Peeking into the future: inferring mechanics in dynamical tissues

Tissue morphogenesis is driven by changes in cell numbers (in turn caused by mitoses, apoptosis, and cell extrusion), collective movements of cells, and changes in cellular mechanical properties together with alterations of the constraints imposed by the environment onto the epithelium [1–10]. Hence, tissue mechanics needs to be addressed to understand the remarkable morphogenetic processes that shape embryonic tissues during development and the outgrowth of tissues in species capable of regeneration [11].

The study of forces in embryo morphogenesis can be traced back to the Entwicklungsmechanik (developmental mechanics movement), which emerged during the 19th century [12]. Later, D'Arcy Thompson enlightened his time with a revolutionary concept: the size and shape of body organisms could also be interpreted as a map of the acting and driving mechanical forces [13], in the same way in which Faraday conceptualized that iron particles in his experiments were a map of the invisible magnetic fields underneath [14] (Supplementary Figure S1A–C). Thus, D'Arcy Thompson transformed our way of perceiving living organisms: The geometrical features of organisms, tissues, and cells reflect not only gene regulatory networks but also a vivid manifestation of mechanical forces at work (Figure 1A). This idea re-emerged and crystallized into a mathematical method to infer mechanical stresses in tissues during the late part of the last century.

Tissue shape is not solely a product of biochemical signaling, but results from the collective interactions between constituent cells [15–19]. Changes in the mechanical interactions between cells and their environment/substrate, reflected in tissue stiffness, can promote epithelial-mesenchymal transitions, which promote tissue repair and development [20–23]. In tumorigenesis, increased stiffness can lead to tumor progression and metastasis [21,24,25]. Spatial and temporal changes in stiffness can influence the cell-to-cell stresses that arise in tissues. Interestingly, tissue stress distribution is not temporally invariant during complex biological processes such as development [26–28]. For example, tissue stress fluctuations have been shown to play a key role in facilitating tissue reorganization in the developing zebrafish [29]. Mechanical stress also plays a critical role in tissue regeneration by promoting cell functions such as proliferation, differentiation and migration [30–32]. Mechanical stimuli can improve bone fracture healing by promoting the growth and differentiation of bone-forming cells [30].

This mini-review¹ first describes the computational methods of static stress inference, their advantages over the standard experimental stress determination, and their limitations. Next, we discuss how these limitations lead to the alternative computational methods of dynamic stress inference, which essentially use the topology of the tissue in the ‘present’ combined with its ‘future’ state to predict the stresses therein. We explore the similarities and differences between the dynamic and static counterparts and highlight their advantages and disadvantages. We have summarized all stress inference algorithms that exist to the best of our knowledge in Table 1. We conclude this review with what we believe are exciting future perspectives of stress inference in tissues.

Computational estimation of the mechanical forces at play in biological systems was originally called Force inference, even though the calculated magnitudes are scalars. Here, as pointed out by other authors [35,39,40], we will use the term stress inference. This inexpensive technique approximates the relative stresses and pressures operating in a given system using a microscopy image as sole input, exploiting the information in the tissue topology to obtain the apparent distribution of stresses therein [35,41–46].

The contact angle among cells can serve as a proxy for cell-cell junctional tension, a concept imported from wetting phenomena to study cell interfaces [47] (Figure 1B). Thus, it can be used for stress determination between cells in monolayers, as it depends on the relative forces along the interfaces [48,49]. The cell-cell junctional tension might arise from different sources, such as actomyosin contraction due to myosin accumulation at the cortex or cell-cell adhesion through cadherin binding [2,50]. The membrane stress will be equal at a tricellular junction where all cells meet with the same contact angle (⁠$α=β=γ=120∘$⁠) (Figure 1C, upper panel). On the other hand, when there is a near-straight angle between two membranes, the stress in the third one will be almost zero (Figure 1C, lower panel). As determining the contact angles at each junction is paramount, membrane segmentation needs to be precise to avoid angle misrepresentation. New image segmentation software is appearing [51–53], and constant efforts are being made to standardize their applicability [54].

Therewith, the intracellular pressure can be inferred using the Young-Laplace Law (Figure 2A) by combining the intercellular stress and the local cell membrane curvature. Conceptually, more convex cells will have a higher pressure than concave ones (Figure 2B). Pressure differences arise from an interplay between the hydrostatic pressure in the cell's environment and the osmotic pressure due to the cell's semipermeable membrane. Only recently have the osmotic contributions to the intracellular pressure been determined in vivo in zebrafish [55].

