In the past decade measurements utilizing high energy electron-nucleus
scattering at Thomas Jefferson Laboratory (JLab), have identified two-nucleon Short Range Correlations (2N-SRC), studied their structure and related them to the underlying basic short range Nucleon-Nucleon (NN) interaction. When the distance between the two nucleon centres is about 1 fm, the local nuclear matter density of the SRC-pair is several times larger than that of the central density in nuclei and comparable to that expected in neutron stars. SRC-pairs in nuclei are therefore a form of cold dense nuclear matter that can be approached and studied in the laboratory; SRCs become important in atomic nuclei when high densities are locally reached. The significant value of SRCs observed recently could be important to understand the physics of neutron stars, in spite of the small probability of SRCs in an netron-neutron system. As a matter of fact, a small concentration of protons inside neutron stars is compensated to large extent by a significantly larger probability of SRCs in proton-neutron systems as compared to neutron-neutron systems (a Â factor of about 18). Moreover, future investigations of SRCs may help to elucidate the possible role of quark transitions between different nucleons.
Cosenza Biagio (Università degli Studi di Salerno)
Rendering algorithms range from off-line, computationally costly, physically-based, realistic techniques, to fast, approximate and plausible interactive ones. Ray tracing is a popular rendering technique aiming for high realism, and is a basis for global illumination algorithms. Interactive ray tracing has seen enormous progress in recent years. Highly optimized packet-based ray tracing implementations allow the computation of millions of ray-triangle intersection per second, and fully exploit modern multi-core CPUs, or GPUs. However, complex scenes and lighting, and high-quality renderings with anti-aliasing are still not feasible at interactive speed, and only possible when using compute clusters. In these scenarios, good load balancing is crucial in order to exploit the computational power, and not to suffer from communication overhead and synchronization barriers. We introduce a Parallel Ray Tracing architecture that exploits several parallelization techniques in order to reach interactive performance. First, it exploits SIMD vector instructions and more coherent memory accesses by tracing more rays at once (packet of rays). Second, we take advantage of the multi core availability by using multiple threads and scheduling packets of rays to different threads. Finally, we afford the distributed memory parallelization by using several techniques. In particular, we introduce a method, which uses a cheap GPU rendering technique to compute a cost map: An estimation of the per-pixel cost when rendering the image using ray tracing. Using this information, we improve load balancing, task scheduling, and work stealing strategies.
Tiribocchi Adriano (University of Bari)
A numerical study about nematic cells will be presented. In particular we focus our attention on a two-domain hybridly aligned nematic device that shows unexpected bi-stability properties. The flow properties are analyzed by solving the continuity, the Navier-Stokes and the order parameter equations. A hybrid approach, that consists of coupling a lattice Boltzmann algorithm, necessary for solving the Navier-Stokes and the continuity equations, with a finite-difference scheme, used to solve the equation for the order parameter, is employed. Moreover the switching dynamics in the presence of an external electric field is studied, which shows an unexpected bistable behavior. An analysis about the relations between the topological defects and the two-domain structure will be discussed.
Meliani Zakaria (Katholieke Universiteit Leuven)
Transverse stratification is a common intrinsic feature of astrophysical jets. There is growing evidence that jets in radio galaxies consist of a fast low density outflow at the jet axis, surrounded by a slower, denser, extended jet. The inner and outer jet components then have a different origin and launching mechanism, making their magnetization, associated energy flux and angular momentum content different as well. Their interface will develop differential rotation, where disruptions may occur. We here investigate the stability of magnetized, rotating, two-component relativistic outflows typical for jets in radio galaxies. For this purpose, we parametrically explore the long term evolution of radially stratified jets numerically. With grid-adaptive relativistic magnetohydrodynamic simulations, augmented with approximate linear stability analysis, we revisit the interaction between the two jet components. We study the influence of dynamically important magnetic fields, with varying contributions of the inner component jet to the total kinetic energy flux of the jet, on their non-linear azimuthal stability. We demonstrate that two-component jets with high kinetic energy flux, and an inner jet magnetization which is lower than the external jet magnetization are subject to the development of a relativistically enhanced, rotation-induced Rayleigh-Taylor type instability. This instability appears to play a major role in decelerating the inner jet and the overall jet decollimation. This novel deceleration scenario can partly explain the radio source dichotomy, relating it directly to the efficiency of the central engine in launching the inner jet component. The FRII/FRI transition could then occur when the relative kinetic energy flux of the inner to the external jet grows beyond a certain treshold.
