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.

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.

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'.

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.

Highlighted Article: Segregation of the germ layer progenitors in the zebrafish gastrula is driven by directed cell migration, rather than differential tissue surface tension.

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.

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.

Forces driving epithelial wound healing

A fundamental feature of multicellular organisms is their ability to self-repair wounds through the movement of epithelial cells into the damaged area. This collective cellular movement is commonly attributed to a combination of cell crawling and "purse-string" contraction of a supracellular actomyosin ring. Here we show by direct experimental measurement that these two mechanisms are insufficient to explain force patterns observed during wound closure. At early stages of the process, leading actin protrusions generate traction forces that point away from the wound, showing that wound closure is initially driven by cell crawling. At later stages, we observed unanticipated patterns of traction forces pointing towards the wound. Such patterns have strong force components that are both radial and tangential to the wound. We show that these force components arise from tensions transmitted by a heterogeneous actomyosin ring to the underlying substrate through focal adhesions. The structural and mechanical organization reported here provides cells with a mechanism to close the wound by cooperatively compressing the underlying substrate.

CellFIT: a cellular force-inference toolkit using curvilinear cell boundaries

Mechanical forces play a key role in a wide range of biological processes, from embryogenesis to cancer metastasis, and there is considerable interest in the intuitive question, "Can cellular forces be inferred from cell shapes?" Although several groups have posited affirmative answers to this stimulating question, nagging issues remained regarding equation structure, solution uniqueness and noise sensitivity. Here we show that the mechanical and mathematical factors behind these issues can be resolved by using curved cell edges rather than straight ones. We present a new package of force-inference equations and assessment tools and denote this new package CellFIT, the Cellular Force Inference Toolkit. In this approach, cells in an image are segmented and equilibrium equations are constructed for each triple junction based solely on edge tensions and the limiting angles at which edges approach each junction. The resulting system of tension equations is generally overdetermined. As a result, solutions can be obtained even when a modest number of edges need to be removed from the analysis due to short length, poor definition, image clarity or other factors. Solving these equations yields a set of relative edge tensions whose scaling must be determined from data external to the image. In cases where intracellular pressures are also of interest, Laplace equations are constructed to relate the edge tensions, curvatures and cellular pressure differences. That system is also generally overdetermined and its solution yields a set of pressures whose offset requires reference to the surrounding medium, an open wound, or information external to the image. We show that condition numbers, residual analyses and standard errors can provide confidence information about the inferred forces and pressures. Application of CellFIT to several live and fixed biological tissues reveals considerable force variability within a cell population, significant differences between populations and elevated tensions along heterotypic boundaries.

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.

DRhoGEF2 regulates cellular tension and cell pulsations in the Amnioserosa during Drosophila dorsal closure

Coordination of apical constriction in epithelial sheets is a fundamental process during embryogenesis. Here, we show that DRhoGEF2 is a key regulator of apical pulsation and constriction of amnioserosal cells during Drosophila dorsal closure. Amnioserosal cells mutant for DRhoGEF2 exhibit a consistent decrease in amnioserosa pulsations whereas overexpression of DRhoGEF2 in this tissue leads to an increase in the contraction time of pulsations. We probed the physical properties of the amnioserosa to show that the average tension in DRhoGEF2 mutant cells is lower than wild-type and that overexpression of DRhoGEF2 results in a tissue that is more solid-like than wild-type. We also observe that in the DRhoGEF2 overexpressing cells there is a dramatic increase of apical actomyosin coalescence that can contribute to the generation of more contractile forces, leading to amnioserosal cells with smaller apical surface than wild-type. Conversely, in DRhoGEF2 mutants, the apical actomyosin coalescence is impaired. These results identify DRhoGEF2 as an upstream regulator of the actomyosin contractile machinery that drives amnioserosa cells pulsations and apical constriction.

