A Morphoelastic Shell Theory of Biological Invagination in Embryos

Abstract

The embryo of Volvox globator, a monolayer spheroidal cell sheet, undergoes an inversion to release its flagella during the late stage of its development. This inversion, known as the type-B inversion, initiates from the equator. Other types of inversions also exist, such as the inversion from the anterior pole of Volvox carteri and the bowl-shaped inversion of Pleodorina. These inversions can be regarded as axisymmetric processes, during which complex fold patterns are generated. The invagination of the cell sheet plays a crucial role in embryonic development, and our aim is to understand this process from an interdisciplinary point of view, with a particular focus on the mechanical aspects. In this work, we first develop a morphoelastic shell theory for general deformations of biological shells, incorporating both active and passive biomechanical effects, as well as membrane and bending effects. By means of asymptotic analysis, we establish an analytical framework to study axisymmetric deformations of morphoelastic shells focusing mainly on the membrane effects. For illustrative purposes, we apply this framework to investigate the invagination of Volvox globator embryo. The underlying active stretches driving this process are derived analytically by inverse analysis of experimental data through the morphoelastic shell model. We highlight a two-order remodeling strategy that generates the observed invagination pattern: the Gaussian morphing of the cell sheet creates the first fundamental form of the stress-free folded patterns, while a secondary remodeling generates the membrane tension necessary to balance the external pressure and the second fundamental form of the invaginated pattern. This remodeling strategy unveils the complex interplay between geometry, mechanics, and biological processes during Volvox globator embryogenesis.

1 Introduction

Embryogenesis consists in the formation and the development of complex supracellular structures through the coordination of cell movements during the biological processes of growth, proliferation and differentiation [1]. The orchestration of morphogenetic movements is often directed by mechanical information through morphogens signalling, which often controls cell polarity [2].

Folding is a distinctive feature of embryonic transformation allowing to acquire a three-dimensional structure driven either by differential growth [3] or by active contractility [4]. In both cases, folding emerges as a mechano-adaptation process with the formation of invaginations/evaginations as the cellular tissue behaves as an elastic membrane [5, 6]. In many biological system models, folding may lead to the inversion of the cellular assembly, in order to fulfil a biological function, such as the assembly of the adult exoskeleton in the Drosophila disc [7] and the telencefalic develoment in mammals and teleosts [8]. Inversion is also commonly observed in plants, being used in adult organisms to facilitate pollen dispersal [9] and in embryos to bring flagellar units at the external surface for enabling locomotion ability [10].

The multicellular green alga Volvox is considered as a prototypical biological system model for embyogenetic inversion. During its embryonic development, the spheroidal monolayer cell sheet undergoes five representative stages of mophogenetic movements as illustrated in Fig. 1: (A-B) initial stage to invagination stage of the cell sheet at the equator, (C-E) posterior inversion, and (F-I) anterior peeling [11]. This type of inversion is called the “type-B” inversion. There also exist other types of inversions including the “type-A” inversion, which starts the inversion from the anterior pole, and the bowl-shaped inversion, whose initial configuration is a bowl-shaped geometry [12], see the illustration in Fig. 2. Other multicellular organisms undergo similar inversion processes from shell-like initial geometries, such as Choanoeca flexa, that uses it for acquiring feeding and swimming abilities [13]. This work proposes a novel mechanical approach to study the coordination of morphogenetic movements during biological inversion of axisymmetric shell-like cell sheets. For this purpose, we perform a rigorous asymptotic derivation of a morphoelastic shell theory that is applied to a nonlinear elastic tissue, possibly undergoing growth and remodelling. For illustrative purposes, we apply the model to study the invagination of the type-B inversion of Volvox globator embryo, discussing the model predictions against existing experimental data.

Fig. 1

The type-B inversion of Volvox globator embryo. Stage A is the initial state, and stage B is the invaginated state. Stages C-E are regarded as posterior inversion process and stages F-I are regarded as anterior peeling process. Adapted from [11] under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0)

Fig. 2

The type-A inversion of Volvox carteri (A1-A8) and the bowl-shaped inversion of Pleodorina (B1-B8). The former is adapted from [11] under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0). The latter is adapted from [12] under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/)

The invagination process of the Volvox embryo is important since it is analogous to morphological transformations in higher organisms, such as gastrulation in animals, which is critical for proper embryonic development. By studying this in a simpler organism like Volvox, we can gain insights into the mechanical principles that might also apply to more complex systems. The invagination of the cell sheet during the inversion of the Volvox globator has been extensively studied using elastic models [14–17], giving insights on the spatial distribution of the growth functions mainly during the invagination process. In particular, starting from an educated assumption on the growth functions, existing approaches either prescribe the extrinsic parameters or derive them through an inverse optimization problem. In the present study, we provide a more general approach in which we obtain directly the growth functions solving a physics-driven inverse problem fed by available experimental data on the embryo morphology throughout all the stages.

This allows to bridge the gap with developmental biologists, who study these morphogenetic process mainly by experimental techniques, e.g. [18–21], but require physics-driven information on the generation of mechanical stress due to growth or active processes, and its possible influence in shaping the tissue development.

Thus, our work aims at deriving a physics-informed morphoelastic shell model, which combines nonlinear elasticity, growth theory and asymptotic techniques, to study the axisymmetric inversion of cell sheets using existing morphological data in the related biological literature. For other types of active shell models, readers may refer to some recent works [22, 23].

The paper is organized as follows. In Sect. 2, we derive a morphoelastic shell model using asymptotic analysis and we specify its membrane limit. In Sects. 3 and 4, we use the morphoelastic membrane model to derive an analytical framework to study the axisymmetric deformations of morphoelastic membranes. In Sect. 5, this framework is applied to study the invagination of the Volvox globator embryo. Sect. 6 finally contains a summary of the main achievements and a few concluding remarks.

Notation

Throughout this paper, boldface letters, for example $\mathbf{x}$, $\mathbf{F}$, represent vectors or second-order tensors. The summation convention for repeated indices is adopted. In a summation, Greek letters $\alpha , \beta , \ldots$ run from 1 to 2, whereas Latin letters $i, j, k, \ldots$ run from 1 to 3. A comma preceding indices means differentiation. Let $\left (\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right )$ be the standard basis vectors in the three-dimensional Euclidean space $\mathbb{R}^{3}$. For a scalar-valued function of a tensor $W(\mathbf{F})$, its derivative with respect to $\mathbf{F}$ is defined to be $\partial W/\partial \mathbf{F}:=\partial W/\partial F_{j i} \mathbf{e}_{i} \otimes \mathbf{e}_{j}$; higher-order derivatives are defined in a similar way. The divergence of a tensor $\mathbf{S}$ is defined by $\mathrm{Div}(\mathbf{S}):=\partial S_{i j}/\partial x_{i} \mathbf{e}_{j}$.

2 Derivation of the Morphoelastic Shell Model

In this section, we derive the morphoelastic shell model from the three-dimensional governing equations and boundary conditions.

2.1 Three-Dimensional Governing System

Let us consider a thin shell of constant thickness $2h$ that occupies a domain $\Omega \times [-h, h]$ in its reference configuration, where $\Omega$ is the middle surface of the shell, see the geometric illustration in Fig. 3. The middle surface $\Omega$ is parametrized by two curvilinear coordinates $\theta ^{\alpha}, (\alpha =1,2)$, and the position vector of a point on $\Omega$ is expressed as $\mathbf{X}^{(0)}=\mathbf{X}^{(0)}(\theta ^{1}, \theta ^{2})$. The tangent vectors along the coordinate lines are defined by $\mathbf{G}_{\alpha}=\mathbf{X}^{(0)}_{,\alpha}$, where here and henceforth a comma denotes differentiation. The contravariant basis vectors, denoted by $\mathbf{G}^{\alpha}$, satisfy the relations $\mathbf{G}^{\alpha} \cdot \mathbf{G}_{\beta}=\delta ^{\alpha}_{\beta}$, where $\delta _{\beta}^{\alpha}$ is the Kronecker delta operator. The unit normal vector to the middle surface is defined by $\mathbf{N}=\mathbf{G}_{1} \times \mathbf{G}_{2}/|\mathbf{G}_{1} \times \mathbf{G}_{2}|$, where × means cross product. By setting $\mathbf{G}_{3}=\mathbf{G}^{3}=\mathbf{N}$, $\{\mathbf{G}_{i}\}$ and $\{\mathbf{G}^{i}\}$ form two sets of right-handed bases. The curvature tensor is defined from the differentiation of the unit normal vector as

$$\mathit{\kappa }=-\frac{\partial \mathbf{N}}{\partial {\mathbf{X}}^{(0)}}=-{\mathbf{N}}_{,\alpha }\otimes {\mathbf{G}}^{\alpha }.$$

