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ABSTRACT: Cross-linked semiflexible polymer networks are omnipresent in living cells. Typical examples are actin networks in the cytoplasm of eukaryotic cells, which play an essential role in cell motility, and the spectrin network, a key element in maintaining the integrity of erythrocytes in the blood circulatory system. We introduce a simple mechanical network model at the length scale of the typical mesh size and derive a continuous constitutive law relating the stress to deformation. The continuous constitutive law is found to be generically nonlinear even if the microscopic law at the scale of the mesh size is linear. The nonlinear bulk mechanical properties are in good agreement with the experimental data for semiflexible polymer networks, i.e., the network stiffens and exhibits a negative normal stress in response to a volume-conserving shear deformation, whereby the normal stress is of the same order as the shear stress. Furthermore, it shows a strain localization behavior in response to an uniaxial compression. Within the same model we find a hierarchy of constitutive laws depending on the degree of nonlinearities retained in the final equation. The presented theory provides a basis for the continuum description of polymer networks such as actin or spectrin in complex geometries and it can be easily coupled to growth problems, as they occur, for example, in modeling actin-driven motility.
Physical Review E. 04/2013; 87:042721.
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ABSTRACT: Inspired by experiments on the actin driven propulsion of micrometer sized beads we develop and study a minimal mechanical model of a two-dimensional network of stiff elastic filaments grown from the surface of a cylinder. Starting out from a discrete model of the network structure and of its microscopic mechanical behavior we derive a macroscopic constitutive law by homogenization techniques. We calculate the axisymmetric equilibrium state and study its linear stability depending on the microscopic mechanical properties. We find that thin networks are linearly stable, whereas thick networks are unstable. The critical thickness for the change in stability depends on the ratio of the microscopic elastic constants. The instability is induced by the increase in the compressive load on the inner network layers as the thickness of the network increases. The here employed homogenization approach combined with more elaborate microscopic models can serve as a basis to study the evolution of polymerizing actin networks and the mechanism of actin driven motion. Comment: 19 pages, 7 figures
11/2010;
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ABSTRACT: Eukaryotic cells and intracellular pathogens such as bacteria or viruses utilize the actin polymerization machinery to propel themselves forward. Thereby, the onset of motion and choice of direction may be the result of a spontaneous symmetry-breaking or might be triggered by external signals and preexisting asymmetries, e.g. through a previous septation in bacteria. Although very complex, a key feature of cellular motility is the ability of actin to form dense polymeric networks, whose microstructure is tightly regulated by the cell. These polar actin networks produce the forces necessary for propulsion but may also be at the origin of a spontaneous symmetry-breaking. Understanding the exact role of actin dynamics in cell motility requires multiscale approaches which capture at the same time the polymer network structure and dynamics on the scale of a few nanometers and the macroscopic distribution of elastic stresses on the scale of the whole cell. In this chapter we review a selection of theories on how mechanical material properties and growth processes interact to induce the onset of actin based motion. Comment: 16 pages, 14 figures, chapter in book "Cell mechanics: from single scale-based models to multiscale modelling"
09/2009;
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ABSTRACT: Thanks to their geometrical organization at the cell level, soft biological tissues can be modelled from the mechanical point of view as multidimensional networks of elastic bars. The length of the bars is supposed to be small with respect to the size of the macroscopic medium. We introduce a detailed description of the overall structure accounting both for the tensions due to the bars and for the moments between pairs of bars. Using quasi-periodicity hypotheses, we apply a discrete homogenization technique, [1]. We derive a continuous homogenized mechanical law in the large transformation setting. We describe the basic principles of this approach that was first introduced in the cardiac modelling context in [2]. In a last step we present its implementation in a finite element framework. We comment some aspects of our numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
PAMM 11/2007; 7(1):1121601 - 1121602.
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ABSTRACT: To address the advantages and drawbacks of quantitative polarized light microscopy for the study of myocardial cell orientation and to identify its contribution in the field.
Quantitative polarized light microscopy allows to measure the orientation of myocardial fibers into the ventricular mass. For each pixel of a horizontal section, this orientation is the mean value of the directions of all myosin filaments contained in the thickness of the section for each pixel of the section and is accounted for by two angles, the azimuth angle, which is the angle of the fiber in the plane of the section, and the elevation angle, which measures the way the fiber escapes from the section. The azimuth is accurately measured, and its range of definition is complete from 0 degrees to 180 degrees . The elevation angle can be defined only in the range 0 degrees to 90 degrees . It is accurately measured between 20 degrees and 70 degrees . From 0 degrees to 20 degrees , there is a systematic bias raising the measured values, and from 70 degrees to 90 degrees , the angle is not accurately measured.