Cheptsov Alexey (HLRS - University of Stuttgart)
OPATM-BFM is a MPI-parallel physical-biogeochemical simulation model practically applied for short-term forecasts of key biogeochemical variables (e.g. chlorophyll, salinity and other) for a wide range of coastal areas, among others Mediterranean Sea. Obtaining maximal performance (in terms of the execution time) and scalability (in terms of speed-up gained due to running on the increasing number of computing nodes) is mandatory for OPATM-BFMâ€™s practical usability for tasks of the real complexity. Moreover, the application poses great challenge for different HPC architectures with regard to both optimal utilization of resources for performing the identified complicated tasks of environmental simulation and development of algorithms enabling such an efficient usage. In this work, we developed a hybrid OpenMP+MPI communication pattern that allowed us to improve parallel efficiency of the OPATM-BFM as compared to the current pure MPI implementation and increase the OPATM-BFM usability for a wide range of practical tasks of the environmental simulation.
Schabauer Hannes (University of Vienna)
We discuss a method for efficiently solving a complex symmetric (non-Hermitian) generalized EVP $Ax = \\lambda Bx$ in parallel. Although non-Hermitian EVPs do not occur as frequently in practice as real symmetric or complex Hermitian problems, there are many important applications where they arise. In particular, our motivation arises from optoelectronics, where reduced accuracy requirements provide an opportunity for trading accuracy for execution time. The conventional approach, as implemented for example in zgeev (LAPACK), is to treat complex symmetric problems as general complex and therefore abstaining from utilizing special algebraic properties for efficient codes to solve the corresponding EVP. The proposed procedure comprehends the cardinal steps (a) reduction to standard form, (b) tridiagonalization, (c) solution of the resulting tridiagonal problem, and (d) backtransformation of eigenvectors (cf. Gansterer et al, 2008). Potential benefits of this procedure are reduced RAM requirements and faster execution.
Glutamic Acid structures at Ag(100)
Smerieri Marco (Universita' degli Studi di Genova)
Amino acids adsorption at metal and oxide surfaces is a model system for chemisorption of biofunctional molecules. To date most investigations concentrated on the simplest species (cysteine, glycine, alanine, lysine, aspartic acid) on Cu , but also on Ni , Ag  and non-metallic substrates. In general, these systems form ordered adlayers with chemisorbed molecules in the anionic (H2N.CRâ€™Râ€™â€™.COO-) or in the zwitterionic (NH3+.RCH.COO-) form. On roughened Ag surfaces at T=300 K  amino acids adsorption as zwitterions was suggested. In contrast, (S)-glutamic acid (GA)  is found to bind on Ag(110) mainly in the anionic form. Due to the weak molecule-metal interaction, it organises in extended islands with geometry depending on evaporation parameters. We extended the investigation to GA adsorption on Ag(100) using a combined low temperature STM and periodic ab initio molecular dynamics (MD) approach followed by complete geometry optimisation. Experimentally we find that GA molecules self-assemble in ordered structures depending on the preparation conditions: high density, extended monolayer islands with two different geometries are observed for exposures at 250 K
 S.M. Barlow and R. Raval, Surf. Sci. Rep. 50, 201(2003).
 T.E. Jones and C.J. Baddeley, Lagmuir 22, 148 (2006 ).
 T.E. Jones et al., Langmuir 21, 9468 (2005).
