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Motility-induced fracture reveals a ductile-to-brittle crossover in a simple animal’s epithelia

Nature Physics volume 17, pages 504–511 (2021)Cite this article

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Abstract

Epithelial tissues provide an important barrier function in animals, but these tissues are subjected to extreme strains during day-to-day activities such as feeding and locomotion. Understanding tissue mechanics and the adaptive response in dynamic force landscapes remains an important area of research. Here we carry out a multi-modal study of a simple yet highly dynamic organism, Trichoplax adhaerens, and report the discovery of abrupt, bulk epithelial tissue fractures induced by the organism’s own motility. Coupled with rapid healing, this discovery accounts for dramatic shape change and physiological asexual division in this early-divergent metazoan. We generalize our understanding of this phenomenon by codifying it in a heuristic model focusing on the debonding–bonding criterion in a soft, active living material. Using a suite of quantitative experimental and numerical techniques, we demonstrate a force-driven ductile-to-brittle material transition governing the morphodynamics of tissues pushed to the edge of rupture.

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Fig. 1: Elastic–plastic shape-change phenomena in Trichoplax adhaerens.
Fig. 2: Physiological epithelial tissue fractures in the T. adhaerens at a cellular resolution.
Fig. 3: A heuristic in silico tissue model captures yielding and ductile–brittle transitions.
Fig. 4: Organismal motility-induced forces cause local tissue fractures in T. adhaerens.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The computer codes used in this paper are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank A. Bhargava and N. Eiseman for their contributions in the initial stages of the project, and S. Armon, P. Vyas, H. Soto-Montoya, W. Gilpin, H. Vaidehi Narayanan and members of the Prakash Lab at Stanford University for useful suggestions. We thank V. Chikkadi, S. Bandyopadhyay, S. Shekhar, N. Clarke and C. Lowe for helpful scientific discussions; K. Uhlinger for help with marine algae cultures; L. Buss for the generous gift of animals; R. Konte for help with the illustrations, C. Espenel, A. Khan and J. Mulholland (Cell Sciences Imaging Facility (CSIF), Stanford University) for help with microscopy. M.S.B. was supported by the National Science Foundation Graduate Research Fellowship (DGE-1147470) and the Stanford University BioX Fellows Program. This work was supported by NSF CCC grant (DBI-1548297) (M.P.), the CZI BioHub Investigator Program (M.P.) and the Howard Hughes Medical Institute (M.P.).

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Author notes

  1. Vivek N. Prakash

    Present address: Department of Physics, University of Miami, Coral Gables, FL, USA

Authors and Affiliations

  1. Department of Bioengineering, Stanford University, Stanford, CA, USA

    Vivek N. Prakash & Manu Prakash

  2. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Matthew S. Bull

  3. Chan Zuckerberg Biohub, San Francisco, CA, USA

    Manu Prakash

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Contributions

All authors designed the research. V.N.P. performed the experiments with assistance from M.S.B. and M.P.; V.N.P. and M.S.B. analysed the data; M.S.B. developed the theoretical framework and performed simulations. All authors wrote and reviewed the manuscript.

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Correspondence to Manu Prakash.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Gijsje Koenderink and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Quantification of organismal shape change and area.

A, (i) Time-lapse images of the asexual reproduction process; the animal changes shape from a more circular to an elongated shape. (ii) Quantification of these shape changes: Circularity (4π(Area)/(Perimeter2)) decreases over time. B, (i) Time-lapse images of ventral epithelial fractures and healing; holes appear and disappear (heal). (ii) Quantification of organismal area and fracture hole area. The fracture hole (red) opens up for less than half an hour. The organismal area (blue) fluctuates due to the folding and unfolding of the animal, but the maximum value remains the same over ~10 hours, indicating that the growth timescales via cell division are much longer.

Extended Data Fig. 2 Ventral and dorsal epithelial fractures in T. adhaerens.

A, Ventral hole: All imaging channels corresponding to Fig. 2b (iv) are shown separately (cell membrane: green, acidic granules in lipophil cells: red). The bright-field channel reveals that the dorsal layer is still intact. B, Dorsal hole: (i) The brightfield channel corresponding to (ii) the dataset shown in Fig. 2d. (iii) All the imaging channels corresponding to third panel in (ii) are shown separately (cell membrane: green, acidic granules in lipophil cells: red). The dorsal fractures are through holes inside the bulk tissue of the animals. The ventral fracture event released broken fragments of ventral epithelial cells and lipophil cells (and their acidic granules), and since the dorsal epithelium is sticky, these granule fragments got stuck on the dorsal surface. This is the reason we see acidic granules on the dorsal epithelium in (iii).