Identifying same-cell contours in image stacks: a key step in making 3D reconstructions

Identification of contours belonging to the same cell is a crucial step in the analysis of confocal stacks and other image sets in which cell outlines are visible, and it is central to the making of 3D cell reconstructions. When the cells are close packed, the contour grouping problem is more complex than that found in medical imaging, for example, because there are multiple regions of interest, the regions are not separable from each other by an identifiable background and regions cannot be distinguished by intensity differences. Here, we present an algorithm that uses three primary metrics-overlap of contour areas in adjacent images, co-linearity of the centroids of these areas across three images in a stack, and cell taper-to assign cells to groups. Decreasing thresholds are used to successively assign contours whose membership is less obvious. In a final step, remaining contours are assigned to existing groups by setting all thresholds to zero and groups having strong hour-glass shapes are partitioned. When applied to synthetic data from isotropic model aggregates, a curved model epithelium in which the long axes of the cells lie at all possible angles to the transection plane, and a confocal image stack, algorithm assignments were between 97 and 100% accurate in sets having at least four contours per cell. The algorithm is not particularly sensitive to the thresholds used, and a single set of parameters was used for all of the tests. The algorithm, which could be extended to time-lapse data, solves a key problem in the translation of image data into cell information.

Biaxial mechanical testing of human sclera

The biomechanical environment of the optic nerve head (ONH), of interest in glaucoma, is strongly affected by the biomechanical properties of sclera. However, there is a paucity of information about the variation of scleral mechanical properties within eyes and between individuals. We thus used biaxial testing to measure scleral stiffness in human eyes. Ten eyes from 5 human donors (age 55.4+/-3.5 years; mean+/-SD) were obtained within 24h of death. Square scleral samples (6mm on a side) were cut from each ocular quadrant 3-9 mm from the ONH centre and were mechanically tested using a biaxial extensional tissue tester (BioTester 5000, CellScale Biomaterials Testing, Waterloo). Stress-strain data in the latitudinal (toward the poles) and longitudinal (circumferential) directions, here referred to as directions 1 and 2, were fit to the four-parameter Fung constitutive equation W=c(e(Q)-1), where Q=c(1)E(11)(2)+c(2)E(22)(2)+2c(3)E(11)E(22) and W, c's and E(ij) are the strain energy function, material parameters and Green strains, respectively. Fitted material parameters were compared between samples. The parameter c(3) ranged from 10(-7) to 10(-8), but did not contribute significantly to the accuracy of the fitting and was thus fixed at 10(-7). The products cc(1) and cc(2), measures of stiffness in the 1 and 2 directions, were 2.9+/-2.0 and 2.8+/-1.9 MPa, respectively, and were not significantly different (two-sided t-test; p=0.795). The level of anisotropy (ratio of stiffness in orthogonal directions) was 1.065+/-0.33. No statistically significant correlations between sample thickness and stiffness were found (correlation coefficients=-0.026 and -0.058 in directions 1 and 2, respectively). Human sclera showed heterogeneous, near-isotropic, nonlinear mechanical properties over the scale of our samples.

Cinemechanometry (CMM): A method to determine the forces that drive morphogenetic movements from time-lapse images

Although cell-level mechanical forces are crucial to tissue self-organization in contexts ranging from embryo development to cancer metastases to regenerative engineering, the absence of methods to map them over time has been a major obstacle to new understanding. Here, we present a technique for constructing detailed, dynamic maps of the forces driving morphogenetic events from time-lapse images. Forces in the cell are considered to be separable into unknown active driving forces and known passive forces, where actomyosin systems and microtubules contribute primarily to the first group and intermediate filaments and cytoplasm to the latter. A finite-element procedure is used to estimate the field of forces that must be applied to the passive components to produce their observed incremental deformations. This field is assumed to be generated by active forces resolved along user-defined line segments whose location, often along cell edges, is informed by the underlying biology. The magnitudes and signs of these forces are determined by a mathematical inverse method. The efficacy of the approach is demonstrated using noisy synthetic data from a cross section of a generic invagination and from a planar aggregate that involves two cell types, edge forces that vary with time and a neighbor change.

From genes to neural tube defects (NTDs): insights from multiscale computational modeling

The morphogenetic movements, and the embryonic phenotypes they ultimately produce, are the consequence of a series of events that involve signaling pathways, cytoskeletal components, and cell- and tissue-level mechanical interactions. In order to better understand how these events work together in the context of amphibian neurulation, an existing multiscale computational model was augmented. Geometric data for this finite element-based mechanical model were obtained from 3D surface reconstructions of live axolotl embryos and serial sections of fixed specimens. Tissue mechanical properties were modeled using cell-based constitutive equations that include internal force generation and cell rearrangement, and equation parameters were adjusted manually to reflect biochemical changes including alterations in Shroom or the planar-cell-polarity pathway. The model indicates that neural tube defects can arise when convergent extension of the neural plate is reduced by as little as 20%, when it is eliminated on one side of the embryo, when neural ridge elevation is disrupted, when tension in the non-neural ectoderm is increased, or when the ectoderm thickness is increased. Where comparable conditions could be induced in Xenopus embryos, good agreement was found, an important step in model validation. The model reveals the neurulating embryo to be a finely tuned biomechanical system.

