Theoretical model for cellular shapes driven by protrusive and adhesive forces

The forces that arise from the actin cytoskeleton play a crucial role in determining the cell shape. These include protrusive forces due to actin polymerization and adhesion to the external matrix. We present here a theoretical model for the cellular shapes resulting from the feedback between the membrane shape and the forces acting on the membrane, mediated by curvature-sensitive membrane complexes of a convex shape. In previous theoretical studies we have investigated the regimes of linear instability where spontaneous formation of cellular protrusions is initiated. Here we calculate the evolution of a two dimensional cell contour beyond the linear regime and determine the final steady-state shapes arising within the model. We find that shapes driven by adhesion or by actin polymerization (lamellipodia) have very different morphologies, as observed in cells. Furthermore, we find that as the strength of the protrusive forces diminish, the system approaches a stabilization of a periodic pattern of protrusions. This result can provide an explanation for a number of puzzling experimental observations regarding cellular shape dependence on the properties of the extra-cellular matrix.     

Cyclic loading on the red cell membrane in a shear flow: a possible cause of hemolysis

The fluid force acting on single human red cells in a high shear flow was analyzed. A two-dimensional elliptical microcapsule as a model of the deformed red cells was adopted to numerically calculate the distributions of the shear forces on both sides of the cell membrane. It is theoretically shown that the cell membrane undergoes an unsteady cyclic loading under the rotational motion around the interior. The mechanism leading to blood cell trauma is examined by repeatedly loading the continuously moving cell membrane.     

Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Cultured vascular endothelial cells undergo significant morphological changes when subjected to sustained fluid shear stress. The cells elongate and align in the direction of applied flow. Accompanying this shape change is a reorganization at the intracellular level. The cytoskeletal actin filaments reorient in the direction of the cells' long axis. How this external stimulus is transmitted to the endothelial cytoskeleton still remains unclear. In this article. we present a theoretical model accounting for the cytoskeletal reorganization under the influence of fluid shear stress. We develop a system of integro-partial-differential equations describing the dynamics of actin filaments, the actin-binding proteins, and the drift of transmembrane proteins due to the fluid shear forces applied on the plasma membrane. Numerical simulations of the equations show that under certain conditions, initially randomly oriented cytoskeletal actin filaments reorient in structures parallel to the externally applied fluid shear forces. Thus, the model suggests a mechanism by which shear forces acting on the cell membrane can be transmitted to the entire cytoskeleton via molecular interactions alone.     

The hydrodynamic components in joints: proof of a membrane-cistern system within the calcified zone of the femur head cartilage

The theoretical postulate of a hydrodynamic system for the transmission of forces at the hip joint led to the discovery of a two-membrane system. Within the calcified cartilaginous zone of the femoral head there exists a system of small spaces and cisterns which can only be understood in the postulated sense as an overall acting hydrodynamic system. A correlation between the destruction of the calcified zone, i.e. the hydrodynamic system, and arthrosis is indicated. Ongoing studies and deliberations will help to discover the true model of the transmission of forces and impulses and to solve the problems of treating arthrosis.     

Spectrin, red cell shape and deformability. II. The antagonistic action of spectrin and sialic acid residues in determining membrane curvature in genetic spectrin deficiency in mice

In a companion paper, the shapes of spectrin deficient mouse erythrocytes were described; in contrast to previous assumptions, spherules with tethered microvesicles rather than true "spherocytes" were found. Thence, spectrin deficient mouse erythrocytes are endowed with an excess of surface area for the given volume but the membrane is assuming a highly positive curvature. Observations during and after the action of enzymes cleaving the red cell surface charge (Neuraminidase, Trypsin, Chymotrypsin) showed that the previously positive membrane curvature, as well as the tendency of the membrane to flow into fingerlike protrusions was completely abolished. The erythrocytes of the spectrin deficient, desialylated mouse erythrocytes assumed a variety of shapes, often discocytic or even stomatocytic, i.e. their membrane presented with negative curvature. However, while these desialylated membranes could be easily deformed (elongated) by shear flow they did not recoil elastically into any definitive configuration after removal of the deforming forces. It is concluded from these observations that spectrin (acting on the inner interface between membrane and cytoplasm) and sialic acid residues (acting on the outer interface between membrane and plasma) exert antagonizing effects on membrane curvature and membrane bending elasticity. Sialic acid residues, strongly charged and situated on the outer side of the cell, produce positive membrane curvature; this observation can most readily be explained by assuming that this mechanical effect is caused by repulsive coulombic forces expanding the outer half of the bilayer. To explain the effect of the spectrin-complex in counteracting positive or in producing negative membrane curvature, a similar expansive coulombic force acting between the highly charged residues has been postulated. Thence, a model for explaining the overall elastic behaviour of the normal mammalian red cell is developed which is based on the assumption of elastic interactions of proteinacous membrane components coupled to the lipid bilayer of the membrane.     

