Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

Computational models were used to explore the idea that morphogenesis is regulated, in part, by feedback from mechanical stress according to Beloussov’s hyper-restoration (HR) hypothesis. According to this hypothesis, active tissue responses to stress perturbations tend to restore, but overshoot, the original (target) stress. To capture this behavior, the rate of growth or contraction is assumed to depend on the difference between the current and target stresses. Stress overshoot is obtained by letting the target stress change at a rate proportional to the same stress difference. The feasibility of the HR hypothesis is illustrated by models for stretching of epithelia, cylindrical bending of plates, invagination of cylindrical and spherical shells, and early amphibian development. In each case, an initial perturbation leads to an active mechanical response that changes the form of the tissue. The results show that some morphogenetic processes can be entirely self-driven by HR responses once they are initiated (possibly by genetic activity). Other processes, however, may require secondary mechanisms or perturbations to proceed to completion.     

Cyclic Bending Contributes to High Stress in a Human Coronary Atherosclerotic Plaque and Rupture Risk: In Vitro Experimental Modeling and Ex Vivo MRI-Based Computational Modeling Approach

Many acute cardiovascular syndromes such as heart attack and stroke are caused by atherosclerotic plaque ruptures which often happen without warning. MRI-based models with fluid-structure interactions (FSI) have been introduced to perform flow and stress/strain analysis for atherosclerotic plaques and identify possible mechanical and morphological indices for accurate plaque vulnerability assessment. In this paper, cyclic bending was added to 3D FSI coronary plaque models for more accurate mechanical predictions. Curvature variation was prescribed using the data of a human left anterior descending (LAD) coronary artery. Five computational models were constructed based on ex vivo MRI human coronary plaque data to assess the effects of cyclic bending, pulsating pressure, plaque structure, and axial stretch on plaque stress/strain distributions. In vitro experiments using a hydrogel stenosis model with cyclical bending were performed to observe effect of cyclical bending on flow conditions. Our results indicate that cyclical bending may cause more than 100% or even up to more than 1000% increase in maximum principal stress values at locations where the plaque is bent most. Stress increase is higher when bending is coupled with axial stretch, non-smooth plaque structure, or resonant pressure conditions (zero phase angle shift). Effects of cyclic bending on flow behaviors are more modest (21.6% decrease in maximum velocity, 10.8% decrease in flow rate, maximum flow shear stress changes were < 5%). Computational FSI models including cyclic bending, plaque components and structure, axial stretch, accurate in vivo measurements of pressure, curvature, and material properties should lead to significant improvement on stress-based plaque mechanical analysis and more accurate coronary plaque vulnerability assessment.     

Muscle contraction controls skeletal morphogenesis through regulation of chondrocyte convergent extension

Convergent extension driven by mediolateral intercalation of chondrocytes is a key process that contributes to skeletal growth and morphogenesis. While progress has been made in deciphering the molecular mechanism that underlies this process, the involvement of mechanical load exerted by muscle contraction in its regulation has not been studied. Using the zebrafish as a model system, we found abnormal pharyngeal cartilage morphology in both chemically and genetically paralyzed embryos, demonstrating the importance of muscle contraction for zebrafish skeletal development. The shortening of skeletal elements was accompanied by prominent changes in cell morphology and organization. While in control the cells were elongated, chondrocytes in paralyzed zebrafish were smaller and exhibited a more rounded shape, confirmed by a reduction in their length-to-width ratio. The typical columnar organization of cells was affected too, as chondrocytes in various skeletal elements exhibited abnormal stacking patterns, indicating aberrant intercalation. Finally, we demonstrate impaired chondrocyte intercalation in growth plates of muscle-less Sp(d) mouse embryos, implying the evolutionary conservation of muscle force regulation of this essential morphogenetic process.Our findings provide a new perspective on the regulatory interaction between muscle contraction and skeletal morphogenesis by uncovering the role of muscle-induced mechanical loads in regulating chondrocyte intercalation in two different vertebrate models.     

Mechanical stresses as factors in the autoregulation of morphogenesis

Since morphogenetic processes are nonlinear, feedback must be essential for their regulation. Two concepts of feedback in morphogenesis are developed: 1) inhibition by diffusing morphogenetic substances and 2) action of mechanical strain resulting from morphogenetic movements. The data on the role of mechanical strain in formation of integral structure of embryonic tissues, primary demarcation of embryonic epithelia and mesenchymal rudiments and bending of epithelial layers are presented. Evolutionary aspects of morphogenetic role of mechanical strain and its possible use in applied biotechnology are discussed.     