(1)

Accordingly, the scalar mean and Gaussian curvatures are defined from $\boldsymbol{\kappa}$ as:

$$H=\frac{\mathrm{tr}(\mathit{\kappa })}{2},K=det(\mathit{\kappa }).$$

(2)

Fig. 3

(a) A shell structure with constant thickness $2h$ in the reference configuration. $\mathbf{X}$ is a point on the shell body, $\mathbf{X}^{(0)}$ is the projection of $\mathbf{X}$ onto the middle surface $\Omega$, and $\mathbf{N}$ is the unit outward normal vector of the middle surface. (b) The curvilinear coordinates $(\theta ^{1}, \theta ^{2})$ on the middle surface of the shell and $\{\mathbf{G}_{1}, \mathbf{G}_{2}, \mathbf{N}\}$ are the associated covariant basis. $\mathbf{V}$ and $\pmb{\nu}$ are the binormal vectors respectively for a general curve on the lateral surface and for the curve $\partial \Omega$ of the middle surface, defined as the cross product of the local tangent vector and normal vector $\mathbf{N}$

The position vector of a generic particle within the shell in the reference and current configurations are denoted by $\mathbf{X}$ and $\mathbf{x}$ respectively. Accordingly, $\mathbf{X}$ can be written as

$$ \mathbf{X}=\mathbf{X}^{(0)}(\theta ^{1}, \theta ^{2})+\zeta \mathbf{N} (\theta ^{1}, \theta ^{2}),\quad -h \leq \zeta \leq h, $$

(3)

where $\zeta$ is the normal coordinate along the thickness direction. Thus, the deformation gradient has the following expression

$$ \mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}=(\nabla \mathbf{x}) \boldsymbol{\mu}^{-1}+\frac{\partial \mathbf{x}}{\partial \zeta} \otimes \mathbf{N}, $$

(4)

where $\nabla =\mathbf{G}^{\alpha} \partial /\partial \theta ^{\alpha}$ is a two-dimensional tangent operator on the middle surface and

$$\mathit{\mu }=\mathbf{1}-\zeta \mathit{\kappa },$$

(5)

with $\mathbf{1}=\mathbf{G}^{\alpha} \otimes \mathbf{G}_{\alpha}$ denoting the rank-two identity tensor on the tangent plane of the undeformed middle surface. The determinant of $\boldsymbol{\mu}$ is denoted by

$$ \mu (\zeta )=\det (\boldsymbol{\mu})=1-2 H \zeta +K \zeta ^{2}. $$

(6)

Following the multiplicative decomposition proposed by Rodriguez et al. [24], the total deformation gradient $\mathbf{F}$ is rewritten as the product of a remodelling tensor $\mathbf{U}$ and an elastic deformation tensor $\mathbf{A}$, such that $\mathbf{F}=\mathbf{A} \mathbf{U}$. To prevent confusion with the previously mentioned basis vectors, the notation $\mathbf{G}$ typically used for the growth tensor is replaced with $\mathbf{U}$. Since we aim to deal with biological matter, the elastic deformation is further assumed isochoric, imposing the constraint

$$ \det (\mathbf{A})-1=0. $$

(7)

The shell is also assumed to behave as a hyperelastic body, having a strain energy density function $W=W(\mathbf{A})$ per unit volume of the remodelled state. The nominal stress tensor is therefore

$$ \mathbf{S}=\frac{\partial (J W)}{\partial \mathbf{F}}-pJ\mathbf{F}^{-1}=J \mathbf{U}^{-1} W_{\mathbf{A}} (\mathbf{F} \mathbf{U}^{-1})-p J \mathbf{F}^{-1}, $$

(8)

where $W_{\mathbf{A}}=\partial W/\partial \mathbf{A}$, $J=\det (\mathbf{U})$ and $p$ is the Lagrange multiplier enforcing the incompressibility constraint (7). In terms of $\mathbf{S}$, the equilibrium equation and boundary conditions can be written as

$$\begin{aligned} &\mathrm{Div} (\mathbf{S})=0,\quad \mathrm{in}~\Omega \times [-h, h], \end{aligned}$$

(9)

$$\begin{aligned} &\mathbf{S}^{T}(- \mathbf{N})|_{\zeta =-h}=\mathbf{q}^{-},\quad \mathbf{S}^{T} \mathbf{N}|_{\zeta =h}=\mathbf{q}^{+}, \end{aligned}$$

(10)

$$\begin{aligned} &\mathbf{S}^{T} \mathbf{V}=\mathbf{q},\quad \mathrm{on}~\partial \Omega _{q} \times [-h, h], \end{aligned}$$

(11)

$$\begin{aligned} &\mathbf{x}=\mathbf{b},\quad \mathrm{on}~\partial \Omega _{0} \times [-h, h], \end{aligned}$$

(12)

where $\mathbf{N}$ and $\mathbf{V}$ denote the unit outward normal of the top surface and the lateral surface, respectively, and $\partial \Omega _{q}$ and $\partial \Omega _{0}$ denote the traction and position boundaries of the middle surface, respectively.

2.2 The Recurrence Relations for the 2D Shell Theory

We now sketch the asymptotic derivation of the morphoelastic shell theory from the three-dimensional boundary value problem (Eqs. (9)–(12)), incorporating both the stretching and the bending effects.

Assuming that the deformation fields are smooth enough, we may expand the displacement and the Lagrange multiplier as

$$ \begin{aligned} & \mathbf{x}(\mathbf{r}, \zeta )=\mathbf{x}^{(0)}(\mathbf{r})+\zeta \mathbf{x}^{(1)}(\mathbf{r})+\frac{1}{2} \zeta ^{2} \mathbf{x}^{(2)}( \mathbf{r})+\frac{1}{6} \zeta ^{3} \mathbf{x}^{(3)}( \mathbf{r})+\cdots , \\ & p(\mathbf{r}, \zeta )=p^{(0)}(\mathbf{r})+\zeta p^{(1)}(\mathbf{r})+ \frac{1}{2} \zeta ^{2} p^{(2)}(\mathbf{r})+\cdots , \end{aligned} $$

(13)

where expansion coefficients are defined by $(\bullet )^{(i)}=\partial ^{i} (\bullet )/\partial \zeta ^{i}|_{ \zeta =0}$. The tensors of $\mathbf{F}$, $\mathbf{A}$, $\mathbf{U}$ and $\mathbf{S}$ are also expanded in an identical manner. Substituting these expansions into the kinematic equation (4), the incompressibility constraint (7) and the constitutive relation (8), we obtain the relations among the expansion coefficients at any order. We call the expressions for the expansion coefficients of $\mathbf{S}$ as the shell constitutive relations.

We later exploit the three-dimensional equilibrium equations and the boundary conditions to derive the relations between the higher order expansion coefficients of the displacement (e.g., $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$) and the leading order term $\mathbf{x}^{(0)}$ at the middle surface. These expressions are called the recurrence relations. First, by equating the coefficient of $\zeta ^{0}$ in (9) and the coefficient of $\zeta ^{1}$ in (7) to be zero, we obtain

$$ \begin{aligned} & \nabla \cdot \mathbf{S}^{(0)}+\mathbf{S}^{(1) T} \mathbf{N}=0, \\ & \mathrm{tr}(\mathbf{A}^{(0)-1} \mathbf{A}^{(1)})=0, \end{aligned} $$

(14)

which is a set of linear algebraic equations for $\mathbf{x}^{(2)}$ and $p^{(1)}$ and that is analytically solved. Similarly, $\mathbf{x}^{(3)}$ and $p^{(2)}$ can be solved through linear algebraic equations derived by equating the coefficients of $\zeta ^{1}$ in (9) and $\zeta ^{2}$ in (7) to be zero.

