A continuum shell model of the apical actomyosin cytoskeleton and adherens junctions meshwork can produce convergence and extension flows

Using the above model and the numerical technique described in S1 Text, we obtain numerical approximations of the solution of this problem for a geometry Γ closely mimicking the shape of Drosophila embryo. Fig 3 shows an example of a simulation result. The green colour codes σ_a(x), the location where myosin is assumed to be activated along dorsoventral direction A = e_DV ⊗ e_DV, and the arrows are the predicted velocity of the surface displacement of the apical continuum. This predicted flow is strongly dominated by two laterally-located vortices, which have their centre slightly dorsal from the edge of the GB region. They are rotating such that the velocity in the GB is strongly towards the posterior. The vortices are situated in the tissues located dorsally of the germband (future amnioserosa), which will thus be extensively deformed by the flow. This is indeed the case in Drosophila development, although part of the deformations in vivo may be related to PMG invagination [6], which is not included in this simulation (see below).

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Fig 3. Flow field generated by planar-polarised myosin contractility in the GB.

Parameters are η_b/η = 10³, c_f/η = 1/R, where R is the radius of a transverse cut of Γ, see Fig 1d. The GB region is defined as the ventral region posterior to cephalic furrow, and more ventral than a coronal plane z_max = −0.2R, see Fig 1. (a) Global view. The cephalic furrow is represented by a trough, the ventral furrow is not represented as we assume it to have sealed completely at the time corresponding to the simulations. Green, ventral region of the GB where we assume myosin planar-polarised contractility. White arrows, velocity vectors (arbitrary units, not all vectors calculated are represented). (b) Close-up of the region close to cephalic furrow and ventral midline. Every velocity vector calculated is represented, in arbitrary units 10 times larger than in panel (a).

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In order to compare this flow qualitatively with the actual movements of epithelia during Drosophila germband extension, we have calculated the displacement of the centroid of cells in vivo. We have imaged a wildtype Drosophila embryo expressing plasma membrane associated GFP markers using light sheet microscopy (specifically mSPIM, multidirectional selective plane illumination microscopy) to observe the whole embryo volume throughout germband extension in four perpendicular views then reconstructed to four-dimensional movies [6]. Using custom built software (see S1 Text) we mapped apical cell trajectories on one lateral surface of the embryo. In Fig 1a, the displacement between two consecutive frames during the fast phase of GB extension is shown. Such in toto tracking permits an account of the tissue movement across the whole of the embryo simultaneously, compared to previous microscopy techniques which were restricted to the observation of a limited view field [9]. The global features of the flow predicted by our myosin-activity based simulations are also apparent in this tracking. There is an embryo-scale vortex on each lateral side of the trunk part of the embryo, with the same direction of rotation between simulations and observations. The morphogenetic flow is much reduced in the head region, in simulations this reduction is even stronger, and velocities in the head region are less than 5% of the maximum velocity. Discrepancies between the model prediction and the actual flow can be reduced when biophysically relevant mechanical parameters are adjusted, and when the PMG invagination effect is included: this is the objective of the next sections.

Complementary role of planar-polarised myosin and posterior midgut invagination

It has been shown [6, 7, 14] that GB extension is not solely due to the action of planar-polarised myosin within the GB, but also to the pulling force that another morphogenetic movement causes, namely the invagination of endoderm in the posterior region, also called post-midgut (PMG) invagination.

Our present numerical approach does not allow us to simulate deformations of the surface in the normal direction, which would be necessary to simulate the PMG invagination process. However, it is possible to mimick the effect of PMG invagination on neighbouring tissues by simulating an in-plane isotropic contraction of a posterior patch of tissue, Fig 2b. Fig 4e shows that this does generate an extension of the GB area, although the deformation on the dorsal side is greater in this case. Combined with planar-polarised myosin action in the GB, Fig 4b–4d shows that posterior contraction does modify the flow pattern substantially and provides a complementary cause of GB extension, consistently with the experimental studies cited above. In fact, because of the linearity of Eqs (1) and (2), the superposition principle applies and the flows shown in Fig 4b–4d can be written as the weighted sum of the flows with planar-polarised myosin only, Fig 4a, and posterior contraction only, Fig 4e. The location chosen for posterior contraction corresponds to the location of PMG in early GB extension, Fig 1b, rather than the stage for which we have tracked cell flow, Fig 1a. In spite of this, the agreement with the experimental observations, Fig 1a, is improved by the addition of posterior contraction. The flow within the posterior region itself is not relevant to compare, since the numerical method does not allow for the invagination in itself. In neighbouring tissues, posterior contractility modifies the predicted flow on the dorsal side, creating a posterior-ward flow in a small region close to the contractile patch, and reducing the flow rate in the rest of the dorsal side. In particular, the posterior contractility in Fig 4b leads to flow rates of similar relative magnitude to the ones observed experimentally in Fig 1a.