Extended Data Fig. 3 Thread formation, thread fractures and ruptures.

A, The stretched tissue between two regions of an animal that are pulling apart yield in a ductile fashion and form thread-like regions by undergoing local ‘thread fractures’. Here, the subtle difference is in the location of fractures; ventral and dorsal epithelial fractures typically occur in the bulk, whereas these thread fractures occur on the edges. Functionally, these thread fractures are cumulative over time, resulting in rapid reduction (~ few min) in the cross-sectional area and lead to extremely thin (few cell layers thick) threads, which ‘rupture’ (~ few minutes) when pulled apart continuously. This mechanism plays an important part in the asexual reproduction process in these animals (Fig. 1b). B, Zoom-in versions of (A), with additional brightfield channel overlay on the images. C, Zoom-in version of panel in (B (iv)) highlighting breaking of the thread.

Extended Data Fig. 4 Ventral and dorsal fracture hole initiation.

A, Ventral fracture holes initiate as micro-fractures that coalesce to form larger holes, which are brittle-like with jagged edges and broken cellular fragments (The different imaging channels are shown separately). B, Dorsal fracture holes initiate as circular holes, which grow in size and remain smooth, indicative of a more ductile dorsal tissue (The different imaging channels are shown separately to facilitate comparison).

Extended Data Fig. 5 Analysis of the metastable states.

A, Key experimental observations: (i) Cells within the ventral layer are a bi-disperse mixture of ventral epithelial cells and lipophil cells. The tissue is also not perfectly confluent with gaps at the ventral surface. (ii) Digital zoom of brightfield 1 μm deep from the ventral surface shows bi-dispersity of cell sizes and shapes (lipophil cell - false colour red, and ventral epithelial cells (VEC) - green). B, Schematic cartoon illustration of model: chosen to simulate the experimentally observed bi-disperse mixture of ventral epithelial cells and lipophil cells shown in (A). The in-silico tissue model is a sticky ball-spring model, where the balls represent cells and the springs represent adhesion bonds between the cells.

Extended Data Fig. 6 (A) Model phenomena in metastable state.

(i) Close up of a quenched state of the sticky-ball model where packing fraction is calculated with a simple Monte Carlo solver using blue colour regions between cells. (ii) Hexatic order - structural order measurement that reveals short range correlations of hexatic orientation for this bi-disperse system. (iii) Residual forces represented spatially through force chains demonstrates rich spatial heterogeneity at the metastable state. Thickness of force chains corresponds to magnitude with red representing force under tension and blue force under compression. B, Phase diagrams in metastable state: We explore a 2D cross-section of the model’s parameter space with three diagnostic quantities: (i) 1 minus the packing fraction. (ii) Residual forces, and (iii) hexatic order. These phase spaces demonstrate how the metastable state properties can be tuned across a wide range of parameter configurations of bidispersity in the epithelial alloy, assuming that the results are independent of micro-configurations. We vary two parameters controlling tissue bidispersity: (i) % Larger cells, and (ii) Size ratio of large cells. In (i) we highlight three important cases (A, B and C) that we will consider in detail in the figures below.

Extended Data Fig. 7 Phase diagrams with uniform forcing.