The mechanical basis of morphogenesis: I. Epithelial folding and invagination

We present a mechanical model for the morphogenetic folding of embryonic epithelia based on hypothesized mechanical properties of the cellular cytoskeleton. In our model we consider a simple cuboidal epithelium whose cells are joined at their apices by circumferential junctions; to these junctions are attached circumferential arrays of microfilament bundles assembled into a “purse string” around the cell apex. We assume that this purse string has the following property: if its circumference is increased beyond a certain threshold, an active contraction is initiated which “draws the purse-string” and reduces the apical circumference of the cell to a new, shorter, resting length. The remainder of the cell is modeled as a visoelastic body of constant volume. Clearly contraction in one cell could stretch the apical circumferences of neighboring cells and, if the threshold is exceeded, cause them “to fire” and contract. The objective of this paper is to demonstrate that our model, based on the local behavior of individual cells, generates a propagating contraction wave which is sufficient to explain the globally coherent morphogenetic infolding of a wide variety of embryonic epithelia. Representative computer simulations, based on the model, are presented for the initial gastrulation movements of echinoderms, neural tube formation in urodele amphibians, and ventral furrow formation in Drosophila.     

Toward a generalised tensegrity model describing the mechanical behaviour of the cytoskeleton structure

The control of many cell functions including growth, migration and mechanotransduction, depends crucially on stress-induced mechanical changes in cell shape and cytoskeleton (CSK) structure. Quantitative studies have been carried out on 6-bar tensegrity models to analyse several mechanical parameters involved in the mechanical responses of adherent cells (i.e. strain hardening, internal stress and scale effects). In the present study, we attempt to generalize some characteristic mechanical laws governing spherical tensegrity structures, with a view of evaluating the mechanical behaviour of the hierarchical multi-modular CSK-structure. The numerical results obtained by studying four different tensegrity models are presented in terms of power laws and point to the existence of unique and constant relationships between the overall structural stiffness and the local properties (length, number and internal stress) of the constitutive components.     

Mechanical effects of the surface ectoderm on optic vesicle morphogenesis in the chick embryo

Precise shaping of the eye is crucial for proper vision. Here, we use experiments on chick embryos along with computational models to examine the mechanical factors involved in the formation of the optic vesicles (OVs), which grow outward from the forebrain of the early embryo. First, mechanical dissections were used to remove the surface ectoderm (SE), a membrane that contacts the outer surfaces of the OVs. Principal components analysis of OV shapes suggests that the SE exerts asymmetric loads that cause the OVs to flatten and shear caudally during the earliest stages of eye development and later to bend in the caudal and dorsal directions. These deformations cause the initially spherical OVs to become pear-shaped. Exposure to the myosin II inhibitor blebbistatin reduced these effects, suggesting that cytoskeletal contraction controls OV shape by regulating tension in the SE. To test the physical plausibility of these interpretations, we developed 2-D finite-element models for frontal and transverse cross-sections of the forebrain, including frictionless contact between the SE and OVs. With geometric data used to specify differential growth in the OVs, these models were used to simulate each experiment (control, SE removed, no contraction). For each case, the predicted shape of the OV agrees reasonably well with experiments. The results of this study indicate that differential growth in the OV and external pressure exerted by the SE are sufficient to cause the global changes in OV shape observed during the earliest stages of eye development.     

Emergent material properties of developing epithelial tissues

Background

Force generation and the material properties of cells and tissues are central to morphogenesis but remain difficult to measure in vivo. Insight is often limited to the ratios of mechanical properties obtained through disruptive manipulation, and the appropriate models relating stress and strain are unknown. The Drosophila amnioserosa epithelium progressively contracts over 3 hours of dorsal closure, during which cell apices exhibit area fluctuations driven by medial myosin pulses with periods of 1.5–6 min. Linking these two timescales and understanding how pulsatile contractions drive morphogenetic movements is an urgent challenge.

