Adhesion strength between cells regulate non-monotonic growth by a biomechanical feedback mechanism

We probe the interplay between intercellular interactions and pressure fluctuations associated with single cells in regulating cell proliferation using simulations of a minimal model for three-dimensional multicellular spheroid (MCS) growth. The emergent spatial variations in the cell division rate, that depends on the location of the cells within the MCS, is regulated by intercellular adhesion strength (fad). This in turn results in non-monotonic proliferation of cells in the MCS with varying adhesion strength, which accords well with experimental results. A biomechanical feedback mechanism coupling the fad and cell-dependent pressure fluctuations relative to a threshold value (pc) determines the onset of a dormant phase, and explains the non-monotonic proliferation response. Increasing fad from low values enhances cell proliferation because pressure on individual cells is smaller compared to pc. In contrast, at high fad, cells readily become dormant and cannot rearrange effectively, leading to arrested cell proliferation. Our work, which shows that proliferation is regulated by pressure-adhesion feedback loop, may be a general feature of tumor growth.


Introduction
Regulation of cell division is necessary for robust tissue growth and morphogenesis, with rapid cell proliferation being an integral phenotype characterizing early development and tumor growth. 1,2 Mechanical forces influence cell division and spatial organization of cells through mechano-sensitive processes. [3][4][5][6][7] For instance, spatial constraints due to cell spatial packing limits cell proliferation. [8][9][10][11][12][13] The spatiotemporal arrangement of cells in response to local force (or stress) fluctuations, arising from intercellular interactions and how it controls cell proliferation, is unclear. Studies suggest 11,14-17 that the cross-talk between cell mechanical interactions and proliferation is mediated by cell-cell adhesion receptors.
An important mediator of inter cellular adhesive force is cadherin, which plays a critical role in morphogenesis, tissue healing, and tumor growth. 18,19 The importance of cadherins is documented through their fundamental importance in maintaining multicellular structure during morphogenesis. 20,21 Amongst the family of cell adhesion molecules, E-cadherin is the most abundant, and is expressed in most metazoan cohesive tissues. 22,23 Mechanical coupling between the cortical cytoskeleton and cell membrane involves the cadherin cytoplasmic domain, with forces exerted across the cell-cell contacts being transmitted between cadherin extracellular domain and the cellular cytoskeletal machinery through the cadherin/catenin complex. 24,25 Thus, to understand how adhesive forces control the spatial organization of cells within 3D multicellular spheroid (MCS) and determine proliferation, we study the impact of varying intercellular adhesion strength on cell proliferation arising from a pressure dependent biomechanical feedback using a computational model.
We simulate an agent based minimal model (see 26 for a review) for 3D MCS, where individual cells grow, move in response to intercellular forces and local pressure fluctuations, undergo division, as well as dynamically enter into a dormant phase. We show that variation in cell-cell adhesion strength, f ad , results in non-monotonic growth of the MCS into the surrounding medium. We discover that the pressure experienced by individual cells due to mechanical interactions with its neighbors quantitatively explains the non-monotonic proliferation pattern. The f ad parameter is a proxy for cell-cell adhesion strength, which could vary due to either changes in the cadherin expression level, cadherin clustering, 27 increasing time of contact between cells, 27 or "mechanical polarization", where cells reorganize their adhesive and cytoskeletal machinery to suppress actin density along cell-cell contact interfaces. 28,29 By building on the integral feedback mechanism that couples cell dormancy and local pressure, 6,7,11 we show that the pressure exerted by cells on one another explains the experimentally observed non-monotonic cell collective growth as f ad is varied from a low to high value. The proposed mechanism could be a general attribute for the growth of MCS.

Methods
We simulate the collective movement of cells using a minimal model of an evolving tumor embedded in a matrix using an agent-based 3D model. [30][31][32][33] The cells, embedded in a highly viscous material mimicking the extracellular material, are represented as soft, spherical objects that interact via direct elastic (repulsive) and adhesive (attractive) forces.
