Physical approach of biological problems

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Group leader : Pierre Sens

 Read the scientific activity report. (pdf 31Ko, last update 26th, march 2010)

A rapid inspection of orders of magnitude involved in cell components show that they are very similar to those relevant to “Soft Matter Physics”. There are however two important differences: biological systems are clearly out of equilibrium and molecular specificity can be strongly relevant. These simple remarks convince us that on the one hand Soft Matter Physics can provide a quantitative description of cellular systems, and that on the other hand biological systems raise an interesting number of new and challenging physical questions. For these reasons we concentrate our efforts towards understanding physical features of cell morphology and dynamics. This project is meaningful only with strong interactions with biologists.

Cells contain a very large number of components, but if we focus on mechanical properties, only a few classes of component are relevant: the cytoskeletal networks, molecular motors, phospholipid membranes and the large class of adhesion molecules such as integrins or cadherins. Therefore we study each of these components, keeping in mind the importance of the non-equilibrium behavior. In some cases, this requires the introduction of new physical concepts such as “active” membranes, “active” gels or  “isothermal ratchet”, which is a model to describe molecular motors by the Brownian motion of a particle switching between two different states.

A good physical understanding requires quantitative comparison between theory and experiments by systematically varying controlled parameters: for that reason, we work in close collaboration with the experimental groups both in our laboratory and in the Curie Subcellular structure and cellular dynamics Unit (UMR 144). We contribute to the theoretical description of polymerization-based motion using biological models such as the bacteria Listeria and the keratocytes type of cells, but also biomimetic systems such as plastic beads and oil drops properly treated. We also describe  certain  aspects  of  cell behavior such as phospholipidic nanotube pulling by  molecular  motors,  cell motility, cell division and mechano-transduction. As an additional example, we show on Figure 1 the results of calculations on spontaneous oscillations  of assemblies of molecular motors


Fig. 1 Motor oscillations: Oscillations of assemblies of molecular motors obtained from stochastic simulations. The model is sketched in (a) which shows the potentials seen by the motors. The oscillations in the position of the filament interacting with the motors, the power spectrum and the histogram of the positions of the filament are then shown for two sets of parameters.Fig. 1 Motor oscillations: Oscillations of assemblies of molecular motors obtained from stochastic simulations. The model is sketched in (a) which shows the potentials seen by the motors. The oscillations in the position of the filament interacting with the motors, the power spectrum and the histogram of the positions of the filament are then shown for two sets of parameters.


We  have  now  reached  a  reasonable  physical  understanding   of   single components and are consequently extending our analysis  to  the  interaction between  components, that is to  the  multi-cellular  level  and  the  mechanical  properties  of tissues. We have for example  performed  numerical  simulations  of  tissues competing for space (see Figure 2 below).

Fig. 2 Tissue competition: Numerical simulation of the competition between two tissues.Fig. 2 Tissue competition: Numerical simulation of the competition between two tissues.

At last, we keep close contact with the evolution  of  statistical  physics. In particular  we  have  derived  a  general  relation  between  the  linear response of a system to any  external  perturbation  and  its  fluctuations. This relation is valid as soon as the system is  described  by  a  Markovian dynamics. It will  be  useful  for  discussing  fluctuations  of  biological systems.

Key publications

  • Year of publication : 2015

  • An essential question of morphogenesis is how patterns arise without preexisting positional information, as inspired by Turing. In the past few years, cytoskeletal flows in the cell cortex have been identified as a key mechanism of molecular patterning at the subcellular level. Theoretical and in vitro studies have suggested that biological polymers such as actomyosin gels have the property to self-organize, but the applicability of this concept in an in vivo setting remains unclear. Here, we report that the regular spacing pattern of supracellular actin rings in the Drosophila tracheal tubule is governed by a self-organizing principle. We propose a simple biophysical model where pattern formation arises from the interplay of myosin contractility and actin turnover. We validate the hypotheses of the model using photobleaching experiments and report that the formation of actin rings is contractility dependent. Moreover, genetic and pharmacological perturbations of the physical properties of the actomyosin gel modify the spacing of the pattern, as the model predicted. In addition, our model posited a role of cortical friction in stabilizing the spacing pattern of actin rings. Consistently, genetic depletion of apical extracellular matrix caused strikingly dynamic movements of actin rings, mirroring our model prediction of a transition from steady to chaotic actin patterns at low cortical friction. Our results therefore demonstrate quantitatively that a hydrodynamical instability of the actin cortex can trigger regular pattern formation and drive morphogenesis in an in vivo setting.

