A combination of active forces is required to drive ventral furrow invagination

Our quantification and digitisation of the wild type invagination starts during the fast phase of cellularisation and shows that it finishes in the mesoderm before the ectoderm, with the mesodermal cells beginning to radially lengthen before cellularisation is complete in the ectoderm, see Fig. 3 and Video V1. The cell lengths are quantified in Fig. 3d, which shows that prior to t = −10.43 min mesodermal cells increase their radial length at an average rate of 0.8 µm/min which corresponds to the same value measured by Lecuit et al. [17] for the fast phase of cellularisation. Fig. 3d also shows that the rate of growth of the ectodermal cells during cellularisation is slower than that of mesodermal cells (0.66 µm/min). This is evidence that the origin of differing cell fates is established early, perhaps during the slow phase of cellularisation.

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Figure 3. Modelling ventral furrow invagination: wild type genotype.

(a) Selected multi-photon images (columns) of in vivo transverse cross-sections of a wild type embryo during ventral furrow formation. The embryo is labelled with Sqh-GFP and is oriented with its dorsal surface upward (Fig. 1a). The time interval between successive frames is 45 sec. Time zero was set by using apical-basal cell height profiles on the dorsal side (Fig. S2a). (b) Selected frames (columns) of in silico transverse cross-sections of a wild type genotype. The embryo's 2D geometry is oriented with its dorsal side upward. Undeformed initial configuration (panel b′) corresponds to in vivo deformations at t = −10.43 min (see panel d). (c) In silico force distributions utilised to simulate invagination in b). Total invagination interval [−1.43, 6 min] has been subdivided in the three subintervals [−10.43 min, 1.2 min], [1.2 min,2 min] and [2 min, 6 min] respectively referred to as the first, second and third invagination intervals. Force trends are a discretisation of the in vivo force distribution available in Brodland et al. [1]. Smooth variations in time and position of in vivo force distributions over epithelial regions have been discretised to in silico force distributions that vary with time but not with position over epithelial regions. It is worth noticing that the components constituting the in vivo mechanism of invagination (mesodermal apical constriction, ectodermal basal constriction, ectodermal and mesodermal radial shortening) have been preserved. (d) Time-trends of average radial mesodermal and ectodermal thicknesses of the epithelium. Measurements in vivo have been repeated on a set of three different animals of the same genotype, and average trend and error bars (standard error of the mean) are obtained through statistical analysis of this data set (see Methods). It is worth noticing that mesodermal cells character begins differentiating from the ectodermal one approximately at t = −10.43 min (red vertical line), before which all cells share a communal epithelial fate. In silico, mesodermal radial thickness M (see schematic in panel e) has been measured at different angles of an angular section of the epithelium, which spans approximately 50 degrees astride ventral point V (25 degrees in each direction from V). In silico measurements for ectodermal radial thickness E, instead, stretched approximately up to 120 degrees astride the dorsal point D (60 degrees in each direction from D). (e) Time-trends of furrow's height h at the ventral side (see schematic in panel d).

https://doi.org/10.1371/journal.pone.0034473.g003

At t = −10.43 min, when mesodermal cells are on average 24 microns long, wild type mesodermal cells increase their radial length at approximately 1.20 µm/min, which is ∼3× that seen for ectodermal cells over this same period. This may be due to the fact that mesodermal cells have begun to actively lengthen, whereas ectodermal cells have yet to complete cellularisation. This hypothesis is supported by the evidence shown in Fig. S1 that epithelial cells consume yolk as they increase in area up until t = 0 min.

At t = 0, the ectodermal cells appear fully cellularised and begin to shorten radially, at an approximate rate of 0.1 µm/min. Just prior to this, at t = −1.8 min, the mesodermal cells reach their peak height, before immediately shortening at a rate of 1.8 µm/min, which is 18–times greater than the rate of ectodermal cell shortening (Fig. 3d).