Cell-to-cell and cell-to-matrix damping account for the effects of friction due to other cells, and the extracellular matrix, such as collagen matrix, respectively. The model explicitly accounts for apoptosis, cell growth and division. We also implicitly take into account the ability of individual cells to transition from growth to dormant phase depending on the local pressure. Thus, the collective motion of cells is determined by systematic cell-cell forces and the dynamics due to stochastic cell birth and apoptosis subject to a free boundary condition. The details of the models are presented in the Supplementary Information (SI) which is primarily based on our previous work. [34][35][36][37] Assessment of the values of f ad : We estimated the f ad from the typical strength of cell-cell attractive interactions reported in previous experimental studies in order to assess if they are in the reasonable range. Early experiments showed that the interaction strength between cell adhesion proteoglycans is ∼ 2 × 10 −5 µN/µm 2 . 38 Recently, single cell force spectroscopy has been used to measure the typical forces required to rupture E-cadherin mediated bonds between cells. 39 A typical force-distance curve (FDC), a plot of | F i | versus | r i − r j | from Eq. S.1 and the elastic force, is shown in Fig. 1(A). The plot shows that, for typical cell sizes (∼ 5µm), the minimum force is ≈ 2 × 10 −4 µN , which is fairly close to the values ≈ 1.5 × 10 −4 µN in Fig. 1(B) and 4 × 10 −4 µN reported elsewhere. 38 The inset in Fig. 1 is comparable to experiment, and accord well with the energy 1.25 × 10 −16 Nm required to rupture the adhesive forces between two cells used in the simulations. Note that we did not adjust any parameters to obtain reasonable agreement.
Dependence of cell-cell contact length (l c ) and contact angle (β) on f ad : Based on the center-to-center distance, r ij = |r i − r j |, between cells i and j, and the contact length, l c , the contact angle between two adhering cells, β, can be calculated (see Inset Fig.1(C) for the definition). Let x be the distance from center of cell i to contact zone marked by l c , along r ij . Similarly, we define y as the distance between center of cell j to l c once again along r ij ( Fig. 1(C)). Based on the right triangle between x, R i and l c /2, R 2 i − x 2 = R 2 j − y 2 = (l c /2) 2 and x + y = r ij , we obtain, The probability distributions for l c and β from the simulation as f ad is varied are shown in Figure 1: A) Force on cell i due to j, F ij , for R i = R j = 5 µm using mean values of elastic modulus, poisson ratio, receptor and ligand concentration (see Table I in the SI). F ij is plotted as a function of cell center-to-center distance |r i − r j |. We used f ad = 1.75 × 10 −4 µN/µm 2 to generate F ij . B) Force-distance data extracted from single cell force spectroscopy (SCFS) experiment. 39 Inset shows the work required to separate cell and E-cadherin functionalized substrate in SCFS experiment and two cells in theory, respectively. Minimum force values are indicated by vertical dashed lines. (C) Probability distribution of contact lengths between cells for f ad = 0 (red), f ad = 1.5 × 10 −4 (blue) and f ad = 3 × 10 −4 (black). Inset shows how cell-cell adhesion dictates the angle of contact between cells, β, and the length of contact, l c . (D) Probability distribution of contact angles, β as a function f ad . The color scheme is the same as in (C).

Figs.1(C) and (D). The extent of the overlap between cells increase as f ad increases, leading
to enhanced jamming at higher adhesion strengths. The distribution of β (see Fig.1(D)), whose width increases at higher adhesion strength, shows that the mean angle value decreases with f ad . By noting that β + θe 2 = π 2 (θ e is defined previously 40,41 ) we infer from the results in Fig.1(D) that θe 2 should increase as f ad increases. Indeed, this is precisely what was found in phase transition in the zebrafish blastoderm (see Fig. 4C of Ref. 41 showing decreased connectivity with decreased cell-cell adhesion). The plots in Fig.1 show that the range of cell-cell adhesion strength we use in the simulations is in a reasonable range.
The dependence of the total cell-cell interaction force, F ij as a function of f ad is shown in  Table I in the SI) in order to model rapid tumor growth.
If E i , ν i , k a , and k b are fixed, then tumor evolution is determined by f ad and a specified threshold pressure, p c . If the pressure on the i th cell at a given time (p i (t)) is less than p c the cell grows, and eventually divides. If p i (t) > p c , the cell enters the dormant phase ( Fig. 2(C).
We note parenthetically that p c ≥ 0 is required for describing time dependent tumor growth law observed in experiments (see Fig. S1A in the SI). The existence of p c above which cell division is halted is in agreement with recent experiment. 42 Here, we explore the effects of f ad and p c , on tumor proliferation.

Results
Cell proliferation depends non-monotonically on cell-cell adhesion strength f ad : We first quantify how changing f ad impacts cell division, and hence the overall tumor growth.