Next, we derive the equations for $\mathbf{x}^{(1)}$ and $p^{(0)}$ by averaging the bottom and top traction conditions through the operation

$$ \left [\mu (h) \cdot \mathbf{S}^{T} \mathbf{N}|_{\zeta =h}-\mu (-h) \cdot \mathbf{S}^{T} (-\mathbf{N})|_{\zeta =-h}\right ]/2, $$

and we obtain

$$ \frac{1}{2} \left [\mu (h) \mathbf{S}^{T} \mathbf{N}|_{\zeta =h}+\mu (-h) \mathbf{S}^{T} \mathbf{N}|_{\zeta =-h}\right ]=\mathbf{m}, $$

(15)

where

$$ \mathbf{m}= \frac{\mu (h) \mathbf{q}^{+}-\mu (-h) \mathbf{q}^{-}}{2}. $$

(16)

The factors $\mu (h)$ and $\mu (-h)$ are incorporated to compensate for the area variation across the thickness that results from curvature effects. Then taking the series expansions into (15) and truncating this equation at $O(h^{2})$, we have

$$ (1+K h^{2}) \mathbf{S}^{(0) T} \mathbf{N}-2 H h^{2} \mathbf{S}^{(1) T} \mathbf{N}+\frac{1}{2} h^{2} \mathbf{S}^{(2) T} \pmb{N}+O(h^{3})= \mathbf{m}. $$

(17)

To eliminate $\mathbf{S}^{(2) T} \mathbf{N}$ in the above equation, we consider the coefficient of $\zeta ^{1}$ in (9), which leads to

$$ \nabla \cdot \mathbf{S}^{(1)}+\mathbf{S}^{(2) T} \mathbf{N}+( \pmb{\kappa} \mathbf{g}^{\alpha}) \cdot \mathbf{S}^{(0)}_{, \alpha}= \mathbf{0}. $$

(18)

In view of $\text{(14)}_{1}$ and (18), we can eliminate $\mathbf{S}^{(i) T} \mathbf{N}, (i=1,2)$ in (17), which yields

$$\begin{array}{rl}& {\mathbf{S}}^{(0)T}\mathbf{N}+h^2(K{\mathbf{S}}^{(0)T}\mathbf{N}+2H\mathrm{\nabla }\cdot {\mathbf{S}}^{(0)})\\ & -\frac{1}{2}{h}^2[\mathrm{\nabla }\cdot {\mathbf{S}}^{(1)}+(\mathit{\kappa }{\mathbf{g}}^{\alpha })\cdot {\mathbf{S}}_{,\alpha }^{(0)}]+O\left(h^3\right)=\mathbf{m}.\end{array}$$

(19)

The above equation (19) and the $\zeta ^{0}$ order equation from (7), i.e.

$$ \det (\mathbf{A}^{(0)})=1, $$

(20)

are used to eliminate $\mathbf{x}^{(1)}$ and $p^{(0)}$. Equations (19) and (20) are nonlinear differential-algebraic equations and can be solved using the perturbation method.

2.3 The Morphoelastic Shell Equations and Boundary Conditions

After the recurrence relations, the only unknown is the leading order position vector $\mathbf{x}^{(0)}$, that can be determined from the bottom and top traction conditions. To this end, by performing

$$ \left [\mu (h) \cdot \mathbf{S}^{T} \mathbf{N}|_{\zeta =h}+\mu (-h) \cdot \mathbf{S}^{T} (-\mathbf{N})|_{\zeta =-h}\right ]/(2h), $$

we obtain

$$ \begin{aligned} -2 H \mathbf{S}^{(0) T} \mathbf{N}+(1+K h^{2}) \mathbf{S}^{(1) T} \mathbf{N}-H h^{2} \mathbf{S}^{(2) T} \mathbf{N}+ \frac{h^{2}}{6} \mathbf{S}^{(3) T} \mathbf{N}+O(h^{3})=\bar{\mathbf{q}}, \end{aligned} $$

(21)

where

$$ \bar{\mathbf{q}}= \frac{\mu (h) \mathbf{q}^{+}+\mu (-h) \mathbf{q}^{-}}{2h}. $$

(22)

After lengthy manipulations [25], we can rewrite the above equation (21) in a more compact form as

$$ \nabla \cdot \bar{\mathbf{S}}=-\bar{\mathbf{q}}, $$

(23)

where the $O(h^{3})$ error is omitted for notational simplicity and $\bar{\mathbf{S}}$ represents the average stress tensor, which is defined by

$$\begin{array}{rl}\overline{\mathbf{S}}=& \mathbf{1}{\mathbf{S}}^{(0)}+\frac{1}{3}{h}^2(\mathit{\kappa }-2H\mathbf{1}){\mathbf{S}}^{(1)}+\frac{1}{6}{h}^2\mathbf{1}{\mathbf{S}}^{(2)}\\ =& \frac{1}{2h}{\int }_{-h}^h[\mathbf{1}+\zeta (\mathit{\kappa }-2H\mathbf{1})]\mathbf{S}\mathrm{d}\zeta +O(h^3).\end{array}$$

(24)

Adopting the idea of [26], we decompose (23) into its in-plane and out-of-plane parts as

$$ \begin{aligned}[b] &\mathbf{1} \nabla \cdot \bar{\mathbf{S}}_{t}-\pmb{\kappa}\, \bar{\mathbf{S}}\, \mathbf{N}=-\bar{\mathbf{q}}_{t}, \\ &\nabla \cdot (\bar{\mathbf{S}} \,\mathbf{N})+\mathrm{tr} ( \pmb{\kappa}\, \bar{\mathbf{S}})=-\bar{\mathbf{q}} \cdot \mathbf{N}, \end{aligned} $$

(25)

where the projection tensors are defined as $\bar{\mathbf{q}}_{t}=\mathbf{1} \bar{\mathbf{q}}$ and $\bar{\mathbf{S}}_{t}=\mathbf{1} \bar{\mathbf{S}} \mathbf{1}$. Henceforth, we use the subscript $t$ to define quantities in this tangential manifold.

In summary, we have derived a closed system for $\mathbf{x}^{(i)}, (i=0,1,2, 3)$ and $p^{(j)}, (j=0,1, 2)$, consisting of equations (14), (19), (20) and (25) as well as the equations for $\mathbf{x}^{(3)}$ and $p^{(2)}$, which are not shown for the sake of conciseness. By using this closed system to eliminate higher order coefficients $\mathbf{x}^{(i)}, p^{(j)}, (i=1,2,3; j=0,1,2)$, (25) becomes a vector equation for the leading order coefficient $\mathbf{x}^{(0)}$, which provides the morphoelastic shell equations omitting $O(h^{3})$ terms.

In a similar manner, the 2D shell boundary conditions are derived from their 3D counterparts (cf. (11) and (12)). Following the approach in [25], we introduce two boundary conditions at the position boundary $\partial \Omega _{0}$. One is the prescribed position of the middle surface, and the other is the prescribed average position through the thickness, which are written as

$$\begin{aligned} \begin{aligned} &\mathbf{x}^{(0)}=\mathbf{b}^{(0)},\quad \bar{\mathbf{x}}= \bar{\mathbf{b}}, \end{aligned} \end{aligned}$$

(26)

where $\bar{\mathbf{x}}$ is defined as

$$ \bar{\mathbf{x}}(s)=\frac{1}{2 h} \int _{-h}^{h}\mathbf{x}(s, Z)\,dZ= \mathbf{x}^{(0)}+\frac{h^{2}}{6} \mathbf{x}^{(2)}+O\left (h^{3} \right ), $$

and $\bar{\mathbf{b}}$ is defined similarly.

On the other hand, on the traction boundary $\partial \Omega _{q}$, we adopt the conditions that the traction at the middle surface and the average traction through thickness are prescribed, which are written as

$$\begin{array}{rl}& {\mathbf{S}}^{(0)T}\mathit{\nu }={\mathbf{q}}^{(0)},\frac{1}{2h}{\int }_{-h}^h{\mathbf{S}}^T\mathbf{N}{\mu }^{\ast }(Z)dZ={\overline{\mathbf{q}}}_0,\end{array}$$

(27)

where $\bar{\mathbf{q}}_{0}$ is defined as

$$ \bar{\mathbf{q}}_{0}=\frac{1}{2 h} \int _{-h} ^{h} \mathbf{q}(s, Z) \mu ^{*}(Z) \,dZ. $$

Note $\mu ^{*}(Z)$ is the Jacobian factor for the area element, i.e., on the boundary of the shell $\partial \Omega \times [-h, h]$, we have $d a=\mu ^{*}(Z) \,dZ ds$ where $ds$ is the arclength element of the middle surface boundary [25].

In conclusion, the morphoelastic shell theory encompasses the shell equations (25) along with boundary conditions (26) and (27). It is designed for analyzing large deformations and instabilities of thin biological shells accommodating both stretching and bending effects.

2.4 Simplified Morphoelastic Membrane Theory

Many biological tissues, especially at the earlier stages of embryonic development, are composed by thin cell sheets and can be modeled as soft active membranes. When bending effects can be ignored—such as in very thin, very soft, or inflated structures—a simplified version of the shell theory can be used. This streamlined theory is specifically designed to study membrane deformations.

