Apico-basal intercalations enable the integrity of curved epithelia

Abstract

Non-invasive force inference based on imaging data has significantly advanced our understanding of the mechanical cues driving morphogenesis. In 2D studies of confluent tissues, these methods allow for the computation of forces acting on cells by analyzing their geometrical features. Here, we present a novel approach for 3D force and energy inference in curved epithelia. Specifically, we focus on tubular epithelia, which form the foundation of many vital organs, including the lungs, kidneys, and vasculature. Our technique analyzes the average mechanical behavior of cells along their apico-basal axis and is based on an optimal parametrization of a vertex model aimed at obtaining effective tissue parameters. We apply our method to in silico data to investigate the mechanical consequences of different 3D cellular packing scenarios. Our results reveal that in squamous epithelia, prismatic cellular shapes are mechanically stable. However, in cubic/columnar tubes, prismatic shapes are incompatible with the adhesion required to maintain tissue integrity. In conclusion, this study indicates that in cubic/columnar epithelia, stability can only be achieved if cells undergo apico-basal intercalations and adopt an alternative shape: the scutoid.

Graphical abstract

1. Introduction

During animal development, multiple factors such as cell-cell interactions, cell proliferation, cell shape changes, and biophysical forces contribute to organ formation [1], [2], [3]. In confluent tissues such as epithelia, these processes determine cell packing properties, and their analysis has significantly advanced our understanding of morphogenesis [4], [5]. Traditionally, most studies have been conducted in two dimensions (2D) by analyzing the apical surfaces of monolayer epithelia. However, recent research has underscored the need to investigate 3D cellular packing and morphology [6]. In this context, epithelial cells are typically classified by their width-to-height aspect ratio as squamous (“flat”), cuboidal, or columnar (“tall”) cells [7], [8]. Notably, the role of different 3D cell geometries, in conjunction with their packing properties, in maintaining tissue integrity remains insufficiently explored.

The importance of accurately analyzing 3D cell-cell contacts has been highlighted by recent studies on various biological processes, including the growth of mouse embryonic lung explants [9], the early development of C. elegans [10] and ascidians [11], and the cellular and mechanical basis of self-organization in curved tissues within a confined geometry [12]. A major breakthrough in understanding epithelial 3D packing and morphogenesis came with the discovery of a novel geometrical shape, the scutoid, which plays a key role in epithelial morphogenesis [6], [13], [14], [15], [16], [17], [18], [19], [20]. Scutoids, characterized by apico-basal intercalations of cells, facilitate epithelial packing when tissues are subjected to curvature [14] and are considered a general biophysical phenomenon [21]. Additionally, scutoids have been shown to modulate local pressure increases due to cell division [22]. This cell shape has been identified in epithelial tissues across various metazoans, including mammals [6], [19], [22]. However, key open questions remain: (1) how do apico-basal intercalations contribute to the mechanical stability and integrity of an epithelium? and (2) can a curved tissue maintain its integrity without scutoids? Experimentally addressing these questions by removing or modifying apico-basal intercalations is currently unfeasible. Thus, a computational approach is required to infer mechanical forces in different 3D cellular packing configurations. Specifically, a methodology is needed to first simulate epithelial tissues subjected to curvature and composed of cuboidal/columnar cells with either prismatic or scutoidal packing configurations, and second, to develop a force/energy inference technique to analyze and parameterize the results of these simulations.

A wide range of experimental techniques are available to measure or estimate forces in cells and tissues, including laser ablation [23], [24], [25], [26], [27], functionalized droplets [28], [29], [30], optical and magnetic tweezers [31], [32], [33], [34], molecular sensors [35], and traction microscopy [36], among others [37]. Alternatively, in epithelial monolayers, inference methods provide non-invasive means of estimating forces using imaging data [38], [39], [40], [41], [42], [43], [44], [45], [46]. These techniques leverage the fact that the apical surfaces of cells in these tissues exhibit polygonal-like shapes, allowing their geometrical features to be correlated with mechanical equilibrium conditions. This polygonal-like packing characteristic led to the development of the vertex model, which aims to describe the forces acting on epithelial cells in both 2D and 3D environments [47], [48], [49], [50], [51]. In a 3D context, only recently have some studies attempted to quantify forces (tensions and pressures) using Young's formula [52], [53]. Notably, to date, no force inference study has evaluated the impact of epithelial cell shape on the mechanical stability of tissues.

Here, we implement a novel approach to 3D force inference to examine the role of cellular shapes and their packing in epithelial tissues. Conventional inference methods rely on force balance without incorporating assumptions about cell mechanics, such as elasticity or adhesion. In contrast, our approach assumes the validity of the vertex model's description of cellular mechanics and infers the parameter values that satisfy force balance. Furthermore, we employ a statistical method that quantifies the average cell behavior in terms of energy and force components, providing insights into the relevance of 3D cellular shapes. Specifically, we aim to determine the biophysical parameters that best describe the mechanical properties and stability of tissues depending on packing configurations. We test our inference method on simulated tubular tissues. These epithelia form the foundation of many vital organs, including the lungs, kidneys, and vasculature, and have been shown to exhibit the highest prevalence of scutoidal cells due to curvature effects compared to other epithelial geometries [14] (see Discussion). We generated tubular epithelia through a computational geometry approach that produces two possible cellular shapes: scutoidal shapes, which exhibit apico-basal intercalations, and prismatic-like shapes, where apico-basal intercalations are absent. Additionally, we compare these two scenarios in different tissue types by employing computational tubular models composed of either squamous or cuboidal/columnar cells. The biophysical parametrization of these models reveals that tissue integrity depends on a balance between packing architecture and the width-to-height aspect ratio of cells. Our findings suggest that the adhesive properties characteristic of epithelia require the presence of apico-basal intercalations to stabilize tubular structures composed of cuboidal/columnar cells. In other words, the physical properties of epithelial tissues prevent the existence of cuboidal/columnar epithelial tubes composed solely of prismatic-like shapes.