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Fig 4. PMG invagination can contribute to GB extension.

Parameters are the same as in Fig 3, reprinted in (a), but (b–e) another region (red) is actively contracting in an isotropic way, mimicking the effect of PMG invagination on neighbouring tissues. From (b) to (d), the isotropic PMG contraction intensity is doubled each time. (e), effect of isotropic PMG contraction in the absence of any myosin activity in the GB itself.

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Influence of the ventro-lateral patterning of myosin activation

Although in toto imaging of Drosophila embryos is now possible using SPIM to track the morphogenetic movements which our model attemps to predict [6, 9, 26], see also Fig 1a, a global cartography at the embryo scale of myosin localisation and activation during GB extension is is not available yet. The patterning of gene expression upstream of Myosin II planar-polarised recruitment, on the other hand, is accurately described [13]. We tested several configurations of the extent of the region of GB where myosin is anisotropically localised, Fig 5. We find that the global flows present a common aspect and GB extension is consistently obtained. However, some features vary, and in particular an anterior-wards backflow develops along the ventral midline when the myosin activation zone extends more laterally, Fig 5b. These backflows are attenuated or disappear when the PMG contraction is large enough, by superposition with flows in Fig 4e.

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Fig 5. Influence of the patterning of myosin activation.

(a–b) Flow field generated by planar-polarised myosin contractility in the GB. Parameters are the same as in Fig 3, but the region of planar-polarised myosin recruitment is larger: (a) z_max = 0, (b) z_max = 0.2R (compare to Fig 3 where z_max = −0.2R). The same phenomenology is observed, with a strong posterior-oriented GB extension. Close to the midline, the tissue is not extended and goes towards anterior (slightly for z_max = 0, significantly for z_max = 0.2R). (c) Rate of area change as a function of the DV distance to ventral midline (in units of R), along a transverse cut midway along AP (x = 0), for z_max = −0.2R (purple symbols), z_max = 0 (green symbols), and z_max = 0.2R (cyan symbols). In the region where myosin prestress is nonzero, area decreases, while it increases outside this region. (d) Rate of DV strain as a function of the DV distance to ventral midline (in units of R), along a transverse cut midway along AP (x = 0), for z_max = −0.2R (purple symbols), z_max = 0 (green symbols), and z_max = 0.2R (cyan symbols). Dorsally, there is a DV extension and ventrally a DV contraction. A peak of DV contraction is located at the boundary of the myosin activated region in each case.

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In terms of the surface deformation of the epithelium, it is seen in Fig 5c and 5d, that the flow is qualitatively the same although the location of the boundary of the myosin activated region directly changes the location of the strong gradients of strain rate. Within each region, the simulation reveals nontrivial variations of the strain rate. We note in particular strong peaks of deformation rate close to the boundaries of the myosin activated region (Fig 5d), which are not reported in Drosophila. These peaks are strongly reduced if the myosin prestress is assumed to change smoothly rather than abruptly at the boundary of the region, as will be seen below. Overall, the similarity of the flows observed from z_max = −0.2R (Fig 3) and z_max = 0 (Fig 5a) and their qualitative agreement with experimental observations (Fig 1a) is indicative that the system is robust with respect to the exact pattern of myosin activation.

Influence of the choice of the mechanical parameters

Four parameters appear in Eq (3): the magnitude of myosin prestress σ_a, the friction coefficient c_f, and the two viscosities η_b and η. Two of these parameters, σ_a and η, set respectively the magnitude of stresses and velocities, leaving two free parameters: η_b/η, which is nondimensional; and c_f/η, which is the inverse of the hydrodynamic length Λ.