The simulations were initiated with 100 cells at t = 0, and the evolution of the cells is 10 -4 To investigate how cell spatial arrangement affects intercellular pressure, we calculate the pressure gradient with respect to cell-cell overlap, ∂p 1 ∂h , using ( On equating the repulsive and attractive interaction ∂h , can be rewritten as, because the mean number of near-neighborsn N N increases with f ad , which may be approximately written as G + λf ad (see Fig. S2 A-B; G and λ are constants obtained from fitting simulation data). Similarly, the deviation of the cell-cell overlap from optimal packing (h−h 0 ) increases linearly as E + αf ad (see Fig. S4; E, α are constants). Notice that Eq. 5 can be written as, In this form, the second term depends linearly on f ad and the third is inversely proportional to f ad . Since the enhancement in proliferation is maximized if the total pressure, p t , is minimized, the minimum in the total pressure on a cell is given by the solution to ∂pt ∂f ad = 0. Therefore, the predicted optimal cell-cell adhesion strength is f ad opt = ( GE αλ ) 1 2 = 1.77×10 −4 µN/µm 2 . This is in excellent agreement with the simulation results for the peak in the tumor growth (N (t = 7.5 days) in Fig. 2(D)) at f ad ≈ 1.75 × 10 −4 µN/µm 2 . More importantly, the arguments leading to Eq. 5 show that opposing contributions to the average pressure with increasingn N N and decreasing intercellular pressure at high f ad drives the non-monotonic proliferation.
Pressure dependent mechanical feedback controls spatiotemporal proliferation patterns -Insights from simulations: We now consider how intercellular pressure varies with time and distance from the tumor center to periphery. Pressure experienced by a cell is highly dynamic, with peaks and subsequent drops in p i (t) (Figs. 3A-C). In Fig. 3(A), black arrows highlight a switch in pressure from p i (t) > p c (dormant phase) followed by   Next, we quantified the experimentally measurable 45,46 spatial patterns in pressure experienced by cells within the MCS. The pressure associated with cells at the MCS center exceeds p c (see Fig. 3(D)) leading to suppressed cell growth and division. In contrast, as visualized in Fig. 3(D) and quantified in Fig. 3(E), the average pressure experienced by cells close to the tumor periphery tend to be below p c . The f ad dependent average pressure at r (distance from tumor center) is quantified using, showing that the lowest average pressure at the tumor periphery is at the intermediate value of f ad = 1.5 × 10 −4 (see Fig. 3(E)). Due to the high pressure experienced by cells near the tumor center, we expect a well defined spatial trend in the fraction of proliferating cells, is the number of cells with pressure p i < p c . Indeed, the proliferating cell fraction is low (F c < 0.2) at the MCS center compared to the periphery where it approaches unity (F c → 1), indicating that cells are in the growth/division phase (see Fig. 3(F)).
There is a rapid increase in F c near the tumor periphery for f ad = 1.5 × 10 −4 (see the blue asterisks in Fig. 3(F)). To better understand the spatial dependence on proliferation,  Fig. 3(G)). The simulated spatial proliferation profile is in agreement with experimental results, 45 where increased mechanical stress is correlated with lack of proliferation within the MCS, albeit when stress is exerted externally. We note that even in the absence of an external applied stress, intercellular interactions can give rise to heterogeneity in the spatial distribution of intercellular mechanical stresses and impact spatial proliferation patterns.
Our results show that the non-monotonic cell proliferation behavior emerges because the fraction of cells with pressure less than p c is localized at the MCS periphery. The average pressure, p , experienced by cells as a function of N , at the three values of f ad , is shown in Fig 3(H). N c , defined as the number of cells at which p = p c , exhibits a nonmonotonic behavior (inset in Fig 3(H) where pressure rises rapidly is followed by a more gradual increase in pressure, coinciding with the exponential to power law crossover in the growth in the number of cells (see Fig. S1A).
The average pressures ( p ) as a function of N is well fit by double exponential functions (Fig 3(H)). Hence, by experimentally measuring the cell proliferation rate, we can delineate the pressure sensing mechanism of cells.
Fraction of cells with pressure less than p c controls non-monotonic proliferation: The pressure experienced by a cell as a function of contact with other adhering cells plays an essential role in determining the observed non-monotonic proliferation. For f ad > 0, there is a minimum in the pressure at a non-zero cell-cell overlap distance, h ij (SI Fig. S1B).