 S. Stewart and P.M. Fredericks, Spectrochim. Acta Part A 55, 1641 (1999).
Hawkins Rhoda (Université Pierre et Marie Curie)
Motivated by the biological problem of cell motility in confined environments, we study the flow properties of active gels in a narrow channel geometry using the lattice Boltzmann algorithm to study the continuum equations of motion for active gels.The theory of active gels, which has been developed in recent years, is based on the hydrodynamic theory of liquid crystals with additional terms to account for the out-of-equilibrium character of many biological systems such as actin-myosin. Using the version of these equations valid for polar active gels and a narrow channel geometry we vary the activity, channel width and the boundary conditions for the polarisation (free or fixed parallel or perpendicular to the wall). For completeness we consider contractile and extensile activity. We develop novel simulations of a finite slab of active viscous gel surrounded by a passive isotropic fluid to model a finite biological cell. This slab of active gel is allowed to move within the channel by advection of the active/passive interface. Finally we consider more complicated channel wall geometries mimicking experiments using living cells in fabricated microchannels. We find that the ability of a slab of active gel in a channel to move depends crucially on the activity and the
boundary conditions for the polarisation.
Rampino Sergio (Università di Perugia)
The calculation of the detailed state-to-state quantum reactive probabilities for atom-diatom systems requires either the stationary or time dependent Schroedinger equation for the nuclei wavefunction to be integrated step by step from reactants to products along a continuity variable. From a direct or indirect comparison of the system wavefunction of the reactants with that of the products, one can compute the elements of the scattering S matrix from which all the scattering properties of the system can be derived [1,2]. In this work, mostly carried out during my visit at EPCC, the use of less conventional sets of coordinates  - namely bond lengths (BL) and bond order (BO) coordinates - for the description of the reactive event is explored. These coordinates, despite a more complicated formulation of the Hamiltonian, are more suited to describe exchange processes than traditional orthogonal coordinates. Moreover, bond order coordinates (derived from the homonym concept introduced by Pauling ) have the appealing feature of confining the whole reactive event into a finite inverted volume of space. In order to assess the features of the usage of alternative sets of coordinates, a parallel code has been developed for the simple A + BC collinear reaction where the real wavepacket propagation method of ref.  has been employed . Results are shown for some benchmark symmetric reactions and the fine grain parallelization algorithm adopted for the time propagation is discussed.
 R T Pack, G A Parker, Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory, J Chem Phys 87, 3888-3921 (1987)
 S K Gray, G G Balint-Kurti, Quantum dynamics with real wave packets, including application to three-dimensional (J=0) D + H2 -> HD + H reactive scattering, J Chem Phys 108, 950-962 (1998)
 A Lagana\', S Crocchianti, N Faginas Lago, L Pacifici, G Ferraro, A nonorthogonal coordinate approach to atom-diatom parallel reactive scattering calculations, Collect Czech Chem Commun 69, 307-330 (2002)
 L Pauling, Atomic Radii and Interatomic Distances in Metals, J Am Chem Soc 69, 542-553 (1947)
 S Rampino, G Ferraro, A Lagana\', Collinear reactive probabilities from real wavepacket propagation using Bond Order formalism, in preparation (2009)
Muller Joachim (Technical University at Braunschweig)
A wide range of methods is available to numerically model space plasma processes. Dealing with scales comparable to ion gyration radii (e.g. Mars, Venus or Mercury), a hybrid model is the most convenient choice. It treats the electrons as a fluid, whereas a completely kinetic approach is retained to cover ion dynamics. From the numerical point of view it can be categorized as a particle-mesh code. The Ion representing particles interact with the electron fluid defined on the numerical mesh, where the spatial mesh resolution governs the scale of resolvable processes. In order to highly resolve localized small scale plasma processes, block Adaptive Mesh Refinement (AMR) is implemented. I.e. the computational domain is initially composed out of blocks that can be further refined into 8 sub blocks. However, we relax the common rule that a block must be either completely refined or not at all thereby achieving a higher flexibility and improved mesh adaptation to plasma features of interest. This modification is referred to as hybrid-block-AMR. The MPI parallelization of the simulation code is optimized by using a Hilbert-Space-Filling-Curve for the block decomposition. The blocks along this curve are distributed to the individual processors by means of self tuning algorithms.
