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Direct PNAS a is article This interest. of conflict no and declare data; authors analyzed The paper. M.C. the wrote and M.C.M. D.B., and X.Y., M.L.M., research; D.B., performed X.Y., M.C.M. and M.L.M., M.M., uhrcnrbtos ...... n ...dsge eerh .. .. M.C., D.B., X.Y., research; designed M.C.M. and M.L.M., D.B., X.Y., contributions: Author owo orsodnesol eadesd mi:xingbo Email: addressed. be should correspondence whom To faayigtato oc irsoydt htmyprovide may rheology. that tissue on data information microscopy force way traction unique analyzing a at of suggests diverges finding that can This viscosity transition. stresses tissue liquid–solid effective shear the an of define to function used correlation be show we temporal particular, the In properties. that and material tissues to in stress stresses of relates notions framework distinct unifying con- seemingly a existing, provides cortex for work mechanical by Our the tissue. controlling tuned of in properties interplay shape, adhesion, the cell cell–cell and and examine tractility motility we to cell transition, known between epithelia liquid–solid of a model predict Voronoi self-propelled a Using Significance hr ieo eerhhsfcsdo h oesai pres- homeostatic the on focused has research of line third A also methods inference mechanical of set important second A c .Ls Manning Lisa M. , PNAS b eateto hsc,Nrhatr nvriy otn MA Boston, University, Northeastern Physics, of Department | NSlicense. PNAS Drosophila oebr2,2017 28, November c n .Citn Marchetti Cristina M. and , igds 2,2) hr cellular where 26), (23, disk wing . | o.114 vol. www.pnas.org/lookup/suppl/doi:10. | o 48 no. [email protected]. | 12663–12668 c

BIOPHYSICS AND COMPUTATIONAL BIOLOGY So far, there is no unifying theory for these seemingly distinct retical prediction suggests that TFM combined with mechanical notions of pressure and stress or their relationship to material inference can provide rheological information about the tissue properties. In this paper, we show that a recently proposed self- and could be tested by a new analysis of experimental data. propelled Voronoi (SPV) model of epithelia (Fig. 1) (15) pro- vides a natural framework for unifying these ideas. One of the Results and Discussion benefits of the SPV model is that it explicitly accounts for the SPV Model. The SPV model describes an epithelium as a net- forces that motile cells exert on the substrate. This allows us to work of polygons. Each cell i is endowed with a position vec- develop a generalized mechanical inference method to infer cel- tor ri , and cell shape is defined by the Voronoi tessellation of lular stresses from traction forces and to show that these match all cell positions (Fig. 1), which has been shown to provide a the stresses calculated from instantaneous cell shape, relating good representation of some real epithelia, such as the blasto- TFM data and mechanical inference techniques in motile tissues. derm of the red flour beetle Tribolium castaneum and the fruit Additionally, our method provides absolute values for junctional fly Drosophila melanogaster (32). Like for vertex models, tissue tensions and pressure differences. This is in contrast to equi- forces are obtained from an effective energy functional E({ri }) librium mechanical inference, which yields only relative forces for N cells, given by (7, 15, 26, 33, 34) (21, 23). N There are two additive contributions to the mechanical stress X 2 2 that describe the forces transmitted in a material across a bulk E = Ei , Ei = KA(Ai − A0) + KP (Pi − P0) , [1] plane. The first one represents the flux of propulsive forces i=1 through a bulk plane carried by particles that move across it. with Ai and Pi the cross-sectional area and perimeter of the The second one describes the flux of interaction forces across ith cell, respectively. The first term in Eq. 1 arises from incom- a bulk plane. We demonstrate that the generalized mechanical pressibility of the layer in three dimensions and its resistance to inference measurements probe the latter, which we denote inter- height fluctuations, with A0 a preferred cross-sectional area. The action stresses. The former, which we denote the tissue swim second term represents the competition between cortical