bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Dual Role of Cell-Cell Adhesion In Tumor Suppression and Proliferation Due to Collective Mechanosensing

Abdul N Malmi-Kakkada1, Xin Li1, Sumit Sinha2, D. Thirumalai1∗ 1Department of Chemistry, University of Texas, Austin, TX 78712, USA. and 2Department of Physics, University of Texas, Austin, TX 78712, USA. (Dated: February 10, 2020) Abstract: It is known that mechanical interactions couple a cell to its neighbors, enabling a feedback loop to regulate tissue growth. However, the interplay between cell-cell adhesion strength, local cell density and force fluctuations in regulating cell proliferation is poorly understood. Here, we show that spatial variations in the tumor growth rates, which depend on the location of cells within tissue spheroids, are strongly influenced by cell-cell adhesion. As the strength of the cell-cell adhesion increases, intercellular pressure initially decreases, enabling dormant cells to more readily enter into a proliferative state. We identify an optimal cell-cell adhesion regime where pressure on a cell is a minimum, allowing for maximum proliferation. We use a theoretical model to validate this novel collective feedback mechanism coupling adhesion strength, local stress fluctuations and proliferation. Our results predict the existence of a non-monotonic proliferation behavior as a function of adhesion strength, consistent with experimental results. Several experimental implications of the proposed role of cell-cell adhesion in proliferation are quantified, making our model predictions amenable to further experimental scrutiny. We show that the mechanism of contact inhibition of proliferation, based on a pressure-adhesion feedback loop, serves as a unifying mechanism to understand the role of cell-cell adhesion in proliferation.

I. INTRODUCTION how it feeds back onto cell proliferation remains unclear. Indeed, evidence so far based on experimental and the- Adhesive forces between cells, mediated by cadherins, oretical studies on the mechanism underlying the cross- play a critical role in morphogenesis, tissue healing, and talk between the strength of cell-cell adhesion and pro- tumor growth [1,2]. In these processes, the collective liferation, invasion and drug resistance is not well under- cell properties are influenced by how cells adhere to one stood [15, 18–21]. another, enabling cells to communicate through mechan- We briefly discuss two seemingly paradoxical roles of ical forces [3, 4]. Amongst the family of cadherins, E- E-cadherin in proliferation. E-cadherin depletion lead cadherin is the most abundant, found in most metazoan cells to adopt a mesenchymal morphology characterized cohesive tissues [5]. E-cadherin transmembrane proteins by enhanced cell migration and invasion [22, 23] while facilitate intercellular bonds through the interaction of increased expression of E-cadherin in cell lines with min- extracellular domains on neighboring cells. The function imal expression reverses highly proliferative and invasive of cadherins was originally appreciated through their role phenotypes [24, 25]. Besides suppressing tumor growth, in cell aggregation during morphogenesis [6, 7]. Mechan- there is also evidence that cadherin expression can lead ical coupling between the cortical cytoskeleton and cell to tumor proliferation. We detail the dual role that E- membrane is understood to involve the cadherin cyto- cadherin plays below. plasmic domain [8]. Forces exerted across cell-cell con- E-cadherin downregulation and tumor progression tacts is transduced between cadherin extracellular do- through the EMT mechanism: Loss of E-cadherin expres- main and the cellular cytoskeletal machinery through the sion is related to the epithelial-to-mesenchymal transition cadherin/catenin complex [9]. Therefore, to understand (EMT), observed during embryogenesis [6, 26], tumor how mechanical forces control the spatial organization of progression and metastasis [27, 28]. EMT results in the cells within tissues and impact proliferation, the role of transformation of epithelial cells into a mesenchymal phe- adhesion strength at cell-cell contacts need to be eluci- notype, where adhesive strength between the cells is sig- dated. nificantly decreased, and drives tumor invasiveness and Together with cell-cell adhesion, cell proliferation con- cell migration in epithelial tumors. In this ‘canonical’ trol is of fundamental importance in animal and plant de- picture, down regulation of cell-cell adhesion contributes velopment, regeneration, and cancer progression [10, 11]. to cancer progression [29]. Spatial constraints due to cell packing or crowding are E-cadherin upregulation may promote tumor progres- known to affect cell proliferation [12–17]. The spatiotem- sion: In contrast to the reports cited above, others argue poral arrangement of cells in response to local stress field that EMT may not be required for cancer metastasis [30– fluctuations, arising from intercellular interactions, and 32]. In fact, E-cadherin may facilitate collective cell mi- gration that potentiates invasion and metastasis [33, 34]. Increased E-cadherin expression is deemed necessary for the progression of aggressive tumors such as inflamma- ∗ [email protected] tory breast cancer (IBC) [35] and a glioblastoma mul- bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 2 tiforme (GBM) subtype [36, 37], and invasive ductal carcinomas [38]. In multiple GBM cell lines, increased E-cadherin expression positively correlated with tumor aa growth and invasiveness [36]. In normal rat kidney-52E β (NRK-52E) and non-tumorigenic human mammary ep-

Ri R ithelial cells (MCF-10A), E-cadherin engagement stimu- β j lated a peak in proliferation capacity through Rac1 acti- r vation [39]. By culturing cells in micro-fabricated wells to ijij l control for cell spreading on 2D substrates, VE-cadherin c mediated contact with neighboring cells showed enhanced proliferation [40]. The dual role of cell-cell adhesion, sup- pressing proliferation in some cases and promoting it in other instances, therefore warrants further investigation. b b The contrasting scenario raise unanswered questions R that are amenable to analyses using relatively simple m models of tumor growth: (1) How does the magnitude Rm R hij of forces exerted by cells on one another influence their d R D p

