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Talks of the past month

 Stürzer, Dominik; TU Wien WPI Seminar Room 08.135 Fri, 10. Oct 14, 10:45 Spectral Analysis and Long-Time Behavior of a Linear Fokker-Planck Equation with a Non-Local Perturbation We discuss a linear Fokker-Planck (FP) equation with an additional perturbation, given by a convolution with a massless kernel. In this talk we will prove the existence of a unique normalized stationary solution of the perturbed equation, and show that any solution converges towards the stationary solution with an exponential rate independent of the perturbation. The first step of the analysis consists of characterizing the spectrum of the (unperturbed) FP-operator in exponentially weighted $L^2$-spaces. In particular the FP-operator has a one-dimensional kernel (spanned by the stationary solution), possesses a spectral gap, and solutions of the unperturbed equation converge exponentially to the stationary solution. Then we demonstrate that adding a convolution with a massless kernel to the FP-operator leaves the spectrum (and the spectral gap) unchanged, i.e. the perturbed FP operator is an isospectral deformation of the FP-operator. Finally we are able to give a similarity transformation between the unperturbed and the perturbed FP operator, which proves that the corresponding semigroups have the same decay properties. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Falconi, Marco; Université de Rennes WPI Seminar Room 08.135 Fri, 10. Oct 14, 9:30 Schrödinger-Klein-Gordon system as the classical limit of a Quantum Field Theory dynamics In this talk it is discussed how a non-linear system of PDEs, the Schrödinger-Klein-Gordon with Yukawa coupling, emerges naturally as the limiting dynamics of a quantum system of non-relativistic bosons coupled with a bosonic scalar field. The correspondence of the "quantum" (linear) and "classical" (nonlinear) dynamics, often assumed in physics as an heuristic theorem, is made rigorous. After a brief introduction of the quantum system (on a suitable symmetric Fock space), we identify the classical counterparts of the important objects of the quantum theory: time-evolved observables and states. In the classical context, the S-KG dynamics plays a fundamental role, and the study of its properties might provide a valuable indication of important underlying properties of the quantum system, that are much more difficult to investigate. This is a joint work with Zied Ammari. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Achleitner, Franz; TU Wien WPI Seminar Room 08.135 Thu, 9. Oct 14, 14:30 Travelling waves for a non-local Korteweg–de Vries–Burgers equation We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Ehrnström, Mats; Norwegian University of Science and Technology WPI Seminar Room 08.135 Thu, 9. Oct 14, 11:00 On the Whitham equation (and a class of non-local, non-linear equations with weak or very weak dispersion) We consider a class of pseudodifferential evolution equations of the form $$u_t +(n(u)+Lu)_x = 0$$, in which L is a linear, generically smoothing, non-local operator and n is a nonlinear, local, term. This class includes the Whitham equation, the linear terms of which match the dispersion relation for gravity water waves on finite depth. In this talk we present recent results for this equation and its generalisations, including periodic bifurcation results, existence of solitary waves via minimisation, and well-posedness (local). In particular, although for small waves, small times and small frequencies this equation bears many similarities with the Korteweg—de Vries equation, it displays some very interesting differences for ’large' solutions. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Keraani, Sahbi; Université de Rennes WPI Seminar Room 08.135 Thu, 9. Oct 14, 9:30 On the inviscid limit for a 2D incompressible fluid "In this talk, we will present some results of inviscid limit of the 2D Navier-stokes system with data in spaces with BMO flavor. The issue of uniform (in viscosity) estimates for these equations will be also considered. It is a joint work with F. Bernicot and T. Elgindi." Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Wahlen, Erik; Lunds universitet WPI Seminar Room 08.135 Wed, 8. Oct 14, 14:30 Solitary water waves in three dimensions I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localized in the sense that they converge to the undisturbed state of the water in every horizontal direction. The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili I equation (for strong surface tension) or the Davey-Stewartson equation (for weak surface tension). These model equations play an important role in the theory. This is joint work with B. Buffoni, M. Groves and S.-M. Sun. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Mesognon, Benoit; Ecole Normale Supérieure de Paris WPI Seminar Room 08.135 Wed, 8. Oct 14, 11:45 Long time control of large topography effects for the water waves equations We explain how we can get a large time of existence for the Water-Waves equation with large topography variations. We explain the method on the simplier example of the Shallow-Water equation and then present its implementation for the WW equations itselves. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Duchene, Vincent; Université de Rennes WPI Seminar Room 08.135 Wed, 8. Oct 14, 10:30 Kelvin-Helmholtz instabilities in shallow water Kelvin-Helmholtz instabilities arise when a sufficiently strong shear velocity lies at the interface between two layers of immiscible fluids. The typical wavelength of the unstable modes are very small, which goes against the natural shallow-water assumption in oceanography. As a matter of fact, the usual shallow-water asymptotic models fail to correctly reproduce the formation of KH instabilities. With this in mind, our aim is to motivate and study a new class of shallow-water models with improved dispersion behavior. This is a joint work with Samer Israwi and Raafat Talhouk. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Lannes, David; Ecole Normale Supérieure de Paris WPI Seminar Room 08.135 Wed, 8. Oct 14, 9:30 Internal waves in continuously stratified media Many things are known about the propagation of waves at the interface of two fluids of different densities, for which dispersion plays an important role (it plays a stabilizing role controlling Kelvin-Helmholtz instabilities and balances the long time effects of the nonlinearities). When a flow is continuously stratified, the notion of wave is less clear, as well as the nature of dispersive effects. We show that they are encoded in a Sturm Liouville problem and are therefore of 'nonlocal type'; we also derive simpler, local, asymptotic models. This is a joint work with JC Saut and B. Desjardins. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Weishäupl, Rada Maria; Universität Wien WPI Seminar Room 08.135 Tue, 7. Oct 14, 15:30 Multi-solitary waves solutions for nonlinear Schrödinger systems We consider a system of two coupled nonlinear Schrödinger equations in one dimension. We show the existence of solutions behaving at large time as a couple of scalar solitary waves. The proof relies on a method introduced by Martel and Merle for multi solitary waves for the scalar Schrödinger equation. Finally, we present some numerical simulations to understand more the qualitative behavior of the solitary waves. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Koch, Herbert; Universität Bonn WPI Seminar Room 08.135 Tue, 7. Oct 14, 14:30 Global existence and scattering for KP II in three space dimensions The Kadomtsev-Petviasvhili II equation describes wave propagating in one direction with weak transverse effect. I will explain the proof of global existence and scattering for three space dimensions. The key estimates are bilinear L^2 estimates and a delicate choice of norms. This is joint work with Junfeng Li. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Colin, Mathieu; Université de Bordeaux WPI Seminar Room 08.135 Tue, 7. Oct 14, 11:45 Solitary waves for Boussinesq type systems The aim of this talk is to exhibit specific properties of Boussinesq type models. After recalling the usual asymptotic method leading to BT models, we will present a new asymptotic model and present a local Cauchy theory. We then provide an effective method to compute solitary waves for Boussinesq type models. We will conclude by discussing shoaling properties of such models. This is a joint work with S. Bellec. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Genoud, Francois; Universität Wien WPI Seminar Room 08.135 Tue, 7. Oct 14, 10:30 Bifurcation and stability of solitons for the asymptotically linear NLS The purpose of this talk is to convey the idea that bifurcation theory provides a powerful tool to prove existence and orbital stability of solitons for the nonlinear Schrödinger equation. It is especially useful to obtain results for space-dependent problems, and beyond power-law nonlinearities. This will be illustrated in the case of the asymptotically linear NLS. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Klein, Christian; Université de Bourgogne WPI Seminar Room 08.135 Tue, 7. Oct 14, 9:30 Multidomain spectral method for Schrödinger equations A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schr\"odinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schr\"odinger equation, this code is compared to exact transparent boundary conditions and perfectly matched layers approaches. In addition it is shown that the Peregrine breather being discussed as a model for rogue waves can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Linares, Felipe; Institute for Pure and Applied Mathematics , Rio de Janeiro WPI Seminar Room 08.135 Mon, 6. Oct 14, 16:45 Propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation We will discuss special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u_0in H^{3/4^+}(R) whose restriction belongs to H^k((b,infty)) for some kinZ^+ and bin R we prove that the restriction of the corresponding solution u(cdot,t) belongs to H^k((beta,infty)) for any beta in R and any tin (0,T). Thus, this type of regularity propagates with infinite speed to its left as time evolves. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Szeftel, Jeremie; Laboratoire Jacques-Louis Lions de l'Université Pierre et Marie Curie WPI Seminar Room 08.135 Mon, 6. Oct 14, 15:45 The instability of Bourgain-Wang solutions for the L2 critical NLS We consider the two dimensional focusing cubic nonlinear Schrodinger equation. Bourgain and Wang have constructed smooth solutions which blow up in finite time with the pseudo conformal speed, and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical L^2 mass lies on the boundary of both H^1 open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime. This is a joint work with F. Merle and P. Raphael. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Banica, Valeria; Université d'Évry Val d'Essonne WPI Seminar Room 08.135 Mon, 6. Oct 14, 14:45 Large time behavior for the focusing NLS on hyperbolic space In this talk I shall present some results on global existence, scattering and blow-up for the focusing nonlinear Schrödinger equation on hyperbolic space. This is a joint work with Thomas Duyckaerts. Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (BLOW-2014) (2014) Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

 Desvillettes, Laurent; ENS Cachan WPI Seminar Room 08.135 Wed, 24. Sep 14, 15:20 Some existence and regularity results for cross diffusion equations appearing in population dynamics We present results obtained in collaboration with Ariane Trescases, on generalized versions of the triangular Shigesada-Teramoto-Kawasaki model of population dynamics. This model helps to understand how, since the individuals of species in competition change their diffusion rate, patterns can emerge in large time. Our results extend the range of parameters for which existence on one hand, and regularity on the other hand, is proven. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Fellner, Klemens; Universität Graz WPI Seminar Room 08.135 Wed, 24. Sep 14, 14:05 On reaction-diffusion systems: global existence, convergence to equilibrium and quasi-steady-state-approximation. For general systems of reaction-diffusion equations, such basic questions of mathematical analysis as existence of global classical solutions, convergence to equilibrium and rigorous justification of quasi-steady-state-approximations constitute surprisingly many open problems, which have recently attracted a lot of attention in the mathematical community. In this talk, we present a model systems for asymmetric protein localisation in stem cells as a motivation to study systems of reaction-diffusion equations and recall recent advances in the theory of global solutions and their large time behaviour. Beside the system character, an additional difficulty arises from considering systems, which combine volume and surface diffusion and reactions between volume and surface concentrations. Moreover, we proof rigorously an associated quasi-steady-state-approximation, which is strongly motivated by the biological application background. The most important analytical tools applied are the entropy method and suitable duality arguments. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Laamri, El-Haj; Institut Elie Cartan de Lorraine WPI , OMP 1, Seminar Room 08.135 Wed, 24. Sep 14, 11:30 Global existence for some reaction-di usion systems with nonlinear di usion In this talk, we present new results concerning global existence for some reaction-diff usion systems. This is joint work with Michel Pierre (ENS de Rennes). Note:   Click here for further information Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Latos, Evangelos; University of Mannheim WPI Seminar Room 08.135 Wed, 24. Sep 14, 10:05 Existence and Blow-up of Solutions for Semilinear Filtration Problems We examine the local existence and uniqueness of solutions to the semi-linear filtration equation, with positive initial data and appropriate boundary conditions. Our main result is the proof of blow-up of solutions. Moreover, we discuss about the existence of solutions for the corresponding steady-state problem. It is found that there exists a critical value, above which the problem has no stationary solution of any kind, while below that critical value there exist classical stationary solutions. Exactly this critical value of the parameter acts as a threshold also for the corresponding parabolic problem between blow-up and global existence Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Winkler, Michael; Universität Duisburg-Essen Wed, 24. Sep 14, 9:10 How far can chemotactic cross-diffusion enforce exceeding carrying capacities? We consider variants of the Keller-Segel system of chemotaxis which contain logistic-type source terms and thereby account for proliferation and death of cells. We briefly