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.

Cellular interfacial and surface tensions determined from aggregate compression tests using a finite element model

Although previous studies suggested that the interfacial tension gamma(cc) acting along cell-cell boundaries and the effective viscosity mu of the cell cytoplasm could be measured by compressing a spherical aggregate of cells between parallel plates, the mechanical understanding necessary to extract this information from these tests-tests that have provided the surface tension sigma(cm) acting along cell-medium interfaces-has been lacking. These tensions can produce net forces at the subcellular level and give rise to cell motions and tissue reorganization, the rates of which are regulated by mu. Here, a three-dimensional (3D) cell-based finite element model provides insight into the mechanics of the compression test, where these same forces are at work, and leads to quantitative relationships from which the effective viscosity mu of the cell cytoplasm, the tension gamma(cc) that acts along internal cell-cell interfaces and the surface tension sigma(cp) along the cell-platen boundaries can be determined from force-time curves and aggregate profiles. Tests on 5-day embryonic chick mesencephalon, neural retina, liver, and heart aggregates show that all of these properties vary significantly with cell type, except gamma(cc), which is remarkably constant. These properties are crucial for understanding cell rearrangement and tissue self-organization in contexts that include embryogenesis, cancer metastases, and tissue engineering.

Cell sorting in three dimensions: topology, fluctuations, and fluidlike instabilities

Previous 2D and 3D models concluded that cell sorting requires cytoskeletal fluctuations and is stalled by high tension at heterotypic interfaces. New deterministic and stochastic models show that this is not true in 3D. Sorting in 3D involves both topological untangling and domain coalescence. Coalescence requires fluctuations and low tension, but untangling does not. It occurs by a Plateau-Rayleigh instability of cell threads-deterministically driven by high tension. At high minority-cell fractions, untangling dominates and significant partial sorting can occur without fluctuations.

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.

A three-dimensional finite element model for the mechanics of cell-cell interactions

Technical challenges, including significant ones associated with cell rearrangement, have hampered the development of three-dimensional finite element models for the mechanics of embryonic cells. These challenges have been overcome by a new formulation in which the contents of each cell, assumed to have a viscosity mu, are modeled using a system of orthogonal dashpots. This approach overcomes a stiffening artifact that affects more traditional models, in which space-filling viscous elements are used to model the cytoplasm. Cells are assumed to be polyhedral in geometry, and each n-sided polygonal face is subdivided into n triangles with a common node at the face center so that it needs not remain flat. A constant tension gamma is assumed to act along each cell-cell interface, and cell rearrangements occur through one of two complementary topological transformations. The formulation predicts mechanical interactions between pairs of similar or dissimilar cells that are consistent with experiments, two-dimensional simulations, contact angle theory, and intracellular pressure calculations. Simulations of the partial engulfment of one tissue type by another show that the formulation is able to model aggregates of several hundred cells without difficulty. Simulations carried out using this formulation suggest new experimental approaches for measuring cell surface tensions and interfacial tensions. The formulation holds promise as a tool for gaining insight into the mechanics of isolated or aggregated embryonic cells and for the design and interpretation of experiments that involve them.

Measurement of in vivo stress resultants in neurulation-stage amphibian embryos

In order to obtain the first quantitative measurements of the in vivo stresses in early-stage amphibian embryos, we developed a novel instrument that uses a pair of parallel wires that are glued to the surface of an embryo normal to the direction in which the stress is to be determined. When a slit is made parallel to the wires and between them, tension in the surrounding tissue causes the slit to open. Under computer control, one of the wires is moved so as to restore the original wire spacing, and the steady-state closure force is determined from the degree of wire flexure. A cell-level finite element model is used to convert the wire bending force to an in-plane stress since the wire force is not proportional to the slit length. The device was used to measure stress resultants (force carried per unit of slit length) on the dorsal, ventral and lateral aspects of neurulation-stage axolotl (Ambystoma mexicanum) embryos. The resultants were anisotropic and varied with location and developmental stage, with values ranging from -0.17 mN/m to 1.92 mN/m. In general, the resultants could be decomposed into patterns associated with internal pressure in the embryo, bending of the embryo along its mid-sagittal plane and neural tube closure. The patterns of stress revealed by the experiments support a number of current theories about the mechanics of neurulation.