ten- stress, approximates the contribution from cell motility to the sions from the actomyosin network at the apical surface and cell– osmotic pressure generated by cells immersed in a momentum- cell adhesions from adhesive complexes at intercellular junctions conserving solvent on a semipermeable piston and hence to (26), with P0 a preferred perimeter resulting from this compe- the tissue homeostatic pressure. The tensorial sum of the swim tition. We simulate N cells in a square box of area AT , with stress and the interaction stress is the total stress. The normal A¯ = AT /N the average cell area and with periodic bound- component of the total stress determines whether a tissue will ary conditions. The system is initialized with a set of N random tend to exert extensile or contractile forces on its environment, cell positions, independently drawn from a uniform distribution. which is an important consideration in wound healing and cancer Throughout the simulations, we set A¯ = A0 = 1 unless other- tumorigenesis. wise noted, and KA = KP = 1. An obvious open question, then, is how these stresses vary as Each Voronoi cell is additionally endowed with a constant self- a function of material properties. We find that the normal com- propulsion speed v0 along the direction of polarization nˆi = ponent of the interaction stress is contractile in both the solid (cos θi , sin θi ) describing cell motility. The dynamics of each and the liquid due to the contractility of the actomyosin cortex, Voronoi cell are governed by although much more weakly so in the liquid state. In contrast, the √ normal component of the motility-induced swim stress is always ∂t ri = µFi + v0nˆi , ∂t θi = 2Dr ηi (t), [2] extensile, corresponding to a positive swim pressure, although its magnitude depends on the phase: In a solid the swim pres- where Fi = −∇i E is the force on cell i and µ is the mobility. The direction of cell polarization is randomized by orientational noise sure is negligible, while in the fluid it can be significant. This can 0 0 result in a change in sign of the total mean stress: Indeed, we of rate Dr , with hηi (t)i = 0 and < ηi (t)ηj (t ) >= δij δ(t − t ). find it is always contractile in the solid state but becomes exten- The timescale τr = 1/Dr controls the persistence of single-cell sile deep in the liquid state when cell motility exceeds actomyosin dynamics. As in self-propelled particle (SPP) models, an iso- lated cell performs a persistent random walk with a long-time contractility. 2 Because the transition from contractile to extensile does not translational diffusivity D0 = v0 /(2Dr ) (29, 35, 36). After each coincide with the fluid to solid transition, it is natural to ask time step, a new Voronoi tessellation is generated based on the whether the stress displays any signatures of the fluid–solid tran- updated cell positions. The cell shapes are determined in the pro- sition. We develop a definition for the effective viscosity of the cess and the exchange of cell neighbors occurs naturally through tissue that can be extracted from the temporal correlation of the topological transitions (13). interaction shear stress and find that it diverges as the tissue tran- We showed in ref. 15 that the SPV model exhibits a transi- sits from the liquid state to the solid state. Importantly, this theo- tion from a solid-like state to a fluid-like state upon increas- ing the single-cell motility√v0, the persistence time τr , or the cell shape parameter P0/ A0 that characterizes the competi- tion between cell–cell adhesion and cortical tension. The phase diagram in the (P0, v0) plane is reproduced in Fig. 1B√. The tran- sition is identified by setting the shape index q = hPi / Ai i to the value q = 3.813, where h...i denotes the average over all cells. It was shown in ref. 15 that the transition line located by q = 3.813 coincides with the one based on the vanishing of the effective diffusivity obtained from the cellular mean-square displacement. Note that for fixed system size AT and cell number N , the pre- ferred cell area A0 does not affect the interaction forces or cel- lular shapes. Hence the solid–fluid transition is insensitive to A0, as shown analytically in SI Text. The preferred area A0 only shifts the total pressure of the tissue by a constant.

