Introduction

Hydra is a multicellular fresh-water polyp, which exhibits remarkable regeneration capabilities, making it an excellent model system for studying tissue morphogenesis1,2,3,4,5,6. The Hydra body consists of a single-axis hollow-cylindrical tube (about 5−10 mm long)1, which has a triple-layer structure composed of two epithelial cell layers (endoderm and ectoderm) that are adhered together by an intermediate soft layer of extracellular matrix (called mesoglea) as shown in Fig. 1a. Hydra body shape is maintained by a contractile actomyosin cytoskeleton4,5,6,7, which shows a highly-aligned orientation over supracellular scales (see Fig. 1a): the actomyosin bundles takes longitudinal orientation (along the tube axis) in the ectoderm and circumferential orientation in the endoderm. Regenerating Hydras use their cytoskeleton to regulate their cells and to guide the regeneration process4,5,6,7. When fragments are excised from the adult Hydra body, the supracellular pattern of cytoskeletal bundles is inherited, survives, and finally becomes a part of the new daughter Hydra. These supracellular cytoskeletal bundles provide structural “memory” of the alignment along the body axis4,5,6,7. They generate mechanical forces and direct the alignment of the cytoskeleton in regions where the supracellular order is lacking in the early stage of regeneration.

Fig. 1: Schematic illustration of the triple-layer structure of Hydra fragments and experimental observation of the initial spontaneous bending.
figure 1

a Schematic illustration of the Hydra body plan and the triple-layer structure of Hydra fragments: ectoderm cell layer (yellow), endoderm cell layer (light yellow), and intermediate soft matrix (mesoglea). Two sets of perpendicular supracellular actomyosin bundles (red arrows) are formed on the basal sides of each epithelial layer. Interlayer slides are characterized by the misfit strains \({\epsilon }_{{{{{{{{\rm{s}}}}}}}}}^{(1)}\) and \({\epsilon }_{{{{{{{{\rm{s}}}}}}}}}^{(2)}\) of the two cell layers relative to the mesoglea layer, respectively. b Experimental observations: spontaneous bending of freshly excised Hydra (plate- and rod-like) tissue fragments to quasi-stable (equilibrium) bent shape in several minutes. Reproduced from ref. 7 with permission from Elsevier.

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At the initial stage of the regeneration of Hydra tissue fragments that are excised freshly from adult Hydra body7,8, the fragments bend spontaneously to some quasi-stable (or mechanical-equilibrium) shape in several minutes (see Fig. 1b). Subsequently, the cytoskeleton of the fragment remodels to find a balance between maintaining its pre-existing organization and adapting to the new curved conditions4,5,6,7. After about one hour of the bending-remodeling loop, the excised fragments fold into small hollow spheroids4,5,6,7,8.

In this work, we focus on the modeling and analysis of the initial spontaneous bending of Hydra fragments that happens in the first several minutes after their excision from parent Hydra (see Fig. 1a, b). We propose that such spontaneous tissue bending is an active process driven mechanically by the contractions of the two sets of aligned, supracellular actomyosin bundles. We assume that the short bending process is accompanied by changes only in cell shape and structure, but not in cell division and apoptosis9. The equilibrium bent shape of Hydra tissue fragments is then determined by the balance between the actomyosin contractile forces and the elastic restoring forces due to cell-shape deformation9. The bending dynamics are driven by the active and elastic forces that are balanced by some dissipative forces. In the context of Hydra tissue, we identify three types of dissipation mechanisms: (i) the fragment motion relative to surrounding viscous fluids, (ii) the relative frictional slide between adjacent tissue layers10, and (iii) the cellular dissipation associated with sub-cellular-scale remodeling and cellular-scale rearrangements9,11,12.

Based on the above assumptions, we construct an active laminated-plate (ALP) model that connects the triple-layer structure and the contractile supracellular actomyosin bundles of Hydra fragments with the equilibrium shape and dynamics of their spontaneous bending (see Fig. 1a). We predict that the bent shape of plate-like Hydra fragments characterized by the spontaneous curvature tensor is mostly determined by their anisotropy in both the supracellular actomyosin contractility and the elastic properties of each lamina layer, as summarized in Fig. 2a, b. For the bending dynamics of rod-like tissue fragments, we present a minimal model based on the static ALP model and carry out analysis using an approximate variational method (based on Onsager’s variational principle13,14) and bead-spring simulations. We find that the tissue bending propagates diffusively from the edges into the center (with the bent length being proportional to the square root of time). Whereas, when the bending is close to its final equilibrium shape, the end-to-end (ETE) distance decays exponentially with time toward its equilibrium value. Moreover, we suggest that the interlayer frictional sliding and cellular dissipation set the dominant timescale (of minutes) observed in experiments during the initial spontaneous bending of Hydra fragments4,5,6,7.

Fig. 2: Schematic illustration of the spontaneous bent shape of Hydra fragments.
figure 2

a Schematic illustration for the spontaneous bent shape of Hydra fragments driven by uniformly contracting supracellular actomyosin bundles. The contour length, width, and total thickness of the fragments are denoted by ℓc, b, and h, respectively. b A schematic bent-shape diagram of Hydra plates with different anisotropy in actomyosin contractility (measured by c2/c1) and in elasticity (measured by \({E}_{2}^{(k)}/{E}_{1}^{(k)}\) in each k = 1, m, 2 lamina layer). Here ci characterizes the contractility of the two sets of perpendicular supracellular bundles; \({E}_{1}^{(k)}\) and \({E}_{2}^{(k)}\) are the two principal Young’s moduli of each (k = 1, m, 2) orthotropic lamina layer.

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Results

Spontaneous curvature of Hydra tissue fragments driven by uniform supracellular contraction

To construct a continuum model for Hydra tissue fragments, we first note a structural analogy between Hydra fragments (with triple-layer structure) and composite laminated plates in material science15,16. Moreover, we find that in typical regeneration experiments4,5,6,7, the lateral lengths of Hydra fragments are usually ~100 μm much larger than their thickness h ~ 10 μm. We then formulate an active (thin) laminated-plate (ALP) model17 (see Sec. A of Supplementary Note 1) for Hydra fragments by following the classical (thin) laminated-plate theory16 that extends the classical (thin) plate (CP) theory15 of mono-layer isotropic materials to multi-layer orthotropic materials. In our ALP model, the Hydra fragment is modeled as a triple-layer laminated plate (see Fig. 1a). The equilibrium bent shape of a Hydra fragment at small deformation is characterized by spontaneous curvature tensor and determined by minimizing the deformation energy of the laminate fragment upon internal uniform active contractions applied by two sets of perpendicular supracellular actomyosin bundles. Note that, mathematically, such an ALP model is very similar to the model for elastic lamellae with assorted pre-stresses that are set by deformation of “active” thermo- or photo-sensitive elements in composite materials18, non-uniform in-plane growth19, or absorbing/expelling water20 in plant sheets, etc.