The ratio η_b/η compares the bulk to the shear viscosity: if it is very large, then the flow will be nearly incompressible in surface, meaning that any surface element (and in particular, any cell) will conserve the same area through flow, and hence all deformations will be locally pure shear deformations. If the ratio is small, then the viscous cost of locally changing the apical area will be similar to that of pure shear, and, depending on the global force balance, area changes may dominate. Indeed, if η_b/η is ⅔ (such that the Poisson ratio is zero), then a unixial load (such as a perfectly planar-polarised myosin action could be supposed to produce) will result in an area change only and no pure shear at all, or, in developmental biology terms, in convergence only and no extension. This effect is illustrated in Fig 6, where η_b/η covers the range 10 to 10³. In the latter case, the negative DV strain rate (convergence) exactly balances the positive AP strain rate (extension), whereas in the former case, convergence strongly dominates. The strain rates in all cases are not uniform across the ventral side depending on the AP position, with a marked decrease of the pure shear in the central part of the GB, however other factors are seen to affect this spatial distribution in the rest of the paper. Between η_b/η = 10 and η_b/η = 100, there is a switch from area-reduction dominated flow (area reduction rate 4 times AP strain rate) to a shear-dominated flow (area reduction rate of the same order as AP strain rate). Experimentally in Drosophila embryos, the deformations are not limited to pure shear but involve some area change [14]. By comparison of actual tissue flows in Fig 1a with our numerical result, the order of magnitude of η_b/η can be expected to be of the order of 10 to 100.

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Fig 6. Influence of the bulk viscosity on the convergence and extension of GB.

(a–c) Flows calculated for (a) η_b/η = 10³, (b) η_b/η = 10², (c) η_b/η = 10. While the posterior-wards flow at the GB posterior end is similar, the lateral flow is strongly affected with a much larger ventral-wards convergent flow along DV when the bulk viscosity is reduced to 10 (whereas no qualitative change is seen between 10³ and 10²). The position of the vortex centre is also much modified for low bulk viscosity. (d) Rates of strain in the DV (↕ symbols) and AP (↔ symbols) directions for the three choices of η_b/η (purple, 10³, green, 10², cyan, 10), as a function of AP coordinate x (in units of R) close to the midline (y = 0.2R). In the GB, DV rate is negative (convergence) and AP rate positive (extension). The DV rate of strain is increasingly negative for low bulk viscosity, indicating a stronger convergent flow, while the AP rate increases much less, indicating little change in the rate of GB extension. Posterior to the GB (AP coordinate x ≳ 2R), the DV and AP strain rate values ramp and invert their sign, indicating that the direction of elongation swaps from AP to DV, which corresponds to the splayed velocity vectors seen at the posterior limit of the GB e.g. in panel a. Anterior to the GB (AP coordinate x ≃ −R), the same effect is observed due to the obstacle of the cephalic furrow. (e) Rate of area change for the same choices of η_b/η, confirming that area decreases much more for lower bulk viscosity in the GB region. (f) Rate of area change for the same choices of η_b/η as a function of the DV distance (in units of R) to ventral midline, along a transverse cut midway along AP (x = 0). When η_b/η is sufficiently small to allow area variations, the GB region exhibits area reduction and dorsal region area increase.

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The hydrodynamic length Λ = η/c_f is the characteristic length within which shear stress will be transmitted within the actomyosin. Beyond this length, friction with the exterior will balance the internal stress. It thus indicates the length over which the effects of a local force are felt. We tested the effect of a hydrodynamic length either much larger, much smaller or comparable to the size of the embryo (R, the radius of a transverse section, Fig 1). The results shown in Fig 7 show how the action of myosin gives rise to a more local flow pattern when the hydrodynamic length is small. This localisation is around the areas in which there is a gradient of myosin activity. In areas of uniform myosin activity, the resulting effect is a uniform tension (see a similar effect in models of cells plated on a substrate, e.g. [40, Fig 3b]). It can be seen on Fig 7e and 7f, that indeed the strain variations are more abrupt when hydrodynamic length is small, whereas the long range interactions allowed by a very large hydrodynamic length give rise to an embryo-scale flow. On the whole, flows which reproduce experimental observations better are obtained when Λ is of order 1 or more, that is, the hydrodynamic length is comparable to or larger than the GB width in DV. Indeed, there is no strong localisation of the flow features in our experimental results, Fig 1a. This is consistent with the order of magnitude of 10 to 100 μm found by laser ablation in other systems [44].