2. Methods

2.1. Vertex model: force balance

Our force inference methodology is based on a parametrization of the vertex model [54] and assumes the standard energy functional for each of the vertexes, i, that define the polygonal-like shape of a cell at a given planar surface:

$$E_i=\sum _{\alpha }\frac{K_{\alpha }}{2}{(A_{\alpha }-A_{\alpha }^0\left(t\right))}^2+\sum _{\alpha }\frac{{\mathrm{\Gamma }}_{\alpha }}{2}{L}_{\alpha }^2+\sum _{\langle ij\rangle }{\mathrm{\Lambda }}_{ij}{l}_{ij}=E_i^{\text{elastic}}+E_i^{\text{contract.}}+E_i^{\text{adh.}}$$ (1)

where the sums indexed by α and $\langle ij\rangle$ run, respectively, over the cells, α, and the vertexes, j, that share vertex i. The first term, $E_i^{\text{elastic}}={\sum }_{\alpha }\frac{K_{\alpha }}{2}{(A_{\alpha }-A_{\alpha }^0\left(t\right))}^2$ accounts for the elastic energy of cells ($K_{\alpha }$ being proportional to the Young modulus) due to the difference between the actual cell area, $A_{\alpha }$, and the target area $A_{\alpha }^0\left(t\right)$. The second term, $E_i^{\text{contract.}}={\sum }_{\alpha }\frac{{\mathrm{\Gamma }}_{\alpha }}{2}{L}_{\alpha }^2$, models contributions from the tension associated to the contraction activity of the actomyosin cortical ring, $L_{\alpha }$ being the cell perimeter. Finally, the third term, $E_i^{\text{adh.}}={\sum }_{\langle ij\rangle }{\mathrm{\Lambda }}_{ij}{l}_{ij}$, represents the line tension (adhesion force), with $l_{ij}$ being the length of the edge connecting neighboring vertexes i and j.

By neglecting inertia (low Reynolds number) and including dissipation, the following force balance equation holds,

$$\mathbf{0}=-\mathrm{\nabla }{E}_i-\gamma {\dot{\mathbf{r}}}_i$$

where ${\mathbf{r}}_i$ is the position vector of vertex i and γ is the coefficient that determines the characteristic time scale linked to the dissipation of the mechanical energy, $t_c=\gamma /\left(K_{\alpha }{A}_{\alpha }^0\right)$. Further, if the characteristic time scale of cell area changes, either due to growth or mechanical inputs, is slow compared to $t_c$ (or equivalently if the tissue is at a steady state), the balance of the conservative forces at each cell vertex determines the equilibrium condition:

$$\mathbf{0}\simeq -\mathrm{\nabla }{E}_i={\mathbf{F}}_i^{\text{elastic}}+{\mathbf{F}}_i^{\text{contract.}}+{\mathbf{F}}_i^{\text{adh.}}$$ (2)

In the Euclidian plane, the area, $A_{\alpha }$, and the perimeter, $L_{\alpha }$, of a polygon (i.e., cell α) with n clockwise-ordered vertexes are given by:

$$A_{\alpha }=-\frac{1}{2}\sum _{k=1}^n(x_k^{\alpha }{y}_{k+1}^{\alpha }-x_{k+1}^{\alpha }{y}_k^{\alpha })L_{\alpha }=\sum _{k=1}^n{l}_{k,k+1}^{\alpha }=\sum _{k=1}^n\sqrt{{(x_{k+1}^{\alpha }-x_k^{\alpha })}^2+{(y_{k+1}^{\alpha }-y_k^{\alpha })}^2}$$

where ${\mathbf{r}}_i^{\alpha }=(x_i^{\alpha },y_i^{\alpha })$ represents the Cartesian coordinates of vertex i of cell α, and the following periodic boundary conditions for the polygon that describe a cell apply: ${\mathbf{r}}_{n+1}^{\alpha }={\mathbf{r}}_1^{\alpha }$ and ${\mathbf{r}}_0^{\alpha }={\mathbf{r}}_n^{\alpha }$. Thus, the different force terms in Eq. (2) read,

$${\mathbf{F}}_i^{\text{elastic}}=-\frac{\partial }{\partial {\mathbf{r}}_i}\sum _{\alpha }\frac{K_{\alpha }}{2}{(A_{\alpha }-A_{\alpha }^0\left(t\right))}^2=\frac{1}{2}\sum _{\alpha }{K}_{\alpha }(A_{\alpha }-A_{\alpha }^0\left(t\right))(y_{i+1}^{\alpha }-y_{i-1}^{\alpha },x_{i-1}^{\alpha }-x_{i+1}^{\alpha }){\mathbf{F}}_i^{\text{contract.}}=-\frac{\partial }{\partial {\mathbf{r}}_i}\sum _{\alpha }\frac{{\mathrm{\Gamma }}_{\alpha }}{2}{L}_{\alpha }^2=-\sum _{\alpha }{\mathrm{\Gamma }}_{\alpha }{L}_{\alpha }(\frac{x_i^{\alpha }-x_{i-1}^{\alpha }}{l_{i-1,i}^{\alpha }}+\frac{x_i^{\alpha }-x_{i+1}^{\alpha }}{l_{i,i+1}^{\alpha }},\frac{y_i^{\alpha }-y_{i-1}^{\alpha }}{l_{i-1,i}^{\alpha }}+\frac{y_i^{\alpha }-y_{i+1}^{\alpha }}{l_{i,i+1}^{\alpha }}){\mathbf{F}}_i^{\text{adh.}}=-\frac{\partial }{\partial {\mathbf{r}}_i}\sum _{\langle ij\rangle }{\mathrm{\Lambda }}_{ij}{l}_{ij}=-\sum _{\langle ij\rangle }{\mathrm{\Lambda }}_{ij}(\frac{x_j-x_i}{l_{ij}},\frac{y_j-y_i}{l_{ij}})$$

2.2. Geometrical decomposition of forces: estimation of normal and shear stresses

In order to provide an intuitive interpretation to the directionality of the forces, we implement a geometrical decomposition that estimates normal and shear stresses. Since forces in a vertex model are applied at cell vertexes, the normal and perpendicular directions are not well-defined. To address this, we define mock normal and shear forces based on the angle formed by the edges adjacent to a vertex (Fig. 1A). For a given force type applied to a vertex i, ${\mathbf{F}}_i=(F_{i,x},F_{i,y})$, we compute the normal and shear components, ${\mathbf{F}}_i^{\prime }=(F_{i,\text{n}},F_{i,\text{s}})$ by using the geometric rotational transformation ${\mathbf{F}}_i^{\prime }=R_{\theta }{\mathbf{F}}_i$ where θ is defined as a function of the bisector of the angle formed by the edges $({\mathbf{r}}_{i+1}^{\alpha }-{\mathbf{r}}_i^{\alpha })$ and $({\mathbf{r}}_{i-1}^{\alpha }-{\mathbf{r}}_i^{\alpha })$ of vertex i:

$$\left(\begin{array}{c}{F}_{i,\text{n}}\\ F_{i,\text{s}}\end{array}\right)=\left[\begin{array}{cc}\hfill cos\left(\theta \right)\hfill & \hfill sin\left(\theta \right)\hfill \\ \hfill -sin\left(\theta \right)\hfill & \hfill cos\left(\theta \right)\hfill \end{array}\right]\left(\begin{array}{c}{F}_{i,x}\\ F_{i,y}\end{array}\right)$$ (3)

Finally, the magnitude of the normal/shear forces acting on each cell is calculated by averaging the magnitudes of the normal/shear forces applied to all its vertexes, M, such that $\overline{F_{\text{n}}}=\frac{1}{M}{\sum }_{i=1}^M{F}_{i,\text{n}}$ and $\overline{F_{\text{s}}}=\frac{1}{M}{\sum }_{i=1}^M{F}_{i,\text{s}}$. The average cellular normal and shear forces are then computed by averaging over all cells, N: $\langle F_n\rangle =\frac{1}{N}\sum \overline{F_{\text{n}}}$ and $\langle F_s\rangle =\frac{1}{N}\sum \overline{F_s}$.