  • Tissue fusion eliminates physical voids in a tissue to form a continuous structure and is central to many processes in development and repair. Fusion events in vivo, particularly in embryonic development, often involve the purse-string contraction of a pluricellular actomyosin cable at the free edge. However, in vitro, adhesion of the cells to their substrate favors a closure mechanism mediated by lamellipodial protrusions, which has prevented a systematic study of the purse-string mechanism. Here, we show that monolayers can cover well-controlled mesoscopic nonadherent areas much larger than a cell size by purse-string closure and that active epithelial fluctuations are required for this process. We have formulated a simple stochastic model that includes purse-string contractility, tissue fluctuations, and effective friction to qualitatively and quantitatively account for the dynamics of closure. Our data suggest that, in vivo, tissue fusion adapts to the local environment by coordinating lamellipodial protrusions and purse-string contractions.

  • The mechanical behaviour of cells is largely controlled by a structure that is fundamentally out of thermodynamic equilibrium: a network of crosslinked filaments subjected to the action of energy-transducing molecular motors. The study of this kind of active system was absent from conventional physics and there was a need for both new theories and new experiments. The field that has emerged in recent years to fill this gap is underpinned by a theory that takes into account the transduction of chemical energy on the molecular scale. This formalism has advanced our understanding of living systems, but it has also had an impact on research in physics per se. Here, we describe this developing field, its relevance to biology, the novelty it conveys to other areas of physics and some of the challenges in store for the future of active gel physics.

  • Year of publication : 2014

  • Morphogenesis during embryo development requires the coordination of mechanical forces to generate the macroscopic shapes of organs. We propose a minimal theoretical model, based on cell adhesion and actomyosin contractility, which describes the various shapes of epithelial cells and the bending and buckling of epithelial sheets, as well as the relative stability of cellular tubes and spheres. We show that, to understand these processes, a full 3D description of the cells is needed, but that simple scaling laws can still be derived. The morphologies observed in vivo can be understood as stable points of mechanical equations and the transitions between them are either continuous or discontinuous. We then focus on epithelial sheet bending, a ubiquitous morphogenetic process. We calculate the curvature of an epithelium as a function of actin belt tension as well as of cell-cell and and cell-substrate tension. The model allows for a comparison of the relative stabilities of spherical or cylindrical cellular structures (acini or tubes). Finally, we propose a unique type of buckling instability of epithelia, driven by a flattening of individual cell shapes, and discuss experimental tests to verify our predictions.

  • Cytokinesis is the process of physical cleavage at the end of cell division; it proceeds by ingression of an acto-myosin furrow at the equator of the cell. Its failure leads to multinucleated cells and is a possible cause of tumorigenesis. Here, we calculate the full dynamics of furrow ingression and predict cytokinesis completion above a well-defined threshold of equatorial contractility. The cortical acto-myosin is identified as the main source of mechanical dissipation and active forces. Thereupon, we propose a viscous active nonlinear membrane theory of the cortex that explicitly includes actin turnover and where the active RhoA signal leads to an equatorial band of myosin overactivity. The resulting cortex deformation is calculated numerically, and reproduces well the features of cytokinesis such as cell shape and cortical flows toward the equator. Our theory gives a physical explanation of the independence of cytokinesis duration on cell size in embryos. It also predicts a critical role of turnover on the rate and success of furrow constriction. Scaling arguments allow for a simple interpretation of the numerical results and unveil the key mechanism that generates the threshold for cytokinesis completion: cytoplasmic incompressibility results in a competition between the furrow line tension and the cell poles' surface tension.

  • We study the closure dynamics of a large number of well-controlled circular apertures within an epithelial monolayer, where the collective cell migration responsible for epithelization is triggered by the removal of a spatial constraint rather than by scratching. Based on experimental observations, we propose a physical model that takes into account border forces, friction with the substrate, and tissue rheology. Border protrusive activity drives epithelization despite the presence of a contractile actomyosin cable at the periphery of the wound. The closure dynamics is quantified by an epithelization coefficient, defined as the ratio of protrusive stress to tissue-substrate friction, that allows classification of different phenotypes. The same analysis demonstrates a distinct signature for human cells bearing the oncogenic RasV12 mutation, demonstrating the potential of the approach to quantitatively characterize metastatic transformations.

