Control of the mitotic cleavage plane by local epithelial topology.

For nearly 150 years, it has been recognized that cell shape strongly influences the orientation of the mitotic cleavage plane (e.g., Hofmeister, 1863). However, we still understand little about the complex interplay between cell shape and cleavage-plane orientation in epithelia, where polygonal cell geometries emerge from multiple factors, including cell packing, cell growth, and cell division itself. Here, using mechanical simulations, we show that the polygonal shapes of individual cells can systematically bias the long-axis orientations of their adjacent mitotic neighbors. Strikingly, analyses of both animal epithelia and plant epidermis confirm a robust and nearly identical correlation between local cell topology and cleavage-plane orientation in vivo. Using simple mathematics, we show that this effect derives from fundamental packing constraints. Our results suggest that local epithelial topology is a key determinant of cleavage-plane orientation, and that cleavage-plane bias may be a widespread property of polygonal cell sheets in plants and animals.

Interstitial fluid osmolarity modulates the action of differential tissue surface tension in progenitor cell segregation during gastrulation.

The segregation of different cell types into distinct tissues is a fundamental process in metazoan development. Differences in cell adhesion and cortex tension are commonly thought to drive cell sorting by regulating tissue surface tension (TST). However, the role that differential TST plays in cell segregation within the developing embryo is as yet unclear. Here, we have analyzed the role of differential TST for germ layer progenitor cell segregation during zebrafish gastrulation. Contrary to previous observations that differential TST drives germ layer progenitor cell segregation in vitro, we show that germ layers display indistinguishable TST within the gastrulating embryo, arguing against differential TST driving germ layer progenitor cell segregation in vivo We further show that the osmolarity of the interstitial fluid (IF) is an important factor that influences germ layer TST in vivo, and that lower osmolarity of the IF compared with standard cell culture medium can explain why germ layers display differential TST in culture but not in vivo Finally, we show that directed migration of mesendoderm progenitors is required for germ layer progenitor cell segregation and germ layer formation.

Elongated Cells Drive Morphogenesis in a Surface-Wrapped Finite-Element Model of Germband Retraction.

During Drosophila embryogenesis, the germband first extends to curl around the posterior end of the embryo and then retracts back; however, retraction is not simply the reversal of extension. At a tissue level, extension is coincident with ventral furrow formation, and at a cellular level, extension occurs via convergent cell neighbor exchanges in the germband, whereas retraction involves only changes in cell shape. To understand how cell shapes, tissue organization, and cellular forces drive germband retraction, we investigate this process using a whole-embryo, surface-wrapped cellular finite-element model. This model represents two key epithelial tissues-amnioserosa and germband-as adjacent sheets of two-dimensional cellular finite elements that are wrapped around an ellipsoidal three-dimensional approximation of an embryo. The model reproduces the detailed kinematics of in vivo retraction by fitting just one free model parameter, the tension along germband cell interfaces; all other cellular forces are constrained to follow ratios inferred from experimental observations. With no additional parameter adjustments, the model also reproduces quantitative assessments of mechanical stress using laser dissection and failures of retraction when amnioserosa cells are removed via mutations or microsurgery. Surprisingly, retraction in the model is robust to changes in cellular force values but is critically dependent on starting from a configuration with highly elongated amnioserosa cells. Their extreme cellular elongation is established during the prior process of germband extension and is then used to drive retraction. The amnioserosa is the one tissue whose cellular morphogenesis is reversed from germband extension to retraction, and this reversal coordinates the forces needed to retract the germband back to its pre-extension position and shape. In this case, cellular force strengths are less important than the carefully established cell shapes that direct them. VIDEO ABSTRACT.

How computational models can help unlock biological systems.

With computation models playing an ever increasing role in the advancement of science, it is important that researchers understand what it means to model something; recognize the implications of the conceptual, mathematical and algorithmic steps of model construction; and comprehend what models can and cannot do. Here, we use examples to show that models can serve a wide variety of roles, including hypothesis testing, generating new insights, deepening understanding, suggesting and interpreting experiments, tracing chains of causation, doing sensitivity analyses, integrating knowledge, and inspiring new approaches. We show that models can bring together information of different kinds and do so across a range of length scales, as they do in multi-scale, multi-faceted embryogenesis models, some of which connect gene expression, the cytoskeleton, cell properties, tissue mechanics, morphogenetic movements and phenotypes. Models cannot replace experiments nor can they prove that particular mechanisms are at work in a given situation. But they can demonstrate whether or not a proposed mechanism is sufficient to produce an observed phenomenon. Although the examples in this article are taken primarily from the field of embryo mechanics, most of the arguments and discussion are applicable to any form of computational modelling.