review results and open problems with regard to the fundamental question whether solutions exist globally in time or blow up. The primary focus will then be on the prototypical parabolic-elliptic system [ begin{array}{l} u_t=varepsilon u_{xx} - (uv_x)_x + ru - mu u^2, 0= v_{xx}-v+u, end{array} right. ] in bounded real intervals. The corresponding Neumann initial-boundary value problem, though known to possess global bounded solutions for any reasonably smooth initial data, is shown to have the property that the so-called {em carrying capacity} $frac{r}{mu}$ can be exceeded dynamically to an arbitrary extent during evolution in an appropriate sense, provided that $mu<1$ and that $eps>0$ is sufficiently small. This is in stark contrast to the case of the corresponding Fisher-type equation obtained upon dropping the term $-(uv_x)_x$, and hence reflects a drastic peculiarity of destabilizing action due to chemotactic cross-diffusion, observable even in the simple spatially one-dimensional setting. Numerical simulations underline the challenge in the analytical derivation of this result by indicating that the phenomenon in question occurs at intermediate time scales only, and disappears in the large time asymptotics. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Lorz, Alexander; Laboratoire Jacques-Louis Lions WPI Seminar Room 08.135 Tue, 23. Sep 14, 9:55 Population dynamics and therapeutic resistance: mathematical models Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance phenotype. This phenotype influences birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations in healthy and tumor cells. We extend previous work by demonstrating how qualitatively different actions of cytostatic (slowing down cell division) and cytostatic (actively killing cells) treatments may induce different levels of resistance. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Hirsch, Stefanie; Universität Wien WPI Seminar Room 08.135 Tue, 23. Sep 14, 9:10 A Free Boundary Value Problem for Acto-Myosin Bundles Acto-Myosin bundles are macroscopic structures within a cell that are used for various processes such as transport of nutrients and mechanical stability of the cell. Dietmar Ölz developed a model relating the flows of F-Actin to the effects of cross-link and bundling proteins, the forces generated by myosin-II filaments as well as external forces at the tips of the bundle. In the asymptotic regime where actin filaments are assumed to be short compared to the length of the bundle, a fixed and a free boundary value problem can be derived. In the free boundary value problem the force at the tips is prescribed and the position of the tips can be computed. The model consists of transport equations for the density of actin filaments coupled to elliptic equations for the velocities of these filaments, as well as an ODE for the tip of the bundle. In order to solve this system, fixed point arguments are employed, a strategy which proved successful in solving the corresponding problem with fixed boundary (where the positions of the tips are known, and the force can be computed by post-processing). Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Winkler, Christoph; Universität Wien WPI Seminar Room 08.135 Mon, 22. Sep 14, 16:00 The Flatness of Lamellipodia Explained by the Interaction Between Actin Dynamics and Membrane Deformation The crawling motility of many cell types relies on lamellipodia, flat protrusions spreading on flat substrates but (on cells in suspension) also growing into three-dimensional space. Lamellipodia consist of a plasma membrane wrapped around an oriented actin filament meshwork. It is well known that the actin density is controlled by coordinated polymerization, branching, and capping processes, but the mechanisms producing the small aspect ratios of lamellipodia (hundreds of nm thickness vs. several $\mu$m lateral and inward extension) remain unclear. The main hypothesis of this work is a strong influence of the local geometry of the plasma membrane on the actin dynamics. This is motivated by observations of co-localization of proteins with I-BAR domains (like IRSp53) with polymerization and branching agents along the membrane. The I-BAR domains are known to bind to the membrane and to prefer and promote membrane curvature. This hypothesis is translated into a stochastic mathematical model where branching and capping rates, and polymerization speeds depend on the local membrane geometry and branching directions are influenced by the principal curvature directions. This requires the knowledge of the deformation of the membrane, being described in a quasi-stationary approximation by minimization of a modified Helfrich energy, subject to the actin filaments acting as obstacles. Simulations with this model predict pieces of flat lamellipodia without any prescribed geometric restrictions. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Manhart, Angelika; Universität Wien WPI Seminar Room 08.135 Mon, 22. Sep 14, 15:20 How do Cells Move? Model and Simulation of Actin-dependent Cell Movement Several types of cells use a sheet-like structure called lamellipodium for movement. The main structural components, actin filaments, are connected via cross-linking proteins. Adhesions allow for a connection with the substrate and the contraction agent myosin helps pulling the cell body forward. Additionally the cell has to regulate its filament number locally by nucleation (via branching) of new filaments and degradation (via capping and severing) of existing ones. I will present a continuous model of this structure including the forces created by the described molecular players. The non-linear PDE model is based on an variational approach and approximated using the finite element method with non-standard finite elements. The simulation can reproduce stationary and moving steady states, describe the transition between the two, mimic chemotaxis, describe interaction with an obstacle and simulate turning cells. In particular I will also show how this model can be applied to fish keratocytes. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Schappacher-Tilp, Gudrun; Universität Graz WPI Seminar Room 08.135 Mon, 22. Sep 14, 14:05 Modelling actin-myosin-titin interaction in a half sarcomere In this talk we consider a structural three fillament model of muscle contraction in half-sarcomeres. The proposed model is based on (i) active force production based on cross-bridge interactions and (ii) force produc- tion based on the elongation of titin. While cross-bridge interaction is de- scribed by a deterministic system of reaction-convection equations forces attributed to titin are random variables due to protein unfolding. More- over, titin is acting as an activatable spring able to bind to actin upon activation. We provide an intriguingly simple approach to predict forces based on titin elongation in a half sarcomere and analyse the impact of actin-titin interaction on force predictions. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Campbell, Kenneth; University of Kentucky WPI Seminar Room 08.135 Mon, 22. Sep 14, 11:30 Myocardial strain rate modulates the speed of relaxation in dynamically loaded twitch contractions Slow myocardial relaxation is an important clinical problem in about 50% of patients who have heart failure. Prior experiments had suggested that the slow relaxation might be a consequence of high afterload (hypertension) but clinical trials testing this hypothesis have failed; lowering blood pressure in patients with slow relaxation does not help their condition. We performed new experiments using mouse, rat, and human trabeculae and showed that it is not afterload but the strain rate at end systole that determines the subsequent speed of relaxation. To investigate the molecular mechanisms that drive this behavior, we ran simulations of our experiments using the freely available software MyoSim (http://www.myosim.org). This software simulates the mechanical properties of dynamically activated half-sarcomeres by extending A.F.Huxley’s cross-bridge distribution technique with Ca2+ activation and cooperative effects. We discovered that our experimental data could be reproduced using a relatively simple framework consisting of a single half-sarcomere pulling against a series elastic spring. Further analysis of the simulations suggested that quick stretches speed myocardial relaxation by detaching myosin heads and thereby disrupting the cooperative mechanisms that would otherwise prolong thin filament activation. The simulations therefore identify myofilament kinetics and tissue strain rate as potential therapeutic targets for heart failure attributed to slow relaxation. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Vincenzo Lombardi; University of Florence WPI Seminar Room 08.135 Mon, 22. Sep 14, 10:05 The muscle as a motor and as a brake Force and shortening in a contracting striated muscle are generated by the dimeric motor protein myosin II pulling the actin filament towards the centre of the sarcomere during cyclical ATP-driven working strokes. The motors in each half-sarcomere are arranged in antiparallel arrays emerging from the two halves of the thick myosin filament and mechanically coupled via their filament attachments. The co-operative action of this coupled system, including the interdigitating actin filaments and other elastic and regulatory proteins, is the basic functional unit of muscle. When the sarcomere load is smaller than the maximum force developed in isometric contraction (T0), the myosin array works as a collective motor, converting metabolic energy into mechanical work at a rate that increases with reduction of the load. When an external load larger than T0 is applied to the active muscle, the sarcomere exerts a marked resistance to lengthening, with reduced metabolic cost. Thus the chemical and mechanical properties of the half-sarcomere machine during generation of force and shortening, when muscle works as a motor, are quite different from those during the response to a load or length stretch, when it works as a brake. Sarcomere-level mechanics and X-ray interferometry in single fibres from frog skeletal muscle have provided detailed information about the mechanical properties of the various components of the half-sarcomere and about kinetics and structural dynamics of the myosin motors as they perform different physiological tasks. The high stiffness of the myosin motor resulting from the analysis of the compliance of half-sarcomere elements indicates that in isometric contraction 20-30% of myosin motors are attached to actin and generate force by a small sub-step of the 11 nm working stroke suggested by the crystallographic model (Fusi et al. 2014, J. Physiol. 