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Fig 7. Friction with vitelline membrane and/or cytosol and yolk modifies the flow pattern.

(a–d) Global flow pattern. Parameters are the same as in Fig 3, but the hydrodynamic length varies between Λ = 10R (a), Λ = R (b, reprinted from Fig 3), Λ = R/10 (c), and Λ = R/100 (d). In all cases the GB extends posteriorly. In the cases of small hydrodynamic length, convergence and extension flow occur mostly in regions where there is a gradient of contractility. Overall, friction renders the effect of actomyosin activity more local to regions where they exhibit a variation, hence vortex structures are more localised next to these regions with higher friction and have less influence in regions of uniform actomyosin activity (dorsally or close to ventral midline e.g.) (e) Rates of strain in the AP direction for the four choices of Λ/R (purple, 10R, green, 1, cyan, 1/10 and orange, 1/100), as a function of AP coordinate x (in units of R) close to the midline (y = 0.2R). The overall magnitude of strain decreases with increased friction, as an increasing part of the energy provided by myosin activity needs to overcome friction in addition to deforming the cell apices. The rate of strain is uniformly positive (elongation) only when Λ is close to unity or smaller, else a region of shortening appears in the central part of GB. (Note that since area change is close to zero, the DV strain value is very close and opposite to the value of AP strain everywhere.) (f) Rate of strain in the DV direction for the four choices of Λ, as a function of AP coordinate x (in units of R) close to the midline (y = 0.2R). DV rate of strain always peaks close to the boundary of the GB area where myosin is active, in relative terms the peak is more pronounced for large frictions. Dorsally, there is always a positive DV rate of strain, indicating a DV elongation due to the pull of the neighbouring converging GB. This is matched with an AP shortening of a similar magnitude. Ventrally, the negative rate of strain (indicating convergence) is observed to decay when the hydrodynamic length becomes small, in that case the DV narrowing is limited to a narrow band at the DV edge of the GB. This localisation effect of small hydrodynamic length is also seen for the dorsal DV elongation, but to lesser extent.

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The cephalic furrow acts as a guide for morphogenetic movements

In simulations, Fig 3, it is seen that the flow follows the cephalic furrow in a parallel way. Thus we wondered whether the presence of the CF could be important for the flow pattern observed. To test this, we performed the same simulation on two different meshes, one featuring the CF and the other without it. In the absence of a CF, the flow at the anterior boundary of the GB does not deviate laterally but continues towards the anterior, Fig 8e. The flow field is thus much more symmetrical than in when the CF is included in the model, Fig 8b. From the mechanics, we expect that if both the geometry of the embryo and the myosin localisation patterns are symmetric, then the flow will be symmetric too (see Fig 8d for a verification of this). In real embryos, two sources of asymmetry arise: the localisation of the posterior boundary of the region of planar-polarised myosin recruitment pattern, and the invagination in the posterior midgut. Blocking it however does not completely suppress the posterior-ward extension of GB [7]. Our results suggest that another asymmetry could originate from the geometry of the embryo, with the CF acting as a barrier resisting flow towards the anterior. Indeed, if one introduces this geometric feature at one end of the otherwise perfectly symmetrical embryo, the flow is strongly asymmetric towards the posterior end, Fig 8a. In experimental embryos, the local influence of the CF on the flow is visible in tracked in toto images, Fig 1a, where a reduction of the magnitude of the velocity is seen in the row of cells posterior to the CF. The numerical simulations bring the additional understanding that, in addition to this local effect, this mechanical feature has a global influence on the morphogenetic flow and orients GB extension towards the posterior.

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Fig 8. Cephalic furrow (CF) can guide GB extension to be mostly posterior-wards.

Lateral view of flow fields generated by myosin contractility in the presence (a–c) or absence (d–f) of a cephalic furrow. (a, d) With a hypothetical symmetric planar-polarised myosin activity, the presence of CF orients the flow towards the posterior whereas it is perfectly symmetric in its absence. (b, e) With a realistic asymmetric planar-polarised myosin activity, the presence of CF still has a major role in orienting the flow the towards posterior. Although the asymmetric myosin patterning induces a asymmetric flow in the absence of the CF, the flow is not biased towards the posterior. (c, f) The flow created by PMG invagination is much less sensitive to the presence of CF.