Fig. 1. (A) Illustration of the SPV model, where cells are represented by Developing and Validating Traction-Based Mechanical Inference. It polygons obtained via a Voronoi tessellation of initially random cell posi- is well established that in a model tissue described by the energy tions, with a self-propulsion force applied at each cell position. (B) Phase dia- 1 i gram in the (P , v ) plane based on the value of the shape parameter q (color Eq. , the mechanical state of cell is characterized by a local 0 0 (i)int scale), with the phase boundary (red crosses) determined by q = 3.813. stress tensor σαβ given by (21, 23, 37)

12664 | www.pnas.org/cgi/doi/10.1073/pnas.1705921114 Yang et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 cinsrs rmisatnoscl hp utain through fluctuations shape cell Eq. instantaneous from stress action ainadipiil eedo elmtlt,prmtrzdby parameterized motility, cell on depend speed implicitly and lation 30), (29, colloids active on work recent following where, Eq. contract, to tends it neighbors, within its network cell. actomyosin from the the of off contractility cut means the with is stress consistent cell Contractile contractile extensile. the is is if cell conven- cell that the the the when used when positive have negative is We and stress cells. cellular two the by that Eq. shared tion of is side edge each right-hand that the on term in second shown as clockwise, versed vertices joining vector Eq. summa- in The components. tion Cartesian denote indexes Greek while indexes with tension, cell-edge the age al. et Yang more likely and known not is exper- complicated. functional in energy use the limited where of iments is method this numerically, implemented Both where of values instantaneous the Eq. contribution by motility The given cell cells. the nonmotile to for proportional vanishes is and stress exten- swim for negative The and stress. stress sile contractile for positive interaction is the as i.e., convention stress, same the follows stress Eq. swim in the sign that negative The expression. virial a from culated force propulsive of flux the describes contributions as of forces sum propulsive the and not as dynamics interactions written be from the do can stress by configurations cellular governed local the cellular are Eq. but where by energy, described cells, tissue motile the minimize of layer a model. (SI Voronoi Eq. the small use the are following because differences the in the the model and that in Text) verified Voronoi have present We the not model. for constraints vertex introduces case construction the Voronoi not however, is, where elsae ocluaeteitrcincontribution interaction Eq. definitions, the calculate to from tensions shapes simple and cell is pressures it instantaneous known, the is extract functional directly to energy the where simulations In Text) (SI as tissue the in stress by governed dynamics system the Eq. by but minimization, energy by Π σ oeta navre oe h neato tesa endin defined as stress interaction the model vertex a in that Note u oli ooti h itiuino ellrsrse in stresses cellular of distribution the obtain to is goal Our i αβ int rmtelclsrse n a hnoti h oa mean total the obtain then can one stresses local the From 2. 3a 3a = A v = iligwa ecl hp-ae tess hl easily While stresses. shape-based call we what yielding , Π −2K ab 0 side h tesatn ntetsu onay This boundary. tissue the on acting stress the indeed is i i i n persistence and A , and stehdottcclua rsueand pressure cellular hydrostatic the is stecl–elitraeta eaae cells separates that interface cell–cell the is j 1 T , A 3a k 3a (A X σ ... , P i αβ usoe l de fcell of edges all over runs ( u eed mlctyo elmtlt through motility cell on implicitly depends but , give 3b, i i i ) nti ae sdsusdi h Introduction, the in discussed as case, this In 2. Π int A − r banda vr iese ftesimu- the of step time every at obtained are olblclsand cells label to i i σ A σ = = σ αβ i αβ ( 0 αβ ( i a ) i ) −Π − ) int , = and swim τ ∂ ∂ T r σ , σ A i hsmto ietyifr h inter- the infers directly method This . E ˆ l ab δ Π ab αβ ( b i αβ i = ) i , 2 = h atrof factor The S1A. Fig. αβ hnteprmtro cell of perimeter the when swim int σ = and + − αβ + 3a K l µA 2A ab T v = = T P 1 σ 0 ab sago prxmto for approximation good a as /|l v a αβ i µ [(P ( ab i σ 0 i A , n ) = αβ ab int n ab X 1 b swim α i T htaedtrie not determined are that i j , ∈i eew s Roman use we Here |. r cosabudr,cal- boundary, a across ∂ c − ∂ + β X i i ... , l T and , E 3a i ab , P σ ab α αβ 0 swim , A sdet h fact the to due is olblvertices, label