On the origins of the mitotic shift in proliferating cell layers.

BACKGROUND: During plant and animal development, monolayer cell sheets display a stereotyped distribution of polygonal cell shapes. In interphase cells these shapes range from quadrilaterals to decagons, with a robust average of six sides per cell. In contrast, the subset of cells in mitosis exhibits a distinct distribution with an average of seven sides. It remains unclear whether this 'mitotic shift' reflects a causal relationship between increased polygonal sidedness and increased division likelihood, or alternatively, a passive effect of local proliferation on cell shape. METHODS: We use a combination of probabilistic analysis and mathematical modeling to predict the geometry of mitotic polygonal cells in a proliferating cell layer. To test these predictions experimentally, we use Flp-Out stochastic labeling in the Drosophila wing disc to induce single cell clones, and confocal imaging to quantify the polygonal topologies of these clones as a function of cellular age. For a more generic test in an idealized cell layer, we model epithelial sheet proliferation in a finite element framework, which yields a computationally robust, emergent prediction of the mitotic cell shape distribution. RESULTS: Using both mathematical and experimental approaches, we show that the mitotic shift derives primarily from passive, non-autonomous effects of mitoses in neighboring cells on each cell's geometry over the course of the cell cycle. Computationally, we predict that interphase cells should passively gain sides over time, such that cells at more advanced stages of the cell cycle will tend to have a larger number of neighbors than those at earlier stages. Validating this prediction, experimental analysis of randomly labeled epithelial cells in the Drosophila wing disc demonstrates that labeled cells exhibit an age-dependent increase in polygonal sidedness. Reinforcing these data, finite element simulations of epithelial sheet proliferation demonstrate in a generic framework that passive side-gaining is sufficient to generate a mitotic shift. CONCLUSIONS: Taken together, our results strongly suggest that the mitotic shift reflects a time-dependent accumulation of shared cellular interfaces over the course of the cell cycle. These results uncover fundamental constraints on the relationship between cell shape and cell division that should be general in adherent, polarized cell layers.

Modeling cell elongation during germ band retraction: cell autonomy versus applied anisotropic stress.

The morphogenetic process of germ band retraction in Drosophila embryos involves coordinated movements of two epithelial tissues - germ band and amnioserosa. The germ band shortens along its rostral-caudal or head-to-tail axis, widens along its perpendicular dorsal-ventral axis, and uncurls from an initial 'U' shape. The amnioserosa mechanically assists this process by pulling on the crook of the U-shaped germ band. The amnioserosa may also provide biochemical signals that drive germ band cells to change shape in a mechanically autonomous fashion. Here, we use a finite-element model to investigate how these two contributions reshape the germ band. We do so by modeling the response to laser-induced wounds in each of the germ band's spatially distinct segments (T1-T3, A1-A9) during the middle of retraction when segments T1-A3 form the ventral arm of the 'U', A4-A7 form its crook, and A8-A9 complete the dorsal arm. We explore these responses under a range of externally applied stresses and internal anisotropy of cell edge tensions - akin to a planar cell polarity that can drive elongation of cells in a direction parallel to the minimum edge tension - and identify regions of parameter space (edge-tension anisotropy versus stress anisotropy) that best match previous experiments for each germ band segment. All but three germ band segments are best fit when the applied stress anisotropy and the edge-tension anisotropy work against one another - i.e., when the isolated effects would elongate cells in perpendicular directions. Segments in the crook of the germ band (A4-A7) have cells that elongate in the direction of maximum external stress, i.e., external stress anisotropy is dominant. In most other segments, the dominant factor is internal edge-tension anisotropy. These results are consistent with models in which the amnioserosa pulls on the crook of the germ band to mechanically assist retraction. In addition, they suggest a mechanical cue for edge-tension anisotropy whereby cells do not globally orient their internal elongation axis towards the amnioserosa, but instead orient this axis perpendicular to the local principal stress direction.