Pulling the heuristic model out of equilibrium for given parameters reveals two distinct behaviours: (i) a direct yielding through new edge formation and (ii) an intermediate yielding through reshuffling of the cell-cell junction network via higher order STZs. A, First, we consider the parameter of the maximum junctional length before failure, at fixed maturation time, and bi-dispersity case A from Extended Data Fig. 6B (12% larger cells, 1.7 size ratio). At small lbreak the model yields to new edge formation at smaller loading. However, at a crossover ~ lbreak = 1.7, for low loading gradients, the dominant yielding occurs through coordinated neighbour exchanges similar to STZs. This crossover value can be understood as the edge-forming energy exceeding the average STZ energy barrier (which is independent of lbreak). The subsequent three panels in (A)ii-iv, are each one channel of the RGB parameter space displayed on the left. From left, red is the number of STZ transformations (or CCC), green is the number of edge forming transformations (or non-CCC), and blue is a helpful diagnostic, the packing fraction, for identifying holes within the tissue (light blue is low packing). B, Considering model tissues at different maturation times (at fixed threshold breaking length, and bi-dispersity case A from Extended Data Fig. 6B (12% larger cells, 1.7 size ratio) reveals a different crossover between a more ductile and a more brittle behaviour under distributed driving. At short maturation times, the formation of a new cell-cell junction is nearly instantaneous, so in a regime of lbreak = 1.95, the yielding is driven through STZ-like (or CCC) transformations. However, when the maturation time reaches ~ 10x the dynamical timescale, the cell-cell junctions are too slow to reform and the tissue yields through new-edge formation and holes. This rapid de-lamination can be seen through the split channels (B)ii-iv in the reduction of STZs, the increase in new edges and the reduction of the packing fraction. C, To study the steady-state dependence of the model’s Herschel-Buckley scaling on τmature, we complete a series of fits (ii) in the extension versus time curve. The asymptotic value of this fit represents the long-time behaviour of this yielding rate. Plotting the strain rate at steady-state versus the applied stress reveals a Herschel-Buckley-like relationship with a finite yield stress and a 3 scaling behaviour. Over the range explored τmature has only limited impact on the flow behaviour of this tissue.

Extended Data Fig. 8 Phase diagrams at different bi-dispersity cases.

A, Case B from Extended Data Fig. 6B (12% larger cells, 1.1 size ratio), (B) case C from Extended Data Fig. 6B (6% larger cells, 2 size ratio). There are only subtle changes in the ductile–brittle transition at the different bi-dispersity cases.

Extended Data Fig. 9 Non-affine motion analysis.

Leveraging the knowledge of the microstate in numerics to develop techniques for learning about the material properties of real tissue from experimental signatures. A, We employ a measure of non-affine motion called \({D}_{min}^{2}\). This measure subtracts actual motion of particles from the local affine transformation inferred from its neighbours’ motion (ii). Large values of disagreement between actual motion and the affine projection are signaled in red (i). B,(i) We display the strain rate field calculated from interpolation of the displacements into a grid. Colour corresponds to orientation angle of the vector ranging from 0 to 2π. (B)(v) displays the pair correlation function showing rapid attenuation of the characteristic spacing over distances greater than 0.4 simulation units. The red dotted line signals the neighbourhood size used to calculate the metric in Fig. 3. Panels (B)(ii-iv) demonstrate the effect of the neighborhood size with with 50% variation on either side of that shown in the figure (B)(iii). Above is 50% smaller neighbourhood and below is 50% larger. The trade-off between sensitivity and spatial resolution is balanced by choosing the middle values B(iii). (B)(vi) shows the map between known simulation STZs and the measured \({D}_{min}^{2}\). B(vii) We find that this measure is a nice diagnostic for otherwise unobservable motions with 80% of the observed large \({D}_{min}^{2}\ge 1{0}^{-3}\) corresponding to a known neighbourhood exchange via a STZ-like transformation.

Extended Data Fig. 10 Experimental time lapse images of tensile fractures.

Here, we display more information on the same dataset as in Fig. 4(a). A, Time-lapse images showing the evolution of tensile-induced ventral fractures and their healing over a period of ~ 25 mins. B, PIV-derived velocity magnitude contours (with mean speed subtraction) of data in panel (A) at t = 12s. The fracture hole opens up in a dead velocity zone in the middle of the tissue being pulled from above and below - indicating a tensile force- induced fracture (blue color indicates zero velocity zones). C, Time-series plot of the fracture hole area in (A). The tension-induced fracture is an extremely fast process, with the hole reaching its maximum area in about 2 mins. The healing process is completed at about 12 mins. The green markers indicate timepoints corresponding to the snapshots displayed above.

Supplementary information

Supplementary Information

Supplementary Text, Methods, Discussion, Video descriptions, references and Figs. 1 and 2.

Supplementary Video 1

Asexual reproduction by fission in Trichoplax adhaerens. Time-lapse quasi-dark field imaging of animals in lab culture conditions using a digital single-lens reflex (DSLR) camera. A single animal ‘splits into two’ by a binary fission process in about one hour. Video playback is sped up, and time stamp represents hours and minutes. Scale bar, 3 mm.

Supplementary Video 2

Physiological tissue fractures in the ventral epithelium of Trichoplax adhaerens. Time-lapse quasi-dark field imaging of animals in lab culture conditions using a DSLR camera. The ventral epithelium sustains fracture holes, which heal completely in about one hour. Video playback is sped up, and time stamp represents hours and minutes. Scale bar, 1 mm.