The recurrence relations for $\mathbf{x}^{(1)}$ and $p^{(0)}$, corrected to $O(h)$ of (19) and (20) are:

$$ \mathbf{S}^{(0) T} \mathbf{N}=\mathbf{m},\quad \det (\mathbf{A}^{(0)})=1. $$

(28)

The morphoelastic shell equations, corrected to $O(h)$ of (25), are:

$$\begin{array}{rlr}& \mathbf{1}\mathrm{\nabla }\cdot {\mathbf{S}}_t^{(0)}-\mathit{\kappa }{\mathbf{S}}^{(0)}\mathbf{N}=-{\bar{\mathbf{q}}}_t,& \mathrm{\nabla }\cdot (\mathbf{1}{\mathbf{S}}^{(0)}\mathbf{N})+tr(\mathit{\kappa }{\mathbf{S}}_t^{(0)})=-\overline{\mathbf{q}}\cdot \mathbf{N}.\end{array}$$

(29)

The leading order traction condition is expressed as:

$${\mathbf{S}}^{(0)T}\mathit{\nu }={\mathbf{q}}^{(0)},\mathrm{on}\partial {\mathrm{\Omega }}_q.$$

(30)

This set of equations, from (28) to (30), is the membrane theory which will be applied in the following.

3 Analytical Framework for the Axisymmetric Deformations of Morphoelastic Membranbes

In the following, we use the above morphoelastic membrane theory to build an analytical framework for studying axisymmetric deformations of morphoelastic membrane.

3.1 Geometry Before and After the Deformation

For the undeformed middle surface which is axisymmetric, a material point on it is represented as

$$ \mathbf{X}^{(0)}=R^{(0)}(s) \mathbf{e}_{r}+Z^{(0)}(s) \mathbf{e}_{Z}. $$

(31)

where $s$ is the arclength variable in the initial configuration and $\{\mathbf{e}_{r}, \mathbf{e}_{\phi}, \mathbf{e}_{z}\}$ are the polar cylindrical basis. To facilitate analysis, we introduce a new set of curvilinear coordinates $(s, \phi , \zeta )$ representing the arclength, the azimuthal angle and the thickness variables, respectively. The corresponding covariant basis vectors on the reference middle surface are defined as

$$ \begin{aligned} &\mathbf{G}_{1}=\frac{\partial \mathbf{X}^{(0)}}{\partial s}=R^{(0) \prime}(s) \mathbf{e}_{r}+Z^{(0) \prime}(s) \mathbf{e}_{z}, \quad \mathbf{G}_{2}=\frac{\partial \mathbf{X}^{(0)}}{\partial \phi}=R^{(0)}(s) \mathbf{e}_{\phi}, \\ &\mathbf{G}_{3}= \frac{\mathbf{G}_{1} \times \mathbf{G}_{2}}{\left |\mathbf{G}_{1} \times \mathbf{G}_{2}\right |}= \frac{-Z^{(0) \prime}(s)}{\sqrt{E}} \mathbf{e}_{r}+ \frac{R^{(0) \prime}(s)}{\sqrt{E}} \mathbf{e}_{z} =\mathbf{N}, \end{aligned} $$

(32)

where the components of the reference first fundamental form are

$$ \begin{aligned} &E=\mathbf{G}_{1} \cdot \mathbf{G}_{1}=(R^{(0) \prime}(s))^{2}+(Z^{(0) \prime}(s))^{2},~ F=\mathbf{G}_{1} \cdot \mathbf{G}_{2}=0,~ \\ &G=\mathbf{G}_{2} \cdot \mathbf{G}_{2}=(R^{(0)}(s))^{2}. \end{aligned} $$

(33)

The components of the reference second fundamental form are give by

$$ \begin{aligned} &L=\frac{\partial ^{2} \mathbf{X}^{(0)}}{\partial s^{2}} \cdot \mathbf{N}= \frac{-Z^{(0) \prime}(s) R^{(0) \prime \prime}(s)+R^{(0) \prime}(s) Z^{(0) \prime \prime}(s)}{\sqrt{E}}, \\ &M=\frac{\partial ^{2} \mathbf{X}^{(0)}}{\partial s \partial \phi} \cdot \mathbf{N}=0, \\ &N=\frac{\partial ^{2} \mathbf{X}^{(0)}}{\partial \phi ^{2}} \cdot \mathbf{N}=\frac{R^{(0)}(s) Z^{(0) \prime}(s)}{\sqrt{E}}. \end{aligned} $$

(34)

Since $\{\mathbf{G}_{i}\}, (i=1,2,3)$ are already orthogonal basis, we can get the contravariant basis directly as

$$ \mathbf{G}^{1}=\frac{\mathbf{G}_{1}}{E},\quad \mathbf{G}^{2}= \frac{\mathbf{G}_{2}}{G},\quad \mathbf{G}^{3}=\mathbf{G}_{3}. $$

(35)

More generally, a material point inside the undeformed shell body is described as

$$ \mathbf{X}=\mathbf{X}^{(0)}+\zeta \mathbf{N}, \quad -h \leq \zeta \leq h. $$

(36)

The associated covariant basis vector are

$$ \begin{aligned} &\hat{\mathbf{G}}_{1}=\frac{\partial \mathbf{X}}{\partial s}= \mathbf{G}_{1}+\zeta \frac{\partial \mathbf{N}}{\partial s},\quad \hat{\mathbf{G}}_{2}=\frac{\partial \mathbf{X}}{\partial \Theta}= \mathbf{G}_{2}+\zeta \frac{\partial \mathbf{N}}{\partial \Theta}, \quad \hat{\mathbf{G}}_{3}=\frac{\partial \mathbf{X}}{\partial \zeta}= \mathbf{N}=\mathbf{G}_{3}. \end{aligned} $$

(37)

The contravariant basis vectors $\hat{\mathbf{G}}^{j}$ are defined as

$$ \hat{\mathbf{G}}_{i} \cdot \hat{\mathbf{G}}^{j}=\delta _{i}^{j}, $$

(38)

where $\delta _{i}^{j}$ is the Kronecker delta. (Remark: These are not needed for a membrane-type theory.)

Correspondingly, for the current axisymmetrically deformed shell, the position of the material point is represented as

$$ \mathbf{x}=r(s, \zeta ) \mathbf{e}_{r}+z(s, \zeta ) \mathbf{e}_{z}. $$

The covariant basis vectors are

$$ \begin{aligned} &\hat{\mathbf{g}}_{1}=\frac{\partial \mathbf{x}}{\partial s},\quad \hat{\mathbf{g}}_{2}=\frac{\partial \mathbf{x}}{\partial \Theta}, \quad \hat{\mathbf{g}}_{3}=\frac{\partial \mathbf{x}}{\partial \zeta}. \end{aligned} $$

(39)

The deformed middle surface is represented as

$$ \mathbf{x}^{(0)}=r^{(0)}(s) \mathbf{e}_{r}+z^{(0)}(s) \mathbf{e}_{Z}, $$

(40)

where $r^{(0)}=r(s, 0)$ and $z^{(0)}=z(s, 0)$. The covariant basis vectors on the deformed middle surface are defined by

$$ \begin{aligned} &\mathbf{g}_{1}=\hat{\mathbf{g}}_{1}|_{\zeta =0}=r^{(0) \prime}(s) \mathbf{e}_{r}+z^{(0) \prime}(s) \mathbf{e}_{z}, \\ &\mathbf{g}_{2}=\hat{\mathbf{g}}_{2}|_{\zeta =0}=r^{(0)}(s) \mathbf{e}_{\phi}, \\ &\mathbf{g}_{3}=\hat{\mathbf{g}}_{3}|_{\zeta =0}=r^{(1)}(s) \mathbf{e}_{r}+z^{(1)}(s) \mathbf{e}_{z}, \end{aligned} $$

(41)

where $r^{(1)}=\partial r(s, \zeta )/\partial \zeta |_{\zeta =0}$ and $z^{(1)}=\partial z(s, \zeta )/\partial \zeta |_{\zeta =0}$. Moreover, the components of the spatial first and second fundamental forms of the deformed middle surface are given by

$$ \begin{aligned} &e=(r^{(0) \prime}(s))^{2}+(z^{(0) \prime}(s))^{2},\quad f=0,~\quad g=(r^{(0)}(s))^{2}, \\ &l= \frac{-z^{(0) \prime}(s) r^{(0) \prime \prime}(s)+r^{(0) \prime}(s) z^{(0) \prime \prime}(s)}{\sqrt{e}}, \quad m=\boldsymbol{0},\quad n=\frac{r^{(0)}(s) z^{(0) \prime}(s)}{\sqrt{e}}. \end{aligned} $$

(42)

The contravariant basis vectors $\hat{\mathbf{g}}^{j}$ and $\mathbf{g}^{j}$ of the deformed shell are defined similarly as (38).