From observation alone it is not possible to ascertain whether the mesodermal and ectodermal shortening observed is due to an active force generated within the cells or is passive. This because an observed shortening could be due to its neighbour cells lengthening, which would be a passive response, or to the cell itself shortening using its internal cellular mechanisms, which would be an active response. To resolve this issue we used our biomechanical model (Fig. 2) to investigate quantitatively what active cellular forces are required to produce the movements observed. In the model, mesodermal apical constriction, mesodermal shortening, ectodermal basal constriction and ectodermal shortening are all simulated as independent active forces. They can be localised both temporally and spatially within the embryo and switched on and off in any combination.

In previous work we have measured the temporal–spatial distribution of forces during ventral furrow invagination in Drosophila [1]. Here we have discretised these forces so that they could be implemented in a step–wise fashion (as shown in Fig. 3c), rather than in a spatial and temporal continuum. This change from continuous to a discrete set of forces is important for our systematic perturbation analysis and hypothesis testing, which is the focus of this paper.

The mesodermal apical constriction forces were induced to act first. Then at t = −1.23 min mesodermal shortening forces were switched on along with forces causing basal constriction and shortening of the ectoderm, while mesodermal apical constriction were reduced in intensity. The discrete timings and relative forces implemented in this model of ventral furrow formation are shown in Fig. 3c, and shown in Video V2. As can be seen from the experiment (Fig. 3b) this simulation replicates the main features of wild type invagination. Moreover, as shown in Fig. 3, it is also in good quantitative agreement with experiment as measured by two key parameters: the average thickness of the mesoderm (Fig. 3d) and height h of the furrow (Fig. 3e); see also Fig. 1 for a schematic view of these parameters. Although the effect of implementing the active shape changes in a discrete manner means that changes occur relatively abruptly in the model, the mesoderm thickness in the model follows a very similar path to that of the experiment (Pearson's  = 0.75, see Table 1) deviating on average a few standard errors from the in vivo data (RMSSD = 3.93). Relative differences between the in silico and the in vivo values in the final invagination configuration amount to less than 5% of and h, and less than 15% for the average thickness of the ectoderm (Fig. 3d–e). As shown in Fig. 3d, within the ectoderm, the model only mirrors the experimental values after it has reached its maximum thickness at t = 0 (Pearson's  = 0.86) deviating on average only one and a half standard errors from the in vivo data (RMSSD = 1.45). Before this time point the in vivo and in silico data differ possibly because the model does not include growth of the ectoderm associated with on-going cellularisation (for t<0 the ectodermal region in our model already undergoes apical-basal shortening whereas in vivo, the apical-basal length of ectodermal tissue is still increasing, see Fig. 3d). This is reflected by a very low Pearson's  = 0.19, when computed over the invagination interval, although in silico predictions still follow quite closely the experimental data (RMSSD = 1.26). Fig. 3e shows that height of the furrow in the simulation matches the height obtained in the experiment very well (Pearson's  = 0.87), although again the discrete nature of the implementation of the active shape changes causes the height profile to be less smooth than the experiment (RMSSD = 4.26).

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Table 1. Goodness-of-fit analysis for in silico versus in vivo curves.

https://doi.org/10.1371/journal.pone.0034473.t001

Agreement between mutant phenotypes of simulation and experiment validates the model

A more stringent test of this biomechanical model is to use it to predict mutant phenotypes lacking in one or more of the force components actively involved in invagination, and to compare these quantitatively with experimental data. For this analysis, we used three mutants: bnt, to test the role of forces elsewhere in the embryo; arm, to explore the role of cell–cell adhesion and the transmission of tension; and cta/t48 to test the role of apical constriction itself. We consider each in turn.

