In this chapter, we analyse non-uniform bending of single-layer cell tissues—epithelia, surrounding organs throughout the body. Dimensionally reduced model is suggested, which is equivalent to membranes with bending stiffness: the total elastic energy of the tissue is a combination of stretching and bending energies. The energy, suggested in this chapter, is a piecewise function, the branches of which correspond to a specific deformation regime: compression, pure bending and stretch.
- embryonic tissue
- adherens junctions
- quasi-convex energy densities
- isometric deformations
- short maps
- tissue mechanics
- lower semi-continuity
- hexagonal elements
Morphogenesis, the evolutionary process of shape and structure development of organisms or their parts, is driven by certain types of cell shape changes or, in other words, deformations. One type of such deformations is the apical constriction—visible shrinkage of the apical side of cells, leading to bending of epithelia—cell sheets, which surround organs throughout the body. First occurring at early stages of embryogenesis, the apical constriction results initially flat epithelia to be bent obtaining three-dimensional form, which, depending on physiological context and morphogenetic stage, leads to different consequences. It has been first hypothesized in  that in various developmental systems the apical constriction may drive the bending of epithelia. Using bulky mechanical construction, the hypothesis of Rhumbler is first practically tested in . The testing mechanism consists of 13 identical bars (cell walls) that are kept apart by means of stiff tubes connecting their centres (cell kernels), and are held in a row by rubber bands connecting their ends (cell membrane). In the first stage, the rubber bands at both sides are stretched equally and the mechanism is in straight equilibrium. Shortening the rubber bands at, for instance, the upper side uniformly in each segment (cell), the mechanism is bent on the side of the greater tension, as it could be expected. Thus, Rhumbler’s hypothesis could take place.
A more illustrative, computer model for verifying Rhumbler’s hypothesis is suggested in , where bars and bands are replaced by virtual cells. Prescribing certain mechanical properties to the cell membrane and cytoplasmic components, and assuming that as a result of deformations the volume of a single cell is preserved, which is implicitly done by Rhumbler and Lewis, a model of cuboidal epithelia folding is suggested, based on the local behaviour of individual cells. The model is demonstrated on the example of ventral furrow formation in
Much is known about the causes of the apical constriction, but some issues still remain unexplored [5–8]. The most studied causes include contraction of actin filaments—fibrous network localized at the cortex of the cell, by interacting with motor protein myosin—the most known converter of chemical energy into mechanical work. Under influence of myosin, actin fibres contract leading to shrinkage of the cell in the area of their localization [9, 10]. The contraction size and principal direction of the fibres, that is, the microscopical deformation of a single cell, mainly depend on the tissue type: actin-myosin network contraction may deform columnar (occurring, for instance, in digestive tract and female reproductive system) or cuboid (resp. in kidney tubules) cells into trapezoidal-, wedge- or bottle-shaped cells. The deformation size strongly depends also on the emplacement of deforming cells: cells with different placement are constricted differently, so that macroscopically one observes localized wrinkles (see Figure 2).
Because of the mechanical nature of cell shape change, in particular, apical constriction, its theoretical study first of all must rely on mechanical principles and constitutive laws. In early mechanical models, the tissue is modelled as a continuum material, so that the position and behaviour of individual cells are unimportant, that is, only macroscopic deformation of the tissue is studied. Moreover, the height of the cells (viz. the thickness of the tissue) is supposed to be negligible with respect to other measures, such that the deformation of the tissue can be described in terms of its mid-surface. In such models, the actin network is not explicitly accounted and the contraction forces are modelled as a force term acting on the outer surface of the epithelium. The actin network is explicitly accounted in , where thin elastic shell model based on linear Cauchy relations is derived to describe apical constriction in initially non-flat epithelia. The model is tested to simulate apical constriction in initially flat, cylindrical and spherical tissues. Ventral furrow formation in
The physical and geometrical characteristics of individual cells are taken into account by mechanical model suggested in  describing three-dimensional deformations of cell sheet. Both actin-myosin network contractile tensions and cell-cell adhesion stresses are involved to contribute in epithelial tissue bending. Prescribing specific mechanical characteristics to cytoplasmic components and to the kernel, the effective energy of a cell, which is assumed to be a hexagonal prism with constant volume, is expressed in terms of its basal length. As a result of simulations, it is particularly established that when adhesion stresses, distributed on lateral sides of the cell, are large enough, the effective energy has two local minima, that is, there are two equilibrium cell shapes. More particular, depending on position, cell shape may be discontinuously transited from squamous to columnar forms. Figure 4 shows that there is a whole domain in the plane of lateral adhesion and actin-myosin tensions for which the effective energy has two local minima.