Results

We present a novel framework to measure in a continuous manner the mechanical properties of epithelial cells in the natural context of a tissue undergoing morphogenesis. We show that the relationship between apicomedial myosin fluorescence intensity and strain during fluctuations is consistent with a linear behaviour, although with a lag. We thus used myosin fluorescence intensity as a proxy for active force generation and treated cells as natural experiments of mechanical response under cyclic loading, revealing unambiguous mechanical properties from the hysteresis loop relating stress to strain. Amnioserosa cells can be described as a contractile viscoelastic fluid. We show that their emergent mechanical behaviour can be described by a linear viscoelastic rheology at timescales relevant for tissue morphogenesis. For the first time, we establish relative changes in separate effective mechanical properties in vivo. Over the course of dorsal closure, the tissue solidifies and effective stiffness doubles as net contraction of the tissue commences. Combining our findings with those from previous laser ablation experiments, we show that both apicomedial and junctional stress also increase over time, with the relative increase in apicomedial stress approximately twice that of other obtained measures.

Conclusions

Our results show that in an epithelial tissue undergoing net contraction, stiffness and stress are coupled. Dorsal closure cell apical contraction is driven by the medial region where the relative increase in stress is greater than that of stiffness. At junctions, by contrast, the relative increase in the mechanical properties is the same, so the junctional contribution to tissue deformation is constant over time. An increase in myosin activity is likely to underlie, at least in part, the change in medioapical properties and we suggest that its greater effect on stress relative to stiffness is fundamental to actomyosin systems and confers on tissues the ability to regulate contraction rates in response to changes in external mechanics.

     

Toward a Generalised Tensegrity Model Describing the Mechanical Behaviour of the Cytoskeleton Structure

The control of many cell functions including growth, migration and mechanotransduction, depends crucially on stress-induced mechanical changes in cell shape and cytoskeleton (CSK) structure. Quantitative studies have been carried out on 6-bar tensegrity models to analyse several mechanical parameters involved in the mechanical responses of adherent cells (i.e. strain hardening, internal stress and scale effects). In the present study, we attempt to generalize some characteristic mechanical laws governing spherical tensegrity structures, with a view of evaluating the mechanical behaviour of the hierarchical multi-modular CSK-structure. The numerical results obtained by studying four different tensegrity models are presented in terms of power laws and point to the existence of unique and constant relationships between the overall structural stiffness and the local properties (length, number and internal stress) of the constitutive components.     

Surprisingly simple mechanical behavior of a complex embryonic tissue

Background

Previous studies suggest that mechanical feedback could coordinate morphogenetic events in embryos. Furthermore, embryonic tissues have complex structure and composition and undergo large deformations during morphogenesis. Hence we expect highly non-linear and loading-rate dependent tissue mechanical properties in embryos.

Methodology/Principal Findings

We used micro-aspiration to test whether a simple linear viscoelastic model was sufficient to describe the mechanical behavior of gastrula stage Xenopus laevis embryonic tissue in vivo. We tested whether these embryonic tissues change their mechanical properties in response to mechanical stimuli but found no evidence of changes in the viscoelastic properties of the tissue in response to stress or stress application rate. We used this model to test hypotheses about the pattern of force generation during electrically induced tissue contractions. The dependence of contractions on suction pressure was most consistent with apical tension, and was inconsistent with isotropic contraction. Finally, stiffer clutches generated stronger contractions, suggesting that force generation and stiffness may be coupled in the embryo.

Conclusions/Significance

The mechanical behavior of a complex, active embryonic tissue can be surprisingly well described by a simple linear viscoelastic model with power law creep compliance, even at high deformations. We found no evidence of mechanical feedback in this system. Together these results show that very simple mechanical models can be useful in describing embryo mechanics.     

Active reinforcement of externally imposed folding in amphibians embryonic tissues

Although the folding of epithelial layers is one of the most common morphogenetic events, the underlying mechanisms of this process are still poorly understood. We aimed to determine whether an artificial bending of an embryonic cell sheet, which normally remains flat, is reinforced and stabilized by intrinsic cell transformations. We observed both reinforcement and stabilization in double explants of blastocoel roof tissue from Xenopus early gastrula embryos. The reinforcement of artificial bending occurred over the course of a few hours and was driven by the gradual apical constriction and radial elongation of previously compressed cells situated at the bending arch of the concave layer of explant. Apical constriction was associated with actomyosin contraction and endocytosis-mediated engulfing of the apical cell membranes. Cooperative apical constrictions of the concave layer of cells produced a tensile force that extended over the entire surface of the explant and correlated with apical contraction of the concave side cells. In the explants taken from the anterior regions of the embryo, this reinforcement was more stable and the bending better expressed than in those taken from suprablastoporal areas. The morphogenetic role of cell responses to the bending force is discussed.     

