Author response:
Public Reviews:
Reviewer #1 (Public review):
Summary:
In this article, the authors couple a 3d vertex model to the extracellular matrix and include activity through contractile springs at the edge. They study, sequentially, the distribution of shear stresses in liquid and solid spheroids, the correlation between stress and cell shape, and the spatial distribution of stresses. The authors find that stresses are higher in solid spheroids (somewhat unsurprisingly), but that the stress distributions are wider in the fluid spheroids. Moreover, stress and shape are not correlated with each other in solids (that seems to be due to vertex model peculiarities), but they are for liquids. In contrast, for solids, the stresses are concentrated at the interface.
The authors attribute a lot of the phenomenology to strain-stiffening properties of vertex models as being akin to a network model (correctly in my opinion). Then they strain individual cells and confirm this link, though I missed any explanation of how they did this. Would it have to be within a medium for computational consistency?
We thank the reviewer for this helpful comment. The current manuscript already describes this procedure in Sec. II.C, “Cell strain-stiffening with volume-preserving deformations,” where we state that individual cells are taken from the final spheroid configuration and then strained by imposing a prescribed volume-preserving deformation along their principal elongation axis. Figure 4 then compares the original and strained cells and shows the resulting increase in maximum shear stress.
We agree, however, that this point was not explained clearly enough. In the revised manuscript, we will make explicit that this is a single-cell deformation test designed to isolate the intrinsic strain-stiffening response of the vertex-model cell. The cell does not need to remain embedded in a surrounding medium for this specific test, since the goal is not to simulate the full coupled cell–ECM dynamics, but rather to measure how the stress of an individual vertex-model cell changes under imposed strain.
Indeed, single cells can exhibit strain stiffening as presumably can a spheroid. However, given that we are studying strain stiffening in the context of single/few cell breakout, we also plan to measure the stress in the breakout cells in the extended vertex model to determine the extent of strain stiffening given the surrounding medium of fibers and cells.
Finally, they generate an extended vertex model, where they replace the single face linking cells with a double face and mechanoresponsive springs. This allows for stronger coupling of individual cell motion to eventual movement out of the spheroid.
Strengths:
Coupling a three-dimensional vertex model to the extracellular matrix, modelled as a crosslinked fiber model, is a computational tour-de-force. Adding activity through fluctuations at the interface is also of the correct symmetry (stresses), instead of the self-propulsion which has been used by other authors, and which is not compatible with Newton's 3rd law. This also allows for accurate back-and-forth mechanical coupling between the cells and the ECM.
I would like to highlight that deriving vertex model stress tensors in full three dimensions is an open problem due to the complex topology. Any progress is valuable, and decomposing things into tetrahedra like here will allow for connections with, in particular, finite element approaches. Therefore, adding some of these results (eq. 13) to the main text would strengthen the paper in my opinion.
Adding the nonlinear springs to the VM in the 3rd act is a good idea, and a first step to mechanical feedback. One might argue that at this point, removing the vertex model part would even be an option.
Weaknesses:
The paper is written in a very qualitative manner, with all of the model equations and analysis hidden in the supplementary information. I do not understand this choice, as it makes things fuzzy and hard to read. The conclusion is also very long and simply reiterates the previous points.
At the same time, this paper is rather thin on new results and reads more like a handful of new simulations carried out using the method established in [10] (from largely the same authors). Moving some of the actual results to the main text would help, in particular, the 3d stress formulation and the definitions of different measures.
We thank the reviewer for this constructive criticism. We agree that the main text was too qualitative and that placing most of the equations and definitions in the Supplement made the manuscript harder to read. In the revised version, we will move the essential technical material into the main text, including the 3D cell stress formulation, the definitions of maximum shear stress and cell-shape anisotropy, and the stress–shape alignment measure. Longer derivations and implementation details will remain in the Supplement.
We will also shorten and reorganize the Discussion/Conclusion to avoid reiterating previous points. Finally, we will revise the presentation to make the new contributions beyond Ref. [10] clearer: the 3D polyhedral-cell stress formulation, the stress-distribution and spatialpatterning analyses, the single-cell strain-stiffening test, and the extended adhesion-spring model used to distinguish single-cell from multi-cell breakout. These changes should make the paper less qualitative and make the main results more visible in the body of the manuscript.
Vertex models also have a very clear limitation: They cannot model the transition from a confluent to a non-confluent tissue, and individual cells or groups of cells leaving the spheroid. Even having a surface and having significant deformations of the surface are numerically dicey, so the current model is at the edge of what is feasible. The model as written can only do "invasion" by a single cell moving outward, and then another following it a bit (or not).