Supplementary Video 3

Physiological tissue fractures in the dorsal epithelium of Trichoplax adhaerens. Time-lapse quasi-dark field imaging of animals in lab culture conditions using a DSLR camera. The dorsal epithelium sustains fracture holes, which grow in size and do not heal. These animals eventually become long string-like animals over about seven hours. Video playback is sped up, and time stamp represents hours and minutes. Scale bar, 3 mm.

Supplementary Video 4

Ventral tissue fractures at a cellular resolution. Time-lapse confocal imaging of animals in an open dish configuration matching native culture conditions. The ventral epithelium is tagged with a fluorescent cell membrane dye (green), and a lysotracker dye (red) that labels acidic granules in lipophil cells. Videos are looped over ~5 s, and the time stamps on images represent minutes and seconds. Scale bar, 50 μm.

Supplementary Video 5

Ventral tissue fracture healing. Time-lapse confocal imaging of animals in an open dish configuration matching native culture conditions. This video is a continuation of the previous Supplementary Video 4, and illustrates the tissue fracture healing process in the same animal. Videos are looped over 10 s, and the time stamps on images represent minutes and seconds. Scale bar of images, 50 μm.

Supplementary Video 6

Thread rupture at a cellular resolution. Time-lapse confocal imaging of animals in an open dish configuration matching native culture conditions. This video illustrates the final stages of the thread rupture process going all the way down to the breaking of a single cell–cell junction. The video corresponds to the same image sequence presented in Extended Data Fig. 3c. The fluorescence and brightfield channels are overlaid for clarity. Video playback is real time, and time stamp represents minutes and seconds.

Supplementary Video 7

Model results with steady pulling. We display the phase diagram from our heuristic tissue model, which explores a parameter sweep of steady force gradient versus the threshold strain for breaking cell–cell bonds. Next, we sequentially show simulations that demonstrate cases of elastic, ductile and brittle tissue properties—and their corresponding parameters on the phase diagram. The time stamps represent simulation units.

Supplementary Video 8

Model results with unsteady pulling. We show simulations with unsteady pulling of the model tissue, and demonstrate how this captures both fractures and healing. The time stamps represent simulation units.

Supplementary Video 9

Tension force-induced brittle fracture in Trichoplax adhaerens. Time-lapse fluorescence microscopy imaging reveals a tensile force-induced fracture. The ventral epithelium is tagged using a lysotracker dye, which labels acidic granules in lipophil cells. Video playback is real time, and time stamp represents minutes and seconds. Scale bar, 0.5 mm.

Supplementary Video 10

Shear force-induced brittle fracture in Trichoplax adhaerens. Time-lapse fluorescence and bright field microscopy imaging reveals a shear force-induced fracture in the ventral epithelium. The dorsal epithelium is tagged using 0.5 µm sticky, fluorescent microbeads. A PIV analysis is carried out to quantify the internal tissue velocity fields (highlighted by green arrows). Video playback is sped up, and time stamp represents minutes and seconds. Scale bar, 1 mm.

Supplementary Video 11

Non-affine motion analysis on experimental data. Time-lapse fluorescence and bright field microscopy imaging reveals a shear force-induced fracture in the ventral epithelium. The dorsal epithelium is tagged using 0.5 µm sticky, fluorescent microbeads. A particle tracking analysis is carried out to quantify the non-affine motion of the microbeads (with magnitude highlighted by colours). Video playback is sped up, and time stamp represents minutes and seconds. Scale bar, 1 mm.

Supplementary Video 12

Correlation between non-affine motion and internal strain rate, in experimental data. Time-lapse fluorescence and bright field microscopy imaging reveals a shear force-induced fracture in the ventral epithelium. The dorsal epithelium is tagged using 0.5 µm sticky, fluorescent microbeads. Larger values of non-affine motion are thresholded and overlaid (white dots) on contours (jet colour scale) of the internal strain rate calculated from the PIV analysis. Video playback is sped up, and time stamp represents minutes and seconds. Scale bar, 1 mm.

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Prakash, V.N., Bull, M.S. & Prakash, M. Motility-induced fracture reveals a ductile-to-brittle crossover in a simple animal’s epithelia. Nat. Phys. 17, 504–511 (2021). https://doi.org/10.1038/s41567-020-01134-7

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