The proper determination of the membrane's shape is crucial for stress inference. Noisy images might lead to numerical errors that get amplified down the pipeline, reducing the accuracy of the inference. Several methods can be used to estimate the membrane's shape contribution to the geometrical matrix defined in eqn 2. Straight edges, though easy to estimate from microscopy images, have the inconvenience of making the system less robust [35,60]. This can be visualized, for example, by taking an edge and realizing that the force will have the same components at each end, with an opposite sign (Figure 3A). On the contrary, curved edges will have a distinct value at each end of the edge, adding independent information to the system of equations (Figure 3B). In this case, each edge's direction (vector) could be determined from the tangent to the limiting angle to the cell membrane [60].

Early methods inferred cortex stresses and cell pressures simultaneously and used the straight-edge approximation [35,57,58]. These methods assume that the cell's mechanical energy will depend only on the length of the edges and the cell's area to derive the system of equations from a potential energy [39,59]. Having the pressure and stress intermixed in the same expression increases the number of unknowns per equation while maintaining only one equation for each space component. Moreover, tissue boundaries and fourfold vertices give fewer equations than unknowns, leading to an underdetermined system. Due to the constraints mentioned, this type of system has been solved using Bayesian methods [19,45,59]. These constraints are not unavoidable, as was demonstrated by the Cellular Force Inference Toolkit: CellFIT [60]. This method allows considering curved edges and makes stress and pressure inference independent. Pressure is modeled using the Young-Laplace equation. This equation relates the shape and stress of an interface with the difference in pressure between both sides (Figure 2A). As the stress in an edge is required to calculate the pressure, this inference needs to be performed subsequently, effectively decoupling both magnitudes.

Each microscopy image (Figure 4A) is segmented (Figure 4B) and then converted to a system of equations (Figure 4C) when performing stress inference (Figure 4D). Each spatial dimension will contribute an equation per junction and one unknown per membrane: the stress. The most common case is to have triple junctions, i.e. junctions with three connecting membranes [64,65] (Figure 1B). In two dimensions, each new junction will contribute two equations and, at most, three unknowns, as some membrane stresses will be repeated.

In geometrical stress inference, where no movement of the junctions is considered, the right-hand side of eqn 3 is set to the null matrix, i.e. $[b]=0$⁠. If all the junctions are at rest, a possible solution to the system of equations would be that no stresses are acting on the system (⁠$λj=0∀j)$⁠. Two methods are prevalent in the literature to avoid this unrealistic and trivial solution. We and other authors have included a constraint linking all the stresses in the tissue, such as a particular value for the average stress [57,58,60,66]. Fixing a value has the advantage of adding one more equation to the system without any new unknowns, thus increasing its stability. Another method consists of changing the cost function associated with the method, adding a regularizer [67] that penalizes the null solution. More generally, this could allow tailoring of the cost function to the specific needs of the problem at hand by enabling the addition of new terms, such as penalizing wider distributions of stresses.

Stress inference methods do not need to make any assumptions about the specifics of the force generation mechanism. However, stress inference may yield negative results if the inference method allows it or null stresses if not. In static stress inference, this usually means that the particular shape of a membrane is incompatible with the underlying model. Sometimes, it is possible to identify these issues and avoid problematic junctions or membranes. Typical pathological junctions include near right angles, curvy membranes, or junctions higher than three-fold [35,68,69].

Tissues in a quasistatic regime may be encountered during adult tissue homeostasis or late embryonic development. However, tissues are often found in a more dynamic state in early embryonic development where cell motility and rearrangements cannot be neglected [75,76]. During development, morphogenetic flows shape the organisms into functional forms, such as in Drosophila [77–80] and Zebrafish [81–85]. Therefore, accurate prediction of mechanical forces in these tissues requires inference algorithms that work optimally in the presence of significant motion. This modality of stress inference is called Dynamic Stress Inference [35].