FIG. 2. a) Proliferation capacity (PrC) measured as the total lap distance, hij (Appendix A Fig. 7b). For instance, number of cells (N in units of 1000), at t = τmin to 12τmin, at at f ad = 1.5 × 10−4µN/µm2 and 3 × 10−4µN/µm2, the intervals of τmin(= 54, 000 sec), as a function of cell-cell adhe- ad −4 pressure (pi) experienced by cells is zero at hij ∼ 0.4µm sion strength (f ) using pc = 10 MPa. Error bars here, and henceforth, are calculated from the standard deviation. The and 1.3µm respectively. At this minimum pressure, p → 0, the proliferation capacity (PrC) of the cells scale on the right gives t in units of τmin. b) Fractional change i in N (defined in the text) between f ad = 1.75 × 10−4µN/µm2 is maximized because cells are readily outside the dor- ad ad −4 and f = 0 (diamonds), between f = 1.75 × 10 and mant regime, pi/pc < 1. Due to the relationship be- ad −4 f = 3 × 10 (left triangles), as a function of time. Pre- tween cell-cell overlap (hij) and center-to-center distance dicted power law behavior of ∆N(t)/N are plotted as lines. (|ri − rj| = Ri + Rj − hij), our conclusions regarding c) The dependence of the invasion distance (∆r), at 7.5 days hij can equivalently be discussed in terms of the cell-cell (= 12τ ), on f ad. Inset shows a cross section through a min contact length (lc), the angle β (see Fig. 1) and cell-cell tumor spheroid and the distance ∆r from tumor center to ad −4 internuclear distance, |ri − rj|. periphery at t ∼ 12τmin for f = 1.5 × 10 . Color in- dicates the pressure experienced by cells. d) Schematic for Fig. 3a shows the probability distribution of pressure ad the three different regimes of cell-cell adhesion exhibiting dif- at t = 7.5 days at three representative values of f , fering PrCs. E-cadherin molecules are represented as short corresponding to low, intermediate and high adhesion red bonds. These regimes in the simulations correspond to strengths. The crucial feature that gives rise to the non- (i) f ad ≤ 0.5 × 10−4 characterized by low cell-cell adhesion monotonic proliferation and invasion (Figs. 2a -2c) is and low PrC. (ii) 1 × 10−4 ≤ f ad ≤ 2 × 10−4 character- the nonlinear behavior of the pressure distribution be- ized by intermediate cell-cell adhesion and high PrC, and (iii) low p = 1 × 10−4MPa (see shaded portion in Fig. 3a, ad −4 c f ≥ 2.5 × 10 with high cell-cell adhesion and low PrC. quantified in the Inset) as f ad is changed. In the in- set of Fig. 3a, the fraction of cells in the proliferat- ing regime shows a biphasic behavior with a peak at ad ad −4 2 enhanced at intermediate values of f (Fig. 2c). The fc ∼ 1.75 × 10 µN/µm . The fraction of cells in uptick in invasiveness from low to intermediate values the growth phase (pi < pc) peaks at ≈ 38%, between of f ad is fundamentally different from what is expected 1×10−4 µN/µm2 < f ad < 2×10−4 µN/µm2. For both in the canonical picture, where increasing cell-cell adhe- lower and higher cell-cell adhesion strengths, the fraction sion suppresses invasiveness and metastatic dissemina- of cells in the growth phase is at or below 25% on day tion of cancer cells [29, 46]. In contrast, tumor inva- 7.5 of tissue growth. We present the phase diagram of siveness as a function of increasing adhesion or stickiness Σipi the average pressure (hpi = N , color map) on cells as a between cells (as tracked by ∆r(t = 12τ )) initially in- ad min function of f and time (in units of τmin) in Fig.3b. The creases and reaches a maximum, followed by a crossover dotted line marks the boundary hpi = pc, between the to a regime of decreasing invasiveness at higher adhesion regimes where cells on average grow and divide (pi < pc), strengths. We note that the decreased invasive behavior and the opposite limit where they are dormant. At a fixed ad ad at f > fc is in agreement with the canonical picture, value of t, the regime between 1×10−4 ≤ f ad ≤ 2×10−4 where enhanced E-cadherin expression results in tumor shows a marked dip in the average pressure experienced suppression. Schematic summary of the results is pre- by cells. The low pressure regime, hpi < pc, is particu- sented in Fig. 2d. larly pronounced between 2τmin ≤ t ≤ 7τmin. In Fig. 3c, The inset in Fig. 2c shows the highly heterogenous the average pressure at t = 6τmin and 12τmin, as a func- spatial distribution of intercellular pressure (snapshot at tion of f ad, are shown for illustration purposes. Mini- ad −4 2 t = 12τmin), marked by elevated pressure at the core and mum in hpi between f ≈ 1.5 − 2 × 10 µN/µm is decreasing as one approaches the tissue periphery. As cell seen. In our pressure dependent model of contact in- rearrangement and birth-death events give rise to local hibition, there is a close relationship between prolifera- cell density fluctuations, the cell pressure is a highly dy- tion capacity (PrC) and the local pressure on a cell. As namic quantity, see videos (Supplementary Movies 1-3; the number of cells experiencing pressure below critical Appendix A Figs. 8a-c) for illustration of pressure dy- pressure increases, the PrC of the tissue is enhanced. It namics during the growth of the cell collective. Spatial is less obvious why the average cell pressure acquires a ad ad distribution of pressure is important to understanding minimum value at f = fc (Fig. 3b-c). We provide a the non-monotonic proliferation behavior, as we discuss plausible theoretical explanation below. in more detail below. Pressure Gradient Drives Biphasic Prolifera- Fraction of Dormant Cells Determine Non- tion Behavior: The finding in Fig. 2a could be un- monotonic Proliferation: To understand the phys- derstood using the following physical picture. PrC is de- ical factors underlying the non-monotonic proliferation termined by the number of cells with pressures less than behavior shown in Fig. 2a, we searched for the growth pc, the critical value. If the pressure on a cell exceeds control mechanism. We found that the pressure expe- pc, it becomes dormant, thus losing the ability to divide rienced by a cell as a function of cell packing in the and grow. The average pressure that a cell experiences 3D spheroid (Appendix A Fig. 7b) plays an essential depends on the magnitude of the net adhesive force. At role in the observed results. For f ad > 0, a mini- low values of f ad (incrementing from f ad = 0) the cell mum in pressure is observed at non-zero cell-cell over- pressure decreases as the cells overlap with each other bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 5

a proliferate. From these arguments it follows that N(t) should increase (decrease) at low (high) f ad values with a maximum at an intermediate f ad. The physical picture given above can be used to construct an approximate the- ory for the finding that cell proliferation reaches a maxi- ad ad mum at f = fc (Fig. 2a). The total average pressure (pt) that a cell experiences is given by pt ∼ n¯NN hp1i wheren ¯NN is the mean number of nearest neighbors, and hp1i is the average pressure a single neighboring cell exerts on a given cell. We appeal to simulations to es- ad timate the dependence ofn ¯NN on f (see Appendix C Figs. 9a-b). For any cell i, the nearest neighbors are de- 10-5 fined as those cells with non-zero overlap (i.e. h > 0). b 16 ij 2 To obtain hp1i, we expand around h0, the cell-cell over- 14 lap value where both the attractive and repulsive interac- 4 p less than p 12 c tion terms are equal (hij| el ad ), corresponding to the Fij =Fij min 10

t/ 6 overlap distance at which p = 0 (Appendix A Fig. 7b). 8 Thus, by Taylor expansion to first order, hp i can be writ-

(MPa) 1

p ∂p 8 6 ten as hp i ≈ 1 (h − h ). Here, h = h¯ , is the mean p greater than p 1 ∂h 0 ij c ad 4 cell-cell overlap, which depends on f (see AppendixC 10 Fig. 10). We note that the variation in h with respect fad 2 0 c 12 to Ri and Rj, as well as other cell-cell interaction pa- 0123 fad( N/ m2) 10-4 rameters is small compared to cell size. Estimating the 10-4 dependence of h−h0 on adhesion strength (see Appendix C Fig. 11), an approximate linear trend is observed. At 1.6 c higher adhesion strengths, cells find it increasingly diffi- cult to rearrange themselves and pack in such a way that 1.4 intercellular distances are optimal for proliferation. (MPa)

p In order to calculate the pressure gradient with respect 1.2 ∂p1 to cell-cell overlap, ∂h , we use

3 ad 1 |F | Jh 2 − Bf h p = ' , (1) 1 A Ch 0.8 ∂p1 J J 0123 ad 2 ' 1 |h=h0 = 1 , (2) f ( N/ m ) -4 10 ∂h 2Ch 2 2 2Ch0 FIG. 3. a) The probability distribution of pressure experi- to separate the dependence on cell-cell overlap and ad- enced by cells within a 3D tissue spheroid on day 7.5 for three ad hesion strength. In Eqs. 1 and 2 J, B and C are in- different values of f . The shaded region represent pressure ad −4 ad dependent of both h and f and can be obtained experienced by cells below pc = 1 × 10 MPa, at f = 0. from Appendix A Eqs. A2 and A3, and the defi- Fraction of cells (FC ) with p < pc is shown in the Inset. b) Phase plot of average pressure experienced by cells as a func- nition of Aij. The resulting expressions are: J = ad 2 1−ν2 q tion of time and f . The scale for pressure is on the right. 3 1−νi j 1 1 1/[ ( + ) + ], B = [πRiRj/(Ri + 4 Ei Ej Ri(t) Rj (t) c) The average pressure hpi experienced by cells at t = 6τmin 1 rec lig rec lig (left triangles; ∼ day 3.75 of growth) and t = 12τmin (squares; Rj)] × 2 (ci cj + cj ci ), and C = πRiRj/(Ri + ∼ day 7.5 of growth) for different values of f ad. Rj). On equating the repulsive and attractive interac- ad 2 ∂p1 tion terms, we obtain h0 ≈ K(f ) , implying ∂h = J √ 1 , where, K = (B/J)2. Thus, the total pres- because they are deformable (|ri − rj| < Ri + Rj). This 2C Kf ad is similar to the pressure in real gases (Van der Waals sure (pt) experienced by a cell is pt ∼ n¯NN hp1i = ∂p1 picture) in which the inter-particle attraction leads to a n¯NN (h − h0) ∂h , decrease in the average pressure. As a result, in a cer- ad tain range of f , we expect the number of cells capable ad ad J 1 pt = (G + βf )(E + αf )( √ ). (3) of dividing should increase, causing enhanced prolifera- 2C Kf ad tion. At very high values of f ad, however, the attraction becomes so strong that the number of nearest neighbors The mean number of near-neighborsn ¯NN increases of a given cell increases (Appendix C Figs. 9a-b). This with f ad and to a first approximation can be written leads to an overall increase in pressure (jamming effect), as G + βf ad (see AppendixC Fig.9a-b; G, β are con- resulting in a decrease in the number of cells that can stants obtained from fitting simulation data). Simi- bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 6