to ( + ) l ab β i T σ l , ab αβ with , ( ab i σ ) P swim σ αβ ( steedge the is = k i αβ 1/2 ( ) i 5 int − ) T j int ensures . i ab and P nthe in sstill is stra- is The . ˆ 0 l [3b] [3a] ab )], [7] [6] [4] [5] v k is 0 . qiiru ehnclifrnemtosepesteinterac- the force express tion methods inference mechanical Equilibrium pi nnra,ser n omldfeec opnns Below be components. similarly difference normal can and contributions shear, swim normal, in and split interaction the of Each stress stress local normal of and terms σ in mean expressed usefully the most Both are components dimensions. two in components rheolog- the tissue. characterize the to of used properties be as ical can such TFM Eq. measurements by liq- mechanical in provided the Thus, those defined in states. solid properties as the mechanical tissue, and distinct uid correlations the displays tissue temporal in the the stress that conditions, examining mean boundary By the periodic model. of sim- with SPV we layer the tissues, cell using confluent confluent motile a of ulate properties Tissue. the mechanical of the Properties Rheological Characterizes Stress 2 (Fig. simulations com- the resulting stresses from the shape-based exactly the puted that with showing agree by stresses traction-based method coarse-grained the dated (12). cell epithelium (MDCK) within kidney pressures and (25), derm junctions include cell Examples at decom- compati- bodies. be tensions is can into interactions and cell–cell assumptions posed whose such epithelia any make mechani- with isotropic, not ble The an does (18). is inference material tissue cal elastic (MSM), the linearly that microscopy and assumption stress homogeneous, the monolayer upon rests intercellu- with which the obtained Eq. from stress in different from lar generally given from inferred is stress inference traction-based is stresses the mechanical that procedure emphasize to We stresses. coarse-graining refer based the We of Text. outline An tensions grid and a Pressures with diameter. cell grid, a square of a (Π order over the of averaged acces- spacing experimentally forces experi- uses traction in that approach realistic sible this not of coarse- a again version implemented grained and is developed have which we Therefore, vertex, ments. each at tractions Text). mechan- (SI the inference for the ical minimization requires least-squares a which of overdetermined, implementation system the rendering Eq. ables, using yields calculated to then counting contribution is straint stresses interaction cellular The local unknowns. the independent of ber tensions tensions cortical edge the where force the invert we equations cells balance motile of layer epithelial equations nonequilibrium the inverting by cellu- obtained the of (Eqs. geometry measured network the lar and tensions edge and vertex sures the to respect vertex with each energy at tissue force the traction the of position define gradient segmented TFM. we the by model, as from obtained SPV forces pressures traction the and In and boundaries cell tensions of images infers in mechan- inference accessible traction-based is ical proposed that the information Specifically, only the experiments. approximate using to stresses attempts that interaction monolayers motile for method ina h etro ahgi element, grid each of center the at tion n h testensor stress The vali- have we model, SPV the of framework the Within h qain eeoe hsfrrqiekoldeo the of knowledge require far thus developed equations The inference mechanical new a develop we reason this For ha stress shear , i , T i ) r hncluae yivrigtefreblneequa- balance force the inverting by calculated then are t a σ F Drosophila n = a ,d = PNAS −∇ = σ −∇ T s 2 1 .Pesrsadeg esosaethen are tensions edge and Pressures S6–S8). n omlsrs difference stress normal and , i a F (σ fajcn el (Eq. cells adjacent of σ E | grid a αβ F xx hc aacsteitrcinforce interaction the balances which , E oebr2,2017 28, November a igds 2) n ai–ab canine Madin–Darby and (26), disk wing 4N ({Π ({Π ssmercadhstreindependent three has and symmetric is ± tec etxi em fclua pres- cellular of terms in vertex each at T σ oc aac qain for equations balance force Drosophila ab i i yy }, }, σ ), aebe rte stesmof sum the as written been have {T {T i i = }) = }) s F = a ({Π coem()admeso- and (9) ectoderm | 2 1 t t grid a o.114 vol. ,rdcn h num- the reducing S7), (σ , C i }, xy . and {T + σ F ab | σ 9 d ). yx o a For 0. = }) o 48 no. with , straction- as ). eshow we 6, 3a ostudy To 2N con- A . | 12665 vari- [10] F [9] [8] SI a .