Function-dependent shape characteristics of the human skull

Using the FEM-program ANSYS 5.4, we have shaped a model of the human skull in which the flow of forces and the relative location and magnitudes of stresses are investigated. Forces are applied from below through the tooth row of the upper jaw. An ample volume is provided for the transmission of these bite forces upward to the roof of the braincase, where bearings counteract the forces from below. Within this volume, no other morphological features are considered than two cone-shaped orbits and a nasal channel which has a rounded, triangular cross section, extending upward between the orbits. Under loads (= bite forces) acting simultaneously in the directions and relative sizes of realistic bite- and chewing forces, there occurred stress concentrations inside the model which resemble closely the morphological characteristics of the human skull. The most remarkable pathways of stresses correspond to Toldt's and Benninghoff's nasal, zygomatic and pterygoid pillars. Aside from these stress concentrations, stress-free regions become visible at places, where the skull shows excavations: the vaulted palate with canalis incisivus, the canine fossa, superior and inferior orbital fissure, or cavities like the maxillary sinuses and cavum cranii. Behind the posterior molars and the pterygoid, the stresses disappear abruptly, and in the side wall of the nasal cavity a maxillary hiatus remains without stresses. A flow of forces comparable to, but not at the exact position of the zygomatic arch extends from the highly stressed zygomatic bone rearward and upward. In a later step of simulation, somewhat deeper, at the place of the really existing zygomatic arch, a series of small forces was applied, which correspond to the resultant force that is created by the redirection of the pull of the m. masseter into the temporal fascia. This--biologically reasonable--manipulation of the model leads to a reduction of the forces in the zygomatic bone, and to a downward shift of the zygomatic arch and its isolation from the skull's side wall by a deep, stress-free temporal fossa. The similarity between the stress flow in the model and the shape of the skull seems to indicate that the skull, like the bones of the postcranial skeleton, develops its shape in dependence from the mechanic stressing through the process of causal histogenesis. In view of experimental results, the possibility cannot be ruled out, that the safety factors in the skull deviate from those in the postcranial skeleton.     

Flow around cells adhered to a microvessel wall. I. Fluid stresses and forces acting on the cells

To evaluate the fluid forces acting on cells adhered to a microvessel wall, we numerically studied the flow field around adherent cells and the distribution of the stresses on their surfaces. For simplicity, the cells were modeled as rigid particles attached to a wall of a circular cylindrical tube regularly in the flow direction, in a row or two rows. It was found that not the detailed shape of the model cells but their height from the vessel wall is a key determinant of the fluid forces and torque acting on them. In both arrangements of one row and two rows, the axial spacing between neighboring adherent cells significantly affects the distributions of the stresses on them, which results in drastic variations of the fluid forces with the axial spacing and the relative positions with respect to their neighboring cells. The drag force acting on an adherent cell in the vessel was evaluated to be larger than the value in the 2D chamber flow at the same wall shear stress, mainly due to much larger variations of the pressure distribution on the cell surface in the vessel flow.     

Lateral organization of membranes and cell shapes.

The relations among membrane structure, mechanical properties, and cell shape have been investigated. The fluid mosaic membrane models used contains several components that move freely in the membrane plane. These components interact with each other and determine properties of the membrane such as curvature and elasticity. A free energy equation is postulated for such a multicomponent membrane and the condition of free energy minimum is used to obtain differential equations relating the distribution of membrane components and the local membrane curvature. The force that moves membrane components along the membrane in a variable curvature field is calculated. A change in the intramembrane interactions can bring about phase separation or particle clustering. This, in turn, may strongly affect the local curvature. The numerical solution of the set of equations for the two dimensional case allows determination of the cell shape and the component distribution along the membrane. The model has been applied to describe certain erythrocytes shape transformations.     