Fig. 1.

A: Normal and shear forces. To estimate the normal and shear components of a force F_i acting upon vertex i, we implement a geometrical decomposition that defines the normal direction along the bisector of the angle α formed by the adjacent edges r_i+1 − r_i and r_i−1 − r_i (a rotation of an angle θ with respect to the external coordinate system). B: Simulation of Voronoi and frusta tubes. (Left) Side view of a tube. From the apical (inner) radius, R_a, to the basal (outer) radius, R_b, the simulation approach is based on the radial projection of geometrical features of the cells (arrow). (Right) In Voronoi tubes, the cell seeds are projected, while in frusta tubes, the cell vertexes are projected (see text). C: Simulation Examples. In our in silico experiments we consider tubes that mimic squamous (s_b ≃ 1.5) and cuboidal/columnar (s_b = 4) cells that pack following either a frusta-like (Left) or a scutoidal-like (Right) geometry. The green and red circles indicate the apical (inner) and basal (outer) radii, respectively. The shapes and packing of highlighted cells (white contours in tubes) are shown at different apico-basal coordinates (from bottom to top, s = 1,2,4). Since each realization of frusta and Voronoi tubes was generated from the same initial cell distribution at the apical surface, s = 1 (apical surface), the cells have identical shapes in both cases. Notice that in Voronoi tubes, at the apical surface, the red and yellow cells are nearest neighbors while the green and blue cells are not, whereas at the basal surface, the nearest-neighbor relationship is reversed (blue and green cells become nearest neighbors while red and yellow are not). In frusta tubes, the neighboring relationships among cells remain unchanged along the apico-basal axis as intercalations are precluded. D: Quantification of cellular packing. 3D histograms of cells neighbors, $P(n_a,n_b)$, for Voronoi (s_b = 1.5, Center; s_b = 4, Right) and frusta (Left) tubes, sample size of 10 tubes. The absence of apico-basal intercalations in frusta tubes results in identical polygonal distributions at the apical and basal surfaces. In Voronoi tubes, as the surface ratio increases, the distribution widens, indicating an increase in the number of apico-basal intercalations (scutoid-like cell shapes). E: Tubular geometry: force inference. Given an epithelial tube of length L and apical and basal radii R_a and R_b, our tomographic approach to force inference assumes equilibrium is achieved at each surface of constant radius between R_a to R_b. F: Elastic null-stress plane. In a planar epithelial monolayer cells acquire a prism-like shape with a characteristic area in each plane from apical to basal A₀ (grey mesh). When the tissue is subjected to bending (tubular geometry) the cells deform and acquire a frustum-like shape. Cell volume conservation implies that cells are under compression in the apical plane and under tension in the basal plane. The elastic null-stress plane is defined by the value of the apico-basal coordinate for which the area in that plane is A₀.

2.3. Simulations of Voronoi and frusta tubular models

An in-house Matlab code was used to model tubular epithelia following two different approaches that lead to distinct cellular packing configurations: Voronoi and frusta tubes. Voronoi tubes were generated following a similar approach as described in [14]. The inner (apical) surface of the tubes (i.e., hollow cylinders) was populated with a number of seeds (i.e., points) randomly distributed. Subsequently, the Voronoi algorithm was applied to the seeds to generate the corresponding Voronoi cells. To mimic the actual apical organization of epithelial tubes (see [6], [15]), the Lloyd algorithm was iteratively applied 7 times [55]. The Lloyd algorithm updates the location of the Voronoi seeds by relocating them to the center of mass of each Voronoi cell; subsequently, the Voronoi algorithm is reapplied to the updated Voronoi seeds, resulting in a more homogeneous Voronoi tessellation in terms of cellular size.

Once the 2D Voronoi cells of the apical surface were generated, the 3D cellular shape was built as follows. The outer (basal) surface of the tubes was generated with a radius $R_b=s_b{R}_a$ and the Voronoi cell seeds were radially projected from the inner to the outer surface (Fig. 1B). At each intermediate radius, $R\in [R_a,R_b]$, the intersection of the radial projections with the radial surface (surface with constant radius) defines the location of the Voronoi seeds, and the resulting Voronoi tessellation renders the shape of the cells at that surface. For a given seed, the set of 2D Voronoi cells across all radial surfaces, from $R_a$ to $R_b$, defines the 3D cellular shape (Fig. 1B).

For frusta tubes, we proceed as previously described to define the apical shape of the cells. However, instead of projecting the cell seeds, we projected the cell vertexes (Fig. 1B). Thus, the cell shape of a given cell in a particular radial surface is described by the intersection of the projected vertexes. To create the 3D shape of an individual cell, we proceed as previously described: combining the set of 2D cells across all radial surfaces, from $R_a$ to $R_b$. Fig. 1C shows examples of tubular epithelia simulated using these alternative approaches. Our results are typically based on the analysis of 10 realizations of each tubular configuration with $N=200$ cells and the following properties: $s_b=R_b/R_a=1.46875\simeq 1.5$ (referred to as squamous cells), $s_b=4$ (referred to as cuboidal/columnar cells), see also Table 1.

Table 2.

Parameter inference: finite size effects. Ground truth and estimated vertex model parameters values as a function of the ratio ρ = N_p/N.

Parameter	Ground Truth	ρ = 0.7	ρ = 0.57	ρ = 0.51
		(N = 20, Np = 14)	(N = 30, Np = 17)	(N = 45, Np = 23)
$A_0$	1.000	1.023 (δ = 2.3%)	1.015 (δ = 1.5%)	1.001 (δ = 0.1%)
Λ	0.040	0.043 (δ = 5.5%)	0.042 (δ = 5%)	0.041 (δ = 2.5%)
Γ	0.02	0.022 (δ = 10%)	0.021 (δ = 5%)	0.020 (δ = 0%)

Table 1.

Cellular packing properties in frusta and Voronoi tubes. Average number of neighbors (in-plane polygonal order, 〈n〉), standard deviation (σ_n), and coefficient of variation (CV) in the apical and basal surfaces of frusta and Voronoi tubes. Apico-basal intercalations lead to greater disorder in the polygonal distribution of cuboidal/columnar cells.

