Collective dynamics of cancer cells confined in a confluent monolayer of normal cells

Author’s Accepted Manuscript Collective dynamics of cancer cells confined in a confluent monolayer of normal cells Shao-Zhen Lin, Bo Li, Guang-Kui Xu, Xi-Qiao Feng www.elsevier.com/locate/jbiomech

PII: DOI: Reference:

S0021-9290(16)31328-8 http://dx.doi.org/10.1016/j.jbiomech.2016.12.035 BM8068

To appear in: Journal of Biomechanics Accepted date: 20 December 2016 Cite this article as: Shao-Zhen Lin, Bo Li, Guang-Kui Xu and Xi-Qiao Feng, Collective dynamics of cancer cells confined in a confluent monolayer of normal c e l l s , Journal of Biomechanics, http://dx.doi.org/10.1016/j.jbiomech.2016.12.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Collective dynamics of cancer cells confined in a confluent monolayer of normal cells Shao-Zhen Lina, Bo Lia*, Guang-Kui Xub, Xi-Qiao Fenga* a

Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China b Department of Engineering Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China [email protected] [email protected] *

Corresponding authors.

Abstract Tumorigenesis often involves specific changes in cell motility and intercellular adhesion. Understanding the collective cancer cell behavior associated with these specific changes could facilitate the detection of malignant characteristics during tumor growth and invasion. In this study, a cellular vertex model is developed to investigate the collective dynamics of a disk-like aggregate of cancer cells confined in a confluent monolayer of normal cells. The effects of intercellular adhesion and cell motility on tumor progression are examined. It is found that the stresses in both the cancer cells and the normal cells increase with tumor growth, resulting in a crowded environment and enhanced cell apoptosis. The intercellular adhesion between cancer cells and normal cells is revealed to promote tumor growth and invasion. The tumor invasion dynamics hinges on the motility of cancer cells. The cancer cells could orchestrate into different collective migration modes, e.g., directional migration and rotational oscillations, dictated by the competition between cell persistence and local coordination. Phase diagrams are established to reveal the competitive mechanisms. This work highlights the role of mechanics in regulating tumor growth and invasion.

Keywords Tumor growth; Cancer invasion; Cell motility; Collective cellular dynamics

1. Introduction Cancer cells feature abnormal growth, proliferation, invasiveness and metastasis to distant organs (Hanahan and Weinberg, 2011). During growth and division, cancer cells produce growth factor ligands by themselves or stimulate the circumambient normal cells for 1

supplying back with various growth factors. Owing to these capabilities, tumors could grow from small lesions to larger ones. As tumor grows in a constrained space in vivo, compressive stresses are often engendered in both tumor itself and the surrounding normal tissue (Xue et al., 2016). Such compressive stresses may alter gene expression, cause cell apoptosis, destroy stromal cell functions, compress or even collapse intratumoral vessels, and induce genotypic and phenotypic changes related to tumor malignancy (Jain et al., 2014). Furthermore, compressive stresses may impede the delivery of anticancer drugs and impair therapeutic effects (Stylianopoulos et al., 2012). As a primary tumor grows and deteriorates pathologically, cancer cells may undergo such physiological changes as epithelial-to-mesenchymal transition (Diepenbruck and Christofori, 2016; Park et al., 2016), angiogenesis (Jain et al., 2014), and accumulated genetic transformation (Yang et al., 2004), which are critical for the invasion and metastasis of malignant tumors (Chaffer and Weinberg, 2011). Tumor invasion and metastasis induce a secondary tumor in a distant organ, enhancing the difficulty of tumor therapy. It has been reported that more than 90% of cancer-related mortality is ascribed to tumor metastasis (Wirtz et al., 2011). Therefore, tumor invasion, which serves as the first step of the ‘invasion– metastasis cascade’, is of significant importance in anticancer treatment. Tumor invasion occurs via the migration of single cells or cell clusters (Friedl et al., 2012). Cell migration is a highly integrated process that orchestrates embryogenesis (Bertet et al., 2004), contributes to epithelial homeostasis (Eisenhoffer et al., 2012), and drives progression in diseases such as cancer and atherosclerosis (Luster et al., 2005; Friedl and Gilmour, 2009). Single cell migration is attributed to a cyclic biophysically integrated process involving cell polarization, attachment formation and disassembly, and cell contraction (Lauffenburger and Horwitz, 1996). Cell polarization is regarded as the first step and the driving force of single cell migration, and is mediated by complex regulatory pathways. In a polarized cell, different molecular processes take place at the leading edge and the tailing end, resulting in directional vesicle trafficking from the tailing end to leading edge, microtubules orientation, and specific localization of some organelles (Ridley et al., 2003). Polarized cells are characterized by the extension of lamellipodia or filopodia at the leading edge to guide migration. Experiments have evidenced that such polarized migration of a polarized cell is related to cell memory on its historical movement, also referred to as cell persistence (Selmeczi et al., 2005; Takagi et al., 2008; McCann et al., 2010). Besides, collective motion involves local coordination between neighboring cells (Angelini et al., 2011; Reffay et al., 2011; Vedula et al., 2012; Doxzen et al., 2013; Bi et al., 2015). Therefore, cell persistence, with the effects of local 2

coordination between neighbors together, affects the migration of cancer cells during tumor invasion. Understanding the coordinating mechanisms behind collective tumor invasion would benefit cancer diagnose and treatment. In this paper, we focus on tumor growth and local invasion, with the aim to reveal the initial features of primary tumors at the early stage. The cellular dynamics in the growth and invasion of a two-dimensional (2D) epithelial tumor is investigated by using a dynamic vertex model, which integrates the feedback mechanisms between cell migration and motile forces. This model allows us to study the effects of cancer cell invasiveness on the collective dynamics of tumor monolayer.

