Force-mediated cellular anisotropy and plasticity dictate the elongation dynamics of embryos.

We developed a unified dynamic model to explain how cellular anisotropy and plasticity, induced by alignment and severing/rebundling of actin filaments, dictate the elongation dynamics of Caenorhabditis elegans embryos. It was found that the gradual alignment of F-actins must be synchronized with the development of intracellular forces for the embryo to elongate, which is then further sustained by muscle contraction-triggered plastic deformation of cells. In addition, we showed that preestablished anisotropy is essential for the proper onset of the process while defects in the integrity or bundling kinetics of actin bundles result in abnormal embryo elongation, all in good agreement with experimental observations.

INTRODUCTION

The remarkable self-remodeling and force generation capability of cytoskeleton have been recognized to play pivotal roles in processes such as single () or collective (, ) cell migration, mechanosensing (), and cell blebbing (–). This is particularly true in embryo development of organisms like Drosophila () and Caenorhabditis elegans () where, after dorsal intercalation and ventral closure (), the embryo can undergo a fourfold elongation (driven by contractile forces generated in seam and muscle cells) () (refer to Fig. 1A). After decades of study, substantial progress has been made in identifying key genes (as well as understanding their functions) involved in this process. For instance, unc-112 and spc-1 were found to be critical for the proper organization of actin-myosin assembly in body-wall muscle (), as well as the actin cytoskeleton in dorsoventral cells (). On the other hand, kinases like let-502/ROCK, mrck-1, and pak-1 are known to regulate the activity of myosin motors that provide the driving force for the elongation process (). Mutants with deficiencies in these genes/kinases all fail to execute normal embryo elongation.

Fig. 1

Schematic diagrams of Caenorhabditis elegans embryo elongation.

(A) Cartoons showing two stages of elongation driven by seam cell and muscle contractions. (B) Illustration of the microstructure of the embryo wall (treated as a thin-walled cylinder). Ventral closure increases the hydrostatic pressure (P) in the cavity and causes prestretches. After that, contraction generated in seam cells causes the shrinking of the embryo wall in the circumferential direction and eventually drives its axial elongation. Such elongation is further sustained by contraction of body-wall muscles in the second stage. (C) Illustration of our model. The seam cell is treated as an isotropic neo-Hookean material with shear modulus μse while the response of dorsoventral cell is assumed to be determined by the ground matrix (with shear modulus μdv) and embedded actin bundles (characterized by k1, k2, and κ). In the first stage, embryo elongation is driven by the active contractile force Fseam (growing at rate and resulting in circumferential stress σa) in seam cells and realignment of actin fibers (with rate characterized by ζ) in dorsoventral cells. In the second stage, severing of actin bundles (at rate ks) induced by muscle contraction Fmus and their rebundling (at rate kb) lead to plastic deformation of the embryo wall (i.e., a shortening in the rest length in the circumferential direction) that supports further extension of the embryo. During the elongation process, active stress along the axial direction within the embryo wall (with magnitude equaling to ασa) is also assumed to be generated by seam cell contraction.

Recently, advanced techniques such as laser ablation () and total internal reflection fluorescence structured illumination microscopy imaging () have also been used to monitor changes in the cytoskeleton during embryo development. For example, it was reported that bending and subsequent severing/rebundling of actin filaments will be induced by muscle contraction, leading to apparent plastic deformation of cells (). In addition, measurements also suggested that strong anisotropy exists in the elongated embryo wall (). On the basis of this information, different phenomenological or continuum [like nonlinear elastic (, ) or viscoplastic ()] models have been proposed to explain the observed extension curve or estimate the magnitude of active stresses inside the embryo.

Despite these efforts, several important issues remain unsettled. First of all, it has been observed that actin fibers in dorsoventral cells become more and more aligned as elongation progresses, indicating that embryo anisotropy is developed gradually. However, such processes have not been considered in existing studies. In addition, exactly how the interplay among actin plasticity, development of cellular forces, and cytoskeleton remodeling dictates the elongation dynamics of embryos remains unclear. In this work, we developed a dynamic model to answer these fundamental questions.