Finite element simulations of the active stress in the imaginal disc of the Drosophila Melanogaster

During the larval stages of development, the imaginal disc of Drosphila Melanogaster is composed by a monolayer of epithelial cells, which undergo a strain actively produced by the cells themselves. The well-organized collective contraction produces a stress field that seemingly has a double morphogenetic role: it orchestrates the cellular organization towards the macroscopic shape emergence while simultaneously providing a local information on the organ size. Here we perform numerical simulations of such a mechanical control on morphogenesis at a continuum level, using a three-dimensional finite model that accounts for the active cell contraction. The numerical model is able to reproduce the (few) known qualitative characteristics of the tensional patterns within the imaginal disc of the fruit fly. The computed stress components slightly deviate from planarity, thus confirming the previous theoretical assumptions of a nonlinear elastic analytical model, and enforcing the hypothesis that the spatial variation of the mechanical stress may act as a size regulating signal that locally scales with the global dimension of the domain.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Interaction of auxin, light, and mechanical stress in orienting microtubules in relation to tropic curvature in the epidermis of maize coleoptiles

Summary Changes in the orientation of cortical microtubules (longitudinal vs. transverse with respect to the long cell axis) at the outer epidermal wall of maize coleoptile segments were induced by auxin, red or blue light, and mechanical stresses (cell extension or compression produced by bending). Immunofluorescent techniques were used for the quantitative determination of frequency distributions of microtubule orientation. Detailed kinetic studies showed that microtubule reorientations are temporally correlated with the simultaneously measured changes in growth rate elicited by auxin, red light, or blue light. Growth inhibition induced by depletion of endogenous auxin produces a longitudinal microtubule pattern that can be changed into a transverse pattern in a dose-dependent manner by applying exogenous auxin. A mid-point pattern with equal frequencies of longitudinal and transverse microtubules was adjusted at 2 mol/1 auxin. Bending stress applied under these conditions adjusts permanent, maximally longitudinal and transverse microtubule orientations at the compressed and extended segment sides, respectively, quantitatively mimicking the responses to differential flank growth during phototropic and gravitropic curvature. During tropic curvature the changes in microtubule pattern reflect the distribution of growth rather than the distribution of auxin. The microtubule pattern responds to auxin-dependent growth changes and mechanical stress in a synergistic manner, confirming the functional equivalence of these factors in affecting microtubule orientation. Similar results were obtained when segment growth was altered by blue or red light instead of auxin in the presence or absence of mechanical stress. It is concluded from these results that growth changes, elicited by auxin, light, etc., and mechanical stress affect microtubule orientation through a common signal perception and transduction chain.Abbreviations IAA indole-3-acetic acid (auxin) - MT cortical microtubule     

Protein friction and filament bending facilitate contraction of disordered actomyosin networks

We use mathematical modeling and computation to investigate how protein friction facilitates contraction of disordered actomyosin networks. We simulate two-dimensional networks using an agent-based model, consisting of a system of force-balance equations for myosin motor proteins and semiflexible actin filaments. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We then use simulations of elementary two-filament systems to show that filament bending flexibility can facilitate contraction on the microscopic scale. Finally, we show that actin filament turnover is necessary to sustain contraction and prevent filament aggregation. Simulations with and without turnover also exhibit contractile pulses. However, these pulses are aperiodic, suggesting that periodic pulsation can only arise because of additional regulatory mechanisms or more complex mechanical behavior.     

A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

Stresses in peripheral arteries following stent placement: a finite element analysis

The success of stents to restore blood flow in atherosclerotic peripheral arteries is low relative to coronary arteries. It has been shown that joint flexion induces a mechanical environment that makes stent placement in these arteries highly incompatible, and damage and destruction of stents has been recorded. However, the effect of this environment on the stresses in the arteries is unknown. It is hypothesised that the stresses induced in arteries as a result of this mechanical environment could be sufficient to explain the relatively low success rates. To investigate this hypothesis, a finite element model of the stent-artery interaction was developed. Following stent expansion, bending was simulated by applying a displacement boundary condition to the artery. It is found that high stresses occur at the proximal/distal ends of the stent. As high stress and vascular injury are hypothesised to cause restenosis, the results presented here suggest that the mechanical environment of peripheral arteries could be the predominant cause of high restenosis rates.