ad −4 larly, the deviation of the cell-cell overlap from h0 is ap- observed at the intermediate value of f = 1.5 × 10 proximately E + αf ad (see AppendixC Fig. 11; E, α as one approaches the tumor periphery. We calculate the are constants). Notice that Eq.3 can be written as, average pressure dependence on r using, Ωp = (Gα + βE) + βαf ad + GE , where Ω = ( J√ )−1. t f ad 2C K ad Σ p δ[r − (|R~ − ~r )] In this form, the second term depends linearly on f and hp(r)i = i i CM i . (4) ad the third is inversely proportional to f . As described in Σiδ[r − (|R~ CM − ~ri)] the physical arguments, enhancement in proliferation is maximized if pt is as small as possible. The minimum in Due to the high pressure experienced by cells near the the total pressure experienced is given by the solution to tumor center, a low fraction (Fc < 0.2) of cells are in ∂pt growth phase at small r < 50µm while the majority of ∂f ad = 0. Therefore, the predicted optimal cell-cell ad- ad GE 1 −4 2 cells can grow at large r (see Fig. 4c). A rapid increase in hesion strength is fopt = ( ) 2 = 1.77 × 10 µN/µm . αβ Fc is observed approaching the tumor periphery for the This is in excellent agreement with the simulation results intermediate value of f ad = 1.5×10−4 (see the blue aster- ad −4 2 (fc ≈ 1.75 × 10 µN/µm ) for the peak in the prolif- isks in Fig. 4c). To understand the rapid tumor invasion eration behavior (N(t = 7.5 days) in Fig. 2a). More at intermediate value of f ad, we calculated the average importantly, the arguments leading to Eq. 3 show that cell proliferation rate, Γ(r), at distance r from the tumor the variations in the average pressure as a function of f ad center, Γ(r) = N(r,t)−N(r,t−δt) . Here, N(r, t) is the num- drives proliferation. δt ber of cells at time t = 650, 000s and δt = 5000s is the Cell Packing and Spatial Proliferation Patterns time interval. The average is over polar and azimuthal Are Dictated by Cell-Cell Adhesion: Mechanosen- angles, for all cells between r to r + δr. Closer to the sitivity i.e. specialized response to mechanical stimula- tumor center, at low r, Γ(r) ∼ 0 indicating no prolifer- tion is common to cells in many organisms. Exposure ative activity. However, for larger values of r, prolifer- to stresses in living tissues can module physiological pro- ation rate rapidly increases approaching the periphery. cesses from molecular, to cellular and systemic levels [47]. We found that the cell proliferation rate is similar for Here, we quantify the spatio-temporal behavior of the in- different f ad at small r, while a much higher prolifera- tercellular pressure, the parameter that encodes the me- tion rate is observed for the intermediate value of f ad at chanical sensitivity of cell proliferation to the local envi- larger r (see Fig.4d). This spatial proliferation profile is ronment, and delineate its emergent properties in a grow- in agreement with experimental results, where increased ing cell collective. The average pressure, hpi, experienced mechanical stress is correlated with lack of proliferation by cells as a function of the total number of cells at vary- within the spheroid core, albeit in the context of exter- ad ing f is shown in Fig 4a. Nc, defined as the number of nally applied stress [48]. We show, however, that even cells at which hpi = pc, exhibits a biphasic behavior, sup- in the absence of an external applied stress, intercellular porting the maximum number of cells at an intermediate interactions give rise heterogeneity in the spatial distri- f ad. This provides further evidence that in a growing bution of mechanical stresses. collection of cells, at intermediate f ad, cells rearrange and pack effectively in such a manner that the average pressure is minimized. For all three adhesion strengths DISCUSSION considered, an initial regime where pressure rises rapidly is followed by a more gradual increase in pressure, coin- Internal Pressure Provides Feedback In E- ciding with the exponential to power law crossover in the cadherin’s Role In Tumor Dynamics: We have growth in the number of cells (see Fig. 7a in Appendix shown that the growth and invasiveness of the tumor A). hpi as a function of N are well fit by double exponen- spheroid changes non-monotonically with f ad, exhibit- tial functions. We propose that the intercellular pressure ing a maximum at an intermediate value of the inter-cell in 3D cell collectives should exhibit a double exponen- adhesion strength. The mechanism for this unexpected tial dependence on N. However, the precise nature of finding is related to a collective effect that alters the pres- the intercellular pressure depends on the details of the sure on a cell due to its neighbors. Thus, internal pres- interaction between cells. sure may be viewed as providing a feedback mechanism ad With recent advances in experimental techniques [48, in tumor growth and inhibition. The optimal value of fc 49], it is now possible to map spatial variations in inter- at which cell proliferation is a maximum at t >> τmin, cellular forces within 3D tissues. Hence, we study how is due to pc, the critical pressure above which a cell en- the spatial distribution of pressure and proliferation is ters dormancy. Taken together these results show that influenced by cell-cell adhesion strength. We find that the observed non-monotonic behavior is due to an inter- the cells at the tumor center experience higher pressures, play of f ad and pressure, which serves as a feedback in above the critical pressure pc (see Fig. 4b), independent enhancing or suppressing tumor growth. We note that of f ad. In contrast, the average pressure experienced the main conclusion of our model, on the non-monotonic ad ad by cells close to the tumor periphery is below the critical proliferation behavior with a maximum at f ≈ fc , is value pc. The pressure decreases as a function of distance independent of the exact value of pc (see SI Fig. S1), al- r from the tumor center, with the lowest average pressure ternative definitions of pressure experienced by the cells bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 7

10-4 as well as the cell-cell interaction (see SI Figs. S2-S3; see 1.8 SI Section I and II for more details). a 1.6 The growth mechanism leading to non-monotonic pro- liferation at times exceeding a few cell division cycles 1.4 is determined by the fraction of cells, FC , with pres- 1.2

sure less than pc. The growth rate, and hence FC , de- (MPa) ad 1 4000 pends on both pc as well as f . This picture, arising in p 0.8 our simulations, is very similar to the mechanical feed- c