Firstly, for Hydra fragments consisting of isotropic lamina layers, the total bending energy density (per area) is found to be17 (see Sec. A of Supplementary Note 1)

$${F}_{{{{{{{{\rm{b}}}}}}}}}=\frac{1}{2}{D}_{{{{{{{{\rm{p}}}}}}}}}({c}_{{{{{{{{\rm{x}}}}}}}}}^{2}+2\nu {c}_{{{{{{{{\rm{x}}}}}}}}}{c}_{{{{{{{{\rm{y}}}}}}}}}+{c}_{{{{{{{{\rm{y}}}}}}}}}^{2})-({M}_{{{{{{{{\rm{p}}}}}}}}}^{(1)}{c}_{{{{{{{{\rm{x}}}}}}}}}+{M}_{{{{{{{{\rm{p}}}}}}}}}^{(2)}{c}_{{{{{{{{\rm{y}}}}}}}}}).$$
(1)

Here Dp is the effective flexural stiffness. The Poisson ratio ν is assumed to be constant. cx and cy are the two principal curvatures of the neutral surface located at z = z0. \({M}_{{{{{{{{\rm{p}}}}}}}}}^{(1)}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(1)}{z}_{0}\), \({M}_{{{{{{{{\rm{p}}}}}}}}}^{(2)}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(2)}({z}_{0}-{h}_{{{{{{{{\rm{m}}}}}}}}})\), and \({\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(1)},\,{\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(2)}\, < \, 0\) are the active torques and contractile forces generated by the two sets of supracellular actomyosin bundles, respectively. The position of the neutral surface is given by (see Sec. B of Supplementary Note 1)

$${z}_{0}=\frac{{E}_{* }^{(2)}\left[{({h}_{2}+{h}_{{{{{{{{\rm{m}}}}}}}}})}^{2}-{h}_{{{{{{{{\rm{m}}}}}}}}}^{2}\right]+{E}_{* }^{(m)}{h}_{{{{{{{{\rm{m}}}}}}}}}^{2}-{E}_{* }^{(1)}{h}_{1}^{2}}{2\left[{E}_{* }^{(1)}{h}_{1}+{E}_{* }^{(m)}{h}_{{{{{{{{\rm{m}}}}}}}}}+{E}_{* }^{(2)}{h}_{2}\right]}.$$
(2)

with \({E}_{* }^{(k)}\equiv {E}^{(k)}/(1-{\nu }^{2})\); E(k) and hk (k = 1, m, 2) being Young’s modulus and thickness of the ectoderm, mesoglea, and endoderm, respectively. Note that in comparison to the bending energy with a quadratic form of principal curvatures in CP theory15, terms coupling active actomyosin torques linearly with curvatures21,22 also appear in Eq. (1), resulting in non-zero spontaneous curvatures.

Minimizing \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{b}}}}}}}}}=\int\,dxdy{F}_{{{{{{{{\rm{b}}}}}}}}}\) gives the spontaneous curvatures:

$${c}_{{{{{{{{\rm{x0}}}}}}}}}=\frac{{c}_{1}-\nu {c}_{2}}{1-{\nu }^{2}},\quad {c}_{{{{{{{{\rm{y0}}}}}}}}}=\frac{{c}_{2}-\nu {c}_{1}}{1-{\nu }^{2}},$$
(3)

with \({c}_{i}={M}_{{{{{{{{\rm{p}}}}}}}}}^{(i)}/{D}_{{{{{{{{\rm{p}}}}}}}}}\) (i = 1, 2) characterizing the contractility of the two sets of supracellular actomyosin bundles, respectively. From Eq. (3), we see that the spontaneous bending (with non-zero cx0 and/or cy0) of Hydra fragments is driven by the supracellular actomyosin contraction (represented by non-zero \({\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(i)}\)) that occurs away from the neutral surface at z = z0. Moreover, we find from Eq. (3) that the principal curvatures (and the equilibrium bent shape) are mainly determined by the contractility anisotropy measured by c2/c1. If ν < c2/c1 < ν−1, the Hydra fragments will bend spontaneously to an elliptical cap shape; otherwise, they will bend to a saddle shape as summarized in Fig. 2b.

Secondly, for tissue fragments consisting of strongly anisotropic lamina layers (for example, due to the presence of aligned thick actomyosin bundles in the two epithelial layers), we find that the spontaneous bending in the two principal directions is almost independent (see Sec. A of Supplementary Note 1), and the neutral surfaces of the spontaneous bending lie in different layers for bending along different directions: in ectoderm for bending along the \(\hat{{{{{{{{\bf{x}}}}}}}}}\)-direction and in endoderm for bending along the \(\hat{{{{{{{{\bf{y}}}}}}}}}\)-direction, respectively. In this case, the two principal spontaneous curvatures are given by cx0 ≈ c1 > 0, cy0 ≈ c2 < 0, that is, the fragments bend spontaneously to be saddle shape (see Fig. 2b).

Next, based on the above results, we can connect the triple-layer structure and elastic/geometric properties of Hydra fragments with their equilibrium bent shapes observed in experiments7 (see Fig. 1b). Hydra fragments bend spontaneously to a spherical cap shape toward the inner endoderm side, that is, cx0 ≈ cy0 > 0, from which we draw the following conclusions. (i) The cap-like shape corresponds to the case of more-or-less isotropy in both contractility (of the two contractile bundles) and elasticity (of each lamina layer), as shown in Fig. 2b. (ii) The inward bending corresponds to c1, c2 > 0 (using Eq. (3)) and the case where the neutral surface lies in the ectoderm layer, i.e., z0 < 0, as obtained using \({c}_{1}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(1)}{z}_{0}/{D}_{{{{{{{{\rm{p}}}}}}}}}\, > \, 0\) and \({c}_{2}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(2)}({z}_{0}-{h}_{{{{{{{{\rm{m}}}}}}}}})/{D}_{{{{{{{{\rm{p}}}}}}}}}\, > \, 0\). Furthermore, from z0 < 0 and Eq. (2), we obtain that the inward cap-bending needs to satisfy an inequality \({E}_{{{{{{{{\rm{p,eff}}}}}}}}}^{(2)}{({h}_{2}+{h}_{{{{{{{{\rm{m}}}}}}}}})}^{2}\, < \, {E}_{* }^{(1)}{h}_{1}^{2}\) with \({E}_{{{{{{{{\rm{p,eff}}}}}}}}}^{(2)}\equiv {E}_{* }^{(2)}-({E}_{* }^{(2)}-{E}_{* }^{({{{{{{{\rm{m}}}}}}}})}){h}_{{{{{{{{\rm{m}}}}}}}}}^{2}/{({h}_{2}+{h}_{{{{{{{{\rm{m}}}}}}}}})}^{2}\). Therefore, at given geometries (given thicknesses, hk), the presence of a soft mesoglea layer with \({E}_{* }^{({{{{{{{\rm{m}}}}}}}})}\, < \, {E}_{* }^{(2)}\) makes the inequality and z0 < 0 easier to satisfy, thus facilitating inward tissue bending toward the endoderm side. Such a mesoglea-softening mechanism is similar to that of connecting a stiff spring to a soft spring in series. However, when \({E}_{* }^{(1)}{h}_{1}^{2}\, < \, {E}_{* }^{(2)}[{({h}_{2}+{h}_{{{{{{{{\rm{m}}}}}}}}})}^{2}-{h}_{{{{{{{{\rm{m}}}}}}}}}^{2}]\), such a mesoglea-softening mechanism is impossible to ensure inward bending.