	Frusta tubes	Voronoi tubes
	Apical /Basal	Apical (s = 1)	Basal (s ≃ 1.5)	Basal (s = 4)
$\langle n\rangle$	6.0	6.0	6.0	6.0
σn	0.8	0.8	0.8	1
$CV=\frac{{\sigma }_n}{\langle n\rangle }$	0.13	0.13	0.13	0.17

The existence and effect of apico-basal intercalations in Voronoi tubes is shown by the 3D histogram of cell neighbors in Fig. 1D, i.e., the probability $P(n_a,n_b)$ of cells having $n_a$ neighbors in the apical surface and $n_b$ neighbors in the basal surface [15]. In frusta tubes, $P(n_a,n_b)=P\left(n_a\right){\delta }_{n_b,n_a}=P\left(n_b\right){\delta }_{n_a,n_b}$ (${\delta }_{i,j}$ being the Kronecker delta). That is, the histogram only contains entries along the diagonal since there are no changes in the nearest-neighbor relationship from apical to basal. However, in Voronoi tubes, the entries of the 3D histogram outside the diagonal indicate apico-basal intercalations: some cells either gained or lost neighbors between the apical and basal surfaces.

2.4. Dimensionless units

In terms of the characteristic length scale based on the cellular size, $l_c=\sqrt{\langle \mathcal{A}{\alpha }^a\rangle }$, where $\langle \mathcal{A}{\alpha }^a\rangle$ is the average area of cells at the apical surface ($\langle \mathcal{A}{\alpha }^a\rangle =\langle \mathcal{A}{\alpha }^0\rangle$ when considering only 2D cellular packing, as in control simulations; see Appendix), and the characteristic relaxation/dissipation time of the mechanical energy, $t_c=\gamma /(K_{\alpha }\langle {\mathcal{A}}_{\alpha }^a\rangle )$, we implement a dimensionless form of Eq. (1).

Using these definitions, the dimensionless elastic parameter is set as $\widetilde{K_{\alpha }}=1$, while the adhesion and contractility parameters are given by $\tilde{\mathrm{\Lambda }}ij=\mathrm{\Lambda }ij/(K_{\alpha }{\langle \mathcal{A}{\alpha }^a\rangle }^{3/2})$ and $\tilde{\mathrm{\Gamma }}\alpha ={\mathrm{\Gamma }}_{\alpha }/(K_{\alpha }\langle {\mathcal{A}}_{\alpha }^a\rangle )$, respectively.

Furthermore, when applying the tomographic approach toward force inference (see below), the dimensionless target cell area in the elastic null-stress surface (see below) is given by $\widetilde{\mathcal{A}{\alpha }^0}=\mathcal{A}{\alpha }^0/\langle \mathcal{A}{\alpha }^a\rangle$. Thus, we perform an additional change of units such that $\widetilde{\tilde{\mathcal{A}{\alpha }^0}}=1$ (i.e., $l_c^{\prime }=\sqrt{\widetilde{\mathcal{A}{\alpha }^0}}$), leading to $\widetilde{\tilde{K\alpha }}=1$, $\widetilde{\tilde{\mathrm{\Lambda }ij}}=\tilde{\mathrm{\Lambda }}ij/({\widetilde{A_{\alpha }^0}}^{3/2})$, and $\widetilde{\tilde{\mathrm{\Gamma }\alpha }}=\tilde{\mathrm{\Gamma }}\alpha /\widetilde{A_{\alpha }^0}$.

For the sake of simplicity, we assume that all quantities are expressed in dimensionless units and omit the single and double over-hat symbols.

2.5. Linear programming: tomographic force inference

Disregarding boundary effects, in a tissue comprising N cells, the equilibrium condition, Eq. (2), provides $2\times 2N$ scalar equations (on average, each cell has six vertexes shared by three cells) at any given time. Regarding the number of unknowns, in a tissue composed of cells of the same type (i.e., with supposedly identical mechanical properties), the number is just four ($A^0$, K, Γ, and Λ). Moreover, these parameters can be reduced to three (see Dimensionless units above), making the system of equations in Eq. (2) overdetermined and solvable.

Given that the resulting force balance equations described by Eq. (2) are linear in the unknowns (i.e., the parameters Λ, Γ, etc.), we solve the overdetermined system using a linear programming optimization approach and employ an $l_1$-minimization method [56]. The $l_1$-minimization aims to find the minimum $l_1$-norm solution of a linear system $\mathbb{A}\mathbf{X}=\mathbb{B}$, where $\mathbf{X}\in {\mathbb{R}}^m$ (i.e., m unknowns), $\mathbb{B}\in {\mathbb{R}}^p$ ($m<p$, where p represents the total number of vertexes), and $\mathbb{A}\in {\mathbb{R}}^{p\times m}$ [57]. In our study, we implement the “L1-Norm Minimization” function and the “simplex” algorithm in Wolfram Mathematica [58] to determine the optimal values of the unknowns.

The tomographic force inference approach for 3D tubular tissues (Fig. 1E) assumes equilibrium at every value of the apico-basal coordinate (i.e., radial surface) and solves Eq. (2) simultaneously for all surfaces, from $R_a$ to $R_b$. The adhesion parameter (energy per unit length) and $A_0$ (target cell area in the elastic null-stress plane, see below) are assumed to be the same for all values along the apico-basal coordinate, while the contractility parameter, Γ, is allowed to vary along the apico-basal axis.

Thus, in a 3D context, the number of equations and unknowns are given by $2\times 2N\times N_{\text{surfaces}}$ and $2+N_{\text{surfaces}}$ (i.e., $A^0+\mathrm{\Lambda }+N_{\text{surfaces}}\times \mathrm{\Gamma }$), respectively, where $N_{\text{surfaces}}$ is the number of apico-basal surfaces (i.e., radial “slices”) considered.

In the analyses presented in this study for squamous and cuboidal/columnar tissues, $N_{\text{surfaces}}$ is 6 and 7, respectively. As shown in the Appendix, our control simulations indicate that the results remain robust for different values of $N_{\text{surfaces}}$.