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Simulation of wild-type GB extension

Using the qualitative study of the influence of each parameter (η_b/η, c_f/η) and of the patterning of myosin activation, we propose a choice of simulation settings which can be expected to reproduce the features of GB extension flows. Indeed, the equations being linear, the superposition principle guarantees that the effect of each of these causes simply add up. In Fig 9a, the magnitude of PMG strain rate is as in Fig 4b, η_b/η is chosen as in Fig 6b, the hydrodynamic friction as in Fig 7b. The myosin prestress pattern is close to the one in Fig 5a, but with a graded decrease of prestress when reaching the boundary of myosin activated region. Fig 9b–9e show the rate of strain along lines of interest on the surface. Compared to similar plots where myosin prestress is not graded close to the boundary, they have the same global features but no peaks close to the myosin activation boundary: compare e.g. Figs 5d and 9e. This is indeed due to the gradation of myosin prestress, since a simulation where only this parameter is altered restores the peaks (S1 Fig).

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Fig 9. Simulation of GB extension with proposed parameters for the WT Drosophila and comparison with an example of real data.

(a) Flow calculated for a proposed choice of parameters η_b = 10²η, c_f/η = 1/R, and a proposed choice of myosin prestress distribution and polarisation, as shown in colour code. The intensity of the prestress is graded at the boundary of the myosin-activated region. (b–e) Rates of strain predicted by the simulation (purple symbols) along specific AP or DV-oriented lines, and measured from the example of tracked data presented in Fig 1a (tissue strain rate calculated at cell centroids, gray dots; LOESS regression with 25-micron window, black curve). (b) Rate of strain in the AP direction as a function of AP coordinate x close to the midline (y = 0.2R). (c) Rate of strain in the DV direction as a function of AP coordinate x close to the midline (y = 0.2R). (d) Rate of area change as a function of AP coordinate x close to the midline (y = 0.2R). (e) Rate of strain in the DV direction as a function of the DV distance to ventral midline, along a transverse cut midway along AP (x = 0).

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In Fig 9b–9e, we also plot real data of tissue strain based on the velocity of cell centroids on the surface of an embryo at one time point of GB extension, shown in Fig 1a [15] (see also Methods). In order to do this comparison, we use the known value R = 90 μm, and adjust manually the time unit of the simulations (one scaling parameter). It is seen that the spatial dependence of each quantity of interest follow similar patterns of variations in the simulation and the real data.

Along the AP direction, Fig 9b and 9c, there is a systematic shift of the position of the first peak value, which is located immediately posterior of the CF (x = −100 μm) in simulations, and straddles it in real data. This may be due to the incompleteness of the mechanical model of the CF, which in simulations appears as an immoveable fold, while in reality this fold has some limited movement, and deepens over time during GB extension as cell apices enter it [45]. In spite of these differences, the magnitude and shape of the peaks of strain associated with the CF are accurately reproduced in the simulations.

Further into the GB, −25 μm ≲ x ≲ 175 μm, simulations and real data exhibit a plateau of negative DV strain (convergence), Fig 9c, and positive AP strain (extension), Fig 9b, with little area variations, Fig 9d. as already discussed in Fig 6d. At both the posterior and anterior (CF) end of the GB, there are regions in which the sign of the rate of strain invert. This is consistently observed in experiments and simulations. In these regions, there is thus a local convergence-extension process in the direction orthogonal to the one in the GB. In real data, part of the convergence in AP is due to the loss of tissue as it goes into the CF and deepens it [45], and there is a corresponding peak of area decrease. In simulations, the AP convergence at the two ends of the GB is driven by the push from the AP extending GB, and the DV extension to the resistance to area changes, see also Fig 6d.

Along DV, Fig 9e, the current tracking technique of 3D SPIM data offers a range limited to ca. 90 degrees in DV. Over this range, the agreement in terms of the magnitude of the DV strain is rather good, but its rate of variation along the DV direction is not accurately predicted. This may be linked with an inacurate hypothesis on the gradation of myosin prestress close to the boundary of the GB, since S1e Fig exhibits a very different slope from the simulation in Fig 9. Dorsally, simulations and experiments agree on the existence of an AP convergence and a DV extension. It is consistent with observations in [9] that cells in this dorsal region extend in the DV direction while not rearranging and incurring little area change.