RESULTS

Dynamic model for embryo elongation

Given that the body cavity of Caenorhabditis elegans is filled with organs (, ) and fluid and capped by dorsal and ventral epidermis, we proceed by treating the embryo as a thin-walled cylinder (having a rest length L0, thickness t0, and radius r0) with both ends closed and incompressible medium inside. Microscopically, the embryo wall consists of four strips of cells: two dorsoventral cell and two seam cell strips (Fig. 1B). As demonstrated in various experiments (), myosin is mostly active in the seam cell strips whose contraction causes the shrinking of the wall in the circumferential direction, pushes internal fluid to both ends of embryo, and eventually drives its elongation. Similar to the configuration of arterial layers reinforced by collagen fibers, pronounced alignment of actin filaments (along the circumferential direction) has been observed in dorsoventral cells during this process. Therefore, the well-known G-O-H model (), developed by Gasser, Ogden, and Holzapfel for capturing the influence of dispersed collagen fibers on the mechanical behavior of arterial layers, was adopted to describe possible anisotropic response of the dorsoventral cell strips where the strain energy density is given bywith Wm and Wf representing contributions from the ground matrix (i.e., all the other cellular components except F-actins, having modulus μ) and actin fibers, respectively. Here, k1 and k2 describe the mechanical properties of filaments with k1 being modulus-like and k2 being dimensionless. I1 = λz2 + λθ2 + λr2 is the first invariant of the right Cauchy-Green deformation tensor, with λz, λθ, and λr being the stretch of material in the axial, circumferential, and radial direction, and I4 = λθ2 indicates that fiber alignment tends to happen along the hoop direction (refer to Supplementary Materials A1). The degree of anisotropy is characterized by the dispersion parameter κ. Specifically, as illustrated in (), κ will decrease from 1/3 to 0 when the orientation of filaments changes from being totally random to perfectly aligned. Note that volume conservation of the embryo wall itself requires I1 = λz2 + λθ2 + (λzλθ)−2. Last, since the orientations of F-actins in seam cells are approximately isotropic-like (), the term Wf was removed in describing the response of seam cell strips (refer to Supplementary Materials A2 for more details). Once W is known, the so-called Cauchy stresses can then be obtained as and .

Before the onset of body-axis elongation, ventral closure will elevate the hydrostatic pressure (P) inside the embryo, resulting in prestretches [λz, , , ] and [λz, , , ] in the dorsoventral and seam cell strips, respectively (Fig. 1B). Note that these strips are assumed to undergo the same axial stretch because no apparent sliding between them has been observed. Once seam cells start to contract, the active force grows gradually () and dorsoventral cells will be stretched in the circumferential direction, while the seam cell strip itself will shrink circumferentially (, ). Given that actin assembly and turnover occur over tens to hundreds of seconds (, ) while it takes hundreds of minutes for embryo elongation to complete, it is conceivable that such compression increases the overlapped portion of parallel actin filaments in seam cells, allows more myosin bindings to take place, and ultimately leads to an increased active contraction. Here, for simplicity, the contractile force Fseam [with initial value Fseam(t = 0) = F] generated in seam cells is assumed to grow aswhere is the total number of binding sites brought by overlapped F-actins, with β representing this number provided by unit compression and being the rest circumferential length (where is the length percentage) of the seam cell strip, and So refers to the number of sites that have been occupied by myosin with dSo/dt = γ(St − So). Essentially, each myosin is assumed to be capable of binding to F-actins at rate γ and then generating a force fm. The apparent active hoop stress inside the embryo wall therefore is . Myosin contraction was also found to generate an active stress in the axial direction, albeit with a lower magnitude (). Therefore, an axial stress ασa (with 0 < α < 1) is also assumed to exist in the seam cell strip here. Equilibrium of the embryo undergoing quasi-static elongation implieswhere r, , and are the radius of the deformed embryo and the circumferential length percentage of the seam and dorsoventral cell strip (Supplementary Materials A3), respectively. In addition, volume conservation of the embryo requireswith V being the normalized volume (refer to Supplementary Materials A4). Note that the potential energy of the system can be expressed as where and represent contributions from active contractile forces in the circumferential and the axial direction (acting at the seam-dorsoventral cell interface and both ends of the seam cell strip, respectively; refer to Supplementary Materials A5 for details). Evidently, E is a function of κ (i.e., the degree of material anisotropy in dorsoventral cell strip). Therefore, we proceed by assuming that gradual alignment of actin fibers will take place in dorsoventral cells, effectively changing the value of κ, to minimize the total potential energy. Specifically, the rate of change of κ is taken to bewhere ζ is a parameter characterizing how fast filament reorientation can occur (Fig. 1C). It must be pointed out that since the alignment of F-actins progresses slowly (over a time scale of hours as will be demonstrated later), the quasi-static equilibrium assumption of the embryo wall (i.e., Eqs. 3 to 5) should be reasonable.