N 2000 back as a control mechanism for tissue growth proposed 0.6 by Shraiman [15, 21]. In his formulation, the tissue is 0.4 0 densely packed (perhaps confluent) so that cellular rear- -4 -4 ad =0 rangements does not occur readily, and the tissue could 0.2 f 10 10 ad =3 ad =1.5 f be treated as an elastic sheet that resists shear. For 0 f N this case, Shraiman produced a theory for uniform tis- 0 2000c 4000N 6000 8000 10000 12000 10-4 sue growth by proposing that mechanical stresses serves 2.5 as a feedback mechanism. In our case, large scale cell rearrangements are possible as a cell or group of cells could go in and out of dormancy, determined by the 2 pi(t)/pc. Despite the differences between the two studies, the idea that pressure could serve as a regulatory mecha- 1.5 nism of growth, which in our case leads to non-monotonic

dependence of proliferation on cell-cell adhesive interac- p(r) tion strength, could be a general feature of mechanical 1 feedback[15]. Cell-matrix Interactions: Biphasic cell migration 0.5 controlled by adhesion to the extracellular matrix has b been previously established. In pioneering studies, Lauf- 0 fenburger and coworkers [50, 51] showed using theory 0 20 40 60r( m) 80 100 120 140 and experiments that speed of single cell migration is 1.2 biphasic, achieving an optimal value at an intermediate c strength of cell-substratum interactions. Similarly, inva- 1 sion of melanoma cells into a matrix was also found to be biphasic [52], increasing at small collagen concentra- 0.8 c tions and reaching a peak at an intermediate value. At F 0.6 much higher values of the collagen concentration, the in- vasion into the matrix decreases giving rise to the bipha- 0.4 sic dependence. A direct link between the survival of genetically induced glioma-bearing mice and the expres- 0.2 sion level of CD44, which is a cell surface marker, has recently been reported [53]. The authors showed that 0 0 20 40 60 80 100 120 140 the survival depends in a biphasic manner with increas- r( m) ing CD44 expression. Simulations using the motor clutch 0.03 model established that the results could be explained in 0.025 d terms of the strength of cell-substrate interactions. In contrast to these studies, our results show that adhesion 0.02 strength between cells, mediated by E-cadherin expres- 0.015 sion, gives rise to the observed non-monotonic behavior (r)

in the tumor proliferation (Fig. 2a). The mechanism, 0.01 identified here, is related to the pressure dependent feed- 0.005 back on growth [15] whose effectiveness is controlled by cell-cell adhesion strength, f ad. 0 Comparison With Experiments: We consider -0.005 0 20 40 60 80 100 three experiments, which provide support to our con- r( m) clusions: (i) Based on the observation of enhanced cell migration due to lower E-cadherin levels, it has been FIG. 4. (Caption next column.) proposed that tumor invasion and metastasis follows the loss of E-cadherin expression [24]. However, most breast cancer primary and metastatic tumors express E- bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 8

250 FIG. 4. a) (Figure in previous page) Average pressure ex- 240 Theory perienced by cells, hpi, as a function of the total number, Growth rate E-cad(-) 220 ~21.9 m/day N, of cells; hpi versus N at f ad = 0 (red;triangle), f ad = ratio = 1.87 E-cad(control) 1.5 × 10−4 (blue;asterisk) and f ad = 3 × 10−4 (black;squares) 200 are shown. The corresponding double exponential fits are, m) 180 200 ~11.7 m/day −4 5.8×10−5×N −4 −4.4×10−3×N 160 [1.1 × 10 e − 1 × 10 e ] (red line, 4567 −5 −4 f ad = 0), [1.0 × 10−4e1.7×10 ×N − 9.9 × 10−5e−9.2×10 ×N ] −6 (blue line, f ad = 1.5 × 10−4) and [1.8 × 10−4e7.6×10 ×N − a 4 −4 Experiment

Tumor Diameter( 10 −4 −9.1×10 ×N ad −4 2 1.9 × 10 e ] (black line, f = 3 × 10 ). Inset 150 Growth rate ~395 m/day ratio = 1.31 shows Nc, the number of cells at which hpi = pc. b) The av- erage pressure experienced by cells at a distance r (µm) from 1 ~302 m/day the spheroid center. The dashed line shows the critical pres- 0 sure p = 10−4MPa. c) The fraction of cells with p < p at a 10 20 30 40 c c 100 distance r. d) The average cell proliferation rate at distance r 01234567t(Days) from the center of the spheroid for f ad = 0, f ad = 1.5 × 10−4 and f ad = 3 × 10−4. Colors and symbols corresponding to ad f are the same for a-d. t is fixed at 650, 000sec(∼ 12τmin) b for b-d.

cadherin [54]. To resolve this discrepancy, Padmanaban et. al [38] compared the tumor growth behavior between E-cad(control) E-cadherin negative cells (characterized by reduced E- cadherin expression compared to control) and control E- cadherin expressing cells using three-dimensional (3D) tumor organoids. They report that tumors arising from 100 low E-cadherin expressing cells (E-cad(-)) were smaller c than tumors from control E-cadherin (E-cad(control)) 80 expressing cells at corresponding time points over mul- tiple weeks of tumor growth (see Fig. 5a Lower Inset). 60 Using the experimental data from Ref. [38], we extracted the tumor growth rate for E-cad(-) and E-cad(control) 40 organoids, and observe enhanced growth rate for tumors made up of E-cad(control) cells. The longest tumor di- Normalized cell number % mension for E-cadh(control) tumor organoids expanded 20 1.3 times faster compared to E-cad(-) tumor organoids 0 (see Bottom Inset, Fig. 5a). From our tumor simula- AE-cad(-) BControl CE-cad(+) tions, we predicted the E-cadh(control) tumor spheroids 100 expand 1.8 times faster compared to E-cad(-) tumor 90 spheroids (see Top Inset, Fig. 5a). This shows good 80 d agreement between theory and experimental results. It is 70 worth emphasizing that the simulation results were ob- 60 tained without any fits to the experiments [38], which 50 appeared while this article was already submitted. Com- 40 parison of the growth in tumor diameter between E-cad(- 30 Normalized cell number % ) cells and E-cad(control) (red and blue lines respectively, 20 Fig. 5a Main Panel) over 7.5 days of simulated tumor 10 growth shows similar growth rates until t ∼ 4 days fol- 0 ad ad -4 ad -4 lowed by enhanced growth rates for E-cad(control) cells f =0 f =1.5 10 f =3 10 at t > 4 days. The fit for the simulated tumor diame- FIG. 5. (Caption next column.) ter growth rate is obtained by analyzing data at t > 4 days (see Top Inset, Fig. 5a). The tumor size growth rate over multiple days is best fit by a linear function, larger than 20 µm in the experimental tumor spheroids. in agreement between simulation and experiment. The Other factors such as the difference in cell cycle times, overall magnitude of the tumor growth rate is higher in critical pressure which limits growth and spatially asym- the experiments as compared to the simulations. The dif- metric growth of the tumor could also lead to differences ference in the magnitude of tumor growth rate could be in the overall magnitude of the tumor growth rate. How- due to the variation in the sizes of cells. In simulations, ever, both the linear functional form of the tumor growth the maximum cell diameter is 10 µm while cells can be rates and the higher growth rates in E-cad(control) tu- bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 9