Overall, there is a good agreement in qualitative features in strain rates using the parameters that have been chosen based on flow features in the previous sections. That the latter agreement would lead to the former is not trivial, and Figs 6d–6f, 7e and 7f show that similar flows can exhibit different features in terms of strain rate.

Predictability of GB extension flow from myosin distribution

Our simulations are based on the knowledge of the embryo geometry, which includes the global shape of the continuum formed by epithelial cell apices and also the fold formed by the cephalic furrow (CF), and an assumption of the pattern of myosin expression over this surface and its planar polarisation. A mechanical model (Eq (3)) then allows us to predict the stress and flow field over the surface of the whole embryo, using a novel numerical technique that can solve the equations on the embryo surface in three-dimensions [25]. The mechanical model is based on a liquid-like constitutive relation for the actomyosin of the apical continuum, following [21, 39]. We only obtain a snapshot of the flow field that corresponds to a given myosin activation pattern. In the course of GB extension, this pattern is transported by the flow [13], and the flow field evolves in its details while preserving its global features [9]. Because there is no time evolution term in our model, Eq (3), there is a unique instantaneous flow field for a given pattern of myosin prestress σ_a. At a given time point, the results (Fig 9) compare well with corresponding observations of the morphogenetic movements of GB extension Fig 1a, which we track over the three-dimensional surface of the embryo using in toto imaging and tracking software [15]. This agreement suggests that the model captures the essential balance of stresses originating from the contractility of actomyosin. In order to go further into model validation, the next step will be to apply the model to measured embryo shape and myosin localisation patterns, which could in the future be obtained by in toto embryo imaging. This would allow the model to be validated by testing its ability to predict the flow that corresponds to each successive myosin localisation pattern and CF position in the course of GB extension.

We note that in order to obtain a good agreement of model predictions with observations in terms of strain, we have had to use a graded intensity of myosin prestress at the boundary of the planar-polarised actomyosin region, Fig 9, rather than use a sharp change, S1 Fig. It remains to be shown whether such a gradation of planar-polarised actomyosin can be observed experimentally. The DV patterning of myosin localisation, although important for the precise localisation of flow features, is not crucial to achieve a convergence-extension motion with the correct orientation, Fig 5. For a planar-polarised actomyosin region extending far laterally however, our simulations predict backflows close to the midline. Such backflows are not observed experimentally, although mutants such as torsolike that present planar-polarised myosin but not PMG invagination form ectopic folds in the germband [6], which could be due to a buckling phenomenon. The mechanical role of the invaginated mesoderm may have to be taken into account in order to improve our modelling of the midline region, Fig 1d.

In addition to the actomosin patterning, two mechanical parameters govern the flow we predict, the ratio of apical surface resistance to compression relative to its resistance to shear, and the hydrodynamic length. Both have a more general biophysical relevance (see e.g. [44]). The material parameters of the cortical actomyosin in Drosophila are not known, and quantifying them experimentally is challenging [39], since this early step of development requires the presence of the rigid vitelline membrane that surrounds the embryo and prevents direct mechanical measurements from the exterior. Magnetic tweezers have been used [46], but the magnetic particles were not directly associated with the actomyosin cortex and thus measured other mechanical properties than those required by our model. Laser cuts can provide valuable information on the evolution of tension [47] or its anisotropy [6]. By comparison with a model of subcellular actomyosin, its material parameters can be obtained (medial actomyosin relaxation time, shear viscosity and friction coefficient) [44]. These values correspond to a subcellular system while we focus on the tissue scale, which may lead to quantitative differences with our case. Indeed, although the model in [44] is formally identical to Eqs (1) and (2), their friction coefficient includes the friction cost of movements of the actomyosin cortex relative to the cell membrane, whereas both apical membranes and actomyosin cortices flow in GB extension. In spite of this, the hydrodynamic lengths of 14 to 80 μm found for actomyosin cortex of single-cell C. elegans embryos and actomyosin ring of gastrulating zebrafish respectively in [44]. The ratio of bulk to shear viscosity η_b/η that we retain is much larger than the ratio 3 that could be expected from a simple 3D-isotropic modelling of actomyosin [44]. This is important in the model in order for the DV stresses to lead to AP extension as well as DV convergence, since it is the bulk viscosity that couples these effects, Fig 2d. The origin of this large resistance to area variations could be due to the combination of the volume constraint of each cell and of a basolateral regulation of its height, as has been hypothesised in cell-based models [30]. Alternatively, active processes may regulate the apical actomyosin cortex thickness and density.