As seam cell contraction approaches its limit in driving the elongation (~2-fold/100 min in wild type), body-wall muscle will be activated to exert a concentrated contractile force on the dorsoventral cell strip along the axial direction in a cyclic fashion (). It has been found that such force can bend actin fibers and cause their breakage/severing. It was also suggested that severed F-actins can be rebundled, which leads to a shortening in the circumferential length of dorsoventral cells (). To capture such plastic deformation in our model, we assume there are N actin fibers with Ns of them being severed and Nb of them remaining bundled, i.e., N = Ns + Nb. The presence of muscle contraction is approximated by a constant force Fmus, uniformly distributed on N fibers and leading to a maximum shear force of in each of them. Following McCullough et al. (), the work done by this shear force across fiber diameter d is estimated, as . From thermodynamics, this amount of work will lower the barrier energy for F-actin breakage to take place and result in an elevated severing rate as (i.e., the severing rate increases exponentially with the shear work in each filament) with kBT and representing thermal energy and basal severing rate. On the other hand, a severed actin fiber can be rebundled at a constant rate kb (Fig. 1C), leading to a rate equation for Nb as

It is conceivable that the added material stiffness by actin fibers mainly comes from bundled ones. Therefore, the modulus-like parameter k1 is taken to be proportional to Nb as , where is a constant. In addition, once a severed fiber gets rebundled, a shortening in its length (denoted as lover) is expected to take place, effectively resulting in a plastic stretch λp in the dorsoventral cell strip aswhere since there are four muscle quadrants on the embryo wall and (refer to Supplementary Materials A6). Note that such plastic deformation must be excluded from the strain energy expression; that is, the invariants for calculating Wf in Eq. 1 need to be modified as and with (see Supplementary Materials A7). In the present study, the values of different parameters were either selected based on experimental observations in literature or chosen within reasonable ranges (refer to table S1 and related discussions in Supplementary Materials A8 for details).

Preestablished anisotropy and subsequent fiber reorientation are essential for proper embryo elongation