With this method, we validated Streeter's conjecture concerning the architecture of the left ventricle. We formulated a pretzel conjecture about the fiber architecture of the whole ventricular mass during fetal period. In our model, elaborated by visual analysis of registered maps of orientation, the fibers run like geodesics on a nested set of 'pretzels'. Next, the validity of the helical ventricular myocardial band model of Torrent-Guasp has been examined. It appears that the band model does not account for the patterns observed in our data during the fetal period. However, after the major events of postnatal cardiovascular adaptation, our data can neither discard nor confirm Torrent-Guasp's model.
Present limitations of quantitative polarized light analysis can neither confirm nor discard the existing models of fiber orientation in the whole ventricular mass after the neonatal period. However, the problems of mathematical and experimental validation of these two models have been posed in a rigorous manner. Non-ambiguous fiber tracking and demonstration of these models will require significant improvement of the definition range of the elevation angle that should be extended to 180 degrees .
European Journal of Cardio-Thoracic Surgery 06/2007; 31(5):915-21. · 2.55 Impact Factor
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ABSTRACT: Graphene sheets can be considered as lattices consisting of atoms and of interatomic bonds. Their bond lengths are smaller than one nanometer. Simple models describe their behavior by an energy that takes into account both the interatomic lengths and the angles between bonds. We make use of their periodic structure and we construct an equivalent macroscopic model by means of a discrete homogenization technique. Large three-dimensional deformations of graphene sheets are thus governed by a membrane model whose constitutive law is implicit. By linearizing around a prestressed configuration, we obtain linear membrane models that are valid for small displacements and whose constitutive laws are explicit. When restricting to two-dimensional deformations, we can linearize around a rest configuration and we provide explicit macroscopical mechanical constants expressed in terms of the interatomic tension and bending stiffnesses.
Journal of Elasticity 06/2006; 84(1):33-68. · 1.11 Impact Factor
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Functional Imaging and Modeling of the Heart, First International Workshop, Helsinki, Finland, November 15-16, 2001, Proceedings; 01/2001
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ABSTRACT: Phospholipidic membranes and vesicles constitute a basic element in real biological functions. Vesicles are viewed as a model system to mimic basic viscoelastic behaviors of some cells, like red blood cells. Phase field and level-set models are powerful tools to tackle dynamics of membranes and their coupling to the flow. These two methods are somewhat similar, but to date no bridge between them has been made. This is a first focus of this paper. Furthermore, a constitutive viscoelastic law is derived for the composite fluid: the ambient fluid and the membranes. We present two different approaches to deal with the membrane local incompressibility, and point out differences. Some numerical results following from the level-set approach are presented.
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ABSTRACT: We derive a constitutive law for the myocardium from the description of both the geometrical arrangement of cardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discrete lattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discrete homogenization technique. The macroscopic constitutive law is obtained through the resolution of a nonlinear self-equilibrum system of the discrete lattice reference cell.
http://dx.doi.org/10.1051/m2an:2003054.
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ABSTRACT: In the recent years, lattice modelling proved to be a topic of renewed interest. Indeed, fields as distant as chemical modelling and biological tissue modelling use network models that appeal to similar equilibrium laws. In both cases, obtaining an equivalent continuous model allows to simplify numerical procedures. We describe an homogenization technique designed for discrete structures that provides a limit continuum mechanics model and, in the special case of hexagonal lattices, we investigate the symmetry properties of the limit constitutive law.
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ABSTRACT: We describe in this paper two applications of Eulerian level set methods to fluid-structure problems arising in biophysics. The first one is concerned with three-dimensional equilibrium shapes of phospholipidic vesicles. This is a complex problem, which can be recast as the minimization of the curvature energy of an immersed elastic membrane, under a constant area constraint. The second deals with isolated cardiomyocyte contraction. This problem corresponds to a generic incompressible fluid-structure coupling between an elastic body and a fluid. By the choice of these two quite different situations, we aim to bring evidence that Eulerian methods provide efficient and flexible computational tools in biophysics applications.
Mathematical and Computer Modelling.