Before closing this section, we would like to present a simple way of finding the bending direction driven by each set of supracellular actomyosin bundles in a laminated tissue: comparing the position of contracting bundles to the location of the neutral surface, z0, given in Eq. (2). For example, in Hydra fragments (see Fig. 1a), the actomyosin bundle at the basal side of the ectoderm layer near z = 0 drives the bending toward the endoderm side if z0 < 0, and toward the ectoderm side if z0 > 0. Similarly, the actomyosin bundle at the basal side of the endoderm layer near z = hm drives the bending toward the endoderm side if z0 < hm, and toward the ectoderm side if z0 > hm. Therefore, in the Hydra tissue, an inward bending (toward the endoderm side) to an elliptical or spherical cap shape requires z0 < 0, a condition that has already been obtained previously from c1, c2 > 0 and Eq. (3).

Spontaneous bending and interlayer slide of rod-like Hydra fragments

If instead of a Hydra plate, one cuts a rod-like Hydra fragment, the total bending energy density (per length) in Eq. (1) reduces to (see Sec. B of Supplementary Note 1):

$${F}_{{{{{{{{\rm{b}}}}}}}}}=\frac{1}{2}{D}_{{{{{{{{\rm{r}}}}}}}}}{c}^{2}-{M}_{{{{{{{{\rm{r}}}}}}}}}c.$$
(4)

Here Dr is the effective flexural stiffness. c is the curvature of the neutral line at z = z0. Mr ≈ τrz0 and τr < 0 are the active torque and contractile force generated by the actomyosin bundle in the ectoderm layer, respectively. The neutral line position z0 is given by the same form as Eq. (2) but \({E}_{* }^{(k)}\) should be replaced by E(k)b with b being the width of the Hydra rod.

Minimization of \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{b}}}}}}}}}=\int\,dx{F}_{{{{{{{{\rm{b}}}}}}}}}\) gives the spontaneous curvature (see Sec. B of Supplementary Note 1): c0 = Mr/Dr ≈ τrz0/Dr. Then using Dr ~ Eh3b, z0 ~ h, and noting that in typical cell experiments9: τr/hb ~ 0.1 kPa, h ~ 10 μm, E ~ 1 kPa, we estimate the spontaneous curvature c0 ~ 1/100 μm−1. Interestingly, for Hydra regenerating from cell aggregates3, the curvature cs0 of the hollow Hydra spheroids formed from an aggregate of a minimal number of Nc ~ 1000 cells2,3 can be estimated to be the same scale cs0 ~ 1/100 μm−1 by calculating the surface area of the spheroid, \(4\pi {c}_{{{{{{{{\rm{s0}}}}}}}}}^{-2} \sim {N}_{{{{{{{{\rm{c}}}}}}}}}{R}_{{{{{{{{\rm{c}}}}}}}}}^{2}\), and using the cell size Rc ~ 10 μm. Moreover, such cell-scale curvature is also known to appear very often in many natural in vivo microenvironment21, e.g., cylindrical-shaped glands and blood vessels23.

Note that the supracellular contraction and the elastic/geometric properties are generally non-uniform, hence the spontaneous curvature c0 can be different at different positions through the fragments. In this case, for a freely suspending Hydra fragment, the equilibrium bent shape corresponds to the equilibrium state with bending energy minimized locally, that is, the local curvature equal to local c0. In contrast, the bent shape of a constrained Hydra rod is more complex. Some non-trivial periodic bent shapes have been obtained, as shown in Fig. S2 of Supplementary Note 1. The mechanistic perspective of the spontaneous bending of tissue fragments during morphogenesis conveyed in this work has been proposed before8,24,25,26, and several classical theories have been developed. Among the earliest theories of this kind, W.H. Lewis proposed (1947)24 a mechanical model of epithelial sheets, consisting of brass bars, tubes, and rubber bands (see Fig. S2 in Sec. C of Supplementary Note 1) for a schematic figure of the model). We discuss this model in Sec. C of Supplementary Note 1 and show the equivalence (the mapping) between Lewis’ model and our ALP model in predicting the spontaneous bending shape of Hydra tissue.

In addition, during the tissue bending, the interlayer slide may occur between each cell layer and the intermediate mesoglea matrix10, as shown in Fig. 1a, and the fragment becomes incoherent27. In this case, we neglect the in-plane contraction, and the total energy \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) takes the following phenomenological form22,27

$${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}=\int\nolimits_{-{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}^{{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}ds\left[\frac{{Y}_{{{{{{{{\rm{s}}}}}}}}}}{2}{\epsilon }_{{{{{{{{\rm{s}}}}}}}}}^{2}-\chi {\epsilon }_{{{{{{{{\rm{s}}}}}}}}}c+\frac{{D}_{{{{{{{{\rm{r}}}}}}}}}}{2}{\left(c-{\tilde{c}}_{0}\right)}^{2}\right].$$
(5)

Here ℓc is the rod contour length, \({\tilde{c}}_{0}\) being the spontaneous curvature of coherent fragments. ϵs denotes the interlayer slide or the misfit strain discontinuity at the cell-mesoglea interfaces, Ys is the stiffness for the interlayer slide, and in the thin rod limit, the misfit interlayer displacement is simply given by ϵsh/2 with h being the rod thickness. χ is the linear coupling coefficient between bending and interfacial slide. Note that in a complete triple-layer model (as shown in Fig. 1a and discussed in Sec. D of Supplementary Note 1 for the bending of Hydra tissue, we should introduce two slide strains or displacements at the two interfaces: ectoderm-mesoglea interface and endoderm-mesoglea interface, respectively. However, for simplicity, we assume in a coarse-grained manner that one strain ϵs is enough to characterize the interlayer slide. The equilibrium bent state of the rod is then characterized by the spontaneous curvature \({c}_{0}\equiv {\tilde{c}}_{0}{D}_{{{{{{{{\rm{r}}}}}}}}}/{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}\) and the equilibrium slide ϵs0 = χc0/Ys with \({\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}\equiv {D}_{{{{{{{{\rm{r}}}}}}}}}-{\chi }^{2}/{Y}_{{{{{{{{\rm{s}}}}}}}}}\) being the normalized flexural stiffness of the rod. Particularly, in the limit of Ys → ∞, we recover the case of coherent fragments with \({c}_{0}={\tilde{c}}_{0}\) and ϵs0 = 0 where the strain is continuous through the thickness.