2.6. Elastic null-stress plane: energy as a function of the apico-basal coordinate

Given an epithelial monolayer with N cells and tissue depth (i.e., cell height) h, if the tissue is in a planar configuration (lacking tension-compression stresses due to tissue bending/curvature), then the average cell volume is $\langle V\rangle =\langle A_0\rangle h$, where $\langle A_0\rangle$ is the target area (Fig. 1F). On the other hand, if the same tissue is shaped into a tubular configuration, with length L and apical and basal radii $R_a$ and $R_b$, respectively, then the average cell volume is given by

$$\langle V\rangle =\frac{\pi L}{N}(R_b^2-R_a^2).$$

If the cell height remains constant, $R_b-R_a=h$, and the cell volume is conserved, it follows that

$$\langle A_0\rangle =\frac{\pi L(R_b+R_a)}{N}=\frac{R_a\pi L(s_b+1)}{N}=\frac{1}{2}\langle A_a\rangle (s_b+1),$$

where $\langle A_a\rangle$ is the cell apical area, and we define the dimensionless apico-basal coordinate $s=R/R_a$, referred to as the surface ratio ($s_b=R_b/R_a$).

The average elastic energy for a given surface ratio reads:

$${\langle E\rangle }_A=\frac{K}{2}\langle {(A-A_0)}^2\rangle =E_{\langle A\rangle }+\frac{K}{2}{\sigma }_A^2,$$

where $E_{\langle A\rangle }=\frac{K}{2}{(\langle A\rangle -\langle A_0\rangle )}^2$ is the elastic energy of a cell with average area $\langle A\rangle$, and ${\sigma }_A^2=\langle A^2\rangle -{\langle A\rangle }^2$ is the cellular area variance. Cell number conservation implies that for a given surface ratio:

$$\langle A\rangle =\frac{2\pi RL}{N}=\langle A_a\rangle s.$$

Consequently,

$$E_{\langle A\rangle }=\frac{K{\langle A_a\rangle }^2}{2}{(s-\frac{1}{2}(s_b+1))}^2.$$

The elastic null-stress plane, where $E_{\langle A\rangle }=0$, is located at:

$$s^{⁎}=\frac{1}{2}(s_b+1)=\frac{\langle A_0\rangle }{\langle A_a\rangle }.$$

From the perspective of planar elastic deformation, if $1\le s<s^{⁎}$, cells are under compression, whereas if $s_b\ge s>s^{⁎}$, cells are under tension. Furthermore, since $\langle L\rangle \sim {\langle A\rangle }^{1/2}={\left(\langle A_a\rangle s\right)}^{1/2}$, the average adhesion and contractile energies, ${\langle E\rangle }_L$ and ${\langle E\rangle }_{L^2}$, respectively, scale as a function of s as:

$${\langle E\rangle }_L=E_{\langle L\rangle }=\mathrm{\Lambda }\langle L\rangle \sim \mathrm{\Lambda }\langle A_a\rangle s^{1/2},$$

$${\langle E\rangle }_{L^2}=\frac{\mathrm{\Gamma }}{2}\langle L^2\rangle =E_{\langle L^2\rangle }+\frac{\mathrm{\Gamma }}{2}{\sigma }_L^2\sim \frac{\mathrm{\Gamma }}{2}s.$$

3. Results

3.1. Force inference reveals mechanical differences between cell geometries

We developed and implemented a force inference approach based on a vertex model parametrization (Methods). Using control simulations, we showed that the proposed inference methodology is robust to finite-size and boundary effects and is also capable of capturing time-dependent mechanical parameters accurately as long as their temporal variation is slow compared to the energy dissipation time scale (Appendix). We leveraged this latter fact to implement a tomographic approach (i.e., plane by plane from the apical to the basal surface) for 3D force inference by exchanging the concepts of space and time. This tool was applied to investigate the mechanical characterization of epithelial tubes based on their cellular packing organization and morphological properties (Methods). Specifically, we simulated tubes composed of either cuboidal/columnar or squamous epithelial cells and varied their packing such that apico-basal intercalations were either allowed (scutoidal cell geometries) or precluded (frusta cell geometries).

We used dimensionless units such that $A_0=1$ at the null-stress plane ($s^{⁎}=(s_b+1)/2$) and assumed that the parameter Λ (energy “cost” per unit length) remains constant along the apico-basal coordinate, s (Methods). However, we allowed the cell contractility parameter to vary along the apico-basal axis, i.e., $\mathrm{\Gamma }(s)$. This assumes that while cells adhere to each other with the same “intensity,” cortical activity may change as a function of the apico-basal coordinate [59]. Fig. 2 shows the estimated contractility parameter as a function of the apico-basal coordinate, s, for frusta and Voronoi tubes using either squamous or cuboidal/columnar cells.

Fig. 2.

Contractility in frusta and Voronoi tubes depending on the cellular packing geometries. Contractility parameter as a function of the surface ratio (apico-basal coordinate) as estimated by the inference method for frusta (cyan) and Voronoi (yellow) geometries in squamous (left) and cuboidal/columnar (right) cells. The error band accounts for the standard deviation (10 samples).

For a given tubular configuration (either squamous or cuboidal/columnar), both cellular geometries (frusta or Voronoi) yielded similar contractility values and trends: Γ decreases as s increases (i.e., from apical to basal) and approaches zero at the basal surface. However, differences emerged between tubular configurations, as cuboidal/columnar cells exhibited higher contractility values. In this regard, we note that the value of $\langle \widetilde{A_{\alpha }^0}\rangle$ differs between squamous and cuboidal/columnar cells (see Methods, Dimensionless units):

$$\frac{{\langle \widetilde{A_{\alpha }^0}\rangle |}_{s_b=4}}{{\langle \widetilde{A_{\alpha }^0}\rangle |}_{s_b=1.5}}\simeq 2.$$

Consequently, using the same dimensionless units for cuboidal/columnar and squamous cells, the maximum contractility parameter (apical surface) is approximately four times larger in cuboidal/columnar cells than in squamous cells.

Regarding the line tension (adhesiveness), squamous cells displayed similar values for Voronoi and frusta geometries:

$${\mathrm{\Lambda }}_{\text{Voronoi}}=0.04\pm 0.01,{\mathrm{\Lambda }}_{\text{frusta}}=0.03\pm 0.01.$$

However, cuboidal/columnar cells showed large differences between packing shapes:

$${\mathrm{\Lambda }}_{\text{Voronoi}}=0.023\pm 0.009,{\mathrm{\Lambda }}_{\text{frusta}}=0.0023\pm 0.002.$$

For the same dimensionless units, the adhesion parameter in Voronoi cuboidal/columnar cells is actually larger than in squamous cells (by a factor of ∼1.85), whereas frusta cuboidal/columnar cells exhibited significantly lower adhesion values (∼0.185 of the value observed in squamous cells).

In summary, the inference method revealed that cortical activity, as characterized by the contractility parameter, is similar in frusta and Voronoi tubes but significantly greater in cuboidal/columnar cells compared to squamous cells. In contrast, for the line tension parameter, scutoid-free (frusta) cuboidal/columnar cells exhibited mechanical stability that depended on extremely low adhesion values —an order of magnitude smaller than either Voronoi or frusta cells in squamous tubes or Voronoi cells in cuboidal/columnar tubes.