BIOPHYSICS AND COMPUTATIONAL BIOLOGY A D

B E

Fig. 2. Comparison of shape-based and coarse- grained traction-based stress. (A–C) Solid state at v0 = 0.5, P0 = 3.3. (D–F) Liquid state at v0 = 0.5, P0 = 3.8. (A and D) Interaction normal stress σ(i)int calculated from the instan- 3 3 n taneous cell shapes obtained from Eqs. 3a and C F 7. Red denotes positive (contractile) stress and 2 2 blue negative (extensile) stress. (B and E) Inter- (i)int action normal stress σn calculated using the coarse-grained traction-based mechanical infer- 1 1 ence by inverting Eq. 9 and using Eq. 3a. The arrows denote the traction forces. (C and F) The coarse-grained traction-based mechanical infer- 0 0 ence is validated by plotting the traction-based Interaction normal stress Interaction normal stress stress against the shape-based stress in the solid Traction-based stress Traction-based stress Interaction shear stress Interaction shear stress (C) and in the liquid (F) state. The data are for -1 -1 -10123-10123400 cells in a square box of side L = 20 with Shape-based stress Shape-based stress Dr = 0.1 and with periodic boundary condition.

we focus on normal and shear stresses. TFM probes the forces atively weaker spatial fluctuations, but much larger mean value exchanged between tissue and substrate, which by force bal- in the solid, where contractile cortical tension exceeds cell–cell ance are determined entirely by intercellular forces and hence by adhesion. interaction stresses. In contrast, the swim components of stress Fig. 3 displays the total mean normal stress (the separate con- and pressure cannot be probed in TFM, but contribute to the tributions from interaction and swim stress are shown in Fig. S2) pressure Π = −σn = Πint + Πswim that the tissue would exert across the solid–liquid transition. The color map shows that the laterally on a confining piston. As we show below, the swim con- total normal stress is contractile in the solid and across the tran- tribution dominates the pressure in the liquid state. sition line (Fig. 3A, black crosses), but changes sign and becomes Using the expression for the local stress obtained from cell extensile deep in the liquid. While the interaction stress is always shapes, the mean interaction normal stress of the tissue can be positive due to cell contractility and consistent with experimen- expressed entirely in terms of area and perimeter fluctuations in tal observations (1–3, 38, the change in sign of the total stress is a virial-like form (SI Text) due to the swim stress that is zero in the solid and always negative in the liquid (Fig. S2), indicating that motility induces extensile int 1 X σn = [2KAAi (Ai − A0) + KP Pi (Pi − P0)] . [11] stresses, tending to stretch the tissue. The total normal stress is AT i analogous to the stress on a wall confining an active Brownian col- loidal fluid (29, 30). We speculate that its change in sign could lead The first term represents the interaction contribution from the to an expansion of the tissue if released from confinement due to pressures within the cells. The second term is the contribution substrate patterning or to surrounding tissue and may contribute from the competition between actomyosin contractility and cell– to epithelia expansion in wound-healing assays. In our model con- cell adhesion that controls the cortical tensions. In our simula- finement is provided by the periodic boundary conditions. tion, the cellular pressure is suppressed by setting A¯ = A0 and the normal stress comes mainly from the cortical tensions. Eq. 11 The tissue effective shear viscosity diverges at the liquid–solid then provides a way for extracting mechanical information directly transition. While the local shear stress averages to zero in both from cell shape based on snapshots of segmented cell images. the liquid and the solid states, its temporal correlations provide Normal stresses are contractile in the solid phase and may become a distinctive rheological metric for distinguishing the liquid from the solid and identifying the transition. The time autocorrelation extensile deep in the liquid phase. We show in Fig. 2 snapshots of the local interaction normal stress in the solid state (A–C) and function of the interaction shear stress, in the liquid state (D–F). In both the solid and the liquid the int int interaction normal stress is on average contractile (red), with rel- Css (τ) = hσ (t0)σ (t0 + τ)i , [12] s s t0