Bending dynamics of Hydra tissue rods: effects of different dissipation mechanisms

We now consider the dynamic process of the spontaneous bending of Hydra tissue rods in surrounding viscous fluids starting from the initial state to the final equilibrium shape as shown in experiments (see Fig. 1b). The initial state of the Hydra fragment is assumed to be flattened to mimic the state when it is excised from the parent Hydra. The bending dynamics of long soft rods with non-zero spontaneous curvature c0 and contour length ℓc ≫ 2πR0 (with \({R}_{0}={c}_{0}^{-1}\)) have been explored intensively in both theory and experiments28,29. However, in typical Hydra regeneration experiments7, Hydra rods are usually short with ℓc ~ 2πR0. In this case, the total energy \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) of the laminated Hydra rod is given by Eq. (5), and the bending dynamics is driven by the relaxation of stored elastic energy at the initial flattened state. Furthermore, the three types of dissipation mechanisms (arising from viscous drag by the surrounding fluids, the interlayer frictional sliding, and cellular remodeling/rearrangement inside the fragments) during the bending are taken into account by the dissipation function (see Sec. A of Supplementary Note 2)

$$\Phi =\int\nolimits_{-{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}^{{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}ds\left[\frac{1}{2}{\xi }_{{{{{{{{\rm{v}}}}}}}}}{\dot{{{{{{{{\bf{r}}}}}}}}}}^{2}+\frac{1}{2}{\xi }_{{{{{{{{\rm{s}}}}}}}}}{\dot{\epsilon }}_{{{{{{{{\rm{s}}}}}}}}}^{2}+\frac{1}{2}{\eta }_{{{{{{{{\rm{cell}}}}}}}}}bh{(h\dot{c})}^{2}\right].$$
(6)

Here r(s) is the position vector of points (parameterized by arc length s) on the rod, the local curvature c is given by its second derivatives30, and the upper dot hereafter denotes the time derivative. ξv and ξs are the viscous drag coefficient per rod length and friction coefficient for interlayer sliding, respectively. Interestingly, interlayer frictional sliding has recently been found to be able to facilitate long-range force propagation in tissues that usually have multi-layer structures10. ηcell is an effective viscosity characterizing the additional cellular dissipation at sub-cellular and cellular scales.

We then use Onsager’s variational principle13,14 and minimize Rayleighian \({{{{{{{\mathcal{R}}}}}}}}[\dot{{{{{{{{\bf{r}}}}}}}}},{\dot{\epsilon }}_{{{{{{{{\rm{s}}}}}}}}}]={\dot{{{{{{{{\mathcal{F}}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}+\Phi\), yielding a set of highly nonlinear partial differential equation (e.g., Kirchhoff equations)29. However, here we will not solve these nonlinear equations but carry out direct-variational analysis13,14 for the very initial bending stage near the flattened state and the final bending stage close to the equilibrium bent shape (see Sec. B of Supplementary Note 2).

Initially, near the flattened state (see Fig. 3a), we neglect the effects of the interlayer slide as well as cellular dissipation, and take the simplest continuous trial profile of local curvature c(s, t) (see Fig. 3b):

$$c=\left\{\begin{array}{ll}0,\hfill \quad &0\, \le \, s\, < \, {\ell }_{{{{{{{{\rm{c}}}}}}}}}/2-a,\hfill \\ {c}_{0}(s-{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2)/a+{c}_{0},\quad &{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2-a\, \le \, s\, \le \, {\ell }_{{{{{{{{\rm{c}}}}}}}}}/2,\end{array}\right.$$
(7)

in which case the bending dynamic process is characterized by only one time-dependent parameter: the bent length a(t) of the rod. Substituting the trial profile into Eqs. (5) and (6), we obtain the Rayleighian \({{{{{{{\mathcal{R}}}}}}}}\) and its minimization with respect to \(\dot{a}\) gives (see Sec. B of Supplementary Note 2)

$$a\approx {\left(\frac{2{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}{c}_{0}^{2}}{3{\xi }_{{{{{{{{\rm{v}}}}}}}}}}\right)}^{1/2}{t}^{1/2},$$
(8)

that is, the bending starts from the edges and propagates diffusively into the center. Moreover, from Eq. (8) we calculate the normalized ETE distance \(\tilde{\ell }\equiv (\ell -{\ell }_{{{{{{{{\rm{eq}}}}}}}}})/({\ell }_{{{{{{{{\rm{c}}}}}}}}}-{\ell }_{{{{{{{{\rm{eq}}}}}}}}})\) scaling as \(1-\tilde{\ell } \sim {a}^{3} \sim {t}^{3/2}\), where ℓ is the ETE distance with ℓeq being its equilibrium value. Interestingly, although the linear trial curvature profile assumed in Eq. (7) has some visible deviation from the simulated profile (as shown in Fig. 3b), the scaling law predicted above for ETE distance agrees well with simulation results in Fig. 4a and is not sensitive to the form of trial curvature profile; the same scaling is obtained by using a simpler trial step-function profile of curvature. However, we note that the initial diffusive bending process happens only in a very short period, as shown in Fig. 4a. The bending dynamics slow down quickly from the diffusive regime to a subdiffusive regime with \(1-\tilde{\ell } \sim {t}^{0.9}\).

Fig. 3: Bead-spring simulations of the dynamic bending process of a Hydra rod.
figure 3

Simulation snapshots of the dynamic bending process of a Hydra rod (of contour length ℓc = 2πR0) in viscous fluids on a supporting surface: a the rod shape and b the curvature profile. In b, the linear trial profiles of curvature for variational analysis are plotted (dashed lines), which are parameterized by the bent length \(\tilde{a}=a/{R}_{0}\) for the initial stage (red dashed line) and the center curvature \({\tilde{c}}_{{{{{{{{\rm{m}}}}}}}}}={c}_{{{{{{{{\rm{m}}}}}}}}}/{c}_{0}\) for the final stage (blue dashed line), respectively.

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Fig. 4: Temporal evolution of the end-to-end (ETE) distance \(\tilde{\ell }\) for Hydra rods.
figure 4

a The initial bending starting diffusively from the edges and b the final bending relaxing exponentially to the equilibrium shape. Universal scaling relations are found for rods of different contour lengths \(\pi /2\le {\tilde{\ell }}_{{{{{{{{\rm{c}}}}}}}}}\le 3\pi\).