We first examine how the mechanical properties of actin fibers (i.e., their integrity characterized by and orientation distribution reflected by κ) influence embryo elongation before the activation of muscle contraction. In this case, no fiber alignment is allowed and both the contractile force Fseam and the elongation length reach a steady-state value eventually (Fig. 2, A and B). Figure 2 (A and B) also shows that the maximum contraction force generated in seam cells depends on the number of binding sites provided by unit actin overlap (β) while the steady-state length of embryo becomes rather insensitive to this parameter once it is larger than ~1800 μm−1 because of strain stiffening of the embryo wall (i.e., the wall modulus increases rapidly with the strain when the deformation is large). As a result, the extension length becomes very insensitive to the amplitude of seam cell contraction (refer to fig. S1). On the other hand, at a given β, such steady-state elongation was found to be heavily influenced by the strength of actin fibers () and their dispersion (characterized by κ), refer to Fig. 2C. Specifically, as anisotropy increases (i.e., as κ changes from 1/3 to 0), a much more pronounced elongation can be achieved whereas the final embryo length is rather insensitive to (as long as it is not very small). This indicates that the circumferential alignment of fibers could lead to softening of dorsoventral cells in the axial direction (refer to fig. S1C) and eventually a greater elongation of embryos. As illustrated in Fig. 2D, the percentage of elongation increases from ~70% to more than 120% when κ is allowed to evolve according to Eq. 7. One thing we must point out is that the alignment of F-actins was assumed to be driven by minimization of the potential energy E, which is a complicated function of different system variables. For example, a typical landscape of E revealing its dependence on the contractile force Fseam and dispersion parameter κ is given in Fig. 2E. As Fseam gradually grows from zero, a local maximum of E emerges at intermediate values of κ, indicating that the system becomes bistable under such circumstance. Specifically, depending on the initial value of κ, actin filaments in dorsoventral cells can become perfectly aligned (i.e., κ → 0) or randomly oriented (i.e., κ → 1/3) once seam cells start to contract. However, as illustrated by the blue line in Fig. 2F (corresponding to an initial more isotropic-like κ value of 1/6), elongation will be very limited (only ~1.3-fold) in the second scenario. In comparison, more than twofold elongation can usually be achieved if actin fibers are already aligned to a certain degree at the beginning; refer to the red line in Fig. 2F where κ0 = 1/10. These results highlight the importance of preestablished cellular anisotropy in the proper onset of embryo elongation. Actually, this conclusion appears to be consistent with experimental observations in () where orientation distribution of actin filaments in dorsoventral cells is already considerably anisotropic at the early stage of elongation (i.e., 1.3-fold) and such anisotropy becomes more and more pronounced as the process progresses. In addition, it has been observed that the gradual alignment of actin fibers (i.e., the development of anisotropy) occurs slowly during 1.3- to 1.5-fold embryo extension but accelerates in the 1.5- to 2-fold stage (), a trend that is consistent with our model predictions (fig. S2).

Fig. 2

Intracellular contraction and alignment of actin fibers promote embryo elongation.

Note that, unless specified otherwise, the values of all parameters are given in table S1 (wild-type embryo). (A) In the case of no filament reorientation and muscle contraction, the active contraction Fseam reaches a constant level (with magnitude determined by the number of myosin-binding sites β provided by unit overlap of F-actins in seam cells) as elongation progresses. (B) Extension of embryos will also reach a steady state. The eventual embryo length becomes rather insensitive to β when this parameter is larger than ~1800 μm−1. (C) Steady-state embryo elongation as a function of fiber stiffness () and anisotropy (κ) in dorsoventral cells. Here, no filament reorientation was allowed and β = 1800 μm−1. (D) Comparison between the elongation trajectories of embryos with (ζ = 0.5 × 10−5 s−1) and without (ζ = 0) fiber alignment. (E) Normalized potential energy as a function of the contractile force Fseam and κ. Depending on the initial anisotropy in dorsoventral cells, the elongation trajectories of embryos can follow distinct paths. Specifically, a more isotropic-like κ0 value of 1/6 leads to the disappearance of cellular anisotropy (i.e., κ → 1/3 as shown by the blue line) while perfectly aligned actin fibers can be achieved in the end (i.e., κ → 0 as shown by the red line) if κ0 = 1/10. (F) Elongation depends on initial anisotropy. The embryo with initially more isotropic-like dorsoventral cells (i.e., κ0 = 1/6) could only reach a very limited elongation (~1.3-fold). In comparison, more than twofold elongation can be accomplished with κ0 = 1/10.

Rate of body-axis elongation is controlled by gradual alignment of F-actins and the development of intracellular forces