3.2 Three Dimensional Governing System for a Biological Membrane

We present the three-dimensional governing system for a biological membrane with specifications of the constitutive relations and the external loadings. The deformation gradient reads

$$ \begin{aligned}[b] \mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}={}& \frac{\partial \mathbf{x}}{\partial s} \otimes \hat{\mathbf{G}}^{1}+ \frac{\partial \mathbf{x}}{\partial \phi} \otimes \hat{\mathbf{G}}^{2}+ \frac{\partial \mathbf{x}}{\partial \zeta} \otimes \hat{\mathbf{G}}^{3} \\ ={}& \hat{\mathbf{g}}_{1} \otimes \hat{\mathbf{G}}^{1}+\hat{\mathbf{g}}_{2} \otimes \hat{\mathbf{G}}^{2}+\hat{\mathbf{g}}_{3} \otimes \hat{\mathbf{G}}^{3}. \end{aligned} $$

(43)

We now assume that the shell behaves as incompressible neo-Hookean material, so that the strain energy density function is

$$ \phi (\mathbf{F}, \mathbf{U})=J_{U} \phi _{0}(\mathbf{A})=J_{U} C_{0} \left (\operatorname{tr} \mathbf{A}^{T} \mathbf{A}-3\right ), $$

where $C_{0}$ is the shear modulus. We remark that this constitutive choice allows explicit analysis and it is motivated by the sake of physical understanding. Notwithstanding, the proposed approach can be easily extended to account for any nonlinear constitutive law using straightforward arguments. Accordingly, the nominal stress tensor is

$$ \mathbf{S}=J_{U}\, \mathbf{U}^{-1}\left [2 C_{0} \mathbf{A}^{T}-p \mathbf{A}^{-1}\right ]=J_{U} \left (2 C_{0} \mathbf{M}^{-1} \mathbf{F}^{T}-p \mathbf{F}^{-1}\right ), $$

(44)

where the growth metric is defined as $\mathbf{M}=\mathbf{U}^{T} \mathbf{U}$. The elastic deformation tensor $\mathbf{A}$ in (44) is replaced by $\mathbf{A}=\mathbf{F} \mathbf{U}^{-1}$, which aims to formulate the inverse problem for the growth functions/active stretches. The three-dimensional equilibrium equations and boundary conditions are given by

$$ \begin{aligned} & \operatorname{Div}(\mathbf{S})=0, \quad \text{ in } \Omega \times [-h, h], \\ & \mathbf{S}^{T}(-\mathbf{N})=\mathbf{q}^{-}=\mathbf{0}, \quad \text{ at } \Omega \times (-h), \\ & \mathbf{S}^{T} \mathbf{N}=\mathbf{q}^{+}=-P\mathbf{n}=-J_{U} P \mathbf{F}^{-T} \mathbf{N}, \quad \text{ at } \Omega \times h, \\ & \mathbf{S}^{T} \mathbf{G}_{1}=\mathbf{q}=\mathbf{0}, \quad \text{ at } \partial \Omega \times [-h, h] \end{aligned} $$

(45)

where $P>0$ is the external pressure imposed on the outer surface of the biological membrane.

We assume the following initial scaling of the pressure:

$$ P=-2 \frac{h}{L} \bar{P}=-2 h \bar{P}, $$

(46)

where $\bar{P} \leq O(1)$ and $\bar{P}<0$. Here, $h$ is the normalized aspect ratio of the shell and $L$ is its characteristic size and we normalize $L$ to be 1. More specifically, in later sections, we set $P = O(h^{2})$ to have $\bar{P} = O(h)$. On the one hand, this scaling is more biologically relevant, as explained later. On the other hand, it reduces the risk of instability compared with the lower order scaling $\bar{P}=O(1)$.

3.3 Asymptotic Analysis and Recurrence Relations

From the governing system (45), we now derive the corresponding morphoelastic membrane theory based on the results of the last section.

The position of a point on the deformed shell $\mathbf{x}$ and the Lagrange multiplier $p$ are expanded into Taylor series at $\zeta =0$ as

$$ \begin{aligned}[b] &\mathbf{x}=\left (r^{(0)}+\zeta r^{(1)}+\cdots \right )\, \mathbf{e}_{r}+ \left (z^{(0)}+\zeta z^{(1)}++\cdots \right )\, \mathbf{e}_{z}, \\ &p=p^{(0)}+\cdots . \end{aligned} $$

(47)

Accordingly, the deformation gradient $\mathbf{F}$, the remodelling tensor $\mathbf{U}$, the elastic deformation tensor $\mathbf{A}$ and the nominal stress tensor $\mathbf{S}$ are also expanded as

$$ \begin{aligned} &\mathbf{F}=\mathbf{F}^{(0)}+\zeta \mathbf{F}^{(1)}\cdots ,\quad \mathbf{U}=\mathbf{U}^{(0)}+\zeta \mathbf{U}^{(1)}\cdots , \\ &\mathbf{A}=\mathbf{A}^{(0)}+\zeta \mathbf{A}^{(1)}+\cdots ,\quad \mathbf{S}=\mathbf{S}^{(0)}+\zeta \mathbf{S}^{(1)}+\cdots . \end{aligned} $$

(48)

By taking the expansions into (43) and (44), at the leading order the deformation gradient reads

$$ \mathbf{F}^{(0)}=\mathbf{g}_{1} \otimes \mathbf{G}^{1}+\mathbf{g}_{2} \otimes \mathbf{G}^{2}+\mathbf{g}_{3} \otimes \mathbf{G}^{3}. $$

(49)

Similarly, the nominal stress tensor reads

$$ \mathbf{S}^{(0)}=2 C_{0} \left (\mathbf{M}^{(0)}\right )^{-1} \mathbf{F}^{(0) T}-p^{(0)}\left (\mathbf{F}^{(0)}\right )^{-1}, $$

(50)

where $\mathbf{M}^{(0)}=\mathbf{U}^{(0) T} \mathbf{U}^{(0)}$ is the growth metric at the leading order. We assume that growth is also isochoric so that $J_{U}=1$, which implies the volume of the biological membrane is unchanged (as is the case in the work [11]). However, this still permits the change of the area of the membrane so that complex deformation can happen.

The symmetric $\mathbf{M}^{(0)}$ is decomposed into a spectral form, which is assumed to have the form of

$$ \mathbf{M}^{(0)}=\left [\lambda _{1}(s)\right ]^{2} \mathbf{G}_{1} \otimes \mathbf{G}^{1}+\left [\lambda _{2}(s)\right ]^{2} \mathbf{G}_{2} \otimes \mathbf{G}^{2}+\left [\lambda _{3}(s)\right ]^{2} \mathbf{G}_{3} \otimes \mathbf{G}^{3}, $$

(51)

where $\lambda _{i}^{2}, (i=1,2,3)$ are its eigenvalues and $\mathbf{G}_{i}/|\mathbf{G}_{i}|, (i=1,2,3)$ are the corresponding orthonormal eigenvectors. We call $\lambda _{i}, (i=1,2,3)$ the active stretches, which will be determined by solving an inverse problem formulated later on. Taking (49) and (51) into (50), the constitutive relation at the leading order is rewritten as

$$ \begin{aligned}[b] \mathbf{S}^{(0)}=&\left (\frac{2 C_{0} }{E \lambda _{1}^{2}}- \frac{p^{(0)}}{e}\right ) \mathbf{G}_{1} \otimes \mathbf{g}_{1}+ \left (\frac{2 C_{0} }{G \lambda _{2}^{2}}-\frac{p^{(0)}}{g}\right ) \mathbf{G}_{2} \otimes \mathbf{g}_{2} \\ &+\left (2 C_{0} \lambda _{1}^{2} \lambda _{2}^{2}- \frac{p^{(0)}}{|\mathbf{g}_{3}|^{2}}\right ) \mathbf{G}_{3} \otimes \mathbf{g}_{3}, \end{aligned} $$

(52)

where $\lambda _{3}$ is replaced by $\lambda _{3}=\left (\lambda _{1} \lambda _{2}\right )^{-1}$ due to $J_{U}=\lambda _{1} \lambda _{2} \lambda _{3}=1$.