A key challenge in Dynamic Stress Inference is to follow the tissue's evolution reliably through time, yet not all tissue elements have to be tracked. The elements can be divided into two categories: passive elements, which dissipate energy, and active elements, which generate work [57,58]. Brodland and colleagues suggest that the forces generated by active elements, such as the actomyosin network and cell membranes, deform the passive elements, including the cytoplasm, organelles, and extracellular matrix [58]. The inference method requires selecting an optimal level of detail, as the relevant elements need not only be the cellular cortex but could also be subcellular structures [57,58]. The selected structures must be defined in detail in the microscopy to allow tracking throughout the experiment.

The second challenge is also found in static inference and was mentioned above: the need to choose an underlying model for membrane stress. The choice depends on the specifics of the system of interest and the desired level of complexity. All currently available stress inference software uses a derived version of the model described by eqn 2. As written, this equation reflects the mechanical stress along the membrane, which is conceptually similar to the line tension of a vertex model [63]. However, in systems where the contractility of the actomyosin cortex is particularly relevant, an additional term may need to be added to model the stress along the entire circumference of a cell, such as the perimeter term of a vertex model [63].

Finally, the third key challenge relates to the question of the scales involved in the processes studied. In Static Inference, only the spatial scale is relevant, as it relates to the positions of the membranes among themselves and is ultimately used to find the angles. Importantly, in Dynamic Inference, the time scale comes into play, which implies that, in this formalism, the relation between scales of space and time affects the inference.

In summary, to successfully incorporate cell movement information into the inference process, the dynamic inference pipeline requires (1) faithful tracking of relevant elements of the tissue through time, (2) an underlying model for the stress at each membrane, and finally, (3) knowledge of the relationship between the scales involved in the process under study.

To the authors’ knowledge, three different methods have been proposed to deal with time series of data: DLITE [67], video force microscopy (VFM) [57], and ForSys [66]. Only DLITE and ForSys are currently available. In the following sections, we explore the virtues and limitations of these three methods.

The use of movement information, also reviewed in [35], allows tracking of the nodes through time, dealing with the first issue presented at the start of the previous section. DLITE (Dynamic Local Intercellular Tension Estimation) [67] assumes mechanical equilibrium at each junction (eqn 1), similar to CellFIT [60]. DLITE takes membrane tensions in the direction of the edge joining the junction, with no additional terms (eqn 2). Moreover, it uses a regularizer that penalizes small stress values to avoid the null solution. Though time series tracking of tissues is involved, as Roffay et al. [35] pointed out, this is not a dynamic inference method per se. In fact, DLITE only tracks all nodes, edges, and cells through time to use the inferred solutions of the previous time as an initial guess for the current frame (Figure 5A). Importantly, the authors show that this is sufficient to improve performance on in silico data and maintains better robustness over time than CellFIT [67]. Moreover, unlike CellFIT, DLITE is an open-source project, allowing users to examine the details of its implementation.

Brodland et al. [57] reported the first dynamical stress inference of tissues to study the mechanics of ventral furrow invagination in Drosophila. It was called VFM and previously cinemechanometry [58]. They developed this method in a series of papers spanning more than fifteen years [86–89]. VFM successfully identified stresses during ventral furrow failure due to reduced myosin II activity [57]. The authors built a finite element mesh over the tissue of interest's active landmarks (Figure 5B). This meshing need not correspond precisely with all the cell's hallmarks, namely the cell membranes, and could have subcellular details. Brodland et al. chose the mesh so that its nodes correspond with the elements they categorize as active. Therefore, it could be the case that the polygons enclosed by the mesh are not cells but rather subcellular domains (Figure 5B, upper panel).

Most recently, we proposed the ForSys method, which builds upon the advancements in the field and integrates them into an open-source pipeline [66]. This software allows Static and Dynamic Stress Inference in curved geometries, which is achieved by fitting a circle to the segmented edges [66], as in CellFIT and DLITE [60,67].

When tested using in silico tissue movies, this method outperforms other software when the junctions move significantly and negligibly [66]. Furthermore, this method opened a window into using force inference technology in migratory structures. When applied to the mobile Zebrafish lateral line primordium, the method detected zones of high pressure/tension that indicate the presence of rosettes predating the separation of the neuromast organs [66].