3.2. Energy profiles reveal the role played by apico-basal intercalations in cuboidal/columnar epithelia

Once the mechanical parameters of the tubular models were calibrated, we computed the average cellular energy profiles (Methods), Fig. 3. These profiles provide a “map” from the apical to the basal surface, representing the characteristic values of different energy components and illustrating the effect of apico-basal intercalations (cellular geometry) in squamous (Fig. 3A-B) and cuboidal/columnar cells (Fig. 3C-D).

Fig. 3.

Energy inference in squamous and cuboidal/columnar tubular models depending on the cell packing geometry. A-B: Average energy per cell as a function of the surface ratio (apico-basal coordinate) (Left) and individual cell energies at the apical, null-stress, and basal surfaces (Right) in representative frusta (A) and Voronoi (B) squamous tubes sharing the same packing configuration at the apical surface. In the left panels, the error bands correspond to the standard deviation (10 samples), and the green, yellow, and red triangles indicate the values of s at the apical, null-stress, and basal planes, respectively. Color scales in the right panels range from minimum (white) to maximum values. C-D: Same information as in A-B panels for tubes formed by cuboidal/columnar cells. Notice that in frusta tubes, the lack of apico-basal intercalations forces cells to stretch.

For squamous cells, we obtained similar energy profiles for frusta and Voronoi geometries. That is, in squamous epithelia, modifying the cellular geometry from frusta to Voronoi —and consequently altering cellular connectivity [15]— does not confer an energetic advantage. We propose that this could explain why the scutoidal shape has not yet been reported in squamous cells (Discussion). Additionally, the functional behavior of the different energy components aligns with the theoretical expectations (Methods). In particular, the elastic energy reaches a minimum at the apico-basal coordinate:

$$s^{⁎}=\frac{1}{2}(s_b+1)=\frac{\langle A_0\rangle }{\langle A_a\rangle },$$

which defines the elastic null-stress plane.

For the cuboidal/columnar tubular model, the functional behavior of the energy components as a function of s is also in agreement with the theoretical expectations. However, we observed significant quantitative and qualitative differences between packing geometries, originating from the dissimilar values of the line-tension parameter. In frusta geometries, elastic energy dominates from the apical to the basal surface, whereas in Voronoi cells, adhesion becomes dominant around the elastic null-stress plane. Additionally, regarding total energy, the average cellular energy is higher in Voronoi than in frusta tubes. However, this observation must be considered in the context of tissue structural stability. The low adhesiveness required to balance mechanical forces in frusta-columnar tubes is arguably incompatible with the functionality of real tissues (Discussion).

3.3. Apico-basal intercalations buffer shear stresses in cuboidal/columnar epithelia

We further analyzed the average cell force profile to elucidate how apico-basal intercalations shape the stability of the tubes. To that end, we computed the different force terms exerted by the cells and performed a decomposition into normal and shear stresses (Methods), Fig. 4. Normal forces exerted by cells exhibited two regimes as a function of the apico-basal coordinate, s: either an expansive ($\langle F_n\rangle >0$) or compressive ($\langle F_n\rangle <0$) behavior. As expected, the active cellular contractile force is always compressive (i.e., pointing inward, $\langle F_n\rangle <0$) and is more pronounced at the apical surface. Additionally, since the line-tension parameter is positive, $\mathrm{\Lambda }>0$, the adhesion force favors cellular compression throughout the apico-basal axis.

Fig. 4.

Force inference in squamous and cuboidal/columnar tubular models depending on the cell packing geometry. A/B: Average normal (top) and shear (middle) forces per cell as a function of the surface ratio (apico-basal coordinate) in frusta (A) and Voronoi (B) tubes. The error bands correspond to the standard deviation from 10 samples, and the green, yellow, and red triangles indicate the values of s at the apical, null-stress, and basal surfaces, respectively. C/D: Same results as in A and B for tubes representing cuboidal/columnar epithelia. In all cases, the plots in the bottom row depict the distribution (PDF) of net forces exerted on all cell vertexes, demonstrating that force balance is achieved, F = (F_x,F_y)≃0, regardless of cellular packing.

Regarding differences in normal forces between packing configurations for squamous (Fig. 4A-B) and cuboidal/columnar epithelia (Fig. 4C-D), the normal force is dominated by elastic terms (i.e., cell volume conservation) in all cases. However, in cuboidal/columnar epithelia, frusta packing results in weaker normal forces near the basal surface compared to Voronoi packing.

For shear stress, we first note that its sign does not have a particular physical meaning (Methods). In squamous epithelia, the differences between frusta and Voronoi packing are minimal, similar to the case of normal stresses, and the elastic component dominates over adhesion and contractile forces. Additionally, shear stresses are at most two orders of magnitude smaller than normal stresses. In contrast, in cuboidal/columnar epithelia, the elastic component remains dominant, but quantitative differences arise depending on cellular packing.

In frusta tubes, the maximum shear force (at the basal surface) is approximately three times larger than in Voronoi tubes. This results from increased cell stretching in frusta tubes compared to scutoidal cells (see Fig. 3C-D). Consequently, the balance between shear and normal forces, $|\langle F_s\rangle /\langle F_n\rangle |$, is larger in frusta packing at the basal surface, where shear stresses reach a maximum:

$${|\langle F_s\rangle /\langle F_n\rangle |}_{\text{frusta}}\simeq \mathcal{O}({10}^1)\times {|\langle F_s\rangle /\langle F_n\rangle |}_{\text{Voronoi}}$$

Finally, estimation of the net forces exerted at cell vertexes (Fig. 4, bottom row) reveals that force balance is achieved independently of cellular packing and geometry.

4. Discussion

Herein, we have introduced a novel 3D force/energy inference approach that focuses on two main aspects: obtaining the average cellular behavior in tissues and parameterizing a vertex model to determine the effective biophysical parameters of cells. Our methodology employs a tomographic approach that determines force equilibrium plane by plane along the apico-basal axis. The parametrization, based on mapping to the vertex model, allows us to elucidate elastic, adhesive, and contractile force components. In this context, we exploited the fact that the force equilibrium condition is linear in these force parameters to implement a linear programming optimization approach.

While cells and tissues are inherently 3D structures, technical difficulties in obtaining accurate imaging data have long hindered the development of force inference methods in 3D. Recent advances in microscopy and machine-learning-assisted segmentation [19] have significantly progressed the field of 3D force inference [60]. However, all current methods still face limitations. Recent approaches assume the embryo as a “foam” in equilibrium [52], with further improvements achievable using simulation-based inference [61]. Still, these methods primarily provide relative values for cellular pressure and surface tension.