Another interesting question to ask is what controls the speed of elongation. Note that there are two important time scales (associated with the alignment of actin fibers in dorsoventral cells and the development of contraction force in seam cells, respectively) in our model. We proceed by examining how the elongation dynamics is influenced by the two corresponding rates: ζ (characterizing how fast actin fibers can reorientate) and (representing the normalized initial growth rate of hoop stress) with St, i being the number of available myosin binding sites once the initial contractile force Fi is applied and Eθ,0 referring to circumferential modulus of dorsoventral cells at rest state (refer to eq. S26 in Supplementary Materials B1). Heatmap of the time for the embryo to reach twofold elongation (t2F), as a function of ζ and , is shown in Fig. 3A. Clearly, our results suggest that the gradual alignment of F-actins must be synchronized with the development of intracellular forces, i.e., both ζ and should be of the order of 10−5 s−1, for embryo elongation to progress normally [i.e., to achieve twofold extension in around 100 min ()]. Mismatches in ζ and could lead to an elongation process that is either too slow or too fast to reach the desired extension (Fig. 3B). It must be pointed out that the value of ζ (~10−5 s−1) estimated here is consistent with the observation that it takes hours for the alignment of F-actins in cells under cyclic stretch to complete (, ).

Fig. 3

The elongation speed of embryos is controlled by the rates of actin fiber alignment (ζ) and growth of active hoop stress () in dorsoventral and seam cells, respectively.

The values of all other parameters are given in table S1 (wild-type embryo). (A) Heatmap of the time for the embryo to reach twofold elongation (t2F), as a function of ζ and . (B) Actual elongation trajectories of the embryo under different combinations of ζ and indicated in (A).

Muscle contraction–induced cellular plasticity can trigger the bifurcation into further extension and retraction of embryos

Since the plastic deformation of cells was directly related to severing and rebundling of actin fibers in the present model, we can then examine how the breakage and bundling of F-actins regulate the accumulation of cellular plasticity and eventually influence macroscopic embryo elongation. For example, it was found that without plasticity, the elongation driven by the gradual development of anisotropy and myosin contraction will reach a limited level (~2.5-fold in Fig. 4A). On the other hand, if severing and rebundling of actin filaments take place at similar rates, then these activities will effectively reduce the rest length of dorsoventral cells in the circumferential direction without weakening filament stiffness (characterized by ) because severed filaments will get rebundled immediately (i.e., Nb will remain largely unchanged) in this case. The circumferential shrinkage of embryo wall will then drive the cavity fluid to flow to two ends of the embryo and eventually cause its further elongation (up to ~4-fold, as illustrated in Fig. 4A). This finding is in agreement with the observation that the embryos of unc-112(RNAi) mutants can only achieve ~2-fold elongation (compared to the 4-fold extension in wild types) because muscle is inactivated in these mutants and hence no plastic deformation (i.e., shrinkage in the rest length of dorsoventral cells in the circumferential direction) will be triggered in the embryo wall ().

Fig. 4

Muscle contraction, activated at 100 min, governs the second stage of embryo elongation.

Parameter values adopted here are given in table S1 unless specified otherwise. (A) Muscle contraction–induced actin plasticity (i.e., severing and rebundling of F-actins) is essential for the further elongation of embryos, where Fmus is taken to be 0.5 or 0 nN for with or without actin plasticity (as shown in insets), respectively. (B) Elongation trajectories of the embryo under different severing and bundling rates (ks and kb are both expressed in s−1) of F-actins. Ending points of the first and second stage of elongation (where the stretch of embryo is given by λz,1 and λz,2) are indicated by the blue dashed lines. The inset illustrates that, if actin severing is very fast, then severed filaments will not have time to get rebundled, leading to the loss of anisotropy in the embryo wall and ultimately its retraction, rather than further extension. (C) Heatmaps of the normalized length change of embryos before and after the second stage. A negative value of (λz,2 − λz,1)/λz, means that the embryo actually retracts during the second stage.

If the severing rate is too high (i.e., much larger than the bundling rate), then most severed actin filaments will not have time to get rebundled, resulting in a decreased fiber stiffness (i.e., a decreased number of bundled filaments Nb) and consequently a reduced cellular anisotropy. Recall that, as shown in Fig. 2C, an embryo with low filament stiffness and anisotropy can only reach very limited elongation. Therefore, in this case, muscle contraction–induced severing of actin filaments will actually lead to shortening of the embryo (Fig. 4, B and C), rather than promote its elongation. This bifurcated effect of actin severing and bundling on embryo elongation or retraction is best illustrated by the heatmaps in Fig. 4C. Figure 4C also shows that the shortening of embryos can also be triggered by an impaired actin cytoskeleton [like in spc-1(RNAi) mutants]. In this case, besides having a reduced value, weakened actin fibers will also have a higher severing rate () that further accelerates the disappearance of cellular anisotropy under muscle contraction (refer to Supplementary Materials A8 for details) and eventually embryo retraction.