In mathematical terms, it means that the total energy of the tissue is not
To overcome such difficulties, the membrane energy density is usually substituted by its
Another possibility for incorporating the lack of lower semi-continuity of membrane energy functionals is accounting the bending stiffness of the membrane and adding to the energy the bending contribution . So, it is established in [33, 34] that the total energy of a membrane with some bending stiffness and thickness , that is, the functional
of deformation consisting of membrane and bending contributions is lower semi-continuous for arbitrary as far as is lower semi-continuous. Explicit forms for and as well as sufficient conditions on material constants for which (and therefore ) is lower semi-continuous are derived ibid. Such materials are called
However, the most rigorous and general approach seems to be the derivation of low-order energies from general three-dimensional non-linear elasticity by means of -convergence. The concept was originally developed by De Giorgi, see . It turns out that the -limit of three-dimensional elastic energy functional, when the thickness of the body goes to zero, is always lower semi-continuous and since most commonly used two-dimensional energy functionals, that is, membrane energy, pure bending (Kirchhoff) energy and higher-order terms (von Kármán–like energies), are already derived as such limits (see Section 3), it becomes possible to analyze real deformations of particular tissues.
By suggested improvements, it is supposed to get rid of disadvantages of previous models and take into account the fact that the total energy of the tissue can have two local minima. Thus, we are intended
to account the height of each individual cell, so the model is fully three-dimensional,
to pick up strongly localized internal forces to model the phenomenon of apical constriction more precisely,
to introduce stretching and bending contributions into the total elastic energy of the epithelial tissue, thus making the model more realistic and convenient not only for qualitative but also for quantitative analysis,
to use the quasi-convexification of the total energy, so the real three-dimensional deformations of the epithelial tissues can be identified as minimizers of the total energy by a gradient flow technique.
The former allows making simplified toolboxes for analysing three-dimensional deformations of epithelial tissues. The rough diagram of the improved model looks like in Figure 5, so that the characteristics of each individual cell are important, as well as the actin belt is explicitly involved in the model.
The chapter is organized as follows: In Section 2, some preliminary definitions and main notations are brought to make the chapter independent of outer sources. In Section 3, known results from -convergent energies are shortly described. In Section 4, the quasi-convex envelope of membrane energy densities of general form is derived explicitly and sufficient conditions for their convexity are derived. Moreover, a specific energy is suggested for different deformation regimes that fully satisfies the needs of basic mechanical theory suggestion. Finally, in Section 5 main results of finite element analysis of tissues of particular initial shapes (flat, cylindrical, spherical) are discussed. Overall conclusion completes the body of the chapter.
There are different concepts of convexity in higher dimensions, such as
In other words, a function
In general, convexity implies quasi-convexity, which implies rank-one convexity, and finally rank-one convexity implies separately convexity.
It is evident that for any
Deformation is always denoted by
respectively, in which
In other words, frame indifferent energy densities are invariant against three-dimensional rotations before a deformation is applied and that isotropic energy densities are invariant against two-dimensional rotations after a deformation is applied.
Short deformations result compressive stresses, while isometric deformations do not allow stretching or compression. The case corresponds to tensile or, in general, non-compressive stresses. It is known that for non-compressive stresses, the membrane energy density function is already quasi-convex [41, 42].