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In the final stage of the bending dynamics close to the equilibrium bent state (see Fig. 3a), we take the same strategy as above and also choose a linear trial profile of the curvature c(s, t) (see Fig. 3b) and the interlayer slide ϵs(s, t) as

$$c\, =\, {c}_{{{{{{{{\rm{m}}}}}}}}}+(2s/{\ell }_{{{{{{{{\rm{c}}}}}}}}})({c}_{0}-{c}_{{{{{{{{\rm{m}}}}}}}}}),\\ {\epsilon }_{{{{{{{{\rm{s}}}}}}}}}\, =\, {\epsilon }_{{{{{{{{\rm{s,m}}}}}}}}}+(2s/{\ell }_{{{{{{{{\rm{c}}}}}}}}})({\epsilon }_{{{{{{{{\rm{s0}}}}}}}}}-{\epsilon }_{{{{{{{{\rm{s,m}}}}}}}}}),$$
(9)

for 0 ≤ s ≤ ℓc/2, in which case the bending dynamic process is described by the two time-dependent parameters: the curvature cm(t) and interlayer slide ϵs,m(t) in the rod center at s = 0. Substituting the trial profile into Eqs. (5) and (6), we obtain the Rayleighian \({{{{{{{\mathcal{R}}}}}}}}\) and its minimization yields (see Sec. D of Supplementary Note 2)

$$(\tau +{\tau }_{{{{{{{{\rm{cell}}}}}}}}}){\dot{\tilde{c}}}_{{{{{{{{\rm{m}}}}}}}}}\, =\, 1-(1+{{{{{{{\mathcal{B}}}}}}}}){\tilde{c}}_{{{{{{{{\rm{m}}}}}}}}}+{{{{{{{\mathcal{B}}}}}}}}{\tilde{\epsilon }}_{{{{{{{{\rm{s,m}}}}}}}}},\\ {\tau }_{{{{{{{{\rm{s}}}}}}}}}{\dot{\tilde{\epsilon }}}_{{{{{{{{\rm{s,m}}}}}}}}}\, =\, {\tilde{c}}_{{{{{{{{\rm{m}}}}}}}}}-{\tilde{\epsilon }}_{{{{{{{{\rm{s,m}}}}}}}}},$$
(10)

with \({\tilde{c}}_{{{{{{{{\rm{m}}}}}}}}}={c}_{{{{{{{{\rm{m}}}}}}}}}/{c}_{0}\) and \({\tilde{\epsilon }}_{{{{{{{{\rm{s,m}}}}}}}}}={\epsilon }_{{{{{{{{\rm{s,m}}}}}}}}}/{\epsilon }_{{{{{{{{\rm{s0}}}}}}}}}\). Here \({{{{{{{\mathcal{B}}}}}}}}={Y}_{{{{{{{{\rm{s}}}}}}}}}{\epsilon }_{{{{{{{{\rm{s0}}}}}}}}}^{2}/{c}_{0}^{2}{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}\) characterizes the relative stiffness of interlayer slide and out-plane bending and it is order unity since all energy terms in Eq. (5) are comparable. The three characteristic timescales \(\tau =6{\xi }_{{{{{{{{\rm{v}}}}}}}}}{{{{{{{\mathcal{L}}}}}}}}/{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}{c}_{0}^{4}\), \({\tau }_{{{{{{{{\rm{cell}}}}}}}}}={\eta }_{{{{{{{{\rm{cell}}}}}}}}}b{h}^{3}/{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}\), and τs = ξs/Ys are associated with viscous drag, sub-cellular or cellular dissipation, and interlayer frictional sliding, respectively. \({{{{{{{\mathcal{L}}}}}}}}\) is a nonlinear function of rod contour length ℓc (changing from 0 to order one; see its expression in Eq. (S79) of Supplementary Note 2).

We consider the following two limits of the dynamic equation (10) that are particularly interesting and may be relevant to the bending dynamics during tissue morphogenesis and regeneration. Firstly, in the limits of either coherent Hydra rods or incoherent Hydra rods with small friction, we have τs/(τ + τcell) ≪ 1, and the viscous drag from the surrounding fluids or the cellular dissipation is the dominant dissipation mechanism. In this limit, we find from Eq. (10) the normalized ETE distance \(\tilde{\ell }(t)\) follows

$$\tilde{\ell } \sim \exp \left[-t/(\tau +{\tau }_{{{{{{{{\rm{cell}}}}}}}}})\right],$$
(11)

which has been justified for rods with various contour lengths by our numerical simulations using the bead-spring model and shown in Fig. 4(b). Note that the relaxational time \(\tau =6{\xi }_{{{{{{{{\rm{v}}}}}}}}}{{{{{{{\mathcal{L}}}}}}}}/{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}{c}_{0}^{4}\) shows a nonlinear dependence on the rod contour length ℓc (due to the nonlinear function \({{{{{{{\mathcal{L}}}}}}}}\)): τ is small for short rods ℓc < πR0, but increases very fast with increasing ℓc when ℓc > πR0. For example, the time \(\tau \sim {\xi }_{{{{{{{{\rm{v}}}}}}}}}/{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}{c}_{0}^{4}\) for ℓc = 2πR0 is about ten times larger than that for ℓc = πR0. In addition, we have used the bead-spring model to numerically examine the bending dynamics of Hydra rods under some different boundary conditions (see Figs. S5–S7 in Sec. C of Supplementary Note 2). We find that the bending dynamics predicted above are very robust: the same scaling laws (including the nonlinear contour-length dependence of relaxational time τ due to \({{{{{{{\mathcal{L}}}}}}}}\)) are found for most different boundary conditions, including both the initial diffusive bending and the final relaxational bending processes.