Video force microscopy reveals the mechanics of ventral furrow invagination in Drosophila.

The absence of tools for mapping the forces that drive morphogenetic movements in embryos has impeded our understanding of animal development. Here we describe a unique approach, video force microscopy (VFM), that allows detailed, dynamic force maps to be produced from time-lapse images. The forces at work in an embryo are considered to be decomposed into active and passive elements, where active forces originate from contributions (e.g., actomyosin contraction) that do mechanical work to the system and passive ones (e.g., viscous cytoplasm) that dissipate energy. In the present analysis, the effects of all passive components are considered to be subsumed by an effective cytoplasmic viscosity, and the driving forces are resolved into equivalent forces along the edges of the polygonal boundaries into which the region of interest is divided. Advanced mathematical inverse methods are used to determine these driving forces. When applied to multiphoton sections of wild-type and mutant Drosophila melanogaster embryos, VFM is able to calculate the equivalent driving forces acting along individual cell edges and to do so with subminute temporal resolution. In the wild type, forces along the apical surface of the presumptive mesoderm are found to be large and to vary parabolically with time and angular position, whereas forces along the basal surface of the ectoderm, for example, are found to be smaller and nearly uniform with position. VFM shows that in mutants with reduced junction integrity and myosin II activity, the driving forces are reduced, thus accounting for ventral furrow failure.

Measuring the modulus of silicone hydrogel contact lenses.

PURPOSE: The purpose of this study is to demonstrate a novel method for measuring the modulus of contact lenses in their as-received, variable-thickness form and to determine whether modulus varies with location within commercial lenses and whether it is dependent on lens geometry and temperature. METHODS: The thickness profiles of lenses having powers from -8 diopters (D) to +4 D were measured using a Rehder electronic thickness gauge. Strip-shaped specimens having a width of 5.5 mm were then cut from the lenses. Graphite particles were sprinkled on the specimen surface so that its motions could be tracked using digital image-correlation techniques. The specimens were mounted in a BioTester test system using BioRakes (rather than clamps) and stretched uniaxially until all parts of the lens between the attachment points had elongated by at least 10%. This procedure allowed local modulus values to be determined at 110 locations over the surface of each lens and any property variations within the lenses to be characterized. Tests were performed at 5, 23, and 37°C. RESULTS: Material modulus was found to be essentially constant within any given lens and was independent of the optical power of the lens. Young's Modulus values ranged from 0.3 to 1.9 MPa, depending on the lens manufacturer and product, and some lens materials showed a decrease in modulus with temperature. For the materials tested, those with lower water content had a tendency to exhibit higher moduli. CONCLUSIONS: Testing of the kind reported here is important for assessing the efficacy of current and proposed contact lens materials and designs, especially if such designs make use of variable properties to enhance function or fit.

Combining laser microsurgery and finite element modeling to assess cell-level epithelial mechanics.

Laser microsurgery and finite element modeling are used to determine the cell-level mechanics of the amnioserosa-a morphogenetically crucial epithelium on the dorsal surface of fruit fly embryos (Drosophila melanogaster). In the experiments, a tightly focused laser ablates a subcellular hole (1 microm in diameter) that passes clean through the epithelium. The surrounding cells recoil from the wound site with a large range of initial recoil velocities. These depend on the embryo's developmental stage and the subcellular wound site. The initial recoil (up to 0.1 s) is well reproduced by a base finite element model, which assumes a uniform effective viscosity inside the cells, a constant tension along each cell-cell boundary, and a large, potentially anisotropic, far-field stress--one that far exceeds the stress equivalent of the cell-edge tensions. After 0.1 s, the experimental recoils slow dramatically. This observation can be reproduced by adding viscoelastic rods along cell edges or as a fine prestressed mesh parallel to the apical and basal membranes of the cell. The mesh also reproduces a number of double-wounding experiments in which successive holes are drilled in a single cell.

Strain uniformity in biaxial specimens is highly sensitive to attachment details.