The bnt mutant lacks AP patterning while dorsal–ventral patterning remains intact. During invagination mesodermal average radial thickness increases at an approximate rate of 0.7 µm/min until t = −10.43 min, after which it starts increasing at the higher rate of 1.3 µm/min (Fig. S2). It is worth noting that at approximately t = −7.5 min, the bnt mesodermal cells start lengthening at an even faster speed of approximately 2.3 µm/min, a rate almost double that of wt mesodermal cells (Fig. 4d and Video V3). After this the bnt mesodermal cells shorten radially with a constant rate of approximately 1.3 µm/min until invagination is complete, which is almost 30% less than the rate seen in wt mesodermal cells (Fig. 4d). In the ectoderm, cellularisation proceeds at approximate rates of 0.9 µm/min and 0.5 µm/min respectively before and after t = −10.43 min, until cells reach a maximum height at t = 0 min. Ectodermal cells then shorten at an average speed of 0.2 µm/min, which is double that of the wt embryo.

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Figure 4. Modelling ventral furrow invagination: bnt-mutant genotype.

Selected multi-photon images of in vivo transverse cross-sections of a bicoid,nano,torso-like mutant embryo during ventral furrow formation. bnt-mutant has germ-band extension and posterior mid-gut invagination suppressed. The embryo is labelled with Sqh-GFP and oriented with its dorsal surface upward (Fig. 1a). Time interval between successive frames is 45 sec and instant zero was set by using apical-basal cell height profiles on the dorsal side (Fig. S2b). (a) Selected multi-photon images (columns) of in vivo transverse cross-sections of a bnt-mutant embryo during ventral furrow formation. The embryo is labelled with Sqh-GFP and is oriented with its dorsal surface upward (Fig. 1a). The time interval between successive frames is 45 sec. Time zero was set by using apical-basal cell height profiles on the dorsal side (see panel d). (b) Selected frames (columns) of in silico transverse cross-section of a bnt-mutant genotype. Embryo's 2D geometry is oriented with its dorsal side upward. The undeformed initial configuration (panel b′) corresponds to in vivo deformations at t = −10.43 min (see panel d). (c) In silico force distributions utilised to simulate invagination in b). (d) Time-trends of average radial mesodermal () and ectodermal () thicknesses of the epithelium (see schematic in Fig. 3e). (e) Time-trends of furrow's height h at the ventral side (see schematic in Fig. 3e).

https://doi.org/10.1371/journal.pone.0034473.g004

The bnt mutant was modelled by increasing the width of the active mesoderm to a region comprising 84 degrees astride the ventral point (the mesoderm in all other simulations covers 68 degrees across the ventral point – see Fig. 1). This value is taken from experimental data which shows that Myosin II that is initially localized on the basal side of the epithelium (red–dotted line in Fig. 1a) vanishes, while myosin newly appears on the apical side (on the opposite end of cells) in a region that in the bnt spans a wider angle (84 degrees) than in the wt embryo (64 degrees) as shown in Fig. 1. When t = −7.5, mesodermal cells in bnt mutants lengthen at a faster rate and to a greater extent than mesodermal cells in wt animals. This phenomenology can only be reproduced in silico either by introducing an active lengthening of mesodermal cells at t = −7.5 or by advancing ectodermal shortening at t = −7.5. Since there is no plausible biological mechanism in the literature in support of the first option, the onset of ectodermal radial shortening in the model was advanced to t = −7.5 s with respect to the wt embryo model, doubled in intensity in the first invagination interval and reduced in the second interval of invagination (Fig. 4c). This also correlates to experimental evidence showing the distribution of myosin–II at the basal end of ectodermal cells (Fig. 1a and Video V4).

Comparison of Fig. 4a and 4b shows the evolution of the in silico bnt phenotype which compares very well the experimentally obtained phenotype. Fig. 4d shows that both the increased mesodermal thickness profile, and the decreased furrow height profile of the bnt mutant are matched by the simulation excellently in terms of trend, with Pearson's  = 0.95 for mesodermal trend and Pearson's  = 0.97 for furrow's height curve. The discrete nature of the force profiles still affects the in silico path with RMSSD = 2.85 and 11.9, for the mesodermal and furrow path respectively. The relative difference between the in silico and the in vivo values in the final invagination configuration amounts to less than 4% for and less than 4% for h (Fig. 4d–e). Importantly this data shows that the experimental trends of increased mesodermal width and lower furrow height are mechanically necessary and coupled.