We now derive the higher-order coefficients $(r^{(1)}, z^{(1)})$ and $p^{(0)}$ in terms of the leading-order fields $(r^{(0)}, z^{(0)})$. From (28), we have

$$ \mathbf{S}^{(0) T} \mathbf{N}=\mathbf{m}=\mathbf{0}, \quad \operatorname{det} \mathbf{F}^{(0)}=1, $$

(53)

where $\mathbf{m}=(-P/2)\, \mathbf{g}^{3}=O(h^{2})$ and $\mathbf{m}=\mathbf{0}$ at $O(h)$. Together with the relation $\mathbf{g}_{1} \cdot \mathbf{g}_{3}=0$, $( r^{(1)}, z^{(1)})$ and $p^{(0)}$ are solved as

$$ \begin{aligned} r^{(1)}(s)=-\sqrt{\frac{E G}{e g}} \frac{z^{(0) \prime}(s)}{\sqrt{e}}, ~ z^{(1)}(s)=\sqrt{\frac{E G}{e g}} \frac{r^{(0) \prime}(s)}{\sqrt{e}},~ p^{(0)}=2 C_{0} \lambda _{1}^{2} \lambda _{2}^{2} \frac{EG}{eg}. \end{aligned} $$

(54)

Based on the orthogonality condition $\mathbf{g}_{1} \cdot \mathbf{g}_{3}=0$ and equation (41), it is established that $\mathbf{g}_{1} \cdot \mathbf{x}^{(1)} =0$. Consequently, $\mathbf{x}^{(1)}$ must be orthogonal to the deformed middle surface up to order $O(h)$. Given that the current position is approximated by $\mathbf{x}=\mathbf{x}^{(0)}+\zeta \mathbf{x}^{(1)}$, and considering that the magnitude of $\mathbf{x}^{(1)}$, given by the areal change $\sqrt{EG/(eg)}$, varies along the arclength, the remodelling mechanism within the deformed shell produces thickness variations.

3.4 The Morphoelastic Membrane Model and Its Boundary Conditions

Applying (54) to (52), we have the leading order stress given in terms of the basis $(\mathbf{G}_{i} \otimes \mathbf{G}^{j})$ by

$$ \mathbf{S}^{(0)}=S^{(0)}_{11} \mathbf{G}_{1} \otimes \mathbf{G}^{1}+S^{(0)}_{13} \mathbf{G}_{1} \otimes \mathbf{G}^{3}+S^{(0)}_{22} \mathbf{G}_{2} \otimes \mathbf{G}^{2}, $$

(55)

where

$$ \begin{aligned} &S^{(0)}_{11}=\frac{2 C_{0}}{e} \left ( \frac{C_{11}}{\lambda _{1}^{2}}- \frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}\right )( \mathbf{G}_{1} \cdot \mathbf{g}_{1}), \\ &S^{(0)}_{13}=\frac{2 C_{0}}{e} \left ( \frac{C_{11}}{\lambda _{1}^{2}}- \frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}\right ) ( \mathbf{G}_{3} \cdot \mathbf{g}_{1}), \\ &S^{(0)}_{22}=\frac{2 C_{0}}{g} \left ( \frac{C_{22}}{\lambda _{2}^{2}}- \frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}\right ) ( \mathbf{G}_{2} \cdot \mathbf{g}_{2}). \end{aligned} $$

(56)

We remark that

$$ C_{11}=\frac{e}{E},\quad C_{22}=\frac{g}{G}. $$

Taking the stress tensor (55) into the membrane equations (29), by standard tensor manipulations we get the following morphoelastic membrane equations

$$ \begin{aligned} &S_{11}^{(0) \prime}+\left (S_{11}^{(0)}-\frac{S_{22}^{(0)}}{E} \right ) \frac{G^{\prime}}{2 G}-\frac{L}{E}S_{13}^{(0)}= \frac{\bar{P}}{E} \sqrt{\frac{g}{G}} (\mathbf{G_{3}} \cdot \mathbf{g}_{1}) , \\ &S_{13}^{(0) \prime}+S_{13}^{(0)} \frac{(E G)^{\prime}}{2 E G} + \frac{L}{E} S_{11}^{(0)} +\frac{N}{G} S_{22}^{(0)}=-\frac{\bar{P}}{E} \sqrt{\frac{g}{G}} (\mathbf{G_{1}} \cdot \mathbf{g}_{1}), \end{aligned} $$

(57)

where $\bar{P}=O(h)$. Assuming $\partial \Omega _{q} :={\mathbf{X}}^{(0)}|_{s=s_{0}}$, the boundary conditions from (30) read

$$ \mathbf{S}^{(0) T} \mathbf{G}_{1}=\mathbf{q}^{(0)},\quad \mathrm{at} \quad s=s_{0}. $$

(58)

The equations (57) and (58) govern the axisymmetric deformation of morphoelastic membrane as a function of the active stretches. In the following, we focus on the inverse problem of computing the active stretches required to realize the axisymmetric deformation of a morphoelastic membrane, which will be further applied in Sect. 5 to study the invaginated configuration reported in the biological literatures.

4 Axisymmetric Deformation of a Spherical Morphoelastic Membrane

From the boundary value problem outlined in the previous section, we formulated an inverse analysis to determine the underlying active stretches in a semi-analytical manner for a spherical morphoelastic membrane.

4.1 The Inverse Problem for the Active Stretches

As depicted in Fig. 4, considering an axisymmetric deformation(e.g., the invagination process), a point on the undeformed spherical middle surface is represented by:

$$ \mathbf{X}^{(0)} = \sin s\, \mathbf{e}_{r} + (1+\cos s) \mathbf{e}_{Z}, $$

(59)

so that $R^{(0)}=\sin s$ and $Z^{(0)}=1+\cos s$. The components of the first and second fundamental forms for the undeformed middle surface are

$$ E = 1\quad F = 0,\quad G = \sin ^{2} s,\quad L=-1, \quad M=0,\quad N=- \sin ^{2} s. $$

(60)

From (57), we have two governing equations to be

$$ \begin{aligned} &S_{11}^{(0) \prime}+\left (S_{11}^{(0)}-S_{22}^{(0)}\right ) \cot s+S_{13}^{(0)}= \bar{P}\left (r^{(0)} r^{(0) \prime}+\frac{G^{\prime}}{2 G} r^{(0)} z^{(0) \prime}\right ), \\ &S_{13}^{(0) \prime}+S_{13}^{(0)} \cot s -( S_{11}^{(0)} +S_{22}^{(0)})= \bar{P}\left (-\frac{G^{\prime}}{2 G} r^{(0)} r^{(0) \prime}+r^{(0)} z^{(0) \prime}\right ), \end{aligned} $$

(61)

where $\bar{P}=O(h)$.

Fig. 4

Middle surface of the spherical biological shell at the initial stage with an opening angle $2\alpha _{0} \geq 0$ at one pole. The cylindrical basis consist of $\{\mathbf{e}_{r}, \mathbf{e}_{\phi}, \mathbf{e}_{z}\}$, while $\{\mathbf{G}_{1}, \mathbf{G}_{2}, \mathbf{G}_{3}\}$ are the covariant basis associated with curvilinear coordinates $(s, \phi , \zeta )$, being the arclength, the azimuthal angle, and the thickness variables respectively

By Nother’s theorem [27], this axisymmetric problem is integrable, and by the manipulation of $\text{(61)}_{1}*(-G)+\text{(61)}_{2}*G^{ \prime}/2$, we have an integrable equation as

$$ S_{11}^{(0) \prime}(-G)+S_{11}^{(0)}(-G)^{\prime}+S_{13}^{(0) \prime} \frac{G^{\prime}}{2}+\frac{G^{\prime \prime}}{2} S_{13}^{(0)}=- \bar{P} r^{(0)} r^{(0) \prime}, $$

(62)

where we have applied $-G+\left (G^{\prime}\right )^{2} /(4 G)=G^{\prime \prime} / 2 \text{ and }-G-\left (G^{\prime}\right )^{2} /(4 G)=-1$, based on (60). Then the above equation (62) is integrated into

$$ S_{11}^{(0)}(-G)+S_{13}^{(0)} \frac{G^{\prime}}{2}+\frac{g}{2} \bar{P}=C, $$

(63)

where $C$ is the integration constant. Equation (63) is further written as equations for the active stretches as

$$ \left (\frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}- \frac{C_{11}}{\lambda _{1}^{2}}\right ) \left (2 \sqrt{ \frac{E G}{e g}} \frac{n}{g} \right )=\frac{1}{2} \bar{P}-\frac{C}{g} . $$

(64)

By applying the boundary condition (58), the constant is determined to be

$$ C=\frac{g(\alpha _{0})}{2} \bar{P}. $$

(65)

On the other hand, by further observing the constitutive relation (56) and the governing equation (61), we find that the operation of $\text{(61)}_{1}*(\mathbf{G_{3}} \cdot \mathbf{g}_{1})-( \text{61})_{2}*(\mathbf{G_{1}} \cdot \mathbf{g}_{1})$ gives another algebraic equation for the active stretches, which is

$$ \left (\frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}- \frac{C_{11}}{\lambda _{1}^{2}}\right ) \left (2 \sqrt{ \frac{E G}{e g}} \frac{l}{e} \right )+\left ( \frac{\lambda _{1}^{2} \lambda _{2}^{2}}{C_{11} C_{22}}- \frac{C_{22}}{\lambda _{2}^{2}}\right ) \left (2 \sqrt{ \frac{E G}{e g}} \frac{n}{g} \right )=\bar{P}. $$

(66)

In conclusion, after deriving the inverse problem of the active stretches which consist of the two differential equations in (57), in conjunction with the boundary condition in (58), we have transformed this differential system into two algebraic equations for the active stretches (64) and (66), that will be solved in the following using available biological data.