Secondly, in the limit of incoherent Hydra rods with large friction, we have τs/(τ + τcell) ≫ 1, we obtain from Eq. (10) the ETE distance \(\tilde{\ell }(t)\) following the same exponential form:

$$\tilde{\ell } \sim \exp \left(-t/{\tau }_{{{{{{{{\rm{s}}}}}}}}}\right).$$
(12)

That is, in either limit, the characteristic time for the spontaneous bending of the tissue rod is controlled by the slowest dissipative dynamics and the longest timescale. Particularly for Hydra tissue rods, we can estimate the magnitude of the three timescales as follows. Using \({\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}} \sim E{h}^{3}b\), ξv ~ ηenv, b ~ 10h, Ys ~ Ehb, and ξs = ξh2b, we obtain \(\tau \sim {\eta }_{{{{{{{{\rm{env}}}}}}}}}/E{h}^{3}b{c}_{0}^{4},\,{\tau }_{{{{{{{{\rm{cell}}}}}}}}} \sim {\eta }_{{{{{{{{\rm{cell}}}}}}}}}/E\), and τs ~ ξh/E with ηenv being the viscosity of surrounding fluids and ξ being the interlayer friction coefficient10. We can then substitute typical parameter values and obtain

$$\tau \sim 1{0}^{-3}\,{{{{{{{\rm{s}}}}}}}},\quad {\tau }_{{{{{{{{\rm{cell}}}}}}}}} \sim {\tau }_{{{{{{{{\rm{s}}}}}}}}} \sim 1\,\min ,$$
(13)

where the parameter values are taken from cell experiments9,10: h ~10 μm, c0 ~1/100 μm−1, E ~1 kPa, ηenv ~10−3 Pa ⋅ s, and ξ ~1010 Pa ⋅ s ⋅ m−1. Particularly, we have taken ηcell ~105 Pa ⋅ s, a magnitude measured in the study of mouth opening dynamics of Hydra9, and for spherical cellular aggregates11 and zebrafish embryonic tissues31. Therefore, the spontaneous bending of Hydra rods is controlled by the dissipation due to interlayer frictional sliding and/or the sub-cellular/cellular dissipation that sets the longest characteristic timescales \({\tau }_{{{{{{{{\rm{cell}}}}}}}}} \sim {\tau }_{{{{{{{{\rm{s}}}}}}}}} \sim 1\,\min\) consistent with the time duration of initial spontaneous bending observed in Hydra regeneration experiments (see Fig. 1b)7. In this case, with τcell ~ τs ≫ τ, the approximate bending dynamics is described by Eq. (10) with τ set to be zero.

Discussions

The active laminated-plate (ALP) model constructed here for Hydra tissue fragments connects the triple-layer tissue structure and the contractile supracellular actomyosin bundles of fragments with the equilibrium shape and dynamics of their spontaneous bending. In this model, the equilibrium bent shape of a Hydra fragment is characterized by spontaneous curvature tensor and determined by minimizing the deformation energy of the laminate fragment upon internal uniform active contractions applied by two sets of perpendicular supracellular actomyosin bundles. The concept of spontaneous curvature has been used a long time ago to represent the tendency of lipids to curve in lipid membranes30 and to describe the spontaneous bending of uniformly-heated bimetal plates32, in which the driving forces are chemical heterogeneity and asymmetric thermal expansion, respectively. In contrast, the spontaneous bending of tissue fragments proposed in this work is driven by completely different forces due to contractile supracellular actomyosin bundles.

From the ALP model, we predict that the equilibrium shape of a Hydra fragment is determined by its anisotropy in contractility and elasticity, as shown in Fig. 2. A spherical cap shape observed in Hydra regeneration experiments7 is generated only for the case of more-or-less isotropy in both contractility (of the two contractile bundles) and elasticity (of each lamina layer). The presence of a soft mesoglea (the intermediate extracellular matrix) at given geometries (layer thicknesses) facilitates inward tissue bending toward the endoderm side. Such proper inward bending is essential for the formation of hollow Hydra spheroid and the whole Hydra regeneration process7. Interestingly, the critical roles of the changes in mesoglea during Hydra regeneration have already been examined in experiments since the 1990s33. However, we would like to point out that such a mesoglea-softening mechanism is very limited in ensuring inward tissue bending; we, therefore, propose to include other mechanisms, such as differential interfacial tension, as having been used to explain cell sorting during the regeneration of Hydra from cell aggregates34.

For the bending dynamics of tissue fragments, we consider three specific dissipation mechanisms in the framework of the triple-layer ALP model: viscous drag due to the fragment motion relative to surrounding fluids, the relative frictional slide between adjacent tissue layers, and the cellular dissipation associated with sub-cellular-scale remodeling and cellular-scale rearrangements. Particularly, for rod-like fragments, we present a minimal dynamic model. The model suggests that tissue “viscosity” (specifically, the interlayer frictional sliding and cellular dissipation associated with remodeling/rearrangement at sub-cellular and cellular scales) sets the longest timescales and controls the spontaneous bending dynamics of Hydra fragments in the first several minutes after their excision from parent Hydra. In most one-layer viscoelastic models of tissue9 (see an example of such a model in Sec. D of Supplementary Note 2), tissue “viscosity” is simply employed as a coarse-grained concept and a “lumped” parameter to represent the underlying dissipation mechanisms. In comparison, our ALP model includes the specific tissue structure and gives predictions on the specific dissipation mechanisms. The characteristic timescale (of minutes) for the initial bending dynamics of Hydra fragments is obtained from parameter values measured in other independent experiments. We think this provides indirect evidence for the specific dissipation mechanisms proposed in this work. We suggest that further quantitative experiments should be done in the future to measure the elastic moduli of each tissue layer and identify the dissipation mechanisms underlying the Hydra regeneration process4,5,6,7,12.

The proposal here on the spontaneous bending of Hydra tissue fragments driven by contractile supracellular actomyosin bundles should be generically present in a broad range of cell assemblies and tissue fragments during tissue regeneration or morphogenesis during embryo development4,5,6. For example, the bending of epithelial monolayers induced by actomyosin contraction35,36, out-turning of the inside of the Volvox embryo37, etc. Our ALP model provides a useful framework for the study of tissue regeneration and morphogenesis. It connects the specific structure of the tissues with complex, emergent tissue behaviors, which is contrasted with other theoretical models for active surfaces with internal forces and torques resulting from differential stresses assumed a priori and acting along the cross-section of thin tissue sheets38,39. Moreover, our ALP model can be further extended by following active gel theory40,41 and active nematic elastomer theory18,42 to be an active laminated-nematic-gel-plate (ALNGP) model to study the morphogenesis of other multi-layered tissues with supracellular nematic cytoskeletal orientation and to include the couplings between the organization of the cytoskeleton and the deformation/curvature of the tissue4,5,6. In the ALNGP model, the topological defects can be considered by following the Landau-de Gennes theory of nematic liquid crystals42. The effects of the permeation of water or other small molecules on the dynamics of tissue bending can also be explored41,43. Additional aspects of cell biology, such as cell division/apoptosis and cell morphological changes25,26, or active behaviors such as migration and oscillations, could be incorporated as well24,44. The potential of the ALP model or the extended ALNGP model can be further illustrated by numerical simulations using the finite element method8,26. For example, one can consider the spontaneous bending of circumferential Hydra fragments (open ring geometry)7 where topological defects play a role4,5,6,45, resulting in defective morphology. One can also investigate the bending instability of tissue fragments at large deformations and study the coupling between deformation and nematic order, as well as the dynamics of topological defects during tissue morphogenesis and regeneration4,5,6.