Embryos mutant in arm, which lack β–catenin, an essential component of adherence junctions between epithelial cells, initially behave like wt embryos, with the mesodermal cell apically constricting as normal, and the furrow starting to form, see Fig. 5a. However at t = −4.22 min the cell–cell apical junctions fail and the arching furrow collapses back into contact with the vitelline membrane, see Fig. 5b. As a consequence for t>1 min the cells start rounding up and make digitisation process difficult. At times later than 6 min, the mesoderm buckles into several folds (Video V5).

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Figure 5. Modelling ventral furrow invagination: armadillo-mutant genotype.

(a) Selected multi-photon images of in vivo transverse cross-sections of an armadillo-mutant embryo during ventral furrow formation. armadillo-mutant lacks strong apical junctions, and ventral indentation collapses at t = −4.22 min. The embryo is labelled with Sqh-GFP and oriented with its dorsal surface upward (Fig. 1a). The time interval between successive frames is 45 sec and instant zero was set by using apical-basal cell height profiles on the dorsal side (Fig. S2c). (b) Selected frames (columns) of in silico transverse cross-section of an arm-mutant genotype. Embryo's 2D geometry is oriented with its dorsal side upward. The undeformed initial configuration (panel b′) corresponds to in vivo deformations at t = −10.43 min (see panel d). (c) In silico force distributions utilised to simulate invagination in b). (d) Time-trends of average radial mesodermal () thickness of the epithelium (see schematic in Fig. 3e).

https://doi.org/10.1371/journal.pone.0034473.g005

The arm mutant was modelled by turning off mesodermal apical constriction at t = −4.22 min, once the furrow rises up from the vitelline membrane to simulate the inability to transmit tension between the cells. Fig. 5c shows the force profile, and Fig. 5b illustrates the development of the phenotype during the simulation which shows good qualitative agreement with experiments. Fig. 5d shows the quantitative comparison of the mesoderm thickness between the experiment and the simulation. Agreement with the experimental trend over time is very good (Pearson's  = 0.85), with the relative difference between the in silico and in vivo values in the final instant of invagination amounting to 34% for . The in silico predictions still deviate on average a few standard errors from the in vivo data– due to discrete nature of force profiles utilised (RMSSD = 4.23). The data clearly shows the extent to which model struggles to reproduce quantitatively the in vivo deformations despite giving good qualitative agreement (Video V6).

Embryos doubly mutant for cta and t48 do not form any visible ventral furrow (Video V7). We see a very similar behaviour in our cta/t48 double mutant embryos; the mesoderm does not apically constrict or thicken, and there is no furrow formation, see Fig. 6a. The cta/t48 double mutant is modelled by turning apical constriction off in the model. Fig. 6b shows the development of the phenotype during the simulation, which shows good qualitative agreement with experiment when comparing the radial thinning undergone by the mesodermal regions. The quantitative comparison shows a very acceptable Pearson's  = 0.60 and RMSSD = 0.99, with the relative difference between the in silico and the in vivo data in the final invagination configuration amounting to less than 8% for the (Fig. 6d and Video V8).

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Figure 6. Modelling ventral furrow invagination: cta/t48-mutant genotype.

(a) Selected multi-photon images of in vivo transverse cross-sections of a cta/t48-mutant embryo during ventral furrow formation. In cta/t48-mutants apical constriction is completely abolished. Embryo is labelled with Sqh-GFP and oriented with its dorsal surface upward (Fig. 1a). Time interval between successive frames is 45 sec and instant zero was set by using apical-basal cell height profiles on the dorsal side (Fig. S2d). (b) Selected frames (columns) of in silico transverse cross-section of a cta/t48-mutants genotype. Embryo's 2D geometry is oriented with its dorsal side upward. The undeformed initial configuration (panel b′) corresponds to in vivo deformations at t = −10.43 min (see panel d). (c) In silico force distributions utilised to simulate invagination in b). (d) Time-trends of average radial mesodermal () thickness of the epithelium (see schematic in Fig. 3e).