  • Year of publication : 2013

  • Collective cell motion is observed in a wide range of biological processes. In tumors, physiological gradients of nutrients, growth factors, or even oxygen give rise to gradients of proliferation. We show using fluorescently labeled particles that these gradients drive a velocity field resulting in a cellular flow in multicellular spheroids. Under mechanical stress, the cellular flow is drastically reduced. We describe the results with a hydrodynamic model that considers only convection of the particles by the cellular flow.

  • Epithelial spreading is a common and fundamental aspect of various developmental and disease-related processes such as epithelial closure and wound healing. A key challenge for epithelial tissues undergoing spreading is to increase their surface area without disrupting epithelial integrity. Here we show that orienting cell divisions by tension constitutes an efficient mechanism by which the enveloping cell layer (EVL) releases anisotropic tension while undergoing spreading during zebrafish epiboly. The control of EVL cell-division orientation by tension involves cell elongation and requires myosin II activity to align the mitotic spindle with the main tension axis. We also found that in the absence of tension-oriented cell divisions and in the presence of increased tissue tension, EVL cells undergo ectopic fusions, suggesting that the reduction of tension anisotropy by oriented cell divisions is required to prevent EVL cells from fusing. We conclude that cell-division orientation by tension constitutes a key mechanism for limiting tension anisotropy and thus promoting tissue spreading during EVL epiboly.

  • This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are “dry” systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or “wet” systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed.

  • Year of publication : 2012

  • Cross-linked actomyosin bundles retract when severed in vivo by laser ablation, or when isolated from the cell and micromanipulated in vitro in the presence of ATP. We identify the timescale for contraction as a viscoelastic time τ, where the viscosity is due to (internal) protein friction. We obtain an estimate of the order of magnitude of the contraction time τ ≈ 10-100 s, consistent with available experimental data for circumferential microfilament bundles and stress fibers. Our results are supported by an exactly solvable, hydrodynamic model of a retracting bundle as a cylinder of isotropic, active matter, from which the order of magnitude of the active stress is estimated.

  • {We study theoretically the morphologies of biological tubes affected by various pathologies. When epithelial cells grow, the negative tension produced by their division provokes a buckling instability. Several shapes are investigated: varicose, dilated, sinuous, or sausagelike. They are all found in pathologies of tracheal, renal tubes, or arteries. The final shape depends crucially on the mechanical parameters of the tissues: Young's modulus, wall-to-lumen ratio, homeostatic pressure. We argue that since tissues must be in quasistatic mechanical equilibrium, abnormal shapes convey information as to what causes the pathology. We calculate a phase diagram of tubular instabilities which could be a helpful guide for investigating the underlying genetic regulation.}

  • {Animal tissues are complex assemblies of cells, extracellular matrix (ECM), and permeating interstitial fluid. Whereas key aspects of the multicellular dynamics can be captured by a one-component continuum description, cell division and apoptosis imply material turnover between different components that can lead to additional mechanical conditions on the tissue dynamics. We extend our previous description of tissues in order to account for a cell/ECM phase and the permeating interstitial fluid independently. In line with our earlier work, we consider the cell/ECM phase to behave as an elastic solid in the absence of cell division and apoptosis. In addition, we consider the interstitial fluid as ideal on the relevant length scales, i.e., we ignore viscous stresses in the interstitial fluid. Friction between the fluid and the cell/ECM phase leads to a Darcy-like relation for the interstitial fluid velocity and introduces a new characteristic length scale. We discuss the dynamics of a tissue confined in a chamber with a permeable piston close to the homeostatic state where cell division and apoptosis balance, and we calculate the rescaled effective diffusion coefficient for cells. For different mass densities of the cell/ECM component and the interstitial fluid, a treadmilling steady state due to gravitational forces can be found.}