4.2 The Analytical Results for the Active Stretches

Since the two algebraic equations (64) and (66) contain a small parameter $\bar{P}$ which is of $O(h)$, we perform a regular asymptotic expansion analysis for the active stretches with respect to this small parameter.

Neglecting terms smaller than $\bar{P}$, we assume

$$ \lambda _{1}=\lambda _{10}+\bar{P} \lambda _{11},\quad \lambda _{2}= \lambda _{20}+\bar{P} \lambda _{21}, $$

(67)

and taking them back to (64) and (66), we have at the leading order

$$ \begin{aligned} &\frac{ \lambda _{10}^{2} \lambda _{20}^{2}}{C_{11} C_{22}}- \frac{C_{11}}{ \lambda _{10}^{2}}=0,\quad \frac{ \lambda _{10}^{2}\lambda _{20}^{2}}{C_{11} C_{22}}- \frac{C_{22}}{ \lambda _{20}^{2}}=0. \end{aligned} $$

(68)

Equations of (68) lead to

$$ \begin{aligned} & \lambda _{10}=\sqrt{C_{11}}=\sqrt{\frac{e}{E}}=\sqrt{[r^{(0) \prime}(s)]^{2}+[z^{(0) \prime}(s)]^{2}}, \\ & \lambda _{20}=\sqrt{C_{22}}=\sqrt{\frac{g}{G}}= \frac{r^{(0)}(s)}{\sin s}. \end{aligned} $$

(69)

From the condition of volume-preserving remodeling process, the solution for the active stretch in the thickness direction finally imposes

$$ \lambda _{30}=(\lambda _{10} \lambda _{20})^{-1}. $$

(70)

At this leading order, the active stretches $\lambda _{10}$, $\lambda _{20}$ perform a Gaussian morphing of the deformed surface [28], i.e. the remodelling process realize the target metric of the deformed middle surface via modulation of the local stretching [29].

The equations of the next-to-leading order from (64) and (66) are

$$ \begin{aligned} & \left (2 \gamma +\frac{l}{e}\right ) \frac{\lambda _{11}}{\lambda _{10}}+ \left (2 \gamma +\frac{n}{g} \right ) \frac{\lambda _{21}}{\lambda _{20}}=\frac{1}{4}\sqrt{ \frac{eg}{E G}}, \\ &2 \frac{\lambda _{11}}{\lambda _{10}}+ \frac{\lambda _{21}}{\lambda _{20}}=\frac{(g(s)-g(\alpha _{0}))}{8 n} \sqrt{\frac{eg}{EG}}, \end{aligned} $$

(71)

where $\gamma$ is the mean curvature of the deformed middle surface and by using $f=0$, $m=0$, it is expressed as

$$ \gamma =\frac{1}{2} \left (\frac{l}{e}+\frac{n}{g}\right ). $$

(72)

The two equations of (71) are linear algebraic equations for $\lambda _{11}$ and $\lambda _{21}$, whose solutions are

$$ \begin{aligned} &\frac{\lambda _{11}}{\lambda _{10}}=\sqrt{\frac{eg}{EG}} \frac{(n+2g\gamma ) [g(s)-g(\alpha _{0})]-2 g n}{24 n^{2}}, \\ &\frac{ \lambda _{21}}{{\lambda _{20}}}=-\sqrt{\frac{eg}{EG}} \frac{g}{e} \frac{(l+2e\gamma ) [g(s)-g(\alpha _{0})]-4 e n}{24 n^{2}}. \end{aligned} $$

(73)

From the incompressibility condition, we have

$$ \frac{\lambda _{31}}{\lambda _{30}}=- \frac{\lambda _{11} \lambda _{20}+\lambda _{21}\lambda _{10} }{\lambda _{10} \lambda _{20}}. $$

(74)

Taking the expansion (67) and the analytical results (69) and (73) into (56), we have the non-zero stress components as

$$ \begin{aligned} &\frac{S^{(0)}_{11}}{C_{0}}=\frac{2 \bar{P}}{e} (2 \lambda _{11}+ \lambda _{21}) (\cos s\, r^{(0) \prime}-\sin s\,z^{(0) \prime}), \\ &\frac{S^{(0)}_{13}}{C_{0}}=\frac{2 \bar{P}}{e} (2 \lambda _{11}+ \lambda _{21}) (\sin s\, r^{(0) \prime}+\cos s\,z^{(0) \prime}), \\ &\frac{S^{(0)}_{22}}{C_{0}}=2 \bar{P} \sqrt{\frac{G}{g}} (2 \lambda _{11}+ \lambda _{21}). \end{aligned} $$

(75)

In summary, we have formulated an inverse problem for determining the active remodeling stretches, which initially comprises two nonlinear first-order differential equations with specified boundary conditions. By symmetry assumptions, we have successfully converted this differential system into an algebraic system. This system was subsequently solved using a perturbation expansion. As a result, the active stretches as well as the residual stresses are derived explicitly.

5 Application to the Biological Example

We will utilize the findings from the previous section to illustrate the resulting remodeling strategy during the invagination process of the Volvox embryo. We find that the membrane theory allows to capture the invagination process (see Fig. 1 B) but further consideration of active bending will be required as folding appears (see Fig. 1 C-H). We start by detailing the available morphological data in the related biological literature.

5.1 Data Extraction and Interpolation

The position functions of the current configuration are interpolated from experimental data. Rather than employing local interpolation methods, such as spline interpolation, we opt for a global approximation approach—specifically, the Padé approximation. This method ensures the smoothness of the position function and yields smooth derivatives. The following steps outline our procedure for processing the experimental data:

1. Data extraction is performed using the WebPlotDigitizer software, which is freely available online. This step yields the data set $(r, z)$.

2. The data set $(r, z)$ is then interpolated using MATLAB software, utilizing the ‘cscvn’ command, which is adept at closed curve interpolation. This process generates the parametric functions $r = r(t)$ and $z = z(t)$, where $t$ denotes the chord length.

3. Building on the parametric functions, we calculate the arclength, resulting in the data set $(\tilde{s}, r(\tilde{s}))$ and $(\tilde{s}, z(\tilde{s}))$, with $\tilde{s}$ representing the arclength variable in the current configuration. Utilizing Mathematica, we apply the Padé approximation (employing a rational function) to this data set, ultimately obtaining the parametric functions $r = r(\tilde{s})$ and $z = z(\tilde{s})$.

By applying the above procedure to the biological data in the literature [11], we get the position functions $(r^{(0)}(\tilde{s}), z^{(0)}(\tilde{s}))$ for the invaginated shell (see Fig. 1 B) as the Padé approximants of order $[6/5]$ and $[5/5]$ respectively. The positions are displayed in Fig. 5 and the surface profile $(r^{(0)}(\tilde{s}), z^{(0)}(\tilde{s}))$ is given in Fig. 6.

Fig. 5

The Padé approximation of the position functions. (a) $r^{(0)}(\tilde{s})$ and (b) $z^{(0)}(\tilde{s})$

Fig. 6

The profile of the middle surface with position ($r^{(0)}( \tilde{s})$, $z^{(0)}(\tilde{s})$). Only half of the surface is shown due to symmetry

5.2 The Relation of Arclength Variables

We have obtained analytical solutions for the leading-order and next-order active stretches (cf. (69) and (73)). They incorporate the current position in the form of $(r^{(0)}(s), z^{(0)}(s))$ while the experimental data are in the form of $(r^{(0)}(\tilde{s}), z^{(0)}(\tilde{s}))$ where $\tilde{s}$ is the current arclength variable. These two sets have the relation

$$ r^{(0)}(s)=r^{(0)}(\tilde{s}(s)),\quad z^{(0)}(s)=z^{(0)}(\tilde{s}(s)). $$

Therefore, to align the theoretical framework with the experimental data, we need to discuss the mathematical mapping between the initial and current arclength variables, i.e., $\tilde{s}(s)$. In the absence of more detailed experimental details, the relationship between $s$ and $\tilde{s}$ is a degree of freedom in the current study. Hypothetically, one could determine experimentally by tracing the position of each cell at any given time [18]. Since this has not yet been reported, we assume a biologically consistent relation for $\tilde{s}(s)$ based on other available experimental information.