https://doi.org/10.1371/journal.pone.0034473.g006

Ventral furrow formation is robust to large biomechanical perturbations

To test the robustness of the invagination mechanism to the loss of ectodermal forces we carried out a set of simulations in which we perturbed them systematically. In the first set of simulations we removed ectodermal basal forces, then ectodermal radial shortening forces, and finally all ectodermal forces. Fig. 7a, b, c shows the phenotypes obtained, and Fig. 7d–e shows the timing–force diagrams for the new mechanisms tested. The results show that while ectodermal forces are not essential for furrow formation they do affect the shape of the furrow and the speed at which is it developed. Fig. 7f–g shows that ectodermal forces have very little effect on mesodermal thickness and furrow height. But Fig. 7b–c show that the loss of ectodermal radial shortening causes a hole to be formed in the furrow (a feature sometimes observed in vivo); a similar result is obtained by increasing or anticipating shortening of mesodermal cells (Fig. 8e–g). We showed that the mechanism is robust to ectodermal radial timing perturbations (Fig. S3), ectodermal basal timing perturbations (Fig. S4), ectodermal radial intensity perturbations (Fig. S5) and ectodermal basal intensity perturbations (Fig. S6).

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Figure 7. Ventral furrow invagination robustness to in silico ectodermal perturbations.

The wild type mechanism of invagination (Fig. 3c) has been perturbed by switching on/off component forces acting in the ectoderm (blue and yellow lines). The figure shows that perturbing the wild type mechanism of invagination does not hinder ventral furrow formation, which is ultimately regained in a shorter or longer period depending on the typology of perturbation. It is worth noticing that the perturbation introduced in the invagination mechanism (see panels f–g) does not affect phenotypes corresponding to t<−1.5 min, which remain the same as in the wild type case (Fig. 3a′–a″). (a) Phenotypes relative to the wild type mechanism where no basal ectodermal constriction is activated. (b) Phenotypes relative to the mechanism where no radial ectodermal shortening has been activated. (c) Phenotypes relative to the mechanism where both ectodermal movements have been turned off, resulting in an inactive ectoderm. (d) Force trends utilised to simulate the wild type invagination mechanism with no ectodermal basal constriction – the component that corresponds in the graph to this movement has been suppressed (blue line). (e) Force trends utilised to simulate the wild type invagination mechanism with no ectodermal radial shortening – the component that corresponds in the graph to this movement has been suppressed (yellow line). Similarly, force trends to simulate wild type invagination with no active ectodermal forces (both basal and radial ones) have been obtained by suppressing the lines that correspond in the graph to these two movements. (f) Time-trends of average radial mesodermal () thickness of the epithelium (see schematic in Fig. 3e). (g) Time-trends of furrow's height h at the ventral side (see schematic in Fig. 3e). The blue dotted lines in panels (f) and (g) indicate that the simulation detailed in (b) (where no radial ectoderm has been activated) fully invaginates on a longer period of time with respect to that shown in figure panels.

https://doi.org/10.1371/journal.pone.0034473.g007

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Figure 8. Meso-radial time study.

The quantitative effects of anticipating the onset of mesodermal radial shortening with respect to the wild type case reported in Fig. 3, while keeping the remaining force trends unchanged (Fig. 3c). (a) Force trend curves labelled by , , and illustrate the case where meso-radial movement was respectively advanced at t = −3.48 min, t = −5.8 min, t = −8.12 min and delayed at t = 0.58 min with respect to the wt case (where meso-radial movement onsets at t = −1.2 min, as shown in Fig. 3c). (b–c) Changes in the onset time of this movement with respect to the others have significant impact on furrow's height h, which increases with the anticipation of the movement. The average mesodermal thickness decreases with the anticipation of the movement in that mesodermal cells start shortening earlier. (d–g) Final phenotypes (t = 6 min) corresponding to wild type with mesodermal radial movements respectively delayed by (dotted lines in panels a–c) and advanced by , and .