FIG. 5. a) Main Panel) Kinetics of growth in the diameter cad expression (see Insets of Fig. 5b). of tumor spheroids composed of cells with low E-cadherin ex- pression (red; f ad = 0; E-cad(-)) and intermediate E-cadherin (ii) Maintenance of appropriate E-cadherin expression expression (blue; f ad = 1.75 × 10−4; E-cad(control)) from is required for normal cell development in several species simulations show enhanced tumor growth rate due to higher such as drosophila, zebrafish and mouse [44, 55]. We E-cadherin expression. The tumor diameter growth is lin- consider a study on how the proliferation capacity of pri- ear at t > 4 days, independent of the E-cadherin expression mordial germ cells (PGCs) depend on cell-cell adhesion. level. However, the growth rate of the tumor colony with Even though the total number of PGCs in an embryo is intermediate E-cadherin expression is larger (Top Inset; The- only around 10-30 cells, while there are thousands of cells ory) in agreement with experimental results. Growth rates in the simulations, we compare the percentage change in of the longest tumor dimension in low E-cadherin express- cell numbers that result from modulating E-cadherin ex- ing organoids compared to control organoids with normal E- cadherin expression is shown in the Bottom Inset (Data was pression levels. We make this comparison to propose that extracted from Ref. [38]). b) Probability distribution of the perhaps E-cadherin does modulate proliferation through mitotic fraction, FC , in cell colonies expressed as % for low a pressure dependent mechanistic pathway in tissues, as E-cadherin tumor organoids (red) and normal E-cadherin ex- discussed above. The proliferation behavior of PGCs in pressing tumor organoids (blue) shows enhanced proliferation Xenopus laevis with changing E-cadherin expression lev- capacity when E-cadherin expression is increased. Data was els is shown in Fig. 5c. In this experiment, E-cadherin extracted from Ref. [38]. (Top Inset; Theory) Mean mitotic overexpression, achieved by mRNA injection of GFP E- fraction, hFC i, for low E-cadherin tumor cells (E-cad(-)) and cadherin DELE mRNA (E-cad GFP), leads to ∼ 50% intermediate E-cadherin expressing tumor cells (E-cad (con- reduction (compared to control) in the number of cells trol)). (Bottom Inset; Experiment) Mean mitotic fraction after 1.5 days of post fertilization embryo growth [44]. in low E-cadherin expressing tumor organoids is lower than Similar reduction in the number of cells is observed with control organoids with normal E-cadherin expression [38]. Al- though there is paucity of experimental data it is encourag- E-cadherin knockdown using specific morpholino oligonu- ing that the currently available measurements are in line with cleotides (E-cad MO) [44]. Therefore, an optimal level of our predictions. c) Experimental data for the number of cells cell-cell adhesion exists where proliferation is maximized, normalized by maximum number (as %) at three levels of in agreement with simulation predictions (Fig. 5d). The E-cadherin expression for primordial germ cells (PGC) from section III in SI provides other examples for the role Xenopus laevis ≈1.5 days after ferilization [44]. Both over- of E-cadherin in both tumor suppression and prolifer- expression and knockdown of E-cadherin leads to a decrease ation. (iii) The loss of E-cadherin is considered to be in PGC cell numbers. Control morpholino oligonucleotides a key characteristic in epithelial to mesenchymal transi- (Contr MO) and uninjected show native E-cadherin expres- tions, priming the cells to dissociate from the primary sion while E-cad MO (E-cad(-)) and E-cad GFP (E-cad(+)) tumor and invade surrounding tissues [56]. However, in induces knockdown and overexpression of E-cadherin respec- tively. d) Simulation data for the number of cells normalized a subset of high grade glioblastoma, patients with tumor by maximum number (as %) at three levels of cell-cell adhe- cells expressing E-cadherin correlated with worse progno- ad sion strength (t = 650, 000sec(∼ 12τmin)). Values of f (in sis compared to patients whose tumor cells that did not units of µN/µm2) are shown in parenthesis, as a proxy for express E-cadherin [36]. In this tumor type, heightened experimental E-cadherin expression levels. expression of E-cadherin correlated with increased inva- siveness in xenograft models [36]. These experimental results, which are consistent with our simulations in pro- mor spheroids as compared to E-cad(-) tumor spheroids moting proliferation as f ad is changed from = 0 µN/µm2 ad 2 ad are in agreement with simulation predictions. to f = 1.75 µN/µm = fc (Fig. 2a), suggest an un- To conclusively show that the fraction of proliferating expected role of E-cadherin in promoting tumor growth cells determine the non-monotonic tumor growth behav- and invasion. ior between low E-cadherin expressing tumors and cells Cautionary Remarks: As detailed in the SI section expressing control levels of E-cadherin, we turn to further III, it is difficult to make precise comparisons between analysis of experimental data from Ref. [38]. By staining the simulations and experiments because the growth of tumor cell colonies for PH3 (phospho-histone 3; a mitotic tumors is extremely complicated. For example, other cy- marker), Ref. [38] quantifies the mitotic fraction i.e. the toplasmic and nuclear signaling factors may be important ratio of the number of actively dividing cells to the to- in understanding the role of cell-cell adhesion in modu- tal number of cells. In agreement with predictions from lating the proliferative capacity of cells, [37] as multi- our simulations, Padmanaban et. al [38] report that E- ple signaling pathways are located in direct proximity cadherin expressing tumor cell colonies are indeed char- to the adherens junction complexes [20]. Nevertheless, acterized by a larger mitotic fraction (blue histogram, our results suggest that the mechanism of contact inhi- Fig. 5b) as opposed to low E-cadherin expressing tumor bition of proliferation, based on critical cellular pressure, cells (red histogram, Fig. 5b). We compare the mean could serve as a unifying mechanism in understanding mitotic fraction, hFC i, in E-cad(-) and E-cad(control) in how cell-cell adhesion influences proliferation. The rela- the insets of Fig. 5b between theory and experiment. In tion between proliferation and cell-cell adhesion has im- agreement with our simulation predictions, enhanced mi- portant clinical applications, such as in the development totic fraction is seen in tumor spheroids with normal E- of innovative therapeutic approaches to cancer [57, 58] bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 10 by targeting E-cadherin expression. Under certain cir- teractions are characterized by direct elastic (repulsive) cumstances, inhibiting E-cadherin expression could lead and adhesive (attractive) forces. The total force on the to tumor progression. On the other hand, in cancer cells ith cell is given by, nominally associated with low or negligible E-Cadherin ~ el ad levels, tumor progression and worsening prognosis could Fi = ΣjNN(i)(Fij − Fij )~nij, (A1) result upon increasing E-Cadherin levels (see SI, Section III for further discussion). where ~nij is the unit vector from the center of cell j to cell i. The forces are summed over the nearest neigh- bors (NN(i)) of the ith cell. The form of the elastic el CONCLUSIONS force, Fij (see Eq. A2), and the inter-cell adhesive force, ad Fij (see Eq. A3), are taken from the study of Schaller In this study, we have established that the modula- and Meyer-Hermann [60]. The strength of the adhesive ad tion of cell-cell adhesion strength contributes to contact interaction between cells, f , is measured in units of 2 ad ad inhibition of cell growth and proliferation. Surprisingly, µN/µm ,(Fij ∝ f given by Eq. A3). Force as a func- cell proliferation exhibits a non-monotonic behavior as a tion of cell center-to-center distance is plotted in Fig. 6c. function of cell adhesion strength, increasing till a critical Cell-to-cell and cell-to-matrix damping account for the value, followed by proliferation suppression at higher val- effects of friction due to other cells, and the extracellular ues of f ad. We have shown that E-cadherin expression matrix (ECM) (for example, collagen matrix), respec- and critical pressure based contact inhibition are suffi- tively. The model accounts for apoptosis, cell growth cient to explain the role of cell-cell adhesion on cell pro- and division. Thus, the collective motion of cells is de- liferation in the context of both morphogenesis and can- termined by both systematic cell-cell forces and the dy- cer progression. The observed dual role that E-cadherin namics due to stochastic cell birth and apoptosis under plays in tumor growth is related to a feedback mecha- a free boundary condition [63]. nism due to changes in pressure as the cell-cell interac- Forces Between Cells and Equations of Motion: tion strength is varied, established here on the basis of Each cell is represented as a soft sphere whose radius simulations and a mean field theory. changes in time to account for cell growth. We charac- The pressure feedback on the growth of cells is suf- terize each cell by its radius, elastic modulus, membrane ficient to account for cell proliferation in the simula- E-cadherin receptor, and ligand concentration. Following tions. For cells, however, it may well be that mechan- previous studies [60, 61, 64], we used Hertzian contact ical forces do not directly translate into proliferative ef- mechanics to model the magnitude of the elastic force fects. Rather, cell-cell contact (experimentally measur- between two spheres of radii Ri and Rj, given by, able through the contact length lc, for example) could biochemically regulate Rac1/RhoA signaling, which in h3/2(t) turn controls proliferation, as observed in biphasic pro- F el = ij , (A2) ij 1−ν2 1−ν2 q liferation of cell collectives in both two and three dimen- 3 i j 1 1 4 ( E + E ) R (t) + R (t) sions [39, 40]. i j i j