2. Dynamic vertex model We employ a dynamic vertex model to explore the collective cell behavior of epithelial-like tumors during growth and invasion. In the model, the cells are represented by closely connected polygons, and their dynamics can be determined by the evolution of the polygonal vertices (Nagai and Honda, 2001; Farhadifar et al., 2007; Fletcher et al., 2014; Xu et al., 2016). This method has been used to study the morphogenesis of epithelial monolayers at the ground state. For example, Farhadifar et al. (2007) examined the influences of the mechanical properties and proliferation on the packing of epithelial cells. Recently, the vertex model was adopted to address the phase transition and glass-like behavior of cell monolayers. Bi et al. (2015) investigated the density-independent rigidity transition in biological tissue from physical perspectives. Park et al. (2015; 2016) considered the jamming of cell aggregates in such diseases as asthma and cancer. They revealed the occurrence of jamming transition as a result of the variation of cell–cell adhesion induced by disease. In these previous studies on the basis of the vertex model, a passive potential energy was defined to determine the equilibrium configuration of a cell monolayer. In the present work, our attention will be focused on the cell dynamics of tumor growth and invasion. An active motile force accounting for cell persistence and local coordination is introduced into the basic vertex model to examine the morphomechanics of cell migration during tumor invasion. A similar concept of motile forces has been introduced in a hybrid method that integrates the vertex model and the self-propelled particle model to study collective cell dynamics (Li and Sun, 2014; Bi et al., 2016). Consider a 2D epithelial tumor, modelled as a roughly circular cluster of motile cancer cells surrounded by non-motile stromal (normal) cells, as shown in Fig. 1(a). Periodic boundary conditions are applied to eliminate the influence of boundaries on cell trajectories, 3

given that the tumor is always encompassed by a large number of normal cells in vivo. The forces at each vertex arise from the elasticity and contractility of cells, the cell–cell interfacial tensions, the cell–substrate friction, and system noises, whereas the inertia of cells are omitted (Fletcher et al., 2014). We assume that the cell dynamics is overdamped. Thus, the force balance condition at vertex i reads Fie  Fia  Fir  0 , as shown in Fig. 1(b). Considering the friction force

Fir   dri dt , the Langevin equation that controls time evolutions of cell vertices is expressed as



dri  Fie  Fia , dt

(1)

where ri (t ) is the position of vertex i at time t ,  denotes the viscosity coefficient e a between the cells and substrate, and Fi and Fi are the passive and active forces acting on

vertex i , respectively. e v s The passive force has two parts, i.e., Fi  Fi  Fi , where Fiv and Fis refer to the force

induced by cell area elasticity and cell–cell interfacial tension, respectively. It can be calculated via

Fie  i U v  Us  ,

(2)

where U v and U s are the potential energies of cell area elasticity and cell–cell interfacial tension, respectively. U v is expressed as (Farhadifar et al., 2007) N

Ka 2  AJ  A0 J  , J 1 2

Uv  

(3)

where K a denotes the area stiffness and AJ represents the area of cell J . A0 J denotes the preferred area of cell J . For those cells that have not been activated and do not grow, the preferred area is assumed to be constant, i.e., A0 J  A0 . For the activated cancer cells that can grow and divide, however, A0 J is larger than A0 and increases as cell growth proceeds until its division. The force vector at vertex i induced by cell area elasticity can be derived as

4





1 Ka  AJ  A0 J  k  r j2  r j1  , J Ci 2

Fiv   

(4)

where k denotes the unit vector normal to the epithelial plane, the summation



runs

J Ci

over the three neighboring cells Ci of vertex i , and j1 and j2 represent the neighboring vertices of vertex i in cell J , as shown in Fig. 1(b). The potential energy induced by cell–cell interfacial tension is expressed as (Nagai and Honda, 2001) U s    IJ lij ,

(5)

i, j

where  IJ denotes the interfacial tension between neighboring cells I and J that form an intercellular edge between vertex i and j , and lij  ri  r j is the corresponding edge length. The summation



runs over all pairs of neighboring vertices. In the composite

i, j

cell system, there exist three kinds of intercellular edges, namely, cancer–cancer, normal– cancer, and normal–normal edges, whose interfacial tensions are denoted as  cc ,  nc , and

 nn , respectively. In light of Eq. (5), the force induced by interfacial tension can be derived as

Fis     IJ jVi

ri  r j ri  r j

,

where the summation

(6)



encompasses all neighboring vertices Vi of vertex i .