Implications of a poroelastic cytoplasm for the dynamics of animal cell shape

Two views have dominated recent discussions of the physical basis of cell shape change during migration and division of animal cells: the cytoplasm can be modeled as a viscoelastic continuum, and the forces that change its shape are generated only by actin polymerization and actomyosin contractility in the cell cortex. Here, we question both views: we suggest that the cytoplasm is better described as poroelastic, and that hydrodynamic forces may be generally important for its shape dynamics. In the poroelastic view, the cytoplasm consists of a porous, elastic solid (cytoskeleton, organelles, ribosomes) penetrated by an interstitial fluid (cytosol) that moves through the pores in response to pressure gradients. If the pore size is small (30-60nm), as has been observed in some cells, pressure does not globally equilibrate on time and length scales relevant to cell motility. Pressure differences across the plasma membrane drive blebbing, and potentially other type of protrusive motility. In the poroelastic view, these pressures can be higher in one part of a cell than another, and can thus cause local shape change. Local pressure transients could be generated by actomyosin contractility, or by local activation of osmogenic ion transporters in the plasma membrane. We propose that local activation of Na(+)/H(+) antiporters (NHE1) at the front of migrating cells promotes local swelling there to help drive protrusive motility, acting in combination with actin polymerization. Local shrinking at the equator of dividing cells may similarly help drive invagination during cytokinesis, acting in combination with actomyosin contractility. Testing these hypotheses is not easy, as water is a difficult analyte to track, and will require a joint effort of the cytoskeleton and ion physiology communities.     

Shape memory of human red blood cells

The human red cell can be deformed by external forces but returns to the biconcave resting shape after removal of the forces. If after such shape excursions the rim is always formed by the same part of the membrane, the cell is said to have a memory of its biconcave shape. If the rim can form anywhere on the membrane, the cell would have no shape memory. The shape memory was probed by an experiment called go-and-stop. Locations on the membrane were marked by spontaneously adhering latex spheres. Shape excursions were induced by shear flow. In virtually all red cells, a shape memory was found. After stop of flow and during the return of the latex spheres to the original location, the red cell shape was biconcave. The return occurred by a tank-tread motion of the membrane. The memory could not be eliminated by deforming the red cells in shear flow up to 4 h at room temperature as well as at 37 degrees C. It is suggested that 1). the characteristic time of stress relaxation is >80 min and 2). red cells in vivo also have a shape memory.     

Post-pleistocene reductions in human dental structure: a reappraisal in terms of increasing population density

This paper discusses the well-documented acceleration which occurred in the reduction of human dental structure during the post-Pleistocene. It suggests that the process can be explained in terms of different but related factors inherent in a transition from late-Pleistocene hunting and gathering conditions to ones involving a sedentary life in larger groups. It is postulated that directional selective forces acting during Upper Paleolithic to maintain large tooth size had only a poor influence and that the new techniques in food preparation had little or no effect. The phenomenon is seen as a possible side-effect of a more complex overall reduction in body size, of which dental reduction only represents a small but demonstrable part. It is also suggested that stresses may have played an important role in producing a transitory reduced dimensional expression of the genetic background. An interaction between the variables directly or indirectly affecting body mass is shown in the model proposed which relates to post-Pleistocene. An increase in population density is indicated as being the most decisive biological factor in determining the acceleration in the trend towards reduction. A preliminary version of this paper was presented at the Symposium on Upper Paleolithic/Mesolithic of Europe and the Mediterranean Basin, Pisa, September 8–10, 1984.     

The shape of the mandible in the domestic sheep: a biomechanical analysis using EMG as an estimator of muscle force

Analysis of the maximal loads to which a skeletal element is subjected in vivo offers attractive possibilities of explaining the shape of the element. The underlying assumption is that the element will be constructed in such a way that deformations (strains) which result from mechanical stress will not exceed certain limits and that stresses will be evenly distributed. The sheep mandible shows a number of characteristic morphological features that invite this kind of explanation. We investigated the patterns of activity of the masticatory musculature by multichannel electromyography, expressing the activity of any particular muscle during a given interval as a percentage of the highest activity recorded for the muscle in question. In combination with data on the physiological cross sections of the muscles, which provide indications of the maximal forces which can be exerted by the muscles, three-dimensional patterns of relative muscular forces acting on the mandible can be constructed for successive stages of a masticatory cycle. No absolute forces can be measured or even estimated by this technique. A two-dimensional finite element model of the mandible was designed, by means of which predictions of stress and strain resulting from the muscular loading can be made. Calculations were based on the highest loads that occurred, during the power stroke of rumination. It is concluded that mechanical loading of the mandible provides a partial explanation of the form, and that a more satisfactory model should include other than purely mechanical influences.     