One implication of our finding is that the mechanical where the parameters Ei and νi, respectively, are the pressure dependent feedback may also play a role in or- elastic modulus and Poisson ratio of the ith cell [60]. The gan size control. As tissue size regulation requires fine overlap between cells, if they interpenetrate without de- tuning of proliferation rate, cell volume, and cell death formation, is hij, defined as max[0,Ri+Rj −|~ri−~rj|] with at the single cell level [59], pressure dependent feedback |~ri − ~rj| being the center-to-center distance (see Fig. 1b mediated by cell-cell adhesion could function as an ef- in the Main Text). The elastic repulsive forces tend to ficient control parameter. In principle, cells in tissues minimize the overlap between cells, and could be thought could be characterized by a range of adhesion strengths. of as a proxy for cortical tension [65, 66]. Competition between these cell types, mediated by ad- The magnitude of the attractive adhesive force, F ad, hesion dependent pressure feedback into growth, could ij between cells i and j is given by, be critical in determining the relative proportion of cells and therefore the organ size. 1 F ad = A f ad (crecclig + crecclig), (A3) ij ij 2 i j j i

Appendix A: Methods rec lig where Aij is the cell-cell contact area, ci (ci ) is the E-cadherin receptor (ligand) concentration (assumed to Model: We simulate the collective movement of cells be normalized with respect to the maximum receptor rec lig using a minimal model of an evolving tumor embedded or ligand concentration such that 0 ≤ ci , ci ≤ 1). in a matrix using an agent-based three dimensional (3D) The coupling constant f ad in Eq. A3, with dimensions model [60–62]. The cells, which are embedded in a highly µN/µm2, allows us to rescale the adhesion force, to ac- viscous material mimicking the extracellular material, count for the variations in the maximum receptor and are represented as deformable objects. The inter-cell in- ligand concentrations. Higher (lower) maximum recep- bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 11 a b

4 0.1510 ad -4 0.15 f =1.75 10 fad=3 10-4 -05 1.092 10 t -05 ) 113.4 e 1.063 10 t c 111.3 e ) l -10 2.37 2.141 10 t 3 -5 1.37 a 10 4.668 10 t 0.1 0.1 fad=0

Prob( -06

Prob( 9.73 10 t

N 112.7 e -06 1.562 3 3.038 10 t 0.05 0.0510

103 104 105 0 01234567 l ( m) 0 a a c b 40 50 60 70 80 90 a b 0.15 c 0.15 d 3 4 5 -5 10 -5 10 time(s) 10 0.15 0.15 10 10 ) -4 c 0 ×10 b ) 0 l

) 4

c ad ) l 0.1 0.1 f =0 ad -4 0.1 Prob( f =1.5×10 0.1 Prob( ad -4 Prob( -5

Prob( f =3×10 0.05 3 0.05 -16 N)

N) 10

0.05 ( (

0.05ij -10 ij F F -10 b 1 0 2 01234567 l ( m) 0 0 c 40 50 600.5 70 80 90 01234567 l ( m) 0 Work(J) c 40 50 60 70 80 90 -15

Pressure(MPa) 0 c -5 d -5 Expt Sim. a b -20 10 1 10 c d 9 9.2 9.4 9.6 9.8 10 10-5 0 -5 00 0.2 0.4 0.6 0.8 1 1.2 10 |r -r | ( m) 0.15 0.15 i j Cell-Substrate distance ( m) 0 0 ) c c 0 ) l -5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0.1 h (µm) -5 -16

N) ij Prob( N) 10 Prob( ( (

ij -10 -16 ij F N) F N) 10 -10

( 1 (

0.05ij -10 FIG. 7. a) Number of cells, N(t), over 7.5 days of growth. ij

F 0.05 F -10 1 Initial exponential growth followed by power law growth be- 0.5 havior is seen for threeWork(J) different f ad values. The onset of 0.5 -15 5 5 0 Work(J) power-law growth in N occurs between t = 10 − 2 × 10 01234567 0 0 l ( m) -15 Expt Sim.ad c -2040 50 60 70 80 90 secs. Power law exponent depends on f . b) Pressure, pi, 9 9.2 9.40 9.6 9.8 10 Expt Sim. 0 0.2 0.4 0.6 0.8 1 1.2 -20 |r -r | ( m) experienced by a cell interacting with another cell as a func- i j Cell-Substrate distance ( m) ad c 9 9.2 9.4 9.6 9.8 10 d 0 0.2 0.4 0.6 0.8 1 1.2 tion of overlap distance (h ) for different values of f . F 10-5 -5 ij ij |r -r | ( m) 10 Cell-Substrate distance ( m) i j is calculated for Ri = Rj = 4 µm using mean values of elas- 0 0 tic modulus, poisson ratio, receptor and ligand concentration d (see Table I).

-5

-16 N)

N) 10

( tor and ligand concentration on the cell surface mem- ( ij -10 ij

F ad F -10 1 brane, is accounted for by higher (lower) value of f . It should be noted that the strength of adhesion between 0.5 Work(J) the cells is mediated by both the extracellular portion -15 0 of E-cadherin and how it interacts with the cytoskele- Expt Sim. -20 ton. The cytoplasmic E-cadherin domain, in conjunction 9 9.2 9.4 9.6 9.8 10 0 0.2 0.4 0.6 0.8 1 1.2 with α-catenin, binds to β-catenin, linking it to the actin |r -r | ( m) i j Cell-Substrate distance ( m) cytoskeleton [66]. In the minimal model, all of these com- FIG. 6. a) Probability distribution of contact lengths be- plicated processes that occur on sub-cellular length scales ad tween cells for f ad = 0 (red), f ad = 1.5 × 10−4 (blue) and are subsumed in f . The inter cell contact surface area, ad −4 f = 3 × 10 (black). b) Probability distribution of con- Aij (see Eq. A3), is obtained using the Hertz model pre- ad tact angles, β at varying values of f with the color scheme diction, Aij = πhijRiRj/(Ri + Rj)[60]. as in a). c) Force on cell i due to j, Fij , for Ri = Rj = 5 µm An optimal range of cell-cell packing (hij) exists where using mean values of elastic modulus, poisson ratio, receptor pi is minimized (see Fig. 7b). With the definition pi, and ligand concentration (see Table I in the SI). Fij is plotted growth in the total number of cells (N(t)) is well ap- as a function of cell center-to-center distance |ri−rj |. We used proximated as an exponential N(t) ∝ exp(const × t) at f ad = 1.75 × 10−4µN/µm2 to generate F . d) Force-distance ij short times (t < 105 secs) (see Fig. 7a). At longer data extracted from SCFS experiment [44]. Inset shows the time scales (t >∼ 3 × 105 secs), the increase in the tu- work required to separate cell and E-cadherin functionalized β substrate in SCFS experiment and two cells in theory, respec- mor size follows a power law, N(t) ∝ t . Such a cross tively. Minimum force values are indicated by vertical dashed over from exponential to power-law growth in 3D tumor lines. See AppendixB for further details. bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 12 spheroid size has been observed in many tumor cell lines pc. A given cell, at any time t, can either be in the with β varying from one to three [67–71]. Based on the dormant (D) or in the growth (G) phase depending on power law growth behavior, we expect, ∆N(f ad, t)/N = the pressure on the cell (see Fig 1b). The total pressure ad −4 2 ad ad th [N(fc = 1.75 × 10 µN/µm , t) − N(f , t)]/N(fc = (pi) on the i cell, 1.75 × 10−4µN/µm2, t), for f ad = 0 to be 1 − (3 × 10−6t1.6)/(2.1 × 10−10t2.4). Hence, ∆N(f ad = 0, t)/N = X |Fij| −0.8 4 pi = , (A7) 1−A0t , where A0 = 1.4×10 is a constant. Similarly Aij ad −4 2 −1 j∈NN(i) for ∆N(f = 3×10 µN/µm , t)/N = 1−A1t where 5 A1 = 2.2 × 10 . Due to the local cell density fluctuations is the overall sum of all the normal pressures due to the caused by the birth-death events, the cell pressure p is a i nearest neighbors. If pi > pc (a pre-assigned value of the highly dynamic quantity (see Figs.8a-c). pi(t) plays an critical pressure), the cell becomes dormant, and can no important role in how local forces provide a feedback on longer grow or divide. Note that a cell that becomes dor- cell growth and division. mant at time t does not imply that it remains so at all While cells are characterized by many different types later times because as the cell colony evolves, pi, a dy- of cell adhesion molecules (CAMs), here we focus on E- namic quantity fluctuates, and hence can become less or cadherin. We consider different levels of CAM expres- greater than pc (see Figs.8a-c, Supplementary Movies 1- sion, varying from low (f ad = 0 µN/µm2) to intermediate 3A). It has been shown in vitro that solid stress, defined (f ad = 1.5 µN/µm2) to high (f ad = 3 µN/µm2) values. as the mechanical stress due to solid and elastic elements For a discussion on the appropriateness of the value of the of the extracellular matrix, inhibits growth of multicellu- range of f ad, see Appendix B. We used a distance sorting lar tumor spheroids irrespective of the host species, tissue algorithm to efficiently obtain a list of nearest neighbors origin or differentiation state [72]. This type of growth in contact with the ith cell. For a cell, i, an array with inhibition is mediated by stress accumulation around the distances from cell i to all the other cells is created. We spheroid as a result of the progressive displacement of the then calculated Ri + Rj − |~ri − ~rj| and sorted for cells surrounding matrix due to the growing clump of cells. In j that satisfy the condition Ri + Rj − |~ri − ~rj| > 0, a our model, however, the effect of pressure on cell growth necessary condition for any cell j to be in contact with is driven by local cell-cell contact as opposed to the global cell i. stress exerted by the surrounding matrix. Both in vivo The justification that the inertial forces can be ne- and in vitro, epithelial cells exhibit contact inhibition of glected can be found in our previous study (see Fig. 18 proliferation due to cell-cell interactions [73]. The value in Ref. [63]). If we neglect inertial effects, the equation of the critical pressure used in our work is in the same of motion of the ith cell is, range as experimentally measured cell-scale stresses (10- 200 Pa) [49] and with earlier works using critical cel- ~ lular compression as the mechanism for contact inhibi- ˙ Fi ~ri = , (A4) tion [60, 62]. We have also verified that the qualitative γi results are independent of the precise value of pc, as well 0 0 0 0 as alternative definitions of local pressure (see Sections where, γ = γα β ,visc + γα β ,ad is the friction coefficient i i i I-II in the SI). with