For instance, at f ad = 1.5 × 10 −4 µN/µm 2 and 3 × 10 −4 µN/µm 2 , the mean pressure exerted on the cells is zero at h ij ∼ 0.4µm and 1.3µm respectively. The minimum pressure is a consequence of the balance between adhesive and elastic forces. At the minimum pressure The fraction of cells, F c in the proliferating regime, corresponding to the shaded portion in Fig. 4(A) exhibits a non-monotonic behavior (inset in Fig. 4(A)) with a peak at f ad c ∼ 1.75 × 10 −4 µN/µm 2 . The F c value in the growth phase (p i < p c ) peaks at ≈ 38%, between 1 × 10 −4 µN/µm 2 < f ad < 2 × 10 −4 µN/µm 2 . At lower and higher values of f ad , F c is at or below 25% on day 7.5. The inset in Fig. 4(A) makes it clear that F c , which determines The results in Fig. 2(D) and the inset in Fig. 4(A) suggest a phase diagram, shown in Fig. 4(B), in terms of the average cell pressure ( p = Σ i p i N , color map) and f ad . The dotted line is the boundary, p = p c , between the regimes where cells on an average grow and divide (p i < p c ), and where they are dormant (p i > p c ). At a fixed t, the regime between 1 × 10 −4 ≤ f ad ≤ 2 × 10 −4 shows a marked dip in the average pressure experienced by the cells. The p < p c regime becomes pronounced between 2τ min ≤ t ≤ 7τ min . In Fig. 4(C), the average pressure, p , at t = 6τ min and 12τ min as a function of f ad , has a minimum between

Discussion
Intercellular pressure determines the biomechanical feedback: Our simulations show that non-monotonic proliferation is determined by the fraction of proliferating cells, F C , with pressure less than p c . This picture is reminiscent of the mechanical feedback as a control mechanism for tissue growth proposed by Shraiman 11,17 but with a key difference. In earlier work, cellular rearrangements does not occur readily, and the tissue is treated as an elastic sheet that resists shear 11 with mechanical stresses serving as a feedback mechanism to restrict proliferation. In our case, large scale cell rearrangements are   48 Our simulations suggest that tumor spheroids made up of cells with higher E-cadherin expression (E-cad(control)) grow ≈ 1.9 times faster compared to tumor spheroids made up of low E-cadherin (E-cad(-)) expressing cells (see Fig. 5(A)), in reasonable agreement with experiments. 44 Comparison of the tumor diameter growth between E-cad(-) cells and E-cad(control) (red and blue lines respectively, Fig. 5(A)) shows enhanced growth rates for E-cad(control) cells at t > 4 days. The linear fit for the simulated tumor diameter growth rate is obtained by analyzing data at t > 4 days (dashed lines; Fig. 5(A)).
The growth rate ratio is then calculated by taking the ratio of the slope of the two diameter growth lines.
Padmanaban et. al 44 compared the tumor growth behavior between E-cadherin expressing cells (control) and E-cadherin negative cells (characterized by reduced E-cadherin expression compared to control) using 3D tumor organoids. They conclude that tumors arising from low E-cadherin expressing cells (E-cad(-)) were smaller than tumors from control E-cadherin (E-cad(control)) expressing cells at corresponding time points over multiple weeks of tumor growth (see Fig. 5(B)). By extracting the tumor growth rate for E-cad(-) and E-cad(control) organoids from the experimental data, 44 the longest tumor dimension for E-cad(control) tumor organoids expanded ≈ 1.3 times faster compared to E-cad(-) tumor organoids (see Fig. 5(B)). Both the linear functional form of the tumor growth rates and the higher growth rates in E-cad(control) tumor spheroids as compared to E-cad(-) tumor spheroids are in reasonable agreement with the simulations. It is worth emphasizing that the simulation results were obtained without any fits to the experiments. 44 To ensure that the difference in tumor growth rate is due to differences in cell proliferation, The observed role of E-cadherin in tumor growth could be related to a feedback mechanism due to changes in local pressure as the cell-cell interaction strength is varied. The pressure feedback on the growth of cells accounts for cell proliferation in the simulations.
In actuality, it may well be that mechanical forces do not directly translate into proliferative effects. Rather, cell-cell contact (experimentally measurable through the contact length l c , for example) could biochemically regulate cell signaling (eg: Rac1/RhoA), which in turn controls proliferation, as observed in biphasic proliferation of cell collectives in both two and three dimensions. 49,50 Nevertheless, contact inhibition of proliferation, based on mechanical pressure experienced by single cells can be captured using a pressure based feedback on proliferation in computer simulations. Such mechanical control of proliferation could be a generic mechanism governing how cell-cell adhesion strength, acting on the scale of at best few microns, affects tumor growth that occurs on the scale of millimeters.