I strongly suspect that further progress on 3d cell models will need particle-based models or models where cells are fully meshed surfaces (some of which are in development currently).
However, none of these problems is mentioned anywhere in the text. The authors also do not review the increasingly broad zoology of other models.
We thank the reviewer for raising this important limitation of standard vertex models. We agree that a strictly confluent 3D vertex model is not designed to fully capture the transition from a confluent tissue to freely migrating detached cells, and we will make this limitation explicit in the revised Discussion. However, the standard 3D vertex model can still capture collective spheroid deformation, surface remodeling, and local protrusive deformations prior to complete breakout. Thus, it remains useful for studying the mechanical state of the spheroid and the onset of outward deformation before full cell detachment.
At the same time, we clarify that this very limitation motivated the extended vertex model introduced in Sec. II.D and Supplement G. In this model, cells no longer share interfaces as in a standard confluent vertex model; instead, neighboring cells interact through explicit, tunable cell– cell adhesion springs. This allows us to represent, in a coarse-grained mechanical way, the separation of a boundary cell from the spheroid and the motion of a follower cell behind it. Thus, while the model does not describe full post-detachment migration, it partially addresses the confluent-to-nonconfluent transition at the level needed to study the mechanical onset of breakout.
We will revise the manuscript to make this distinction clearer and state that our goal is to identify minimal mechanical ingredients for incipient breakout—strain stiffening, adhesion weakening, and adhesion anisotropy—rather than to provide a complete model of long-time invasion.
We will also note that the current Introduction already discusses several existing modeling approaches, including cellular automaton simulations, a 2D Voronoi model, phenotypeswitching/ECM-remodeling models, and the prior 3D vertex–fiber framework. However, we agree that this discussion should be broadened, and we will add a more explicit comparison with particlebased, phase-field, cellular Potts, and fully meshed deformable-surface models, which may be better suited for later-stage non-confluent migration.
Reviewer #2 (Public review):
Summary:
The manuscript concerns the mechanisms by which cells in a spheroid embedded in the extracellular matrix can escape, either as single or multiple cells.
Strengths:
Overall, the manuscript is well written and easy to follow. The claims are mostly justified by the data. Some data can be better analyzed and presented to strengthen the conclusion.
Weaknesses:
(1) The description around Figure 2c is not exactly well supported by their results. While values close to 0 for sigma3 dot g3 for solid-like spheroids indicate little correlation between the direction of maximum stress and maximum elongation, this analysis alone does not imply that highly stressed cells are necessarily less globular. The dot product combines the magnitudes of the two vectors and the angle between them. For the distribution graph, it would be useful to have the cumulative frequency equal 1.
We thank the reviewer for pointing this out. We agree that the interpretation of Fig. 2c should be stated more carefully. In our calculation, the vectors used in the dot product are normalized eigenvectors of the stress tensor and the gyration tensor. Thus, the plotted quantity measures only directional alignment between the principal stress direction and the cell elongation axis, not the magnitudes of stress or shape anisotropy. We will revise the text to make this explicit.
We also agree that Fig. 2c alone does not support statements about whether highly stressed cells are more or less globular. It only quantifies alignment between stress and shape directions. To address this, we will add or refer to an additional analysis, such as the correlation between maximum shear stress and cell-shape anisotropy, or the shape-anisotropy distribution conditioned on high-stress cells.
Finally, we agree that the distribution in Fig. 2c should be normalized more clearly. In the revised figure, we will plot the distribution as a probability density or cumulative distribution with total probability equal to one, and we will update the caption accordingly.
(2) One of the central claims of the paper is that morphology alone is not a reliable indicator of mechanical state. Since the authors compute cellular stresses and cellular shape in their simulation (i.e., Figure 3a and b), can the authors directly plot these two quantities for individual cells in solidlike and fluid-like spheroids?
We thank the reviewer for this helpful suggestion. We agree that a direct cell-by-cell comparison of cellular stress and cellular shape would strengthen the central claim that morphology alone is not a reliable indicator of mechanical state. In the revised manuscript, we plan to add scatter plots of maximum shear stress versus cell-shape anisotropy for individual cells in both solid-like and fluid-like spheroids.
(3) There is experimental evidence showing the solid stress inside a spheroid is higher than at the periphery (e.g., https://www.nature.com/articles/ncomms14056). How does this cellular stress relate to these experimental measurements, since they are opposite to what is simulated here (i.e., the authors find max shear stress is lowest in the center and increases towards the boundary, which is opposite to what is measured?