Our approach assumes that cell shape deformation is driven by in-plane forces while forces along the apico-basal axis are negligible. This approximation is well justified by the expected behavior of elastic materials. However, applying our methodology to real experimental tubular epithelia requires caution. The proposed analysis relies on the assumption that when unrolling the cylinder (see Fig. 1F), radial planes preserve vertex-vertex distances and cell areas. If this condition is not met, force estimation at cell vertexes may introduce artifacts. Consequently, our methodology is best suited for computational studies, where it provides insights into a key biological problem: how apico-basal intercalations contribute to the stability and integrity of tubular epithelia. Along these lines, while the methodology could, in principle, be adapted to other epithelial structures, such as spherical or oval epithelia, doing so would require substantial modifications. Specifically, defining force balance in a fully enclosed 3D structure at the vertexes where apico-basal intercalations occur would necessitate the development of a novel energy/force functional for curved surfaces where stable scutoids develop spontaneously —one that, to the best of our knowledge, has not yet been either proposed or tested in epithelial mechanics. Overall, while our approach is theoretically extendable to other epithelial geometries, it is specifically optimized and well-suited for analyzing tubular epithelia.

Importantly, modeling is the only feasible way to investigate our questions of interest, as current experimental techniques cannot selectively modify tissue packing architectures. Since their discovery, scutoids have been recurrently found in cuboidal/columnar epithelia that have been analyzed in 3D. Examples include tissues in mice, zebrafish, Drosophila, cell cultures, and organoids, supporting the generality of this packing shape. However, to the best of our knowledge no study has reported the existence of scutoidal cell shapes in squamous epithelia. Here, we explored the biophysical basis of this phenomenon by comparing tubular tissues with two different surface ratios, $s_b\simeq 1.5$ and $s_b=4$.

The logic underlying this analysis is as follows. It has been reported that a lower value of $s_b$ correlates with a reduced number of scutoidal cells [14], as the shorter height of these cells limits neighbor exchanges along the apico-basal axis. This suggests that tubular epithelia with either frusta or scutoidal cell packing can be energetically compatible and maintain integrity at low values of $s_b$. To simplify, we use $s_b\simeq 1.5$ tubes as a model for squamous tissue to provide a baseline for understanding the results obtained for $s_b=4$ (cuboidal/columnar model). However, in real biological tissues, squamous cells typically exhibit even lower values of $s_b$. Indeed, our simulations reveal that in $s_b\simeq 1.5$ Voronoi tubes, scutoids develop in approximately $35\pm 6\text{\%}$ of cells. Taken together, our findings indicate that in squamous tissues subjected to curvature, frusta and scutoidal cell packing result in similar energy and force profiles (Figs. 3A-B and 4A-B). Thus, we propose that in real tissues, squamous cells do not derive any biophysical advantage (energetically speaking) from remodeling their morphology into a scutoidal shape.

However, in cuboidal/columnar tissues, when apico-basal intercalations are suppressed (i.e., frusta cell shapes), force equilibrium can only be achieved with extremely low levels of cellular adhesion—approximately an order of magnitude lower than in short squamous cells. Such low adhesion is both unrealistic and incompatible with epithelial integrity, where tight cell packing is essential. Furthermore, preventing apico-basal intercalations in cuboidal/columnar cells disrupts the balance between normal and shear forces, further challenging tissue integrity. This imbalance is linked to basal surface deformation in frusta cells when $s_b=4$ (Fig. 3C) but is absent in squamous epithelia ($s_b\simeq 1.5$, Fig. 3A). Conversely, when apico-basal intercalations are allowed in cuboidal/columnar tissues, cellular adhesion values are higher than in squamous cells, and the balance between shear and normal forces is more compatible with tissue homeostasis, as normal forces clearly dominate over shear forces (Fig. 4D).

In a broader context, recent studies on 2D epithelial packing dynamics have shown that an increase in cell junction tension upon contraction and a reduction in tension upon extension can stabilize higher-order (e.g., four-fold) vertices [62]. Extending this to a 3D context, apico-basal intercalations, established by four-cellular junctions along the apico-basal axis, may play a key role in resolving unstable cell geometries resulting from constraints during tissue packing while maintaining tissue integrity, as demonstrated in our study. Although our work sheds light on the role of apico-basal intercalations in static 3D epithelial tubes, the mechanics underlying cell dynamics during morphogenesis—while simultaneously accounting for stationary apico-basal intercalations and cell division—remain largely unknown. Despite recent efforts [20], [22], [63], further research is needed to elucidate the interplay between mechanics and scutoid dynamics to better understand complex processes such as tissue growth and rearrangement, wound healing, cell extrusion, and cell migration.

CRediT authorship contribution statement

Samira Anbari: Software, Methodology, Investigation. Pedro Gómez-Gálvez: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation. Pablo Vicente-Munuera: Visualization, Methodology. Luis M. Escudero: Writing – review & editing, Writing – original draft, Validation, Supervision, Funding acquisition, Conceptualization. Javier Buceta: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

Declaration of Competing Interest

Declare that none of the authors: Have an undisclosed relationship that may pose a competing interest; Have an undisclosed funding source that may pose a competing interest.

Appendix A. Control simulations

We ran control simulations using the vertex model to test the reliability of the force inference method. To that end, we used the TiFoSi package [64], [65]. In addition to the parameters that describe the cell mechanical properties (i.e., the parameters to be inferred), we introduced some level of stochasticity in the duration of the cell cycle to achieve different cellular sizes at a given time. Thus, the duration of the cell cycle, τ, is defined as,

$$\tau =ϵt_{det.}+(1-ϵ)t_{sto.}$$

where $t_{det.}$ is a deterministic time scale that accounts for a mean cell cycle duration and $t_{sto.}$ is a random variable that accounts for the variability of cell cycle duration and that is assumed to follow an exponential distribution:

$$\rho \left(t_{sto.}\right)=\frac{e^{-\frac{t_{sto.}}{t_{det.}}}}{t_{det.}}$$

We set a value of $ϵ=0.8$ for the parameter that weights the stochasticity of the cell-cycle duration (see [64], [65] for details), and we set a dimensionless average cell cycle duration of $\langle \tau \rangle =1.5\cdot {10}^3$ (∼20 hours) in all simulations.

Fig. 6.

Inference of cell energy components: different dominant contributions. A/B: Elastic/contractile energy dominant tissues. In both cases N = 45 and vertex model parameters as shown in Table 3. In A circles correspond to individual cells. In B the color code of cell energy components as in panel A and the columns corresponds to the cases shown in panel A. The scale bars range between the observed minimum and maximum values of the energy in all cases. We notice that negative values of the adhesion energy obtained when Λ < 0 are because the value of the line tension parameter is taken into account in the calculation of E^adh. as stated in Eq. (A.1).