α0β0,visc α0β0 γi = 6πηRiδ , (A5) and Cell Dynamics: Because the Reynolds number for ~ cells in a tissue is small [60, 74], overdamped approxima- α0β0,ad max 1 Fi · ~nij tion is appropriate. The equations of motion are given γi =γ ΣjNN(i)(Aij (1 + ) × (A6) 2 |F~i| below (see Eq. A4). Besides the cell-cell repulsive and 1 0 0 adhesive forces, another contribution to cell dynamics (crecclig + crecclig))δα β , 2 i j j i comes from cell growth, division and apoptosis. Stochas- tic cell growth leads to dynamic variations in the cell-cell being the cell-to-matrix and cell-to-cell damping contri- forces. Cell division and apoptosis induce temporal re- 0 0 butions respectively. Here, the indices α , β represent arrangements in the cell positions and packing. Hence, cartesian co-ordinates. Viscosity of the medium sur- both the contribution of the systematic forces and cell max rounding the cell is denoted by η and γ is the adhesive growth, birth and apoptosis towards cell dynamics are friction coefficient. Additional details of the simulation taken into account in the model, for which we described methods are given elsewhere [63]. Note that because the the unusual dynamics previously [63, 75]. The parame- equations of motion for the coarse-grained model contain ters used in the simulations are given in the SI (see Table the friction term they do not satisfy Galilean invariance. I). We justify the range of f ad explored in our simulations Pressure-dependent Dormancy: A crucial feature in AppendixB. In order to explore the plausible dual role ad in the model is the role played by the local pressure, pi, of E-cadherin on cell proliferation, we vary f , keeping experienced by the ith cell relative to a critical pressure, all other parameters constant. bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 13

10-4 6 Cell 1 0.3 a 5 Cell 2 Cell 3 0.25

Cell 4 ) 4 Cell 5 NN 0.2 (t) i

p 3 a 0.15 Prob(n 2 0.1 1 0.05 0 0 100 200 300 400 500 600 0 time ( 103sec) 0 2 4 6n 8 10 12 14 16 NN -4 10 Cell 1 10 5 Cell 2 Cell 3 b 9 4 Cell 4 b Cell 5 3 8 (t)(MPa) i p 2 NN 7 n

1 6

0 5 0 100 200 3003 400 500 600 time ( 10 sec) 4 -3 0123ad 2 10 1 Cell 1 f ( N/ m ) -4 c Cell 2 10 Cell 3 0.8 Cell 4 0.4 Experiment Cell 5 c Simulation 0.6

(t)(MPa) 0.3 i ) p

0.4 NN

P(n 0.2 0.2 0.1 0 0 100 200 3003 400 500 600 time ( 10 sec) 0 3456789n FIG. 8. a) Pressure experienced by individual cells as a NN function of time at f ad = 0, b) f ad = 1.5 × 10−4 and c) f ad = 3 × 10−4. Dashed black lines indicate the critical pres- FIG. 9. a) Graph showing the probability distribution of sure, pc. Black arrows in a) highlight fluctuations in pi above and below p . the number of nearest neighbors (nNN ) on day 7.5 of tumor c growth. Red, blue and black curves are for f ad = 0, 1.5 × 10−4µN/µm2, and 3×10−4µN/µm2, respectively. b)Average ad number of nearest neighbors,n ¯NN , as a function of f . Lin- Appendix B: Calibration of Cell-Cell Adhesion 4 ad ad ear fit showsn ¯NN ∼ 5.1+(1.6×10 )×f (G+β×f ). Error Strength bars represent the standard deviation. c) Comparison of the probability distribution of the number of nearest neighbors The crucial parameter in the present study is the cell- (nNN ) between experiments and simulations. Experimental cell interaction strength, f ad, which is a proxy for E- data is for Xenopus tail epidermis (n=1,051 cells) [77]. Sim- ad −5 2 cadherin expression. In order to assess if the values used ulation data is for f = 5 × 10 µN/µm . Data in (a) - (c) in our simulations are in a reasonable range, we estimated are from 3 independent simulation runs. f ad from the typical strength of cell-cell attractive inter- actions reported in previous studies. Early experiments showed that the interaction strength between cell adhe- indeed E-Cadherin expression level that changes at dif- sion proteoglycans is ∼ 2 × 10−5µN/µm2 [76]. More ferent stages of embryo development, Baronsky et. al [44] recently, single cell force spectroscopy (SCFS) technique functionalized gold coated substrate with E-cadherin, has been used to measure directly the typical forces re- and measured the force-distance curves between primor- quired to rupture E-cadherin mediated bonds between dial germ cells and the substrate. cells. Several types of cadherins could be present on the Within the range of f ad considered in the simulations, cell surface, in addition to adhesion molecules such as in- a typical force distance curve (plot of |F~i| versus |~ri − ~rj| tegrins, selectins etc [46]. In order to confirm that it is from Eq. A1) is shown in Fig.6c. The plot in Fig. 6c bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 14