Mechanisms Controlling Cell Size and Shape during Isotropic Cell Spreading

Cell motility is important for many developmental and physiological processes. Motility arises from interactions between physical forces at the cell surface membrane and the biochemical reactions that control the actin cytoskeleton. To computationally analyze how these factors interact, we built a three-dimensional stochastic model of the experimentally observed isotropic spreading phase of mammalian fibroblasts. The multiscale model is composed at the microscopic levels of three actin filament remodeling reactions that occur stochastically in space and time, and these reactions are regulated by the membrane forces due to membrane surface resistance (load) and bending energy. The macroscopic output of the model (isotropic spreading of the whole cell) occurs due to the movement of the leading edge, resulting solely from membrane force-constrained biochemical reactions. Numerical simulations indicate that our model qualitatively captures the experimentally observed isotropic cell-spreading behavior. The model predicts that increasing the capping protein concentration will lead to a proportional decrease in the spread radius of the cell. This prediction was experimentally confirmed with the use of Cytochalasin D, which caps growing actin filaments. Similarly, the predicted effect of actin monomer concentration was experimentally verified by using Latrunculin A. Parameter variation analyses indicate that membrane physical forces control cell shape during spreading, whereas the biochemical reactions underlying actin cytoskeleton dynamics control cell size (i.e., the rate of spreading). Thus, during cell spreading, a balance between the biochemical and biophysical properties determines the cell size and shape. These mechanistic insights can provide a format for understanding how force and chemical signals together modulate cellular regulatory networks to control cell motility.     

Prediction of fluid forces acting on a hand model in unsteady flow conditions

The aim of this study was to develop a method to predict fluid forces acting on the human hand in unsteady flow swimming conditions. A mechanical system consisting of a pulley and chain mechanism and load cell was constructed to rotate a hand model in fluid flows. To measure the angular displacement of the hand model a potentiometer was attached to the axis of the rotation. The hand model was then fixed at various angles about the longitudinal axis of the hand model and rotated at different flow velocities in a swimming flume for 258 different trials to approximate a swimmer's stroke in unsteady flow conditions. Pressures were taken from 12 transducers embedded in the hand model at a sampling frequency of 200Hz. The resultant fluid force acting on the hand model was then determined on the basis of the kinetic and kinematic data taken from the mechanical system at the frequency of 200Hz. A stepwise regression analysis was applied to acquire higher order polynomial equations that predict the fluid force acting on the accelerating hand model from the 12 pressure values. The root mean square (RMS) difference between the resultant fluid force measured and that predicted from the single best-fit polynomial equation across all trials was 5N. The method developed in the present study accurately predicted the fluid forces acting on the hand model.     

Analysis of the interaction between vertebral lateral deviation and axial rotation in scoliosis

There is a lack of clear biomechanical analyses to explain the interaction of the lateral and axial deformity of the spine in idiopathic scoliosis. A finite element model which represented an isolated ligamentous spine with realistic elastic properties and idealized geometry was used to analyse this interaction. Three variations of this model were used to investigate two different hypotheses about the etiology of scoliosis and to define the forces required to produce a scoliosis deformity. The first hypothesis is that coupling within a motion segment produces the interaction between lateral deviation and axial rotation. The second hypothesis is that posterior tethering by soft tissues in the growing spine produces the observed interaction. Modeling of both hypotheses failed to produce the clinically observed pattern of interaction. Therefore, to find which biomechanical forces were required to produce an idealized scoliosis, prescribed displacements were applied to the model. Production of a double curve scoliosis of 10 degrees Cobb angles required lateral forces on the order of 20 N acting 40 mm anterior to the vertebral body centers. There do not appear to be any anatomic structures capable of producing such forces. Therefore, it seems unlikely that scoliosis deformity can be explained in terms of forces acting on the spine, and understanding of its origins may come from examination of other mechanisms such as asymmetric thoracic growth, or asymmetric vertebral development.     