From the arclength formula, we know at the middle surface

$$ \frac{\mathrm{d} \tilde{s}}{\mathrm{d} s}=\sqrt{(r^{(0) \prime}(s))^{2}+(z^{(0) \prime}(s))^{2}}=\lambda _{10}(s). $$

(76)

Drawing inspiration from the work of Haas et al. (2021) [17], and in light of the observed experimental behavior (expansion at the anterior and contraction at the posterior of the cell sheet), we postulate a sigmoidal function to describe the active stretch $\lambda _{10}(s)$, which is

$$ \lambda _{10}(s)=-\frac{1}{1+e^{-k(s-s_{0})}}+a, $$

(77)

where $s_{0}$, $k$, $a$ are constants to be determined. Here, $s_{0}$ is the inflection point, $k$ is the steepness parameter that controls the sharpness of the curve, and $a$ is determined by the current arclength of the cell sheet from the experimental data. Correspondingly, the relation of the arclength variables $\tilde{s}(s)$ takes the form

$$ \tilde{s}(s)=a* s-\frac{1}{k} \log \left (1+e^{k(s-s_{0})}\right ). $$

(78)

We present $\tilde{s}(s)$ and its derivative in Fig. 7.

Fig. 7

The arclength relation $\tilde{s}(s)$ (solid blue) and its derivative $\lambda _{10}(s)$ (dashed red). We take the parameters $s_{0}=1.5$, $k=20$, and $a=1.43$

We would like to highlight that the relationship between the arclength variables is a variable parameter within this study. Consequently, different assumptions regarding this relationship yield distinct leading-order and next-order active stretches. We also remark that the contraction of the cell sheet at the posterior part (see $\lambda _{10} \approx 0.5 <1$ for $s>1.5$) is consistent with the existing biological interpretation.

5.3 Results and Discussion of the Invagination Mechanism

Based on the derived morpholeastic theory and the proposed data treatment strategy, we are ready to integrate the mechanical model with the experimental results. From the interpolated functions $(r^{(0)}(\tilde{s}), z^{(0)}(\tilde{s}))$ and the arclength relation $\tilde{s}(s)$, we derive the position functions $r^{(0)}(s)$ and $z^{(0)}(s)$. Then taking them into equations (69), (73) and (75), we calculate the active stretches and the corresponding residual stresses. These active stretches, for varying values of the steepness parameter $k$, are depicted in Fig. 8. At the leading order, as indicated by $\lambda _{10}$, we observe that the anterior region of the structure undergoes nearly uniform expansion, whereas the posterior region experiences almost uniform compression. A pronounced transition occurs near the inflection point at the equator. The active stretch in the circumferential direction $\lambda _{20}$ reveals a similar trend as $\lambda _{10}$ but with a much smoother variation. For the next-order active stretches (scaled by the leading order ones), the variation is concentrated around the inflection point, with negligible magnitude in other regions. Upon examining the influence of the steepness parameter $k$, we observe that $\lambda _{20}$ is only marginally affected near the inflection point, while the stretch ratios $\lambda _{11}/\lambda _{10}$ and $\lambda _{21}/\lambda _{20}$ remain (almost) constant across different values of $k$.

Fig. 8

The distribution of leading-order active stretches and the scaled next-order active stretches under different values of the steepness parameter $k$. (a) $\lambda _{11}/\lambda _{10}$ and (b) $\lambda _{21}/\lambda _{20}$

We illustrate the distribution of residual stress in Fig. 9. Two key observations are noteworthy. Firstly, only the circumferential stress $S^{(0)}_{22}$ is significantly influenced by the parameter $k$, while the normal and shear stresses $S^{(0)}_{11}$ and $S^{(0)}_{13}$ remain unaffected by changes in $k$. The impact of $k$ on $S^{(0)}_{22}$ is localized, primarily near the inflection point. Secondly, the negative sign of $S^{(0)}_{13}$ indicates that the upper part of the cell sheet experiences an inward pull from the lower part. Consequently, when the lower part is removed, such as in the laser ablation experiments described in [30], the upper part is freed and tends to move outward. This theoretical prediction is in agreement with the experimental findings reported by Haas and Hohn (2023) [30] (see Fig. 10).

Fig. 9

The distribution of scaled non-zero residual stresses under different values of the steepness parameter $k$. (a) $S_{11}^{(0)}/(C_{0} |\bar{P}|)$, (b) $S_{22}^{(0)}/(C_{0} |\bar{P}|)$, and (c) $S_{13}^{(0)}/(C_{0} |\bar{P}|)$,

Fig. 10

The experimental evidence for the existence of out-of-plane stress. Adapted from [30] with permission

We want to further discuss the underlying invagination mechanism based on the analytical results. One can see that the leading order active stretches $\lambda _{i0}, (i=1,2)$ in (69) determine the Gaussian morphing of the unstressed deformed surface while the next-order active stretches $\lambda _{i1}, (i=1,2)$ (73) determine the membrane stresses necessary to balance the external pressure and realize the second fundamental form of the target surface. Mathematically, such a two-step remodeling strategy is consistent with Gauss Theorema Egregium for realizing a target surface under an applied pressure.

The leading-order remodeling process indeed results in a stress-free state, whereas the next-order remodeling induces residual stresses to balance the external pressure. From a biological standpoint, the membrane stresses play a vital role in regulating cellular activities, as recently recognized by the biological community [31]. The inversion process involves complex cellular events, and it is postulated that a certain amount of membrane tension is also indispensable for the inversion. In our model, the pressure induces membrane tensions in the form of residual stresses, underscoring its biomechanical significance.

6 Conclusion

By means of rigorous asymptotic expansion, we have derived a morphoelastic shell theory which incorporates both stretching and bending effects. Based on this theory, we have proposed an analytical framework to study the axisymmetric deformations of biological morphoelastic membranes. This framework was further applied to investigate the invagination of the Volvox globator during its late stage of embryonic development. For this purpose, we have performed an inverse analysis to detect the remodelling strategy resulting from the proposed morphoelastic model against existing experimental data. Our analysis explores the invagination mechanism unveiling the interplay of geometry, mechanics, and biology during this stage of embryonic development. By integrating these aspects, we have developed a potential explanation for the invagination mechanism. If morphogens’ diffusion is sufficient to determine the leading-order active stretches-required to realize a target metric-through an anterior-posterior differential activation of the cell sheet via Gaussian morphing, then the interplay between the morphogens’ diffusion and the environmental stress is essential to realize the second fundamental form of the target surface in response to the external pressure. The resulting residual stresses may play a crucial role in regulating complex cellular activities during the invagination process. Although we illustrated the results using a neo-Hookean material, the proposed approach can be easily applied to any constitutive law for a biological materials. Thus, the resulting insights may encourage further biological investigations to elucidate the key mechano-biological factors driving this embryonic shape shifting.

We wish to elaborate further on the applicability of our model. Firstly, extending the results of the morphoelastic shell model up to the order $O(h^{2})$ is required as soon as bending effects are not negligible. For instance, in the case of the spherical inversion of Volvox globator, the membrane approximation is not suitable in the presence of poles or crowns (local extrema of $z^{(0)}$), as noted by Audoly and Pomeau (2000) [32]. In this case, the model can be enriched by including the active bending due to the apical constriction of the cells. Secondly, while shell instability is acknowledged as a potential outcome, it is not the primary focus of this study. The initial phase of remodeling, which alleviates stress, renders subsequent remodeling-induced instability less probable. However, if the external pressure intensifies significantly, one needs to consider the instability problem. Thirdly, the current experimental data is insufficient. Providing position data for each cell would clarify the relationship between the arclength variables, enhancing the model’s precision.

Our study has the potential of wider implications across various disciplines. A notable aspect of this research is the integration of experimental data with a mechanical model to infer the active stretches within the cell sheet, rather than assuming these stretches a priori. This approach, exemplified by the study of Volvox inversion, is adaptable to other types of deformations and offers a novel analytical method for estimating stress distribution within biological tissues based on morphological images. This is particularly relevant in biological and medical fields where imaging data is readily available, but direct stress measurements are challenging. Another feature of the current study is the advancement of shell theory. Diverging from classical shell theories and some refinements, our development of this theory involves no ad hoc assumptions regarding the strain form, and is applicable to arbitrary hyperelastic material, which broadens its applicability. We follow the foundational concepts pioneered by Hui-Hui Dai and his co-workers [26, 33, 34], while also doing further improvement. This morphoelastic shell/membrane theory, especially beneficial for analyzing large deformations in thin biological tissues, can also be adapted for studying finite-strain deformations in elastomers and polymer gels by adjusting the growth tensor accordingly.