ad shows that for typical cell sizes (≈ 5µm) the minimum tumor is f and pc, whose magnitude is determined by force is ≈ 2 × 10−4µN, which is fairly close to the values E-cadherin expression. Here, we explore the effects of −4 −4 ad ≈ 1.5 × 10 µN in Fig.6d and 4 × 10 µN reported f and pc, on tumor proliferation. elsewhere [76]. The inset in Fig.6d shows the work done to overcome the adhesion force mediated by E-cadherin which is the area under the force-distance curve (FDC). Appendix C: Average Number of Nearest Neighbor The work expended is comparable at 0.78 × 10−16Nm of Cells Increases with f ad for primordial germ cells (PGCs in Xenopus laevis em- bryo) to E-cadherin substrate separation experiment and −16 The collective movement of cells (related to prolifer- 1.25 × 10 Nm for cell-cell separation in the model. Be- ative capacity) is determined by cell arrangement and cause the set up in single cell force spectroscopy and the packing within the three dimensional (3D) tissue, which theoretical model are not precisely comparable, it is grat- clearly depends on the adhesion strength. Analyzing the ifying that the magnitude of forces required to separate distribution of number of nearest neighbors (see Fig. 9a), two cells obtained using Eq. (A1) and the measured val- the arrangement of cells in the spheroid has a peak ues are not significantly different. Note that we did not near 6 nearest neighbors at low f ad. Very few cells, adjust any parameters to obtain the reasonable agree- if any, have less than 2 neighbors. Similarly, at f ad = ment. We undertook this comparison to merely point 5 × 10−5 µN/µm2, few cells have more than 9 neighbors out that the range of E-cadherin mediated forces used (see Fig. 9c). As f ad increases, the peak in the nearest in our simulations reflects the typical cell-cell adhesion neighbor distribution moves to higher values, with the strength measured in experiments. distribution also becoming broader (Fig. 9a). With the The timescale associated with single receptor-ligand highest adhesion strength, f ad = 3 × 10−4µN/µm2, the binding is typically 2-10 seconds [62]. There are about average number of nearest neighbors is ≈ 9 cells which is 2 ∼ 10 cadherins/µm on the surface of typical cells [42], consistent with 3D experimental data for mouse blasto- corresponding to ∼3000 cadherins on the cell surface for cyst after 5 − 9 days of growth [78]. a cell with radius of 5 microns. For studies of cell growth We surmise that the dependence of the average number and dynamics on the time scale of days, the fluctuations of nearest neighbors on cell-cell adhesion strengths f ad in at the level of single receptor-ligand binding can therefore ad the simulations is consistent with experimental findings. be neglected [62], thus justifying the use of constant f Cell packing data in 2D epithelial structures, quantified values. The receptor/ligand concentration are sampled by the probability distribution of nearest neighbors [77], from a Gaussian distribution (see SI Section IV for more allow us to compare the simulation results to experiments details). (Fig. 9c). We compare the simulation results for the dis- Given the center-to-center distance, r = |r − r |, ij i j tribution of the number of nearest neighbors (nNN ) with between cells i and j, the contact length, lc, and contact experiment, keeping in mind that our simulation is in 3D. angle β can be calculated. Let x be the distance from In 3D, the average number of nearest neighbors is higher center of cell i to contact zone marked by lc, along rij. than in 2D. There is indeed an increase in the probabil- Similarly, we define y as the distance between center of ity of nearest neighbors from 7 − 10 (Fig. 9c). Moreover, cell j to lc once again along rij (see Fig.1a of Main Text). f ad > 100 dynes/cm2 = 10−5µN/µm2, within an order of Based on the right triangle that is formed between x, Ri magnitude has been reported in experiments [79], which 2 2 2 2 2 and lc/2, Ri − x = Rj − y = (lc/2) and x + y = rij. we point out only to show that the values of f ad consid- This allows us to solve for x, y and hence, ered in our study are reasonable.

s 2 2 2 2 2 2 The average number of nearest neighbors,n ¯NN , in- 4r R − (r + R − R ) ad l =2 ij i ij i j , (B1) creases as f increases (Fig. 9b). The approximate lin- c 2 ad 4rij ear fitn ¯NN ∼ G + βf is used to rationalize the data in Fig. 2a in the Main Text. The fit parameters G, β are β =arctan(2y/lc). (B2) given in the caption. The linear fit is used only for cal- ad The probability distribution for lc and β obtained from culating fc , the optimal value at which proliferation is the simulation for varying values of f ad is shown in a maximum. Figs.6a - 6b. Distribution of hij: The cell-cell overlap, hij, The interaction parameters characterizing the model gives an indication of how closely the deformable cells ad are, the two elastic constants (Ei and νi) and f if we are packed within the 3D spheroid. At low adhesion assume that the combination of receptor and ligand con- strengths, the distribution is sharply peaked at small centrations in Eq. (A3) is a constant. In addition, the hij (see Fig. 10a), implying there is minimal cell-cell in- evolution of cell colony introduces two other parameters, terpenetration. As f ad increases, the cells are jammed. ad −4 2 birth (kb) and apoptotic rates (ka) of cells. The values of For f = 3 × 10 µN/µm , the average cell overlap ¯ ka and kb depend on the detailed biology governing cell hij ≈ 1.6 µm, implying that the center to center distance fate, which we simply take as parameters in the simula- between cells is approximately 6.4 µm (for cells of radii tions. If the elastic constants, and ka, kb are fixed, then 4 µm). Note that the cell overlap distribution becomes ad ¯ the only parameter that determines the evolution of the broader as f increases. The mean overlap, hij, varies bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 15

0.25 0.8 a 0.6 0.2 0.6 0.4 ) ij 0.2 0.4 0.15 o Prob(h -h

0 ij

0 0.2 0.4 h 0.2 0.1

0 0.05 0 0.5 1h ( m) 1.5 2 2.5 3 ij 0 0 1 2 3 ad 2 1.5 b f ( N/ m ) 10-4 ¯ 1 FIG. 11. Deviation of the average hij (hij ) from h0 (cell in-

ij terpenetration distance with minimum possible pressure) for h differing adhesion strength showing approximate linear de- 0.5 ad ad pendence h¯ij − h0 ∼ 0.05 + (491.7) × f (E + α × f ). The value of h0 is determined by the radii of any two in- 0 teracting cells. Two sets of points are shown, blue bars are 0123ad 2 for average radii of Ri = Rj = 3.9µm and the reds are for f ( N/ m ) -4 10 Ri = Rj = 4µm.

FIG. 10. a) Probability distribution of the overlap (hij ) of cells on day 7.5 of tumor growth for three different adhe- ad Foundation (PHY 17-08128 and CHE 16-32756) and the sion strengths. Red, blue and black curves are for f = Collie-Welch Chair through the Welch Foundation (F- 0, 1.5 × 10−4µN/µm2, and 3 × 10−4µN/µm2, respectively. b) 0019). Average interpenetration distance has a quadratic depen- ad 2 dence on adhesion strength - h¯ij ∼ (4350f ) . Data obtained are from 3 independent simulation runs. Error bars represent the standard deviation. quadratically with adhesion strength (Fig. 10b). If we set F el = F ad, we find that h ∼ (f ad)2, and as expected ¯ ad 2 we obtain the fit hij ≈ K(f ) . To calculate the total pressure experienced by a cell (pt) theoretically, as detailed in the Main text, we look ¯ at the deviation of the average cell overlap hij from the ad optimal overlap (h0) as f is changed, where h0 is the overlap distance at which the pressure experienced by a cell is a minimum. This would occur when the repulsive and attractive interaction forces between a pair of cells el ad ¯ balance (F = F ). The deviation, hij − h0, increases as f ad increases, an indication that it is harder for cells to relax to optimal intercellular distances, due to packing frustration. For the purposes of rationalizing the optimal ad ¯ value of f (see Main Text) we write, hij − h0 = E + αf ad, where the parameters E, α are as listed in Fig. 11. We found that h0 depends on radii of the cells that are in contact as well as other parameters,

2 2 1 − νi 1 − νj 2 RiRj ad 2 h0 = 3.6( + ) ( )(f ) . (C1) Ei Ej Ri + Rj ¯ Hence, we estimate hij − h0 by using average quantities. Acknowledgements: We acknowledge Anne D. Bowen at the Visualization Laboratory (Vislab), Texas Advanced Computing Center, for help with video visual- izations. This work is supported by the National Science bioRxiv preprint doi: https://doi.org/10.1101/683250; this version posted February 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 16

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