Biaxial testing has been used widely to characterize the mechanical properties of soft tissues and other flexible materials, but fundamental issues related to specimen design and attachment have remained. Finite element models and experiments were used to investigate how specimen geometry and attachment details affect uniformity of the strain field inside the attachment points. The computational studies confirm that increasing the number of attachment points increases the size of the area that experiences sensibly uniform strain (defined here as the central sample region where the ratio of principal strains E(11)/E(22)<1.10), and that the strains experienced in this region are less than nominal strains based on attachment point movement. Uniformity of the strain field improves substantially when the attachment points span a wide zone along each edge. Subtle irregularities in attachment point positioning can significantly degrade strain field uniformity. In contrast, details of the apron, the region outside of the attachment points, have little effect on the interior strain field. When nonlinear properties consistent with those found in human sclera are used, similar results are found. Experiments were conducted on 6 x 6 mm talc-sprinkled rubber specimens loaded using wire "rakes." Points on a grid having 12 x 12 bays were tracked, and a detailed strain map was constructed. A finite element model based on the actual geometry of an experiment having an off-pattern rake tine gave strain patterns that matched to within 4.4%. Finally, simulations using nonequibiaxial strains indicated that the strain field uniformity was more sensitive to sample attachment details for the nonequibiaxial case as compared to the equibiaxial case. Specimen design and attachment were found to significantly affect the uniformity of the strain field produced in biaxial tests. Practical guidelines were offered for design and mounting of biaxial test specimens. The issues addressed here are particularly relevant as specimens become smaller in size.

Multi-scale finite element modeling allows the mechanics of amphibian neurulation to be elucidated.

The novel multi-scale computational approach introduced here makes possible a new means for testing hypotheses about the forces that drive specific morphogenetic movements. A 3D model based on this approach is used to investigate neurulation in the axolotl (Ambystoma mexicanum), a type of amphibian. The model is based on geometric data from 3D surface reconstructions of live embryos and from serial sections. Tissue properties are described by a system of cell-based constitutive equations, and parameters in the equations are determined from physical tests. The model includes the effects of Shroom-activated neural ridge reshaping and lamellipodium-driven convergent extension. A typical whole-embryo model consists of 10,239 elements and to run its 100 incremental time steps requires 2 days. The model shows that a normal phenotype does not result if lamellipodium forces are uniform across the width of the neural plate; but it can result if the lamellipodium forces decrease from a maximum value at the mid-sagittal plane to zero at the plate edge. Even the seemingly simple motions of neurulation are found to contain important features that would remain hidden, they were not studied using an advanced computational model. The present model operates in a setting where data are extremely sparse and an important outcome of the study is a better understanding of the role of computational models in such environments.

Coordination of Receptor Tyrosine Kinase Signaling and Interfacial Tension Dynamics Drives Radial Intercalation and Tube Elongation.

We sought to understand how cells collectively elongate epithelial tubes. We first used 3D culture and biosensor imaging to demonstrate that epithelial cells enrich Ras activity, phosphatidylinositol (3,4,5)-trisphosphate (PIP3), and F-actin to their leading edges during migration within tissues. PIP3 enrichment coincided with, and could enrich despite inhibition of, F-actin dynamics, revealing a conserved migratory logic compared with single cells. We discovered that migratory cells can intercalate into the basal tissue surface and contribute to tube elongation. We then connected molecular activities to subcellular mechanics using force inference analysis. Migration and transient intercalation required specific and similar anterior-posterior ratios of interfacial tension. Permanent intercalations were distinguished by their capture at the boundary through time-varying tension dynamics. Finally, we integrated our experimental and computational data to generate a finite element model of tube elongation. Our model revealed that intercalation, interfacial tension dynamics, and high basal stress are together sufficient for mammary morphogenesis.

A new cell-based FE model for the mechanics of embryonic epithelia.

In order to overcome a significant stiffening artefact associated with current finite element (FE) models for the mechanics of embryonic epithelia, two new FE formulations were developed. Cell-cell interfacial tensions gamma are represented by constant-force rod elements as in previous models. However, the viscosity of the cytoplasm with its embedded organelles and filament networks is modeled using viscous triangular elements, it is modeled using either radial and circumferential dashpots or an orthogonal dashpot system rather than the viscous triangular elements typical of previous two-dimensional FE models. The models are tested against tissue (epithelium) stretching because it gives rise to significant changes in cell shape and against cell sorting because it involves high rates of cell rearrangement. The orthogonal dashpot system is found to capture cell size and shape effects well, give the model cells characteristics that are consistent with those of real cells, provide high computational efficiency and hold promise for future three-dimensional analyses.