Lamellipodium-driven tissue reshaping: A parametric study

We recently showed that lamellipodia are able to generate forces of the right type to drive convergent extension (CE), an important class of tissue reshaping, in early stage embryos. The purpose of the present work is to quantify the mechanics of this process using parametric analyses. We use finite elements to implement a gamma-mu model in which a net interfacial tension gamma acts along each cell boundary and the cytoplasm exhibits an effective viscosity mu. The stress-strain characteristics of a rectangular patch of model tissue are investigated in terms of the rate r at which lamellipodia form and the relative strength q of their contractions. In tissues that are not constrained in-plane by adjacent tissues, the rate of tissue reshaping is proportional to r the rate of lamellipodium formation and its dependence on q is nonlinear and, near its expected value of 2 highly sensitive to q. Cell elongation, a central characteristic of CE, and stress is found to vary linearly with e the degree of kinematic restraint. Relevant "mechanical pathways" are also identified.

High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus

The polymeric split Hopkinson pressure bar (PSHPB) apparatus is introduced as a means for measuring the high strain rate (1,000-2,500 s(-1)) compressive properties of soft tissues. Issues related to specimen design are discussed, and protocols are presented for specimen preparation. Proposed specimen geometries were validated using high-speed photography. Stress-strain data were obtained for high strain rate compression of bovine muscle tissue to strains as high as 80%. The stress-strain curves were found to be strain rate-sensitive and concave upward, as is typical of soft tissues. Rigor had a significant impact on the material properties between 5 and 24 h post mortem, while at longer times, properties returned essentially to their pre-rigor values. This study presents some of the first published high rate properties of muscle tissue, data that are urgently for advanced modeling of the human body and for evaluation of safety systems for the human body.

Do lamellipodia have the mechanical capacity to drive convergent extension?

Convergent extension (CE), a kinematic motif associated with several important morphogenetic movements in embryos, entails narrowing of a tissue in one in-plane direction and elongation in the other. Although the cell elongation and intercalation which accompany this process have been investigated and relevant genes and biochemical pathways have been studied in multiple organisms, a fundamental question that has not yet been answered is "Do the lamellipodia thought to drive these motions actually have the mechanical capacity to do so?" Here, we address this and a number of related issues using a state-of-the-art computational model which can replicate cell motions, changes in cell shape and tissue deformations. The model is based on the cell-level finite element approach of Chen and Brodland, but has additional features which allow it to model lamellipodium formation and contraction. In studying CE, computational models provide an important complement to molecular approaches because they reveal the "mechanical pathways" through which gene products must ultimately act in order to produce physical movements. The model shows that lamellipodia can drive CE, that they do so through cell intercalations and that the elongated cells characteristic of CE arise only when adjacent tissues resist convergence, a result which we confirm experimentally.

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.

Multiview Robotic Microscope Reveals the In-plane Kinematics of Amphibian Neurulation

A new robotic microscope system, called the Frogatron 3000, was developed to collect time-lapse images from arbitrary viewing angles over the surface of live embryos. Embryos are mounted at the center of a horizontal, fluid-filled, cylindrical glass chamber around which a camera with special optics traverses. To hold them at the center of the chamber and revolve them about a vertical axis, the embryos are placed on the end of a small vertical glass tube that is rotated under computer control. To demonstrate operation of the system, it was used to capture time-lapse images of developing axolotl (amphibian) embryos from 63 viewing angles during the process of neurulation and the in-plane kinematics of the epithelia visible at the center of each view was calculated. The motions of points on the surface of the embryo were determined by digital tracking of their natural surface texture, and a least-squares algorithm was developed to calculate the deformation-rate tensor from the motions of these surface points. Principal strain rates and directions were extracted from this tensor using decomposition and eigenvector techniques. The highest observed principal true strain rate was 28 +/- 5% per hour, along the midline of the neural plate during developmental stage 14, while the greatest contractile true strain rate was--35 +/- 5% per hour, normal to the embryo midline during stage 15.