  • Cells are populated by a vast array of membrane-binding proteins that execute critical functions. Functions, like signaling and intracellular transport, require the abilities to bind to highly curved membranes and to trigger membrane deformation. Among these proteins is amphiphysin 1, implicated in clathrin-mediated endocytosis. It contains a Bin-Amphiphysin-Rvs membrane-binding domain with an N-terminal amphipathic helix that senses and generates membrane curvature. However, an understanding of the parameters distinguishing these two functions is missing. By pulling a highly curved nanotube of controlled radius from a giant vesicle in a solution containing amphiphysin, we observed that the action of the protein depends directly on its density on the membrane. At low densities of protein on the nearly flat vesicle, the distribution of proteins and the mechanical effects induced are described by a model based on spontaneous curvature induction. The tube radius and force are modified by protein binding but still depend on membrane tension. In the dilute limit, when practically no proteins were present on the vesicle, no mechanical effects were detected, but strong protein enrichment proportional to curvature was seen on the tube. At high densities, the radius is independent of tension and vesicle protein density, resulting from the formation of a scaffold around the tube. As a consequence, the scaling of the force with tension is modified. For the entire density range, protein was enriched on the tube as compared to the vesicle. Our approach shows that the strength of curvature sensing and mechanical effects on the tube depends on the protein density.

  • Year of publication : 2011

  • We study theoretically the shapes of a dividing epithelial monolayer of cells lying on top of an elastic stroma. The negative tension created by cell division provokes a buckling instability at a finite wave vector leading to the formation of periodic arrays of villi and crypts. The instability is similar to the buckling of a metallic plate under compression. We use the results to rationalize the various structures of the intestinal lining observed in vivo. Taking into account the coupling between cell division and local curvature, we obtain different patterns of villi and crypts, which could explain the different morphologies of the small intestine and the colon.

  • The magnitude of traction forces exerted by living animal cells on their environment is a monotonically increasing and approximately sigmoidal function of the stiffness of the external medium. We rationalize this observation using active matter theory, and propose that adaptation to substrate rigidity results from an interplay between passive elasticity and active contractility.

  • The detection of sound begins when energy derived from an acoustic stimulus deflects the hair bundles on top of hair cells. As hair bundles move, the viscous friction between stereocilia and the surrounding liquid poses a fundamental physical challenge to the ear's high sensitivity and sharp frequency selectivity. Part of the solution to this problem lies in the active process that uses energy for frequency-selective sound amplification. Here we demonstrate that a complementary part of the solution involves the fluid-structure interaction between the liquid within the hair bundle and the stereocilia. Using force measurement on a dynamically scaled model, finite-element analysis, analytical estimation of hydrodynamic forces, stochastic simulation and high-resolution interferometric measurement of hair bundles, we characterize the origin and magnitude of the forces between individual stereocilia during small hair-bundle deflections. We find that the close apposition of stereocilia effectively immobilizes the liquid between them, which reduces the drag and suppresses the relative squeezing but not the sliding mode of stereociliary motion. The obliquely oriented tip links couple the mechanotransduction channels to this least dissipative coherent mode, whereas the elastic horizontal top connectors that stabilize the structure further reduce the drag. As measured from the distortion products associated with channel gating at physiological stimulation amplitudes of tens of nanometres, the balance of viscous and elastic forces in a hair bundle permits a relative mode of motion between adjacent stereocilia that encompasses only a fraction of a nanometre. A combination of high-resolution experiments and detailed numerical modelling of fluid-structure interactions reveals the physical principles behind the basic structural features of hair bundles and shows quantitatively how these organelles are adapted to the needs of sensitive mechanotransduction.

  • Treating the epithelium as an incompressible fluid adjacent to a viscoelastic stroma, we find a novel hydrodynamic instability that leads to the formation of protrusions of the epithelium into the stroma. This instability is a candidate for epithelial fingering observed in vivo. It occurs for sufficiently large viscosity, cell-division rate and thickness of the dividing region in the epithelium. Our work provides physical insight into a potential mechanism by which interfaces between epithelia and stromas undulate and potentially by which tissue dysplasia leads to cancerous invasion.

  • Year of publication : 2010

  • We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work, we developed a model for a single actin or microtubule filament where hydrolysis occurred according to the vectorial mechanism: the filament could grow only from one end, and was in contact with a reservoir of monomers. Here we extend this approach in two ways–by including the dynamics of both ends and by comparing two possible mechanisms of ATP hydrolysis. Our emphasis is mainly on two possible limiting models for the mechanism of hydrolysis within a single filament, namely the vectorial or the random model. We propose a set of experiments to test the nature of the precise mechanism of hydrolysis within actin filaments.