https://doi.org/10.1371/journal.pone.0034473.g008

Fig. 8 shows the effect of perturbing the onset of mesodermal radial shortening with respect to the wild type case. The effect is quite dramatic, with both the thickness of the mesoderm and the height of the furrow affected, however invagination still completes, with the final phenotype relatively unaffected (Fig. 8d–f). The effect of changing the intensity of the mesodermal radial shortening forces dramatically affects both mesodermal thickness and the height of the furrow, however invagination still completes (Fig. S7). In contrast reducing or increasing the intensity of mesodermal apical constriction has very little effect on the height of the furrow or the thickness of the mesoderm as well as on the shape of the furrow (Fig. S8).

Embryos and genotypes

At mid–cellularisation stage, embryos were de–choryonated 2 minutes at room temperature in 50% bleach, followed by several washes with PBS. They were kept in PBS for several hours without apparent developmental delays or problems.

To genetically abolish anterior–posterior polarity, germ–band extension and posterior midgut invagination, we used the stock w; Sqh–GFP⁴²; bicoid^E1 nanos^L7 torso–like¹⁴⁶/TM3 Sb and analyzed progeny from mothers homozygous for bicoid^E1 nanos^L7 torso–like¹⁴⁶ (in short: Sqh–GFP; bnt embryos). To specifically block apical constriction, we used w, Sqh–GFP⁴²; cta^R10/CyO; Df(3R)^CC1.2/TM3 Sb stocks. The Df(3R)CC1.2 deletion uncovers the t48 locus. 25% of progeny from homozygous cta^R10 parents showed a complete absence of apical constriction and were thus assumed to be Sqh–GFP; cta, t48 embryos. We generated armadillo germ–line clones using the FLP–DFS system². By crossing arm^043A01 FRT101/FM7; Sqh–GFP⁴² females to w, ovo^D, FRT¹⁰¹/Y; flp–138 males, we obtained arm^043A01 FRT¹⁰¹/w, ovo^D, FRT¹⁰¹; flp–138/+ females. These females were heat shocked as larvae for 2 hr at 37°C to induce mitotic recombination in the germ–line and crossed to FM7/+; flp–138/+ males.

Multi–Photon Microscopy

Transgenic Sqh–GFP embryos were mounted vertically in 1% agarose in PBS on Mattek culture dishes, with the posterior end facing the objective. They were imaged on a custom–built two–photon microscope, using a Nikon 40× NA 0.8 objective and simultaneous epi– and trans–detection as described in ³. We used laser lines between 890 and 910 nm. A 488 nm laser with an output power of 15W generated 150 mW laser pulses between 890 and 910 nm at the level of the objective. Embryos imaged under these conditions were able to develop normally afterwards. Imaging depth was 110 µm from the posterior end of the embryo, and images were acquired every 45 seconds with MATLAB. Note that the furrow is wider in bnt embryos.

Synchronization of Embryos and in vivo data analysis

A set of three different animals of the same genotype has been quantified for each genotype in order to obtain all microscopy data shown in this work. In this work we utilised a timer to synchronise time evolution of all the embryos analysed. According to this timer, the origin of time (instant zero, t = 0) is set for each embryo at the point of average maximal radial growth of ectodermal cells, see Fig. 3d. Importantly, this proved a reproducible timer that was independent of events occurring within the ventral furrow [1]. Coresponding in vivo trends have been obtained through statistical analysis of the resulting data set, where error bars are given by standard error of the mean. Since in silico predictions can fit in vivo trends quite well and yet completely miss the exact locations of the data – and vice versa – we also computed goodness-of-fit of in silico vs. in vivo datasets by using both Pearson's test and RMSSD measures (root mean squared scaled deviation) [26], see Table 1. The former assesses how well the in silico curve captures the relative in vivo trend whereas the latter estimates deviation of our in silico predictions from exact locations of in vivo data points: the quantity varies in the interval [0,1] and the closer is to 1 the better the agreement [26]. The quantity RMSSD is obtained by scaling absolute deviation of each in vivo data point by the standard error of the data and quantifies how close the model is to the in vivo data points (the lower RMSSD the better the agreement) [26].