Mechanical effects of cell anisotropy on epithelia

Theoretical, numerical and experimental methods are used to develop a comprehensive understanding of how cell shape affects the mechanical characteristics of two-dimensional aggregates such as epithelia. This is an important step in relating the mechanical properties of tissues to those of the cells of which they are composed. Statistical mechanics is used to derive formulas for the in-plane stresses generated by tensions gamma along cell-cell interfaces in sheets with anisotropic cellular fabric characterized by average cell aspect ratio kappa. These formulas are then used to investigate self-deformation (strain relaxation) of an anisotropic sheet composed of cells of thickness h and having effective viscosity mu. Finite element simulations of epithelia and of isolated cells and novel relaxation studies of specimens of embryonic epithelia reported herein are consistent with the predictions of the theory. In all cases, geometric factors cause the relaxation responses to be more complex than a single decaying exponential.

A computer model for reshaping of cells in epithelia due to in-plane deformation and annealing

Although cell reshaping is fundamental to the mechanics of epithelia, technical barriers have prevented the methods of mechanics from being used to investigate it. These barriers have recently been overcome by the cell-based finite element formulation of Chen and Brodland. Here, parameters to describe the fabric of an epithelium in terms of cell shape and orientation and cell edge density are defined. Then, rectangular "patches" of model epithelia having various initial fabric parameters are generated and are either allowed to anneal or are subjected to one of several patterns of in-plane deformation. The simulations show that cell reshaping lags the deformation history, that it is allayed by cell rearrangement and that it causes the epithelium as a whole to exhibit viscoelastic mechanical properties. Equations to describe changes in cell shape due to annealing and in-plane deformation are presented.

New information from cell aggregate compression tests and its implications for theories of cell sorting

In order to verify theories about the mechanics of cell sorting, tissue spreading and checkerboard pattern formation, it is necessary to measure certain cell properties such as surface tension and adhesiveness. The purpose of this work is to clarify the relationship between these two important properties and to use computer simulations and analytical calculations to extract additional information from parallel plate compression tests. This paper shows that compression tests can be used to determine not only the surface tension between the aggregate and the surrounding medium, but also the effective viscosity of the cell cytoplasm and the interfacial tension that acts between the cells that make up the aggregate. The findings reported here also support a novel, differential interfacial tension-based theory for cell sorting, tissue spreading and checkerboard pattern formation, and pose further challenges to current differential adhesion-based models.

Computer simulations of mitosis and interdependencies between mitosis orientation, cell shape and epithelia reshaping

Finite element-based computer simulations are used to investigate mitosis and how mitosis, cell shape, and epithelium reshaping depend on each other. Frame- and cell-oriented patterns of mitosis with growing and non-growing daughter cells are considered. Previous simulations have shown that applied stresses or strains can reshape cells so that their long axes are aligned in the principal stretch direction. The simulations reported here show that this can produce global alignment of the mitosis cleavage planes. Other simulations reported here show that mitoses with suitably aligned cleavage planes can drive epithelium reshaping. Formulas that quantify these and other dependencies are derived. These formulas provide quantitative relationships against which current hypotheses regarding epithelia reshaping in real biological systems can be evaluated.

The Differential Interfacial Tension Hypothesis (DITH): a comprehensive theory for the self-rearrangement of embryonic cells and tissues

A comprehensive theory, herein named the Differential Interfacial Tension Hypothesis for the self-rearrangement of embryonic cells and tissues is presented. These rearrangements include sorting, mixing and formation of checkerboard patterns in heterotypic aggregates of embryonic cells, and total or partial engulfment, separation and dissociation of tissues. This broadly-based theory accounts for the action of all currently known cytoskeletal components and cell adhesion mechanisms. The theory is used to derive conditions for the cell and tissue rearrangements named above. Finite element-based computer simulations involving two or more cell types confirm these conditions.

Cell-level finite element studies of viscous cells in planar aggregates

A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available.

The mechanics of heterotypic cell aggregates: insights from computer simulations

Finite element-based computer simulations are used to investigate a number of phenomena, including tissue engulfment, cell sorting, and checkerboard-pattern formation, exhibited by heterotypic cell aggregates. The simulations show that these phenomena can be driven by a single equivalent force, namely a surface (or interfacial) tension, that results from cytoskeletal components and cell-cell adhesions. They also reveal that tissue engulfment, cell sorting, and checkerboard-pattern formation involve several discernible mechanical features or stages. With the aid of analytical arguments, we identify the conditions necessary for each of these phenomena. These findings are consistent with previous experimental investigations and computer simulations, but pose significant challenges to current theories of cell sorting and tissue engulfment.