Control simulations were performed in two stages. First, we allowed cells to grow and divide until the tissue reached a given number of cells, N. During the second stage, we stopped cellular growth and either allowed the tissue to mechanically relax to a stable configuration or modulated some mechanical properties as a function of time to drive the system out of equilibrium. Since cell vertexes at the tissue periphery are shared by either one or two cells, in contrast to vertexes in the tissue bulk that are shared by three cells, we first tested the sensitivity of our method to finite-size (i.e., boundary) effects. To do so, we used the same values of the $A_0$, Λ, and Γ parameters but varied the ratio $\rho =N_p/N$, where $N_p$ is the number of peripheral cells. Since $\rho \sim 1/\sqrt{N}$, better inference results are expected as N increases (i.e., as ρ decreases). The results (Table 3) indicate that as ρ approaches ∼0.5, the observed error, δ, in the inferred parameters remains below 3% and stays under 10% even in tissues where boundary effects are dominant ($\rho =0.7$).

Table 3.

Parameter inference: different energy regimes. Ground truth and estimated vertex model parameters values in different scenarios with respect to the dominant energetic contribution.

Parameter	Case A (elastic dominant)		Case B (contractile dominant)
	Ground Truth	Estimation	Ground Truth	Estimation
$A_0$	2	2.001 (δ = 0.05%)	1	1.002 (δ = 0.2%)
Λ	0.04	0.041 (δ = 2.5%)	−0.02	−0.02 (δ = 0%)
Γ	0.02	0.02 (δ = 0%)	0.02	0.02 (δ = 0%)

Fig. 7.

Time-dependent parameter estimation. In the three top panels the horizontal dashed line indicates the value of Γ₀ = 4 ⋅ 10⁻², the black squares stand for the actual values of $\mathrm{\Gamma }\left(t\right)$, and the red circles and the green diamonds the inferred values using either 10 or 5 samples (i.e., frames) during an oscillatory period. Panels A, B, and C show results for three different frequency regimes: ω > 1 (fast contractile pulses), ω < 1 (slow contractile pulses), and ω ≪ 1 (ultra-slow contractile pulses) respectively. The three bottom panels shown, for each frequency regime, the packing configurations of the tissue at maximum compression and expansion during pulses.

Once the parameters $A_0$, Λ, and Γ were computed, we estimated in each simulation the energy components of each cell k:

$$\begin{gathered} E_k^{\text{elastic}}=\frac{1}{2}{(A_k-A^0)}^2 \\ {E}_k^{\text{contract.}}=\frac{\mathrm{\Gamma }}{2}{L}_k^2 \\ {E}_k^{\text{adh.}}=\mathrm{\Lambda }{L}_k \end{gathered}$$ (A.1)

Fig. 5 shows that convergence to ground truth values is achieved as ρ approaches ∼0.5.

Fig. 5.

Inference of cell energy components: finite-size effects. A: Comparison of cellular energies between estimated and ground truth values as a function of the ratio ρ (N and N_p are indicated in Table 2). Circles correspond to individual cells. A spreading away from the diagonal indicates a mismatch between estimated and ground truth values. B: Cell energy components in simulated tissues (color code as in panel A); columns corresponds to the values of ρ indicated in panel A. The scale bars range between the observed minimum and maximum of the energy values in all cases.

We further checked the robustness of the inference method against the relative importance of different energy contributions. To that end, we performed control simulations using different parameter sets while keeping N constant. Table 3 and Fig. 6 show two representative cases where Γ is kept the same but $A_0$ and Λ change such that either the contractile or the elastic energy becomes the dominant energetic contribution (in contrast to the simulations shown in Fig. 5 where the adhesion energy is dominant). The inference results remain in excellent agreement with ground truth values regardless of the dominant energy component.

Finally, we evaluate the reliability of the force inference approach when the mechanical parameters are modulated in time. In this regard, the time scale for mechanical energy relaxation is $t_r\simeq 1$ (dimensionless time units). Thus, force equilibrium is reached at times $T\gg 1$ (i.e., $t_r/T\ll 1$). This suggests that a dynamic modulation of mechanical parameters can be inferred as long as the time scale of their change is slow enough compared to $t_r$.

In order to test this possibility, we conducted a series of vertex model simulations of tissues with cells that display a time-dependent cortex activity that induces a pulsating behavior similar to that found, for example, during cell extrusion [66]. To that end, we modulated in our simulations the cellular contractility parameter, $\mathrm{\Gamma }(t)={\mathrm{\Gamma }}_0(1+\cos (\omega \cdot t))$, and tested the inference capabilities under different frequency regimes. For a given period of the oscillatory behavior of $\mathrm{\Gamma }(t)$, we selected a number of samples (i.e., frames) for our analyses and we solved simultaneously the force equilibrium equations of all frames subjected to the condition that the adhesion parameter, Λ, and the target area, $A_0$, did not change among frames, but the contractility parameter, Γ, did.

The results revealed that regardless of the oscillatory regime, the actual values of Λ and $A_0$ were correctly inferred: ${\mathrm{\Lambda }}^{\text{actual}}=0.04$ and ${\mathrm{\Lambda }}^{\text{inferred}}=0.039$, $A_0^{\text{actual}}=A_0^{\text{inferred}}=1$. As for the time-dependent behavior of $\mathrm{\Gamma }(t)$, when pulses are fast with respect to the mechanical relaxation time ($\omega >1$, Fig. 7A), the tissue cannot reach equilibrium packing configurations and, as a consequence, the inference method captures an effective behavior of $\mathrm{\Gamma }(t)$ rather than its actual dynamics. As expected, such an effective (predicted) behavior of $\mathrm{\Gamma }(t)$ approaches the temporal average, $\overline{\mathrm{\Gamma }(t)}={\mathrm{\Gamma }}_0=0.04$, as ω increases.

As the frequency decreases and the pulses become “slow” ($\omega <1$, Fig. 7B), the estimation of $\mathrm{\Gamma }(t)$ captures more precisely its actual temporal behavior. Finally, in the case of “ultra-slow” pulses ($\omega \ll 1$, Fig. 7C), the tissue reaches an equilibrium packing configuration in every frame and $\mathrm{\Gamma }(t)$ is accurately captured. Furthermore, we also tested whether the number of samples (i.e., sampling rate) modifies the accuracy of the inference method. We concluded that as long as the sampling rate is smaller than the relaxation time, it does not affect the inference accuracy (red and green curves in Fig. 7). Given that the control simulations show that time trajectories are satisfactorily captured, we implemented our tomographic approach towards force inference in 3D tissues by exchanging the concepts of space and time and solving simultaneously apico-basal trajectories.