Digitisation of embryo movies

Custom software was used to generate a reference mesh that was overlaid and adjusted manually to fit the first image of each movie of the embryo. Automated and manual methods were used to position the registration points of the mesh to track the in vivo movements from image to image in each set. This included ccustom software to enable multiple frames to be viewed in both forwards and backwards directions so that the location of the registration nodes could be placed with precision. A combination of automated and manual methods was used to track the fiduciary points from image to image when the edge of the cell was difficult to discern. Image contrast was locally optimized in each frame with image processing techniques (using Adobe Photoshop) in order to bring out contour of the less visible parts of the forming furrow. Placement of fiduciary points in difficult areas of the in vivo images was also further assisted by automatic constant–volume algorithms emulating conservation of cell cytoplasmic volume. Combination of these two techniques and high level of image zooming ensured tracking error was to same level as for the clearest parts of the images.

Reference configuration for deformations of all embryos is set at 10.43 minutes before instant zero (i.e. t = −10.43 min, see above for temporal synchronisation of embryos). An angular coordinate system was also defined at t = −10.43 min on the circular reference configuration of each embryo in order to locate specific areas of the epithelium. The presumptive mesoderm is defined as the tissue that ultimately forms the ventral furrow. By tracking backward in time, this tissue was found to be nominally 18 cells wide in the wt embryo. In the wild type and bnt mutant embryos, this sub–region was respectively found to span approximately ±30° (green lines) and ±40° (blue lines) astride the dorsal–ventral midline. Fig. 1b shows the mesh of registration points utilised for the digitisation procedure of an intermediate–stage frame from the in vivo imaged sequence of a wt embryo: the polygonal mesh (blue lines) has been overlaid on the tissue to track displacements at registration nodes (magenta dots). Polygonal partitions can correspond to single cells, multiple cells or sub–cellular regions, depending on the measurements refinement required in that particular portion of tissue. Yellow and green arrows on Fig. 1b indicate the radial thickness of the epithelium, which was measured on the mesh as the distance between basal and apical node along the same side of a given cell. The height of the furrow (h) (red arrow on Fig. 1b) has been measured as the radial displacement of the most ventral nodal point from its position in the reference frame. Movies of the wild type and mutant embryo can be seen in the Supplementary Material.

Biomechanical Model

The simulations are based on a finite element method which involves modelling the key biomechanical features of cells and the embryo structure, see Fig. 2. In order to build a finite element model of the single epithelial cell, the effects of the complex force fields originating from molecular processes were assumed to be resolved into equivalent forces along cell edges (Fig. 2 a–c). The incompressibility of the inner cytoplasm was simulated by enforcing a constraint that the surface region enclosed by the cell membrane remained constant over time (Fig. 2c–d). Cell–sized regions of the embryo cross–section were partitioned into 5 quadrilateral elements with nodes at the corners of each element (Fig. 2d). A radial subdivision is used to allow individual cells to bend, as they do in vivo. In keeping with current models of morphogenetic movements in embryos [1], [13], [27], [28], the cytoplasm was assumed to be viscous, and its effective viscosity was assumed to arise from the mechanical properties of passive cellular components such as the cytosol and organelles. The forces needed to drive incremental nodal displacements over time increment were discretised in time with an explicit algorithm yielding the following set of equations:(1)where is a so called damping matrix. It parallels the more common stiffness matrix, but is applicable to viscous materials as opposed to elastic ones. To understand the physical meaning of Eqn. (1), consider a small, not necessarily rectangular, block of viscous material and assume that small incremental displacements of its corners are applied over a short time increment . The forces that would need to be applied to the corners are well defined and they depend on the geometry of the block and the viscosity of its material: they are given by Eqn. (1) [29]. Eqn. (1) can also be used together with appropriate boundary conditions to determine the incremental displacements that would occur if specified nodal forces were applied, or it can be used to relate various combinations of forces and displacements to each other, provided that the resulting system of equations is well posed. In a finite element model, equations like (1) are written for each element and they are mathematically assembled together to give global systems of equations that describe how all of the nodal forces and displacements are related to one another. In the present context, this set of equations relates the viscous forces in the cells in the cross–section shown in Fig. 2e to the incremental motions that they undergo.