Membrane-Mediated Interactions Measured Using Membrane Domains

Cell membrane organization is the result of the collective effect of many driving forces. Several of these, such as electrostatic and van der Waals forces, have been identified and studied in detail. In this article, we investigate and quantify another force, the interaction between inclusions via deformations of the membrane shape. For electrically neutral systems, this interaction is the dominant organizing force. As a model system to study membrane-mediated interactions, we use phase-separated biomimetic vesicles that exhibit coexistence of liquid-ordered and liquid-disordered lipid domains. The membrane-mediated interactions between these domains lead to a rich variety of effects, including the creation of long-range order and the setting of a preferred domain size. Our findings also apply to the interaction of membrane protein patches, which induce similar membrane shape deformations and hence experience similar interactions.     

Force Transduction and Lipid Binding in MscL: A Continuum-Molecular Approach

The bacterial mechanosensitive channel MscL, a small protein mainly activated by membrane tension, is a central model system to study the transduction of mechanical stimuli into chemical signals. Mutagenic studies suggest that MscL gating strongly depends on both intra-protein and interfacial lipid-protein interactions. However, there is a gap between this detailed chemical information and current mechanical models of MscL gating. Here, we investigate the MscL bilayer-protein interface through molecular dynamics simulations, and take a combined continuum-molecular approach to connect chemistry and mechanics. We quantify the effect of membrane tension on the forces acting on the surface of the channel, and identify interactions that may be critical in the force transduction between the membrane and MscL. We find that the local stress distribution on the protein surface is largely asymmetric, particularly under tension, with the cytoplasmic side showing significantly larger and more localized forces, which pull the protein radially outward. The molecular interactions that mediate this behavior arise from hydrogen bonds between the electronegative oxygens in the lipid headgroup and a cluster of positively charged lysine residues on the amphipathic S1 domain and the C-terminal end of the second trans-membrane helix. We take advantage of this strong interaction (estimated to be 10–13 kT per lipid) to actuate the channel (by applying forces on protein-bound lipids) and explore its sensitivity to the pulling magnitude and direction. We conclude by highlighting the simple motif that confers MscL with strong anchoring to the bilayer, and its presence in various integral membrane proteins including the human mechanosensitive channel K2P1 and bovine rhodopsin.     

Predicting the hydraulic forces on submerged macrophytes from current velocity, biomass and morphology

Aquatic macrophytes are important in stabilising moderately eutrophic, shallow freshwater lakes in the clear-water state. The failure of macrophyte recovery in lakes with very soft, highly organic sediments that have been restored to clear water by biomanipulation (e.g. in the Norfolk Broads, UK) has suggested that the physical stability of the sediment may limit plant establishment. Hydraulic forces from water currents may be sufficient to break or remove plants. Our aim was to develop a simple model that could predict these forces from plant biomass, current velocity and plant form. We used an experimental flume to measure the hydraulic forces acting on shoots of 18 species of aquatic macrophyte of varying size and morphology. The hydraulic drag on the shoots was regressed on a theoretically derived predictor (shoot biomass × current velocity^1.5). Such linear regressions proved to be highly significant for most species. The slopes of these lines represent species-specific, hydraulic roughness factors that are analogous to classical drag coefficients. Shoot architecture parameters describing leaf and shoot shape had significant effects on the hydraulic roughness factor. Leaf width and shoot stiffness individually did not have a significant influence, but in combination with shoot shape they were significant. This hydraulic model was validated for a subset of species using measurements from an independent set of shoots. When measured and predicted hydraulic forces were compared, the fit was generally very good, except for two species with morphological variations. This simple model, together with the plant-specific factors, provides a basis for predicting the hydraulic forces acting on the root systems of macrophytes under field conditions. This information should allow prediction of the physical stability of individual plants, as an aid to shallow-lake management. Received: 11 March 1999 / Accepted: 18 January 2000     

Direct and Model Free Calculation of Force-Dependent Dissociation Rates from Force Spectroscopic Data

Force spectroscopy allows testing the free energy landscapes of molecular interactions. Usually, the dependency of the most probable rupture force on the force rate or the shape of the rupture force histogram is fitted with different models that contain approximations and basic assumptions. We present a simple and model free approach to extract the force-dependent dissociation rates directly from the force curve data. Simulations show that the dissociation rates at any force are given directly by the ratio of the number of detected rupture events to the time this force was acting on the bond. To calculate these total times of acting forces, all force curve data points of all curves measured are taken into account, which significantly increases the amount of information which is considered for data analysis compared to other methods. Moreover, by providing force-dependent dissociation rates this method allows direct testing and validating of any energy landscape model.