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BIOPHYSICS AND COMPUTATIONAL BIOLOGY and cell–cell adhesion and can be generalized to account for rheology. Moreover, we observed a similar behavior for the tem- cell division, apoptosis, and nematic/polar order of the tissue. poral correlations of traction forces as demonstrated in SI Text. In contrast to equilibrium mechanical inference techniques (21, Therefore, our work suggests that TFM measurement (11) com- 23), our approach does not require cells to be in or close to bined with mechanical inference could provide information on static mechanical balance, and it also provides the absolute tissue rheology. To our knowledge, this has not been attempted scale of the junctional tensions and pressure differences. This yet on experimental data. can for instance be important for testing hypotheses involv- Our work sets the stage for examining the feedback between cell ing mechanosensitive biomolecules. Experimentally, our method activity and tissue mechanics that is apparent in many tissue-level provides unique ways to extract intercellular interaction stresses phenomena. Recent work has shown that mechanical stresses from existing traction force data and segmented cell images. influence cell proliferation in tumor spheroids (28) and regu- The swim stress, on the other hand, cannot be measured using late cell growth in the developing Drosophila wing (43). Regula- TFM as it represents the flux of propulsive forces across a bulk tion of cell motility, as in contact inhibition of locomotion, has plane in the tissue. It contributes to the homeostatic pressure at been proposed to explain stress patterns during collective cell the lateral boundary of the tissue. The sum of the swim stress and migration (44). TFM has revealed the tendency of cells to move the interaction stress approximates the total stress at the tissue along the direction of minimal shear stress, a phenomenon termed boundary, which is generally contractile but can become exten- “plithotaxis” (11). Our model provides a unifying framework for sile when the tissue is deep in the liquid state and cell motility quantifying the relative roles of various cell properties, such as exceeds actomyosin contractility. The exact location of the tran- shape, motility, and growth, on the mechanics of the tissue. sition from contractile to extensile depends on the average cellu- ACKNOWLEDGMENTS. We thank Daniel Sussman for valuable discussions. lar pressure, which we fix by setting the average cell area to the This work was supported by the Simons Foundation through Targeted Grant preferred area, as discussed in SI Text. This change in sign may Award 342354 (to M.C.M. and M.C.) and Investigator Award 446222 (to be observable in wound-healing assays where the transition from M.L.M., M.C., and M.M.) in the Mathematical Modeling of Living Systems; contractile to extensile behavior can result in tissue expansion by the National Science Foundation (NSF) through Awards DMR-1305184 upon removal of confinement by neighboring tissue. (to M.C.M. and X.Y.), DMR-1609208 (to M.C.M.), DMR-1352184 (to M.L.M. We have extracted an effective tissue viscosity from the tem- and D.B.), and the Integrative Graduate Education and Research Trainee- ship (IGERT) Grant DGE-1068780 (to M.C.M. and M.C.); by the NIH through poral correlation of the interaction shear stress. The correlation Grant R01GM117598-02 (to M.L.M.); and by the computational resources pro- time and effective viscosity display a slowing down and arrest at vided by Syracuse University and through NSF Grant ACI-1541396. All authors the transition to the solid, thus serving as a direct probe of tissue acknowledge support from the Syracuse University Soft Matter Program.

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