3. -convergence and -limits of three-dimensional non-linear elastic energy
Rigorous derivation of two-dimensional energy functionals for thin bodies from three-dimensional functionals is of particular interest. There are several ways for dimension reduction, such as asymptotic expansion, Γ-convergence technique, and so on (for general survey, see ). In , a hierarchy of plate models is derived as Γ-limit of three-dimensional elastic energy functional when the thickness of the body . It turns out that the scaling of external force in plays an important role for Γ-limit derivation. Suppose a body occupies the domain , and the applied forces
denotes the rescaled elastic energy of the body, with
The total energy of the body will be
for will provide constitutive mechanical models to analyse deformed three-dimensional shape of initially flat thin bodies.
For , the limiting deformation is not only isometric but is close to a rigid motion. For that reason in , for deformation
are rescaled by
Then, we have
4. Relaxation of epithelial elastic energy by Pipkin’s procedure and by adding the bending contribution
As was mentioned in Section 1, there are two ways to relax the epithelial elastic energy. The first way is to relax the stretching energy for deformation regimes corresponding to compressive stresses. For that reason, Pipkin‘s procedure  seems to be the most common tool.
Suppose that the stretching energy density function is frame indifferent, isotropic and positive except the cases
where it turns to zero, that is, attains its minimum. On elastic energies with two minima, see . This assumption is motivated by bistability of cell shapes during apical constriction (see Figure 4).
We restrict attention to deformations with
and denote . Here , , are the principal stretches, the squares of which are the eigenvalues of the first fundamental form. Therefore
In view of frame indifference of , and thence, , we conclude that .
From quasi-convexity of , in particular, follows its rank-one convexity, therefore since the matrix
is rank one, we have
therefore is convex with respect to . Repeating the argument when
we see that is separately convex.
Under assumptions made, we have if and . Since and are convex along lines parallel to and , therefore on edges connecting the vertexes in the plane. Repeating the argument, we conclude that inside the square . So, for -short maps .
Since is even and convex in , it will attain its minimum with respect to only when . Denote by when . Suppose that is the largest value of for which , . The convexity of in implies that for , . In the same way, using the symmetry of in its arguments, we will arrive at for , . Evidently, for the rest of deformation regimes,is convex.
If we denote by and , the ranges of principal stretches, corresponding to uniaxial tensions in directions of those stretches, respectively , we finally arrive at
Besides opportunity of explicit relaxation, in  the derivation of criteria for convexity and quasi-convexity is described. In view of rank-one convexity, satisfies Legendre-Hadamard inequality from which it follows
For convexity of in corresponding domain instead of the last two equalities, the stronger conditions
must be satisfied. Above
The other way is the adding of pure bending contribution to the elastic energy of the tissue. Since actin-myosin network contraction leads to compressive or non-stretching stresses, we have to incorporate the elastic energy mainly for such deformations. According to Section 3 for non-stretching stresses, we have
(short deformations and compressive stresses)
(isometric deformations, neither stretch nor compressive stresses)
The deformation regime corresponding to tensile stresses leads to , which is already lower semi-continuous. In the case of uniaxial tension, there is no bending and the energy expression will look like the middle rows of (42). Thence, the total energy might be given as follows:
We assume that the tissue is homogeneous and isotropic, so according to Remark 1
which is valid for large deformations of incompressible hyper-elastic membranes. Above, is the first strain invariant, and are material parameters, evaluated experimentally in . For a neo–Hookean material and , in which is the shear modulus and is the bulk modulus. and are determined according to the procedure described above. For that reason, must be represented in terms of the principal stretches and as follows:
in which the incompressibility condition is taken into account.
For any fixed (uniaxial tension in the first principal direction), the corresponding attains its minimum with respect to at the maximal positive root of
Since in the uniaxial tension regime , we finally get
The correspondent tension is defined by
Since is obviously symmetric in its arguments, in the case of uniaxial tension in the second principal direction, we would have
It is evident, that when we substitute in the first case and in the second case, we arrive at undeformed configuration of the membrane, so the relaxed energy is fully consistent.
Forces driving tissue deformation are strongly localized and in general are compressive. Force must possess those properties.