In our previous inverse analysis [1], the displacements are retrieved from in vivo images, and the set of driving forces that satisfy Eqn. (1) were computed. Instead, in the forward analysis used here, a set of forces are applied, and the associated displacements are computed by solving Eqn. (1). We emphasize that for the purpose of the robustness analysis, the forces employed in our forward analysis have a simpler profile than those retrieved in the inverse analysis.

The active components of the cell include microfilaments and microtubules, and they are assumed to generate forces that can be resolved along the edges of the finite elements [28], [30]. Forces along edges that share a common boundary are added together, and applied vectorially to both of the common nodes, see Fig. 2d. In the case of cells containing multiple finite elements, forces along the internal edges are assumed to be zero. When the driving forces acting during a particular time step are added together node by node, they produce a system of driving forces . For the embryo cross–section to be in a state of dynamic equilibrium, these driving forces must just equal the forces generated by deformation of the cytoplasm as a result of incremental displacements . Thus, at each incremental step, the forces are calculated based on the current geometry, they are substituted into the right hand side of Eqn. (1), and the equation is solved for the incremental nodal displacements . These displacements describe node–by–node how the embryo cross–section would change from one moment to the next if acted on by the system of forces . In this way, one can use the model to determine the changes in embryo shape that would result from any user–specified set of driving forces . The embryo simulations were analysed in the same way as the experiment and movies of the simulations were generated for visualization.

The vitelline membrane has been modelled by introducing a rigid boundary condition that prevents apical nodes from moving outside a bounding circle (Fig. 1b and Fig. 2e). Consequently, the sum of the cross–sectional areas of the yolk, epithelium cells and ventral gap (the region where cells pull away from the vitelline membrane) is constrained to be constant. The yolk was modelled as a compressible viscous fluid exerting a pressure on the inner (basal) side of the epithelium to a maximum of 5% of its initial value. The experimental value of the yolk pressure is not known, so this was a system variable.

Invagination forces, viscosity and dynamics

For each cell the viscosity is used to define the damping matrix in Eqn. (1), which sets the magnitude of the forces operating. The problem from a modelling stand point is that the experimental values of viscosities available in the literature vary by eight orders of magnitude in the interval from 6.8×10⁻¹⁷ N min (µm)⁻² (i.e. 4.3×10−3 Pa.s, as in Gregor et al. [31]) to 1.7×10⁻⁹ N min (µm)⁻² (i.e. 10⁵ Pa.s, as in Bittig et al. [32]). This problem can be circumvented by normalizing the range of forces ‘per unit of viscosity’ that quantitatively reproduced in silico the invagination path observed in vivo. The units of these normalised forces are µm² N⁻¹ min⁻¹, and so forces measured in Newton are recovered simply by multiplying by a specific viscosity value. This approach does not change the location, timing or relative magnitude of the forces – the factors central to this systems analysis.

For the purpose of our perturbation analysis, we transformed the forces obtained from our VFM study [1] from a continuous to a discrete set that could be turned on and off. Once the model had been tuned in this way for the wild type simulation, see Fig. 3c, the range of forces were normalised and the same constants were used for all other mutant simulations and the perturbation analysis. In other words none of the parameters were changed when we used the model to simulate the mutant prototypes, the only things that change are the relative forces and timings, which were obtained from experiment and are shown in the figures.