We thank the reviewer for raising this important point. We agree that the comparison with experimental stress measurements in compressed spheroids should be clarified.
The main distinction is that the cited experiments measure local pressure, or isotropic compressive stress, from the volume change of embedded elastic beads. In contrast, Fig. 3 in our manuscript shows the cellular maximum shear stress, which reflects the deviatoric part of the cell stress tensor. These quantities do not necessarily have the same spatial profile: a region can be under high isotropic compression while having low shear stress. The loading conditions are also different. The experiments apply external osmotic/mechanical compression to the whole spheroid, whereas our simulations consider active cell–ECM coupling through contractile linker springs at the spheroid boundary. Thus, the elevated boundary shear stress in our model reflects local cell– ECM force transmission, not internal hydrostatic pressure. We indeed will revise the manuscript to make this distinction explicit, cite this experimental work, and avoid implying that maximum shear stress is directly comparable to measured solid pressure. Where appropriate, we will also discuss the isotropic component of the simulated cell stress tensor as a more direct comparison to pressure-based measurements.
(4) It's worth pointing out that stress fibers aren't really prominent in cells in 3D spheroids. Nonetheless, cells moving on collagen fibers would have stress fibers and utilize contractile actomyosin bundles to generate traction forces.
We thank the reviewer for this clarification. We did not intend to imply that prominent stress fibers are generally present in cells within the interior of 3D spheroids. The relevant statements in the manuscript were meant to refer to strained boundary cells or cells engaging collagen fibers during mesenchymal-like motion. We will revise the wording in Secs. II.C and II.D to make this distinction explicit and avoid suggesting that bulk spheroid cells generally contain prominent stress fibers.
(5) In section 2D, it talks about the result that as the kcc associated with the boundary cell is decreased 10-fold for every 5 percent strain decrease in the fiber target spring length, can this result be shown? I have a hard time seeing where this came from.
We thank the reviewer for this comment. The 10-fold decrease in kcc for every 5% decrease in the fiber target spring length was meant as a phenomenological adhesion-weakening protocol, not as a directly measured law. We agree that this was not made clear enough. In the revised manuscript, we will explicitly state this.
(6) The results of single-cell vs. two-cell breakouts shown in Figure 5 b and c are very qualitative and should be accompanied by some quantitative comparison.
We thank the reviewer for this helpful suggestion. We agree that the current presentation of Fig. 5b,c is too qualitative. In the revised manuscript, we plan to add a quantitative comparison between the single-cell and two-cell breakout cases. Specifically, we plan to track the displacement of the pulled boundary cell, the separation between this leader cell and its neighboring/follower cell, and the distance between the follower cell and the remaining spheroid as the fiber target length is decreased.
Reviewer #3 (Public review):
Summary:
The authors describe a mathematical and computational approach used to compute stresses and cellular deformations in a multicellular spheroid embedded in a fiber network. This approach is then used to predict stress and cellular anisotropy distributions in "solid-like" and "fluid-like" spheroids. Simulations show that shear stresses in solid-like spheroids are large and concentrated at the boundary of the spheroid, yet cells do not align with the direction of the largest shear. Conversely, shear stresses in fluid-like spheroids are smaller and uniformly distributed in the spheroid. In this case, cellular elongation is more likely to be aligned with the direction of the largest shear stress. The model and simulations also predict a nonlinear stress-strain relationship that is indicative of strain stiffening. This strain-stiffening is more pronounced in fluid-like spheroids. In an extension of the preliminary polyhedral vertex model, in which cellular interfaces are shared, the authors incorporate mechanical cell-cell interactions via adhesion springs between neighboring vertices. Using this extension, they show that cell breakout is more likely to occur in fluid-like spheroids, where cells are more likely to elongate and stiffen, allowing for larger forces to be exerted on the surrounding fiber network. Furthermore, the authors state that anisotropic cellcell adhesion is required for multicell streaming during breakout.
Strengths:
The modeling and computational approach used in this research is this work's biggest strength. Treating the embedded spheroid as a set of polyhedra, where each polyhedron represents a single cell, is a mechanically robust, yet still tractable way to model multicellular spheroids in three dimensions. Starting with expressions for constraining cell volume and surface area as well as a surface energy term, the authors derive an expression for an averaged stress tensor for each polyhedron. This allows the authors to approximate the stress in each polyhedral cell that is caused by cellular deformations during mechanical interactions with the extracellular fiber matrix. This is a clever and robust approach that is based on fundamental mechanical principles that allow one to make reasonable predications about the mechanical state of the spheroid under a variety of conditions.