These aforementioned findings are fully consistent with a variety of experimental observations. For example, it was found that muscle contraction triggered the retraction of embryos consisting of mutant cells with spc-1(RNAi) pak-1(tm403) (), whose F-actins are less organized and have higher severing rate. In addition, it was also reported that knockdown of unc-112 (known to impair muscle contraction) resulted in the arrest of elongating embryos at the twofold stage (). By adjusting parameters according to the known effects of these genes [i.e., mainly a decreased and kb, along with an increased , for mutants with spc-1 (RNAi), and a vanishing Fmus for unc-112 knockdown cells; refer to table S1 and Supplementary Materials A8], good agreement between experiment measurements and our predictions was achieved (Fig. 5A). Last, our model also captures the diameter evolution of embryo reported by Vuong-Brender et al. () who measured changes in the diameter of embryos at head, body, and tail regions after the elongation has reached 1.3-fold. Note that, to ensure reasonable comparison, we have averaged the measurement data from those three sites and then normalized the averaged embryo diameter by its value at 1.3-fold in Fig. 5B. The deviations (between theory and experiment) shown in Fig. 5 may come from the fact that, in reality, the embryo assumes a much more complicated 3D geometry (usually with a bigger head and a tapered tail) rather than a perfect thin-walled cylinder adopted in the model.

Fig. 5

Comparisons between model predictions and experiments.

(A) Model outputs agree well with the measured elongation trajectories of wild-type and different mutant embryos in (). (B) The predicted evolution of normalized embryo diameter beyond 1.3-fold elongation also compares favorably with observations from (). Experimental data shown here are the average of measured diameters at the head, body, and tail of wild-type embryos (). Parameter values adopted in the simulations are summarized in table S1.

DISCUSSION

Mechanical force has long been believed to play important roles in regulating the behavior and functioning of living cells. Here, we present a theoretical study to elucidate exactly how active intracellular/intercellular stresses trigger cytoskeleton remodeling in cells, increase the internal hydrostatic pressure (fig. S3) of embryo, and eventually dictate its elongation. Specifically, we showed that the embryo elongation in the first stage (i.e., before muscle contraction is triggered) is driven by the active contraction force in seam cells along with the alignment of F-actins in dorsoventral cells. Synchronization between the development of contraction force and the reorientation of actin fibers is essential for proper embryo extension in this stage. On the other hand, once muscle is activated, the elongation dynamics is dictated by the competition between the stiffness loss and the accumulation of actin plasticity in the embryo wall. In particular, under normal circumstances, the appearance of actin plasticity (induced by the breakage and rebundling of actin fibers) will reduce the rest length of the embryo wall in the circumferential direction and therefore support the further extension of the embryo. However, a weakened F-actin network (triggered by, for example, knockdown of spc-1) or elevated severing rate of actin could prevent severed actin fibers from being rebundled and consequently lead to retraction of the embryo during this stage (Fig. 4C). It must be pointed out that, as shown in fig. S4, varying the values of different parameters (within the reasonable range) will not cause qualitative changes to the elongation trajectory in general, indicating that conclusions obtained here should be rather robust.

Unlike many previous attempts where the analysis was conducted in a static sense, the gradual development of cellular anisotropy (caused by the alignment of F-actins) and its interplay with the development of intracellular contraction were directly considered in our model. In addition, rather than being taken into account phenomenologically (, ), cellular plasticity was directly linked to the severing/rebundling of actin fibers in the present study, allowing us to systematically examine how microscopic details like the integrity and severing/rebundling kinetics of F-actins influence macroscopic embryo elongation as well as connect them to different experiments. For example, we showed that the preestablished anisotropy in the embryo wall is critical for its proper extension, which is consistent with the observation that considerable actin anisotropy already exists at the very early stage of embryo elongation (). In addition, our predictions also show how changes in the severing and rebundling rates of F-actin, as well as the growth rate of myosin contraction force in seam cells, alter the elongation trajectory of embryos, something that can be tested by future experiments.