Pietrzyk Monika (Warsaw University of Technology)
Authors of the contribution: M. Pietrzyk, I. Kanattsikov, K. D\'Mellow
We describe the propagation of few-cycle optical pulses in silica fibers. As a model equation we use the generalized Short Pulse Equation (SPE) proposed recently as an alternative to the description of few-cycle optical pulses when the slowly varying envelope approximation does not apply. We review the properties of the (generalized) SPE and present an effective numerical scheme for its integration based on the multisymplectic approach. We present the effectiveness of the multisymplectic integrator as compared with two well-known pseudospectral (Split-Step and Runge-Kutta) methods. We also show that the multisymplectic integrator can be easily and effectively parallelized which is not the case of the pseudospectral methods.
Janoschek Florian (Universität Stuttgart)
Simulation of human blood flow is a demanding task both in terms of the complexity of applicable models as well as the computational effort. One reason is the particulate nature of blood which in first approximation may be treated as a suspension of red blood cells (RBC) in blood plasma. A second reason is that in realistic geometries typical length scales vary over several orders of magnitude. In recent work, we coupled a simple molecular dynamics algorithm for modeling suspended particles to the lattice Boltzmann method that provides us with a representation of the hydrodynamics of the surrounding fluid. The resulting suspension code is highly versatile and performs well on current supercomputers using up to 1024 CPUs in parallel. Within this code, we describe the complex interactions of RBCs by phenomenological model potentials. Thus we obtain a coarse-grained yet efficient particulate model for hemodynamics. However, investigations regarding the choice of parameters and the validation of the model are still part of ongoing work. We will present an overview of our
preliminary results and relevant details of the implementation.
La Penna Giovanni (National research council)
The first principle molecular dynamics of a model nanowire,(NBu4)2[Pt3(CO)6]8, was investigated by using the modern Quantum-Espresso package. This open-source code, designed for parallel architectures, allows the exploitation of several ps of dynamics for models including several hundreds of atoms in a condensed phase within the approximations of the density functional theory for the ground state electronic state. The combination of the molecular dynamics simulation with the analysis of delocalization indexes allows the description of bonding for nano-scale aggregates involving metal-metal interactions and beyond the concepts usually adopted for small molecules.
Vayssilov Georgi (University of Sofia)
Ceria is well known as a key component of the automotive catalysts and efficient catalysts or support for a series of other catalytic process of great industrial and environmental interest. Some of the most important applications of these systems are related to their reactivity towards CO or CO2 in relation to reduction of pollutions and greenhouse gases. The present model studies included investigation of the relative stability of oxygen vacancies in different positions in a ceria nanocluster (Ce21O42) and preferred location of the Ce(III) ions formed after the removal of O from the clusters. According to the calculated relative energies of the structures, the stability of the O vacancies decreases in the following order: sub-surface layer, surface, edge. The preferable location of the Ce(III) ions are neighbors of the O vacancy when it is on the surface or edge, or farther from it when it is sub-surface. The interaction of the ceria nanoparticle with CO resulted in spontaneous formation of CO2 or of surface carbonate when the local structure of the particle is suitable. Interaction of CO2 with the catalystâ€™s surface also results in formation of carbonate species. The simulated vibrational frequencies of the carbonates are found similar to the experimentally measured frequencies of some of the surface species. The calculations are performed with periodic plane-wave DFT method (PW91 functional, 415 eV cut-off) as implemented in VASP program. The investigation is supported by HPC-Europa2 program at Barcelona Supercomputer Center and partially by Bulgarian National Science Fund with project DOO2-82/08.
Hetenyi Balazs (Technische Universität Graz)
The presentation will concern many-body physics problems and their implementation in a parallel setting. Calculations characterizing the orientational order in solid hydrogen and hydrogen containing solids based on a model which takes into account orientational degrees of freedom will be discussed. A set of calculations regarding localization in the Hubbard model based on a new definition of the total position operator will also be presented.