The method can account for the fine geometrical detail of the embryo apical surface, such as the CF. It is observed experimentally that during GB extension, the cell displacements are much less in the head region than in the trunk, Fig 1a. However, there is no report of a specificity of the cytoskeleton of cells in the head at this stage that could account for a locally enhanced stiffness. Our simulations lead us to propose that the presence of the CF may be the cause for these smaller displacements. Geometric features such as the CF are important for the mechanical equilibrium, since forces are transmitted directionally and any curvature will modify the equilibrium. We show that the CF modifies strongly the flow we predict and acts as a barrier for deformations, Fig 8. This geometrical feature thus guides the convergence–extension flow towards the posterior end of the embryo.

A mechanical scenario for GB extension

Our results confirm that either of the two mechanisms whose elimination was seen to correlate with a reduction of GB extension [6, 7, 11, 14, 42] can be the direct mechanical cause of a flow towards the posterior in the GB. The results in Fig 4 indicate that the GB extends under the effect of the anisotropic prestress of GB planar-polarised actomyosin, but also under the effect of the pull from the invaginating PMG. However, the precise flow patterns differ in these different cases. An important and obvious next step is therefore to obtain experimentally the distribution and polarisation of myosin on the entire embryo surface across time, for example using SPIM, for wildtype and different mutants, which should allow us to quantify the parameters and test the predictive power of the model. In the interval, our theoretical work already sheds light on the fundamental mechanisms at play and how they integrate in the complex 3D geometry of the embryo to yield the morphogenetic events that are observed.

Based on our simulations and the timings reported in the literature, we can indeed articulate a mechanical scenario for GB extension. The endoderm contraction that leads to PMG invagination, the first event correlated with GB extension ([6], and Fig 4B in reference), starts several minutes before the onset of GB extension can be detected. In Kruppel mutants, for which the planar polarisation of myosin in GB is deficient, GB still extends [14], this is mainly due to PMG invagination [6]. The corresponding simulation is shown in Fig 4e, PMG contraction generates a flow towards the posterior that does extend the posterior half of the germband but decays rapidly in space. The presence of the cephalic furrow for this extension does not have a strong influence there, see Fig 8f. Thereafter, from shortly before the onset and in the course of GB extension, myosin becomes increasingly planar-polarised [19]. The direct consequence of this is a lateral flow from dorsal to ventral, causing convergence, that is, a negative rate of strain of GB along the DV direction. Due to a rather large value of η_b, which corresponds to the in-plane compressibility viscosity of actomyosin, this causes GB extension, see Fig 6. Experimental evidence of such an in-plane low compressibility exists [7, 14], although it is not clear whether this is a passive mechanical property of the actomyosin cortex or an active one [7, 41]. In the absence of the cephalic furrow, Fig 8e, this extension occurs evenly in the anterior and posterior directions, in presence of the cephalic furrow, the viscous cost of flowing around the posterior end is much less than the cost of flowing into the furrow, and planar polarisation driven GB extension is biased towards posterior, even if the contribution of PMG invagination is not accounted for, Fig 3.

Cell intercalation may not be necessary for convergence and extension of planar-polarised tissue

The two prominent features in which the existence of cell-cell junctions are important in GB extension in wildtype (WT) Drosophila are the planar-polarised recruitment of myosin, which preferentially enriches DV-oriented junctions, and the medio-lateral cell intercalations [11, 13, 42]. Because the DV-oriented junctions present both the characteristics of being enriched in myosin and of undergoing shrinkage to lead to intercalation, these two effects have so far been studied in association. However, some mutants such as eve that lack planar polarisation of myosin can still exhibit some cell intercalation, although to a much lesser proportion than cell shape changes [14]. Here, using our modelling approach we can envision the reverse case of studying convergence-extension due to planar-polarised myosin activity but without explicit cell intercalation. We show that the anisotropy of planar-polarised myosin activity is sufficient to explain convergence-extension, without the need for an intercalation mechanism.