Methods

Active laminated-plate model (ALP) of Hydra tissue fragments

Here we briefly explain the ALP model of the triple-layer fragments of Hydra tissue. In the ALP model, the tissue fragments are treated as thin orthotropic laminated plates with internal active contraction, as shown in Fig. 1a. Following the classical laminated-plate (CLP) theory for thin orthotropic laminated plates16, the deformation of the Hydra tissue fragment is due entirely to the in-plane displacement \(u(x,y)\hat{{{{{{{{\bf{x}}}}}}}}}\) and \(v(x,y)\hat{{{{{{{{\bf{y}}}}}}}}}\) and the out-plane deflection (bending) ω(x, y) of the neutral surface. The components of the displacement vector are given by uz ≈ ω and

$${u}_{{{{{{{{\rm{x}}}}}}}}}\approx u-(z-{z}_{{{{{{{{\rm{x0}}}}}}}}}){\partial }_{{{{{{{{\rm{x}}}}}}}}}\omega ,\,\,{u}_{{{{{{{{\rm{y}}}}}}}}}\approx v-(z-{z}_{{{{{{{{\rm{y0}}}}}}}}}){\partial }_{{{{{{{{\rm{y}}}}}}}}}\omega .$$
(14)

Note that in contrast with the traditional CLP theory, we here assume more generally that the positions of the neutral surfaces of the thin Hydra-laminated plate are different for the deformation in the x and y directions, and we denote them as zx0 and zy0, respectively. Such a form of the displacement field allows the reduction of the 3D problem to one of studying the deformation of the reference neutral surfaces. Once the neutral-surface displacements (u, v, ω) are known, the displacements of any arbitrary point (x, y, z) in the 3D continuum can be determined.

Particularly, for a Hydra-plate suspended by its center with ω(0, 0) = 0, we take the trial solution of the form

$$u={\epsilon }_{{{{{{{{\rm{x}}}}}}}}}x,\quad v={\epsilon }_{{{{{{{{\rm{y}}}}}}}}}y,\quad \omega =\frac{1}{2}\left({c}_{{{{{{{{\rm{x}}}}}}}}}{x}^{2}+{c}_{{{{{{{{\rm{y}}}}}}}}}{y}^{2}\right),$$
(15)

and in this case, the bending energy is given (after integrating over the thickness z-direction) by14,17

$${{{{{{{{\mathcal{F}}}}}}}}}_{b}\, =\, \int\,dxdy\left[\left(\frac{1}{2}{D}_{{{{{{{{\rm{p,11}}}}}}}}}{c}_{{{{{{{{\rm{x}}}}}}}}}^{2}+{D}_{{{{{{{{\rm{p,12}}}}}}}}}{c}_{{{{{{{{\rm{x}}}}}}}}}{c}_{{{{{{{{\rm{y}}}}}}}}}+\frac{1}{2}{D}_{{{{{{{{\rm{p,22}}}}}}}}}{c}_{{{{{{{{\rm{y}}}}}}}}}^{2}\right)\right.\\ \left.-\left({M}_{{{{{{{{\rm{p}}}}}}}}}^{(1)}{c}_{{{{{{{{\rm{x}}}}}}}}}+{M}_{{{{{{{{\rm{p}}}}}}}}}^{(2)}{c}_{{{{{{{{\rm{y}}}}}}}}}\right)\right].$$
(16)

Here we have used the fact that the two supracellular bundles inside Hydra fragments lie on the two sides of the intermediate mesoglea layer and align along the x- and y-directions in the ectoderm and endoderm layers (see Fig. 1a), respectively. Dp,ij are the bending stiffnesses, defined by \({D}_{{{{{{{{\rm{p}}}}}}}},11}\equiv \int\nolimits_{-{h}_{1}}^{{h}_{2}}dz{\bar{Q}}_{11}{(z-{z}_{{{{{{{{\rm{x0}}}}}}}}})}^{2}\), \({D}_{{{{{{{{\rm{p}}}}}}}},12}\equiv \int\nolimits_{-{h}_{1}}^{{h}_{2}}dz{\bar{Q}}_{12}(z-{z}_{{{{{{{{\rm{x0}}}}}}}}})(z-{z}_{{{{{{{{\rm{y0}}}}}}}}})\), and \({D}_{{{{{{{{\rm{p}}}}}}}},22}\equiv \int\nolimits_{-{h}_{1}}^{{h}_{2}}dz{\bar{Q}}_{22}{(z-{z}_{{{{{{{{\rm{y0}}}}}}}}})}^{2}\) with \({\bar{Q}}_{11}\), \({\bar{Q}}_{12}\), and \({\bar{Q}}_{22}\) being the orthotropic elastic constants (being different in different lamina layers, see Eq. (S5) in Supplementary Information). \({M}_{{{{{{{{\rm{p}}}}}}}}}^{(i)}\) are the active torques (per length), generated by the asymmetric internal contraction and given in the two lamina layers, respectively, by \({M}_{{{{{{{{\rm{p}}}}}}}}}^{(1)}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(1)}{z}_{{{{{{{{\rm{x0}}}}}}}}}\) and \({M}_{{{{{{{{\rm{p}}}}}}}}}^{(2)}\approx {\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(2)}({z}_{{{{{{{{\rm{y0}}}}}}}}}-{h}_{{{{{{{{\rm{m}}}}}}}}})\) with \({\tau }_{{{{{{{{\rm{p}}}}}}}}}^{(i)}\, < \, 0\) being the active contractile forces (per length).

Onsager’s principle as an approximation method for Hydra tissue bending dynamics

In this work, our major purpose is not to uncover the mechanisms for the interlayer slide in Hydra tissue fragments, but to study the effects of interlayer slide accompanying spontaneous bending on their dynamics. Therefore, for a Hydra rod with differential slide and spontaneous curvature along the arc length, we neglect the in-plane contraction ϵ and simply write the total energy \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) in the simple phenomenological form of Eq. (5). Minimizing \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) with respect to c and ϵs give the spontaneous curvature and the equilibrium slide c0 and ϵs0, respectively. Using them, the total energy in Eq. (5) can be rewritten into an alternative form as

$${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}=\int\nolimits_{-{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}^{{\ell }_{{{{{{{{\rm{c}}}}}}}}}/2}ds\left[\frac{{Y}_{{{{{{{{\rm{s}}}}}}}}}}{2}{({\epsilon }_{{{{{{{{\rm{s}}}}}}}}}-{\epsilon }_{{{{{{{{\rm{s0}}}}}}}}}c/{c}_{0})}^{2}+\frac{{\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}}{2}{(c-{c}_{0})}^{2}\right],$$
(17)

with \({\tilde{D}}_{{{{{{{{\rm{r}}}}}}}}}\equiv {D}_{{{{{{{{\rm{r}}}}}}}}}-{\chi }^{2}/{Y}_{{{{{{{{\rm{s}}}}}}}}}\) being the normalized flexural stiffness of the rod.