Estimation of cellular fabric in embryonic epithelia.

Recent computational and analytical studies have shown that cellular fabric-as embodied by average cell size, aspect ratio and orientation-is a key indicator of the stresses acting in an embryonic epithelium. Cellular fabric in real embryonic tissues could not previously be measured automatically because the cell boundaries tend to be poorly defined, significant lighting and cell pigmentation differences occur and tissues contain a variety of cell geometries. To overcome these difficulties, four algorithms were developed: least squares ellipse fitting (LSEF), area moments (AM), correlation and axes search (CAS) and Gabor filters (GF). The AM method was found to be the most reliable of these methods, giving typical cell size, aspect ratio and orientation errors of 18%, 0.10 and 7.4 degrees, respectively, when evaluated against manually segmented images. The power of the AM algorithm to provide new insights into the mechanics of morphogenesis is demonstrated through a brief investigation of gastrulation, where fabric data suggest that key gastrulation movements are driven by epidermal tensions circumferential to the blastopore.

Inferring cellular forces from image stacks.

Although the importance of cellular forces to a wide range of embryogenesis and disease processes is widely recognized, measuring these forces is challenging, especially in three dimensions. Here, we introduce CellFIT-3D, a force inference technique that allows tension maps for three-dimensional cellular systems to be estimated from image stacks. Like its predecessors, video force microscopy and CellFIT, this cell mechanics technique assumes boundary-specific interfacial tensions to be the primary drivers, and it constructs force-balance equations based on triple junction (TJ) dihedral angles. The technique involves image processing, segmenting of cells, grouping of cell outlines, calculation of dihedral planes, averaging along three-dimensional TJs, and matrix equation assembly and solution. The equations tend to be strongly overdetermined, allowing indistinct TJs to be ignored and solution error estimates to be determined. Application to clean and noisy synthetic data generated using Surface Evolver gave tension errors of 1.6-7%, and analyses of eight-cell murine embryos gave estimated errors smaller than the 10% uncertainty of companion aspiration experiments. Other possible areas of application include morphogenesis, cancer metastasis and tissue engineering.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

A videofluoroscopy-based tracking algorithm for quantifying the time course of human intervertebral displacements.

The motions of individual intervertebral joints can affect spine motion, injury risk, deterioration, pain, treatment strategies, and clinical outcomes. Since standard kinematic methods do not provide precise time-course details about individual vertebrae and intervertebral motions, information that could be useful for scientific advancement and clinical assessment, we developed an iterative template matching algorithm to obtain this data from videofluoroscopy images. To assess the bias of our approach, vertebrae in an intact porcine spine were tracked and compared to the motions of high-contrast markers. To estimate precision under clinical conditions, motions of three human cervical spines were tracked independently ten times and vertebral and intervertebral motions associated with individual trials were compared to corresponding averages. Both tests produced errors in intervertebral angular and shear displacements no greater than 0.4° and 0.055 mm, respectively. When applied to two patient cases, aberrant intervertebral motions in the cervical spine were typically found to correlate with patient-specific anatomical features such as disc height loss and osteophytes. The case studies suggest that intervertebral kinematic time-course data could have value in clinical assessments, lead to broader understanding of how specific anatomical features influence joint motions, and in due course inform clinical treatments.

Non-straight cell edges are important to invasion and engulfment as demonstrated by cell mechanics model.