Schmieschek Sebastian (University of Stuttgart)
The understanding of capillarity and wetting effects plays a crucial role for the design and production of microelectromechanical systems (MEMS). Furthermore it is of interest in the examination of porous media like filters or clay. These systems share complex geometrical boundaries. In the last decades a number of simulation methods has been developed in order to achieve accurate models of microfluidic flows. If the intermolecular interactions in a system are neglectable or can be taken into account by a mean field potential, lattice Boltzmann models have proven as a powerful means of simulating multiphase flows in complex geometries. In the scope of this project a multi relaxation time (MRT) collision scheme is integrated with the three-dimensional multi component lattice Boltzmann method (LBM) implementation LB3D. In comparison to the broadly used single relaxation time Bhatnagar Gross Krook (BGK) collision scheme, the MRT approach allows to account relaxation of different physical modes independently, fixing some shortcomings of the BGK method. The code will be discussed focussing on optimizations to keep the increase in calculation cost caused by using MRT low. Numerical tests to verify the accuracy of the new code as well as basic benchmarking results are provided.
Czerwinska Justyna (Institute of Fundamental Technological Research)
The presented work considers optimization of flow properties on nanoscale DNA analyser. It is a combine work with experimentalist who at TU Twente have build a prototype of such device. Simulations performed by Molecular Dynamics consider DNA chain static and dynamic properties in confined nanochannel with the Lennard-Jones liquid. The optimal flow conditions depend on the channel geometry, strength of the flow and DNA elasticity. Numerical simulations agreed very well with experiments and also have gave the insight to the conformation of DNA in nanochannel, which cannot be at present obtained due to the limitations of experimental techniques. Comment: It is not exactly a project which I have applied for but during my stay at SARA successful collaborations and very interesting results in that matter was obtained."
Francesca Di Patti (University of Florence)
We present a simplified scheme of coupled autocatalytic reactions, namely chemical processes where the reaction product is itself a catalyst for the chemical reaction, occurring inside vesicles, small cell-like structures in which the outer membrane takes the form of a lipid bilayer. The role of stochastic fluctuations is elucidated through the use of the van Kampen system-size expansion and the results compared with direct stochastic simulations, performed through the Gillespie algorithm. Regular temporal oscillations are predicted to occur for the concentration of the various chemical constituents, with an enhanced amplitude resulting from a resonance which is induced by the intrinsic graininess of the system. The associated power spectra are determined and compared with the analytical predictions. Starting from this background, we incorporate the notion of space, as an additional explicit ingredient to the model. The idea is to introduce a spatial coarse graining at the level of small micro-cells which are supposed to uniformly cover the volume occupied by the vescicle. In each microscopic cell autocatalytic reactions do occur. Migration between neighbours cells is allowed, an ingredient which in turn amounts to explicitly account for space. As a simplifying ansatz, we ﬁrst image a periodic geometry and focus on a chain of microscopic cells situated at the frontier with the external membrane. This hypothesis is put forward so to restore the translational invariance, which results in a straightforward mathematical treatment. Numerical simulations again show the emergence of spatio-temporal patterns. In the end we will mention the possibility to simulate a more realistic scenario, namely a three dimensional vescicle shape.We present a simplified scheme of coupled autocatalytic reactions, namely chemical processes where the reaction product is itself a catalyst for the chemical reaction, occurring inside vesicles, small cell-like structures in which the outer membrane takes the form of a lipid bilayer. The role of stochastic fluctuations is elucidated through the use of the van Kampen system-size expansion and the results compared with direct stochastic simulations, performed through the Gillespie algorithm. Regular temporal oscillations are predicted to occur for the concentration of the various chemical constituents, with an enhanced amplitude resulting from a resonance which is induced by the intrinsic graininess of the system. The associated power spectra are determined and compared with the analytical predictions. Starting from this background, we incorporate the notion of space, as an additional explicit ingredient to the model. The idea is to introduce a spatial coarse graining at the level of small micro-cells which are supposed to uniformly cover the volume occupied by the vescicle. In each microscopic cell autocatalytic reactions do occur. Migration between neighbours cells is allowed, an ingredient which in turn amounts to explicitly account for space. As a simplifying ansatz, we ﬁrst image a periodic geometry and focus on a chain of microscopic cells situated at the frontier with the external membrane. This hypothesis is put forward so to restore the translational invariance, which results in a straightforward mathematical treatment. Numerical simulations again show the emergence of spatio-temporal patterns. In the end we will mention the possibility to simulate a more realistic scenario, namely a three dimensional vescicle shape.
Ivan Spisso (University of Leicester)