It must be pointed out that the strain energy density is expressed in terms of invariants of the deformation tensor (see Eq. 1) in this study, implying that the embryo has a reference configuration. We believe this is reasonable because a large amount of cells need to be assembled together (via cell-cell adhesion) to form a tissue; therefore, the embryo as a whole should have a preferred shape (despite individual cells can undergo shape fluctuations) and any deviation from this reference state should be associated with energy penalties (i.e., inducing macroscopic stresses in the embryo wall). We also want to emphasize that, besides continuum-level approaches, different cell-based models have also been developed to describe the collective behavior of cells. For example, cells can be treated as particles interconnected by linear and angular springs (). Alternatively, each cell has also been represented by a polygon, with straight interfaces between neighboring cells, in the so-called vertex model (–). In such approach, a set of microscopic parameters need to be introduced to describe the response and possible state change (like polarization and proliferation) of individual cells, as well as how they interact with each other, while the assumption of a reference state/preferred shape for the tissue becomes unnecessary. On the other hand, compared to continuum models, the computational cost of cell-based modeling could be high and connecting simulation results/conditions with actual experiments may not be that straightforward.

Last, it is conceivable that, besides providing a theoretical explanation for Caenorhabditis elegans embryo elongation, the mechanisms and formulation proposed here could also be used to model other processes such as collective cell migration (, ), cell division (), tissue morphogenesis (, ), and enforced deformation of cells () where the presence of active cellular stress and distinct cytoskeleton remodeling/damage have all been observed. In addition, important factors such as the growth and depolymerization of actin filaments, as well as their stretch-induced alignment (), can also be incorporated into the present model to make it more realistic. Investigations along these directions are currently underway.

MATERIALS AND METHODS

Determining the mechanical stretches, embryo diameter, and internal hydrostatic pressure

By substituting Eq. 3 into Eqs. 4 and 5, the force equilibrium conditions reduce to

These two nonlinear equations, along with Eq. 6, enable us to determine the mechanical stretches , , and λz numerically. After that, embryo diameter D and internal hydrostatic pressure P can be determined as

Algorithm for finding numerical solutions

Given that the elongation process is regarded as quasi-static in the present model and the period of muscle contraction is of the order of seconds, we set the time step in our algorithm as ∆t = 3 s (we have also varied the time step to 6 or 1.5 s, but no noticeable difference was observed in the simulation results). For wild-type embryo, the total number of steps for the first or second elongation stage is chosen as N1 = 2000 or N2 = 2800, respectively. In each time step, the equilibrium embryo shape is determined by iteration (using standard Newton-Raphson method). The full algorithm is as follows:

1) Initialize all parameters including initial mechanical stretches , , and λz, (corresponding to σa = 0).

2) Calculate active contraction force in seam cells Fseam according to Eq. 2.

3) If the step count N > N1, calculate the plastic stretch λp according to Eq. 9.

4) Use the standard Newton-Raphson method to find the updated mechanical stretches , , and λz from Eqs. 6, 10, and 11:

a) Calculate the Cauchy stresses , , , and according to and

b) Determine the residues (R1, R2, R3) (i.e., differences between the left- and right-hand sides of Eqs. 10, 11, and 6).

c) Move to step 5 if the magnitude of residues is smaller than a threshold error tolerance value; otherwise, calculate the Jacobian matrix and increments/corrections in the mechanical stretches asupdate the stretches , , and λz and then return to step 4a.

5) Update filament dispersion parameter κ according to Eq. 7 and the number of available myosin binding sites from . In addition, calculate the embryo diameter D and internal hydrostatic pressure P according to Eqs. 12 and 13, respectively.

6) Increase the time by ∆t and step count by 1 and return to step 2 if the step count N < N1 + N2; otherwise, exit the calculation.

The described algorithm was implemented in MATLAB 2020a (The MathWorks Inc., Natick, MA, USA).