Computational models of cell-cell mechanical interactions typically simulate sorting and certain other motions well, but as demands on these models continue to grow, discrepancies between the cell shapes, contact angles and behaviours they predict and those that occur in real cells have come under increased scrutiny. To investigate whether these discrepancies are a direct result of the straight cell-cell edges generally assumed in these models, we developed a finite element model that approximates cell boundaries using polylines with an arbitrary number of segments. We then compared the predictions of otherwise identical polyline and monoline (straight-edge) models in a variety of scenarios, including annealing, single- and multi-cell engulfment, sorting, and two forms of mixing--invasion and checkerboard pattern formation. Keeping cell-cell edges straight influences cell motion, cell shape, contact angle, and boundary length, especially in cases where one cell type is pulled between or around cells of a different type, as in engulfment or invasion. These differences arise because monoline cells have restricted deformation modes. Polyline cells do not face these restrictions, and with as few as three segments per edge yielded realistic edge shapes and contact angle errors one-tenth of those produced by monoline models, making them considerably more suitable for situations where angles and shapes matter, such as validation of cellular force-inference techniques. The findings suggest that non-straight cell edges are important both in modelling and in nature.

Tensile properties of embryonic epithelia measured using a novel instrument.

We present the first measurements of the tensile properties of embryonic epithelia, data that are crucial to understanding the mechanics of morphogenetic movements. Fine wires were glued to the surface of an intact, live embryo using cyanoacrylate glue, after which the epithelium between the wires was separated from the remainder of the embryo by microsurgery. The wires were then separated from each other in 0.1 microm steps under computer control in order to elongate the tissue at a constant true strain rate. Force was determined from the degree of bending in the wires, and a real-time, image-based feedback system corrected for reductions in elongation that would otherwise have been caused by wire flexure. The instrument was used to determine the tensile properties of epidermis and neuroepithelia from early-stage embryos of the axolotl (Ambystoma mexicanum), a type of amphibian. Monolayer specimens as small as 300 by 500 microm were elongated at physiological strain rates of 5-30% per hour, and the effects of developmental stage, epithelium type, specimen origin, direction of elongation and strain rate were investigated. True strains as high as 50% were observed before tearing began and equivalent moduli for the initial, linear portion of the load resultant versus strain curves ranged from 1 x 10(-3) to 8 x 10(-3) N/m.

Automated 3-D reconstruction of the surface of live early-stage amphibian embryos.

Although three-dimensional (3-D) reconstructions of the surfaces of live embyos are vital to understanding embryo development, morphogenetic tissue movements and other factors have prevented the automation of this task. Here, we report an integrated set of software algorithms that overcome these challenges, making it possible to completely automate the reconstruction of embryo surfaces and other textured surfaces from multiview images. The process involves: 1) building accurate point correspondences using a robust deformable template block matching algorithm; 2) removing outliers using fundamental matrix calculations in conjunction with a RANSAC algorithm; 3) generating 3-D point clouds using a bundle adjustment algorithm that includes camera position and distortion corrections; 4) meshing the point clouds into triangulated surfaces using a Tight Cocone algorithm that produces water tight models; 5) refining surfaces using midpoint insertion and Laplacian smoothing algorithms; and 6) repeating these steps until a measure of convergence G, the rms difference between successive reconstructions, is below a specified threshold. Reconstructions were made of 2.2-mm diameter, neurulation-stage axolotl (amphibian) embryos using 44 multiview images collected with a robotic microscope. A typical final model (sixth iteration) contained 3787 points and 7562 triangles and had an error measure of G = 5.9 microm.

Practical aspects of the cellular force inference toolkit (CellFIT).

If we are to fully understand the reasons that cells and tissues move and acquire their distinctive geometries during processes such as embryogenesis and wound healing, we will need detailed maps of the forces involved. One of the best current prospects for obtaining this information is noninvasive force-from-images techniques such as CellFIT, the Cellular Force Inference Toolkit, whose various steps are discussed here. Like other current quasistatic approaches, this one assumes that cell shapes are produced by interactions between interfacial tensions and intracellular pressures. CellFIT, however, allows cells to have curvilinear boundaries, which can significantly improve inference accuracy and reduce noise sensitivity. The quality of a CellFIT analysis depends on how accurately the junction angles and edge curvatures are measured, and a software tool we describe facilitates determination and evaluation of this information. Special attention is required when edges are crenulated or significantly different in shape from a circular arc. Because the tension and pressure equations are overdetermined, a select number of edges can be removed from the analysis, and these might include edges that are poorly defined in the source image, too short to provide accurate angles or curvatures, or noncircular. The approach works well for aggregates with as many as 1000 cells, and introduced errors have significant effects on only a few adjacent cells. An understanding of these considerations will help CellFIT users to get the most out of this promising new technique.