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Stürzer, Dominik; TU Wien  WPI Seminar Room 08.135  Fri, 10. Oct 14, 10:45 
Spectral Analysis and LongTime Behavior of a Linear FokkerPlanck Equation with a NonLocal Perturbation  
We discuss a linear FokkerPlanck (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) FPoperator in exponentially weighted $L^2$spaces. In particular the FPoperator has a onedimensional 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 FPoperator leaves the spectrum (and the spectral gap) unchanged, i.e. the perturbed FP operator is an isospectral deformation of the FPoperator. 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.  

Falconi, Marco; Université de Rennes  WPI Seminar Room 08.135  Fri, 10. Oct 14, 9:30 
SchrödingerKleinGordon system as the classical limit of a Quantum Field Theory dynamics  
In this talk it is discussed how a nonlinear system of PDEs, the SchrödingerKleinGordon with Yukawa coupling, emerges naturally as the limiting dynamics of a quantum system of nonrelativistic 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: timeevolved observables and states. In the classical context, the SKG 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.  

Achleitner, Franz; TU Wien  WPI Seminar Room 08.135  Thu, 9. Oct 14, 14:30 
Travelling waves for a nonlocal Korteweg–de Vries–Burgers equation  
We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a nonlocal diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the tripledeck regularisation (which is an extension of classical boundary layer theory). The resulting nonlocal 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 nonlocal 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 phaseplane analysis, which is not applicable here due to the presence of the nonlocal 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.  

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 nonlocal, nonlinear 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, nonlocal 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 wellposedness (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.  

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 Navierstokes 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."  

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 threedimensional 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 KadomtsevPetviashvili I equation (for strong surface tension) or the DaveyStewartson 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.  

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 WaterWaves equation with large topography variations. We explain the method on the simplier example of the ShallowWater equation and then present its implementation for the WW equations itselves.  

Duchene, Vincent; Université de Rennes  WPI Seminar Room 08.135  Wed, 8. Oct 14, 10:30 
KelvinHelmholtz instabilities in shallow water  
KelvinHelmholtz 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 shallowwater assumption in oceanography. As a matter of fact, the usual shallowwater 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 shallowwater models with improved dispersion behavior. This is a joint work with Samer Israwi and Raafat Talhouk.  

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 KelvinHelmholtz 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.  

Weishäupl, Rada Maria; Universität Wien  WPI Seminar Room 08.135  Tue, 7. Oct 14, 15:30 
Multisolitary 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.  

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 KadomtsevPetviasvhili 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.  

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.  

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 spacedependent problems, and beyond powerlaw nonlinearities. This will be illustrated in the case of the asymptotically linear NLS.  

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.  

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 kgeneralized Kortewegde Vries equation  
We will discuss special regularity and decay properties of solutions to the IVP associated to the kgeneralized 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.  

Szeftel, Jeremie; Laboratoire JacquesLouis Lions de l'Université Pierre et Marie Curie  WPI Seminar Room 08.135  Mon, 6. Oct 14, 15:45 
The instability of BourgainWang 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 loglog regime. This is a joint work with F. Merle and P. Raphael.  

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 blowup for the focusing nonlinear Schrödinger equation on hyperbolic space. This is a joint work with Thomas Duyckaerts.  

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 ShigesadaTeramotoKawasaki 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.  

Fellner, Klemens; Universität Graz  WPI Seminar Room 08.135  Wed, 24. Sep 14, 14:05 
On reactiondiffusion systems: global existence, convergence to equilibrium and quasisteadystateapproximation.  
For general systems of reactiondiffusion equations, such basic questions of mathematical analysis as existence of global classical solutions, convergence to equilibrium and rigorous justification of quasisteadystateapproximations 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 reactiondiffusion 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 quasisteadystateapproximation, which is strongly motivated by the biological application background. The most important analytical tools applied are the entropy method and suitable duality arguments.  