Weaknesses:
The weakness of the manuscript is the exposition. There are significant pieces of critical information missing from the manuscript that would make the presented work significantly more understandable and better support the authors' claims. Most importantly, many necessary details of the model are missing. I was able to get a better understanding of some of these details by reading the authors' earlier work (ref [10] in the submitted manuscript), and for this reason, I do feel that this work has value. However, several descriptions must be added for the paper to be more readily understandable.
These include
(1) A better explanation of what drives motion, in particular in the case where no external fiber network is present.
We thank the reviewer for pointing this out. We agree that the source of motion should be described more clearly. In the embedded simulations, motion arises from overdamped dynamics driven by the forces from the total mechanical energy, including spheroid mechanics, fibernetwork elasticity, and active contractile linker springs at the boundary. The shortening of the linker-spring target lengths provides the active cell–ECM pulling, while effective fluctuations promote cell-shape fluctuations and rearrangements.
When no external fiber network is present, these linker-mediated cell–ECM forces are absent. The spheroid then evolves only through vertex-model mechanical relaxation, surface tension, cell rearrangements, and effective fluctuations. We will clarify that this no-network case is a control for the intrinsic spheroid stress state, not a simulation of ECM-driven invasion.
(2) What physically distinguishes fluid-like spheroids from solid-like spheroids? Simply stating the value of the parameters s0 with no explanation is not sufficient.
We thank the reviewer for pointing out that the physical distinction between solid-like and fluid-like spheroids was not sufficiently explained. We agree that simply stating the values of s_0 is not adequate.
In this 3D vertex model, the target shape index s_0 controls the mechanical cost of cell rearrangements. Below the rigidity transition (s_0 < s_0^), neighbor exchanges are associated with finite energy barriers, leading to slow structural relaxation and solid-like behavior. Above the transition (s_0 > s_0^), these barriers become very small or vanish, allowing cells to readily move past one another and continuously reorganize their local neighborhood structure. The resulting tissue exhibits fluid-like behavior with efficient stress relaxation through cell rearrangements.
This distinction was characterized in detail in Ref. [9], where the bulk 3D vertex model was shown to undergo a rigidity transition at approximately (s_0^*=5.39), based on the decay of the neighbor-overlap function and cell trajectories. The solid-like value used here lies below this transition, whereas the fluid-like value lies above it. We acknowledge that the present manuscript only briefly summarized this point, mainly in Supplementary Material A. In the revised manuscript, we will add a clearer explanation in the main text of how the target shape index controls the state of the spheroid and why the selected values correspond to solid-like and fluidlike regimes.
(3) An explanation of how histograms in Figure 2 are calculated is necessary. Are these histograms based on one simulation or several simulations?
We thank the reviewer for pointing out that this was not sufficiently clear. The histograms in Fig. 2 are obtained by pooling cell-level quantities from multiple independent simulations, not from a single realization. As listed in Table I, we use 30 independent realizations. We plan to state this explicitly in the revised figure caption and main text.
(4) The experimental results are briefly mentioned, but significantly more connection between these results and the numerical results of the cell breakout model is needed.
We agree. In the current manuscript, the experimental data are used mainly to motivate the single-cell and streaming-like breakout modes shown in Fig. 5. We plan to revise Sec. II.D and the Fig. 5 caption to make the connection more explicit: the MEF spheroid experiments show the invasion modes that motivate the model, while the extended vertex model tests minimal mechanical ingredients capable of producing analogous single-cell and follower-cell breakout.
(5) The description of the model that incorporates variable cell-cell attachments and cell breakout is very terse and needs more detail. Moreover, while the description of the results of this model is strong, the figure that illustrates cell breakout (Figure 5) is difficult to interpret. Addressing these and other issues will make the current manuscript, which presents an interesting model and result, much stronger and easier to read.
We thank the reviewer for this constructive assessment. We agree that the extended model with variable cell–cell attachments was described too tersely and that Fig. 5b,c was difficult to interpret in its current qualitative form.
To make Fig. 5 more quantitative, we plan to add measurements comparing the single-cell and two-cell breakout cases. Specifically, we plan to track the displacement of the pulled boundary cell, the separation between this leader cell and its neighboring/follower cell, and the distance between the follower cell and the remaining spheroid as the fiber target length is decreased.