The mechanics of cell sorting and envelopment

Aggregates of embryonic cells undergo a variety of intriguing processes including sorting by histological type and envelopment of cell masses of one type by another. It has long been held that these processes were driven by differential adhesions, as embodied in the famous differential adhesion hypothesis (DAH). Here, we use analytical mechanics to investigate the forces that are generated by various sub-cellular structures including microfilaments, cell membranes and their associated proteins, and by sources of cell-cell adhesions. We consider how these forces cause the triple junctions between cells to move, and how these motions ultimately give rise to phenomena such as cell sorting and tissue envelopment. The analyses show that, contrary to the widely accepted DAH, differential adhesions alone are unable to drive sorting and envelopment. They show, instead, that these phenomena are driven by the combined effect of several force generators, as embodied in an equivalent surface or interfacial tension. These unconventional findings follow directly from the relevant surface physics and mechanics, and are consistent with well-known cell sorting and envelopment experiments, and with recent computer simulations.

Three-dimensional reconstruction of live embryos using robotic macroscope images

To determine the three-dimensional (3-D) shape of a live embryo is a technically challenging task. We show that reconstructions of live embryos can be done by collecting images from different viewing angles using a robotic macroscope, establishing point correspondences between these views by block matching, and using a new 3-D reconstruction algorithm that accommodates camera positioning errors. The algorithm assumes that the images are orthographic projections of the object and that the camera scaling factors are known. Point positions and camera errors are found simultaneously. Reconstructions of test objects and embryos show that meaningful reconstructions are possible only when camera positioning and alignment errors are accommodated since these errors can be substantial. Reconstructions of early-stage axolotl embryos were made from sets of 33 images. In a typical reconstruction, 781 points, each visible in at least three different views, were used to form 1511 triangles to represent the embryo surface. The resulting reconstruction had a mean radius of error of 0.27 pixels (1.1 microns). Mathematical properties of the reconstruction algorithm are identified and discussed.

Cytoskeletal mechanics of neurulation: insights obtained from computer simulations

The morphogenetic movements associated with the process of neurulation have been the subject of much investigation during the last one hundred years. A plethora of experimental evidence has been generated regarding the forces that drive this seemingly simple process, and many theories about the mechanics of the process have been proposed. Recent computer simulations have proved useful for evaluating these theories from a mechanical perspective. In this work, computer simulations are used to investigate several theories about the forces that drive neurulation. A simplified version of a formulation previously presented by the authors provides the mathematical foundation for these simulations. The simulations confirm that forces generated by circumferential microfilament bundles (CMB's) in conjunction with notochord forces can produce the rolling motions characteristic of amphibian neurulation. They also support the notion that redundancies exist in the systems of forces available to drive neurulation shape changes. The shape changes that occur following a variety of surgical and teratogenic interventions are also simulated. These simulations corroborate the role of circumferential microfilament bundles as a primary force generator.

Embryonic tissue morphogenesis modeled by FEM

A three-dimensional, large-strain finite element formulation for the simulation of morphogenetic behaviors in embryonic tissues is presented. It is used to investigate aspects of invagination, neural tube morphogenesis, contraction wave propagation and mechanical pattern formation. The simulations show that the spacing of patterns and the shapes produced by certain morphogenetic movements in epithelial sheets depend only slightly on the properties of the materials which underlie these sheets. Simulations of neural tube closure show that numerous, experimentally-observed features can be produced by contraction of apical microfilament bundles alone. That certain systems of forces are mechanically equivalent and that certain patterns of deformations are equivalent set practical limits on what can be inferred from the simulations.

Furrowing surface contraction wave coincident with primary neural induction in amphibian embryos

We predicted, and have now observed, a surface contraction wave in axolotl (Ambystoma mexicanum) embryos that appears to coincide temporally and spatially with primary neural induction and homoiogenetic induction, and with involution of the chordomesoderm. The wave starts from a focus anterior to the dorsal lip of the blastopore and spreads as an ellipse, until part of it encounters the rim of the blastopore and vanishes there. The remaining arc then continues over the dorsal hemisphere until it reforms an ellipse that decreases in size. About 9 to 12 hours after it begins, the wave vanishes at a focus diametrically opposite its point of origin. The wave involves both local contraction and furrowing in the monolayer ectoderm. To a good approximation, the hemispherical portion of the ectoderm traversed by the wave becomes neuroepithelium, while the ectoderm not traversed by the wave becomes epidermis. The wave might provide a mechanism to determine the time and location at which neuroepithelial differentiation occurs.