592, 1109-1118; Brunello et al. 2014, J. Physiol. 592, 3881-3899). During steady shortening against high to moderate loads (the condition for the maximum power and efficiency), the number of actin-attached motors decreases in proportion to the load, while each attached motor maintains a 5-6 pN force over a 6 nm stroke (Piazzesi et al. 2007, Cell 131, 784-795). The braking action exerted when an active sarcomere resists an increase in load above the isometric force, depends not only on the mechanical properties of the myosin-actin cross-bridges and of the meshwork of cytoskeleton proteins in each half-sarcomere, but also on the rapid attachment to actin of the second motor domain of the myosin dimer that has the first motor domain already attached to actin during the isometric contraction (Brunello et al. 2007, PNAS 104, 20114-20119; Fusi et al. 2010, J. Physiol. 588, 495-510). Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

 Herzog, Walter; University of Calgary WPI Seminar Room 08.135 Mon, 22. Sep 14, 9:10 A New Model for Muscle Contraction In 1953, Hugh Huxley proposed that muscle contraction occurred through the sliding of two sets of filamentous proteins, actin and myosin, rather than through the shortening of the centre filament in the sarcomere. This proposal was supported by the two classic papers in the May issue of Nature 1954 by Andrew Huxley and Hugh Huxley. Andrew Huxley then proposed how this sliding of the two sets of filament occurs in 1957, and this has become known as the “cross-bridge theory” of muscle contraction. Briefly, the cross-bridge theory assumes that there are protrusions from the myosin filaments attaching cyclically to the actin filaments and pulling the actin past the myosin filaments using energy from the hydrolysis of adenosine triphosphate (ATP). This two-filament thinking of contraction (involving actin and myosin) has persisted to this day, despite an inability of this model to predict experimental results on stability, force and energetics appropriately for eccentric (active lengthening) muscles. Andrew Huxley reported on this limitation of his cross-bridge model and predicted in 1980, that studying of eccentric contractions would lead to new insights and surprises, and would produce thus far unknown elements that might affect muscle contraction and force production. Here, I would like to propose a new model of muscle contraction, that aside from the contractile proteins, actin and myosin, also includes the structural protein, titin. Titin will not only be a passive player in this new theory, but an activatable spring that changes its stiffness in an activation- and force- dependent manner, thus contributing substantially more titin-based (passive) force in activated muscles than in passive (non-activated) muscles. I will show evidence that titin binds calcium at various sites upon activation (activation in muscles is associated with a steep increase in sarcoplasmic calcium), thereby increasing its inherent spring stiffness, and that titin may bind its proximal segments to actin, thereby shortening its free spring length, and thus increasing its stiffness and force in a second way. Incorporating this third filament, titin, into the two filament model of muscle contraction (actin and myosin) allows for predictions of experimental observations that could not be predicted before while maintaining the power of the cross-bridge theory for isometric (constant length) and concentric (shortening) contractions. For example, the three filament model naturally predicts the energetic efficiency of eccentric contractions, the increase in steady-state force following eccentric contractions, and the stability of sarcomeres on the descending limb of the force-length relationship. Aside from its predictive power, this new three filament model is insofar attractive as it leaves the "historic” cross-bridge model fully intact, it merely adds an element to it, and its conceptual and structural simplicity makes it a powerful theory that, although not fully proven, is intuitively appealing and emotionally satisfying. Thematic program: PDE models in biology (BIOMATH-2014) (2014) Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)
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