We conclude that a mechanical model that does not involve individual cells but only a continuum standing for the apical actomyosin connected from cell to cell by apical junctions can produce a flow with strong similarities to GB extension. In vivo, cellularisation is of course important for the planar polarisation of myosin in the GB, since the DV-oriented cell-cell junctions are the location at which oriented actomyosin structures assemble. It is observed that acellular embryos do not exhibit myosin polarisation [6]. In the model however, it is possible to introduce planar polarisation independently of cellularisation. This is what we do when defining an anisotropic myosin prestress σ_a. The fact that no further account of the cellularisation of the embryo is necessary in the model suggests that at the tissue scale, one can address morphogenetic questions by considering ensemble displacements. In this approach, the effect of cell intercalation, which is governed by planar-polarised junctional myosin, is thus not directly taken into account, but rather encapsulated in a global tissue strain rate and its associate viscosities η and η_b, which also include the cell deformation [15], Fig 2f.

This tissue strain rate and the corresponding tissue-scale tension σ are related by the constitutive relation, Eq (2), which includes the contractility term σ_a resulting from planar-polarised myosin activity, and is thus the only term bearing a trace of the embryo’s cellularised organisation. The respective values taken by and σ locally depend on the mechanical balance, i.e. both the local myosin activity and the tension transmitted by neighbouring tissue, see Fig 2d. In the context of convergence-extension caused partly by invagination of the PMG, [14], it has been proposed that cell intercalation could relax the stress by allowing cell shape changes in the GB as it is extended by an extrinsic force. Here we propose that the crucial role of cellularisation in planar-polarised tissue convergence-extension lies in the mechanism through which oriented contractile actomyosin junctions are created, and not in the subsequent tissue deformation. This suggests that if planar-polarisation of actomyosin could be achieved through some other mechanism (as is the case in the Drosophila tracheal tubule [23]), convergence and extension could still be obtained in an acellular tissue, thus in the absence of cell intercalation. In cellularised tissue, an important role of cell intercalation could thus be the relaxation of the cell strain generated by the convergence and extension process, as was already suggested [14].

Geometry-governed mechanical balance as a messenger in early morphogenesis

Planar-polarised myosin in the GB is known to generate a global flow at the surface of the embryo. Our simulations show that the global flow which is generated by such mechanical activity is dependent on the pre-existing geometry of the embryo, such as the presence or absence of the cephalic furrow. Thus, a prior morphogenetic movement such as cephalic furrow formation can affect further movements via mechanical interactions only.

This “messaging” proceeds via the establishment of a different mechanical balance depending on the geometry of the embryo, rather than the diffusion of a morphogen, e.g. [48]. In the early embryo, the distance over which these forces are transmitted is likely to be much larger than in later organisms, as there is no extra-cellular matrix structure that will relieve actomyosin from part of the stress. Indeed, we find that the hydrodynamic length is likely to be at least the width of the GB, consistent with laser ablation results [44], which implies direct mechanical interaction at this scale.

This mechanical messaging behaves differently from biochemical messaging. Its speed of propagation is the speed of sound in the force-bearing structure, here, the actomyosin. It does not propagate in an isotropic way but in a more complex directional one, and contains directional information. Regions of interest within the embryo should thus not be treated as isolated systems, since a distant geometric property of the embryo can have a direct impact on the mechanical stress felt locally when intrinsic forces are being generated.

This prompts further development of computational tools such as the one we present. Tangential flows on curved surfaces are also observed in other epithelia (such as follicular epithelium of Drosophila ovaries), but is also relevant to cortical flows in single cells, prior to mitotic cleavage for example. Mechanical approaches of flat epithelia have shed light on many aspects of tissue growth and dynamics [19, 49–51], in particular at the scale of a few cells, which is the relevant one for cell rearrangements. At the other end of the spectrum of tissue dynamics, 3D phenomenological models of shape changes during ventral furrow formation have been proposed [30]. Here we propose a first step in bridging the gap between these approaches, with the objective to be able to address complex morphogenetic events in their actual geometry, and thus to fully account for the influence of current morphology on the mechanical balance that leads to further morphogenetic movements.