  • Stress fibers are contractile actomyosin bundles commonly observed in the cytoskeleton of metazoan cells. The spatial profile of the polarity of actin filaments inside contractile actomyosin bundles is either monotonic (graded) or periodic (alternating). In the framework of linear irreversible thermodynamics, we write the constitutive equations for a polar, active, elastic one-dimensional medium. An analysis of the resulting equations for the dynamics of polarity shows that the transition from graded to alternating polarity patterns is a nonequilibrium Lifshitz point. Active contractility is a necessary condition for the emergence of sarcomeric, alternating polarity patterns.

  • We propose a two-state "soft-motor'' model for the collective behavior of molecular motors which takes into account both the internal motor stiffness and the periodic interaction with the filament. As in the Prandtl-Tomlinson model of tribology, the important parameter of the model is the pinning parameter, which compares the stiffness of the motors to the stiffness of the potential. The model predicts dynamic instabilities in two disconnected regions of parameter space. These parameter ranges correspond to two existing theories of motor assemblies, the rigid two-state model and the crossbridge model. The model also predicts a discontinuity of the slope of the force-velocity relation at small velocities.

  • During the formation of tissues, cells organize collectively by cell division and apoptosis. The multicellular dynamics of such systems is influenced by mechanical conditions and can give rise to cell rearrangements and movements. We develop a continuum description of tissue dynamics, which describes the stress distribution and the cell flow field on large scales. In the absence of division and apoptosis, we consider the tissue to behave as an elastic solid. Cell division and apoptosis introduce stress sources that, in general, are anisotropic. By combining cell number balance with dynamic equations for the stress source, we show that the tissue effectively behaves as a viscoelastic fluid with a relaxation time set by the rates of division and apoptosis. If the system is confined in a fixed volume, it reaches a homeostatic state in which division and apoptosis balance. In this state, cells undergo a diffusive random motion driven by the stochasticity of division and apoptosis. We calculate the expression for the effective diffusion coefficient as a function of the tissue parameters and compare our results concerning both diffusion and viscosity to simulations of multicellular systems using dissipative particle dynamics.

  • Year of publication : 2009

  • We study a model of an active gel of cross-linked semiflexible filaments with additional active linkers such as myosin II clusters. We show that the coupling of the elasticity of the semiflexible filaments to the mechanical properties of the motors leads to contractile behavior of the gel, in qualitative agreement with experimental observations. The motors, however, soften the zero-frequency elastic constant of the gel. When the collective motor dynamics is incorporated in the model, a stiffening of the network at high frequencies is obtained. The frequency controlling the crossover between low-and high-frequency network elasticity is estimated in terms of microscopic properties of motors and filaments, and can be as low as 10(-3) Hz. Copyright (C) EPLA, 2009

  • We propose a mechanism for the formation of contractile rings and the apparition of a flow in the cortical layer of cells undergoing cytokinesis at the end of cell division or during the healing of a wound in the cortex of Xenopus eggs. We generalize the hydrodynamic active gel theory along the lines of thin shell theory of continuum elasticity to describe the cell cortex. As in liquid crystal physics, the flow couples to the orientation of the actin filaments. The cortical flow is driven by an increased density of myosin motors in the cortex, and orients the filaments to form the ring.

  • We propose a mechanism for tumor growth emphasizing the role of homeostatic regulation and tissue stability. We show that competition between surface and bulk effects leads to the existence of a critical size that must be overcome by metastases to reach macroscopic sizes. This property can qualitatively explain the observed size distributions of metastases, while size-independent growth rates cannot account for clinical and experimental data. In addition, it potentially explains the observed preferential growth of metastases on tissue surfaces and membranes such as the pleural and peritoneal layers, suggests a mechanism underlying the seed and soil hypothesis introduced by Stephen Paget in 1889, and yields realistic values for metastatic inefficiency. We propose a number of key experiments to test these concepts. The homeostatic pressure as introduced in this work could constitute a quantitative, experimentally accessible measure for the metastatic potential of early malignant growths.

  • The fluctuation-dissipation theorem is a central result of statistical physics, which applies to any system at thermodynamic equilibrium. Its violation is a strong signature of nonequilibrium behavior. We show that for any system with Markovian dynamics, in a nonequilibrium steady state, a proper choice of observables restores a fluctuation-response theorem identical to a suitable version of the equilibrium fluctuation-dissipation theorem. This theorem applies to a broad class of dynamical systems. We illustrate it with linear stochastic dynamics and examples borrowed from the physics of molecular motors and Hopf bifurcations. Finally, we discuss general implications of the theorem.