5. Finite element analysis of three-dimensional shape of some characteristic deformations of tissues during morphogenesis
In this section, we summarize main results of finite element analysis of a single layer tissue model, the elastic energy of which is given by
In Figure 6, we bring the model of a single-cell (element) and cell-cell junction (in red). All structures (plate and shell) considered in this section entirely consist of such cell groups. In all tissues considered below, the height of a cell and , in which is its side and is the diameter of actin fibres. The ratio of Young‘s moduli of a cell and actin fibres is , and of actin fibres and cell-cell links .
initially flat rectangular plate (Figure 7 (left)),
cylindrical shell (Figure 9 (left)),
spherical shell (Figure 11 (left)).
Elements of the middle part of the rectangular tissue are compressed in apical sides to imitate apical constriction in cells. Increasing the compressing stresses, the tissue is bent and a blaster-shaped pattern is formed as shown in Figure 7 (right). The quantitative picture of the stresses arising in the tissue is drawn in Figure 8.
Next, we consider a cylindrical shell to imitate ventral furrow (generally all tubular patterns) formation. Elements of the top part of the cylinder are constrained in the apical sides by compressing the links standing for apical fibres (see Figure 9). Increasing the compressing stresses, the ventral furrow formation is simulated similar to stages presented in Figure 1. The quantitative picture of the stresses arising in the tissue is drawn in Figure 10.
Finally, a spherical shell is simulated. Cells at the top of a semi-sphere are constrained in apical sides and depending on values of compressing stresses various stages of blastopore formation in archenteron can be described (see Figure 11 (right)). The quantitative picture of the stresses arising in the tissue is drawn in Figure 12.
Analysis based on lower semi-continuous energy functionals reveals real three-dimensional deformations of soft epithelial tissues. Having different forms for different deformation regimes, such as compressive stresses (short deformations), uniaxial tensions collinear to principal directions of the first fundamental form, no stretching or pure bending stresses (isometric deformations) and stretching stresses (large deformations), the resulting total energy is lower semi-continuous, so the existence of its minimizers, that is, real deformations, is ensured. Particular energy density functions are chosen and the explicit form of the total energy functional is obtained, thereby the discretization is made easy.
On the basis of obtained energy functional, a three-dimensional discretized model of epithelial tissues undergoing combined stretching and bending deformations is constructed. Discretization elements correspond to single cells forming the tissue. Actin fibres and cell-cell adhesion links, mainly contributing on the tissue energy, are explicitly embedded in elements. Deformations characteristic to specific embryonic tissues (ventral furrow, neutral tube, neurosphere) observed earlier are described quantitatively increasing contractile stresses in fibres.
We thankfully dedicate the chapter to the blessed memory of our good fellow and colleague, a candidate of physical and mathematical sciences, Hamlet V. Hovhannisyan (1956–2016), who unexpectedly died before he could realize his best scientific ideas.
The theoretical part of the chapter is investigated under the guidance of Doctor, Professor Benedikt Wirth, Institute for Computational and Applied Mathematics, University of Münster, whom we are heartily thankful. The work of As. Kh. and S. O. was made possible in part by a research grant from the Armenian National Science and Education Fund (ANSEF) based in New York, NY, USA.
|Inner (dot) product|
|Rescaled three-dimensional total energy|
|Measure of bounded open set|
|Gradient (nabla) operator|
|Midsurface of the epithelium|
|Rank of the matrix |
|dist||Usual distance in three-dimensional Euclidean space|
|Diameter of actin fibres|
|Three-dimensional elastic energy|
|Rescaled three-dimensional elastic energy|
|Young’s modulus of actin fibres|
|Young’s modulus of a cell|
|Young’s modulus of cell-cell links|
|Thickness of the epithelium or a single cell|
|Side of a single cell|
|Membrane energy density function|
|The quasi-convex envelope of |
|The second fundamental form associated with deformation |
|The first fundamental form associated with deformation |
|Unit normal vector|
|Deformation acting from to|
|SO(3)||The group of all rotations about the origin of three-dimensional Euclidean space|