To investigate the bending dynamics of Hydra rods, we don’t study this highly nonlinear equation that has been solved numerically extensively (particularly for very long thin rods)29, but instead, we carry out approximate variational analysis by assuming some trial shape dynamics of the elastic curve13,14. We assume that, during the bending of the Hydra rod, the interlayer slide ϵs(s, t) and the local curvature c(s, t) take some trial forms (see Fig. 3b in the main text) that are parameterized by a set of time-dependent variables a(t) with −ℓc/2 ≤ s ≤ ℓc/2 being the arc length. From the trial local curvature c(s; a(t)), we can calculate the local tangential orientation angle θ(s) (defined relative to the horizontal x-direction) by

$$\theta (s,t)=\int\nolimits_{0}^{s}c(\hat{s},t)d\hat{s}+\theta (0,t),$$
(18a)

from which we further calculate the tangential unit vector \(\hat{{{{{{{{\bf{t}}}}}}}}}(s,t)\) and position vector r(s, t), respectively, by

$$\hat{{{{{{{{\bf{t}}}}}}}}}(s,t)=\frac{\partial {{{{{{{\bf{r}}}}}}}}(s,t)/\partial s}{\left\vert \partial {{{{{{{\bf{r}}}}}}}}(s,t)/\partial s\right\vert }=(\cos \theta (s,t),\,\sin \theta (s,t)),$$
(18b)
$${{{{{{{\bf{r}}}}}}}}(s,t)=\int\nolimits_{0}^{s}d\hat{s}\left(\cos \theta (\hat{s},t),\,\sin \theta (\hat{s},t)\right)+{{{{{{{\bf{r}}}}}}}}(0,t).$$
(18c)

Then using the obtained rod profile r(s, t), Eqs. (6) and (17), we can calculate the Rayleighian \({{{{{{{\mathcal{R}}}}}}}}[\dot{{{{{{{{\bf{r}}}}}}}}},{\dot{\epsilon }}_{{{{{{{{\rm{s}}}}}}}}}]={\dot{{{{{{{{\mathcal{F}}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}+\Phi\) as a function of \(\dot{{{{{{{{\bf{a}}}}}}}}}(t)\). Minimizing \({{{{{{{\mathcal{R}}}}}}}}\) with respect to \(\dot{{{{{{{{\bf{a}}}}}}}}}(t)\) gives the ordinary differential equation for a(t), from which we solve the bending dynamics approximately.

Bead-spring model for the bending dynamics of Hydra tissue fragments in viscous fluids

In the discrete bead-spring model, we consider the dynamics of Hydra rods determined by the competition between the bending energy and the viscous dissipation due to the motion of the rod relative to the surrounding stationary viscous fluids. We do not consider the energy and dissipation associated with interlayer slides. Moreover, we assume the rod to inextensible. In this case, the total energy of the Hydra rods includes two parts: \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}({{{{{{{{\bf{r}}}}}}}}}_{i}(t))={{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}}+{{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{b}}}}}}}}}\), which are given, respectively, by

$${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}}\, =\, \mathop{\sum }\limits_{i=1}^{N-1}\frac{{Y}_{{{{{{{{\rm{r}}}}}}}}}}{2{\ell }_{0}}{\left({r}_{i,i+1}-{\ell }_{0}\right)}^{2},\\ {{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{b}}}}}}}}}\, =\, \mathop{\sum }\limits_{i=1}^{N-2}\frac{{D}_{{{{{{{{\rm{r}}}}}}}}}}{2{\ell }_{0}}{\left[\left({\hat{{{{{{{{\bf{t}}}}}}}}}}_{i+1}-{\hat{{{{{{{{\bf{t}}}}}}}}}}_{i}\right)-{\phi }_{0}{\hat{{{{{{{{\bf{n}}}}}}}}}}_{i}\right]}^{2},$$
(19)

with ϕ0 ≡ ℓ0c0 = ℓ0/R0 being the equilibrium bending angle. Here ri,i+1 = ∣ri,i+1∣ is the distance between neighboring beads with ri,i+1 = ri+1 − ri(1 ≤ i ≤ N − 1) and ri = (xi, yi)(1 ≤ i ≤ N) being the position vector of the i-th bead. The local tangential unit vector and normal unit vector are given, respectively, by

$${\hat{{{{{{{{\bf{t}}}}}}}}}}_{i}\equiv \frac{{{{{{{{{\bf{r}}}}}}}}}_{i,i+1}}{{r}_{i,i+1}}=(\cos {\theta }_{i},\sin {\theta }_{i}),\quad {\hat{{{{{{{{\bf{n}}}}}}}}}}_{i}=(-\sin {\theta }_{i},\cos {\theta }_{i}),$$
(20)

in which θi (with −π ≤ θi ≤ π) is the local tangential angle of each spring. In our simulations, we take a very large Yr to keep the rod inextensible.

From the total energy \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}({{{{{{{{\bf{r}}}}}}}}}_{i}(t))\) we can obtain the rate of energy change \({\dot{{{{{{{{\mathcal{F}}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}\) as a function of \({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i}(t)\equiv d{{{{{{{{\bf{r}}}}}}}}}_{i}/dt\). The viscous dissipation function is given by

$$\Phi ({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i})=\mathop{\sum }\limits_{i=1}^{N}\frac{1}{2}{\xi }_{{{{{{{{\rm{b}}}}}}}}}{\dot{{{{{{{{\bf{r}}}}}}}}}}_{i}{(t)}^{2},$$
(21)

with ξb ≡ ξvℓ0 being the effective drag coefficient of the chain segments in the surrounding viscous fluids. Then minimizing Rayleighian \({{{{{{{\mathcal{R}}}}}}}}({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i})=\Phi ({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i})+{\dot{{{{{{{{\mathcal{F}}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i})\) with respect to \({\dot{{{{{{{{\bf{r}}}}}}}}}}_{i}\) gives the dynamic equations of the bead-spring chain as

$${\xi }_{{{{{{{{\rm{b}}}}}}}}}\frac{d{{{{{{{{\bf{r}}}}}}}}}_{i}}{dt}={{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{s}}}}}}}}}+{{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{b}}}}}}}}},\quad i=1,2,\ldots N,$$
(22)

where \({{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{s}}}}}}}}}\equiv -\partial {{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}}/\partial {{{{{{{{\bf{r}}}}}}}}}_{i}\) and \({{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{b}}}}}}}}}\equiv -\partial {{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{b}}}}}}}}}/\partial {{{{{{{{\bf{r}}}}}}}}}_{i}\) are the spring force and the bending force, respectively (see the Sec. A of Supplementary Note 2 for their expressions).