There is a sharp condition for scattering of the radial 3-d cubic nonlinear Schrödinger equation that was given by Justin Holmer and Svetlana Roudenko. Following this spirit, we provide some similar results for this equation with potential.

• Thematic program: PDE models in biology (2014)
• Event: Workshop on "Blow up and dispersion in Nonlinear PDEs" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

In this talk, we present new results concerning global existence for some reaction-diffusion systems. This is joint work with Michel Pierre (ENS de Rennes).

• Thematic program: PDE models in biology (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)

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 (2014)
• Event: Workshop on "Mathematical Modelling in Biology and Physiology" (2014)