As with any technique, its falsifiability is an important aspect of stress inference. The standard for validating the inference engine in static and dynamic modalities is based on in silico validations, often generated using a cell-based computational model such as the Vertex model [45,67,69,92]. Various experimental validations can be performed for static inference by fluorescent measurements, such as with myosin intensity [59], antibodies staining [93,94], and flipper probes [95], or through direct manipulation with methods including laser ablation [45,59] and atomic force microscopy (AFM). Complementary to hydrostatic pressure measurement [96,97], methods for osmotic pressure determination have recently emerged [55]. Pressure inference cannot distinguish between these contributing factors; however, we expect the inferred pressure to be dominated by the hydrostatic component, as osmotic pressure differences equilibrate rapidly across the tissue. In contrast, experimental validation of dynamic inference by ground truth generation is rather limited, as methods such as laser ablation or AFM irreversibly change the state of a tissue and are likely to disrupt or alter the processes of interest. Therefore, dynamic stress inference is typically calibrated using computational model simulations.

Thanks to advances in image segmentation, static stress inference can be performed in 3D, as initially reported by Brodland and colleagues [98] and more recently by the Turlier lab [92]. Incorporating the third dimension into dynamic inference will be a significant leap forward. Reliable 3D stress maps that could be used to validate an inference technique are experimentally challenging to produce, especially if the maps need to be time-dependent.

A desirable feature of future stress inference techniques is to couple the inference engine with the computational packages for simulating tissues using cell-based models that rely on mechanical information, such as the vertex model. This model requires the correct parameterization of the line tensions needed to simulate tissues that recapitulate the phenomena they are intended to describe [63]. In this way, the intercellular stresses obtained by stress inference could be used to parameterize a model and generate predictions embodied in model simulations.

Stress inference can uncover mechanical features of systems in various settings. It has been successfully applied to investigate cell division [40], organogenesis [66], and cell-type-specific mechanical anisotropies [39,66,99]. We expect that stress inference will expand its applications to fields such as immunology [100] and cancer biology [101,102] in the coming years, which could greatly benefit from this technique.

While stress and pressure inference allows the determination of local intracellular pressure and intercellular stress, one might expect that this information could be integrated to provide a whole tissue description of the mechanical state of the system. Recently, it was proposed that local and tissue-level mechanical information could be combined with spatial omics [103]. This will allow a deeper understanding of the connection between the mesoscopic scale and the molecular details of the interactions.

Even though a machine-learning approach has recently been used to infer forces from cytoskeletal protein distributions [104], a new avenue to explore could be using machine-learning algorithms to infer stresses. This would involve curating microscopy images and feeding them into a neural network, which in turn could learn to predict stress distribution for a given topology. To our knowledge, this exciting possibility has not been explored even in the static inference formalism.

• Dynamic stress inference provides a first computational approach to determine the mechanical state of tissues and generate predictions that can guide future experiments. This family of inexpensive tools can advance the study of tissues in development and regeneration.

• While static stress inference from microscopy images allows the characterization of a tissue's mechanical state at a single point in time, dynamic stress inference from video microscopy represents the full spatiotemporal distribution of mechanical stresses experienced by the tissue.

• A next logical step will be to combine dynamic inference with 3D geometric inference to ultimately generate 4D stress inference. Another important step will be the coupling of inference with cell-based models, which will allow direct testing of these models in biologically relevant geometries. Finally, machine learning algorithms that allow the training of networks on specific organisms may prove helpful in recognizing specific patterns of stress distributions.

The in silico image where generated using seapipy and ForSys. The seapipy codebase is available on GitHub at https://github.com/borgesaugusto/seapipy and Zenodo [105]. ForSys codebase is available on GitHub at https://github.com/borgesaugusto/forsys and Zenodo [106].

The authors declare that there are no competing interests associated with the manuscript.

A.B. was funded by the BMBF 01GQ1904 grant. O.C. was funded by grant PICT-2019-03828 from the Agencia Nacional de Promoción Científica y Tecnológica of Argentina and by a Biotechnology and Biological Sciences Research Council grant [grant number BB/X014908/1]. The authors would like to acknowledge the two anonymous reviewers for their very constructive comments.

Open access for this article was enabled by the participation of University of Nottingham in an all-inclusive Read & Publish agreement with Portland Press and the Biochemical Society under a transformative agreement with JISC.

A.B. and O.C. wrote and edited the manuscript.

The authors would like to thank Alice Descoeudres, Diego Cattoni, Nicolas Aldecoa, and all the members of the Chara laboratory for their helpful comments on the manuscript.

AFM

atomic force microscopy

DLITE

Dynamic Local Intercellular Tension Estimation

VFM

video force microscopy