Laamri, ElHaj; Institut Elie Cartan de Lorraine  WPI , OMP 1, Seminar Room 08.135  Wed, 24. Sep 14, 11:30 
Global existence for some reactiondiusion systems with nonlinear diusion  
In this talk, we present new results concerning global existence for some reactiondiffusion systems. This is joint work with Michel Pierre (ENS de Rennes).  
Note: Click here for further information  

Latos, Evangelos; University of Mannheim  WPI Seminar Room 08.135  Wed, 24. Sep 14, 10:05 
Existence and Blowup of Solutions for Semilinear Filtration Problems  
We examine the local existence and uniqueness of solutions to the semilinear filtration equation, with positive initial data and appropriate boundary conditions. Our main result is the proof of blowup of solutions. Moreover, we discuss about the existence of solutions for the corresponding steadystate 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 blowup and global existence  

Winkler, Michael; Universität DuisburgEssen  Wed, 24. Sep 14, 9:10  
How far can chemotactic crossdiffusion enforce exceeding carrying capacities?  
We consider variants of the KellerSegel system of chemotaxis which contain logistictype 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 parabolicelliptic 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 initialboundary value problem, though known to possess global bounded solutions for any reasonably smooth initial data, is shown to have the property that the socalled {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 Fishertype equation obtained upon dropping the term $(uv_x)_x$, and hence reflects a drastic peculiarity of destabilizing action due to chemotactic crossdiffusion, observable even in the simple spatially onedimensional 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.  

Lorz, Alexander; Laboratoire JacquesLouis Lions  WPI Seminar Room 08.135  Tue, 23. Sep 14, 9:55 
Population dynamics and therapeutic resistance: mathematical models  
Motivated by the theory of mutationselection 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.  

Hirsch, Stefanie; Universität Wien  WPI Seminar Room 08.135  Tue, 23. Sep 14, 9:10 
A Free Boundary Value Problem for ActoMyosin Bundles  
ActoMyosin 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 FActin to the effects of crosslink and bundling proteins, the forces generated by myosinII 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 postprocessing).  

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 threedimensional 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 colocalization of proteins with IBAR domains (like IRSp53) with polymerization and branching agents along the membrane. The IBAR 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 quasistationary 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.  

Manhart, Angelika; Universität Wien  WPI Seminar Room 08.135  Mon, 22. Sep 14, 15:20 
How do Cells Move? Model and Simulation of Actindependent Cell Movement  
Several types of cells use a sheetlike structure called lamellipodium for movement. The main structural components, actin filaments, are connected via crosslinking 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 nonlinear PDE model is based on an variational approach and approximated using the finite element method with nonstandard 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.  

SchappacherTilp, Gudrun; Universität Graz  WPI Seminar Room 08.135  Mon, 22. Sep 14, 14:05 
Modelling actinmyosintitin interaction in a half sarcomere  
In this talk we consider a structural three fillament model of muscle contraction in halfsarcomeres. The proposed model is based on (i) active force production based on crossbridge interactions and (ii) force produc tion based on the elongation of titin. While crossbridge interaction is de scribed by a deterministic system of reactionconvection 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 actintitin interaction on force predictions.  

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 halfsarcomeres by extending A.F.Huxley’s crossbridge 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 halfsarcomere 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.  

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 ATPdriven working strokes. The motors in each halfsarcomere are arranged in antiparallel arrays emerging from the two halves of the thick myosin filament and mechanically coupled via their filament attachments. The cooperative 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 halfsarcomere 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. Sarcomerelevel mechanics and Xray interferometry in single fibres from frog skeletal muscle have provided detailed information about the mechanical properties of the various components of the halfsarcomere 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 halfsarcomere elements indicates that in isometric contraction 2030% of myosin motors are attached to actin and generate force by a small substep of the 11 nm working stroke suggested by the crystallographic model (Fusi et al. 2014, J. Physiol. 592, 11091118; Brunello et al. 2014, J. Physiol. 592, 38813899). During steady shortening against high to moderate loads (the condition for the maximum power and efficiency), the number of actinattached motors decreases in proportion to the load, while each attached motor maintains a 56 pN force over a 6 nm stroke (Piazzesi et al. 2007, Cell 131, 784795). 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 myosinactin crossbridges and of the meshwork of cytoskeleton proteins in each halfsarcomere, 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, 2011420119; Fusi et al. 2010, J. Physiol. 588, 495510).  

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 “crossbridge theory” of muscle contraction. Briefly, the crossbridge 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 twofilament 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 crossbridge 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 titinbased (passive) force in activated muscles than in passive (nonactivated) 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 crossbridge 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 steadystate force following eccentric contractions, and the stability of sarcomeres on the descending limb of the forcelength relationship. Aside from its predictive power, this new three filament model is insofar attractive as it leaves the "historic” crossbridge 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.  

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