J Vi

a c p R The active force Fi  Fi  Fi  Fi comprises the contributions from active cell c p R contraction Fi , cell motility Fi , and the system noise Fi . The potential energy of cell

contraction is written as (Farhadifar et al., 2007) N 1 U c   K c L2J , J 1 2

(7)

where K c denotes the contractile modulus and LJ the perimeter of cell J . Thus, the contraction force vector is

5

 ri  r j ri  r j2 1 Fic    K c LJ    ri  r j ri  r j2 J Ci 1 

 .  

(8)

p Introduce a motile force Fi to characterize the effect of cell persistence and local

coordination. Specifically, the motile force of cancer cell J is denoted as

FJp   J e pJ ,

(9)

p where  J and e J stand for the motile force intensity and the preferred migration direction

of cell J , respectively. Thus, the motile force on vertex i is

Fip 



J Ci

 J e pJ nJ

,

(10)

where n J is the edge number of cell J . The motile force of each cancer cell is equally allocated to its vertices. In a collective manner, the preferred migration direction of a cell depends not only on cell persistence associated to its migration history (i.e., cell memory effect) but also on local coordination related to the current migration of its neighboring cells. The preferred migration direction of a cancer cell J is thus determined by (Helbing et al., 2000)

e pJ  pJ esJ  1  pJ  e nJ ,

(11)

s n where e J refers to the persistent direction of cell J and e J indicates the concerted

direction arising from local coordination. The parameter pJ   0,1 characterizes the contribution of cell persistence and local coordination to the preferred migration direction. s The cell persistence direction e J of cell J , depending on its history of velocity, is

expressed as (Li and Sun, 2014)

e  s J

 

t

 t 

v J   e   t  d v J   e

   t  

d

,

(12)

where  is the cell memory decay rate and v J   denotes the velocity of cell J at time

 . Equation (11) indicates that the cell memory decays with a persistent time scale  p  1  . The concerted direction is taken as 6

e  n J

v

KC J

v

KC J

K

,

(13)

K

p where C J denotes the neighboring cells of cell J . The motile force Fi can distinguish the

effects of cell memory of historical migration and neighboring cell–cell coordination on the cell motility in a social context. This way is different from that in Bi et al. (2016), where a self-propulsive force with a direction determined by the memory of stochastic noise in the system was proposed to characterize the cell polarization. Equation (11) shows that when pJ  1 , cancer cells tend to maintain their current p s migration direction (i.e., e J =e J ), and when pJ  0 , they update their migration directions p n with reference to their neighboring cells (i.e., e J =e J ). Therefore, the parameter pJ defined

in Eq. (11) indicates the persistence of cancer cell J . For simplicity, here we assume that

pJ  p and  J   for all cancer cells, and  J  0 for all normal cells. To account for the noise in the system arising from fluctuations, we introduce the Gaussian white noise, in terms of a random force, acting on each cell. It is expressed as

FJR   R ξ J ,

(14)

where  R is the intensity of white noise, and ξ J is a unit-variance Gaussian noise on cell

J , which satisfies ξ J  0,

Ip  t   Jq  t    IJ  pq  t  t  ,

(15)

where   t  is the Dirac function and  IJ is the Kronecker delta. Accordingly, the random force acting on vertex i can be calculated as

FJR F  . J Ci nJ R i

(16)

In addition, we non-dimensionalize the motion Eq. (1) and all geometric and physical parameters (see Supplementary Material). In the problem under study, the epithelial cells may rearrange due to such mechanisms as the intercalation (i.e., neighbor exchange), apoptosis and division of cells. Therefore, we consider topological transitions in the polygonal network, including T1 transition, T2 transition, and cell division, as described in 7

Fig. S1 in Supplementary Material.

3. Tumor growth 3.1 Dynamic simulations of tumor growth Assume that a primary tumor grows along the gradient of nutritional transportation from its periphery. Therefore, the likelihood that a cancer cell J enters the mitotic cycle can be expressed by the exponential decay function pJg  g exp  kgd J  , where d J is the minimum distance from cell J to the tumor periphery that directly exposed to normal cells, and g and k g are constants. When a cancer cell J enters the mitotic cycle, we gradually increase its preferred area A0 J to account for cell growth. When the current cell area AJ reaches the threshold of cell division, it will be divided into two daughter cells. In the simulations, we account for the effect of cell cycle period. The dimensionless period of cell cycle is taken as T  T  0  10 . If a cell enters the mitotic cycle at time t and finally divides, its daughter cells could enter the mitotic cycle only after time t  T . In this section, we ignore the motility of cancer cells and set   0 in order to focus on the cellular dynamics of epithelial tumors induced by cancer cell growth. The numerical computations of Eq. (1) allow us to determine the dynamic process of tumor growth. It is found that the tumor grows, expands, and then invades the normal tissue, accompanied by the development of finger-like structures at the tumor margin, as shown in Fig. 2(a). Further growth will engender progressively enhanced compressive stress in both the cancer cells and normal cells, and lead to the invasion of some cancer cells into the surrounding normal cells, as shown in Fig. 2(a). Besides, despite of the cell apoptosis that may happen due to the accumulated compressive stress, the cancer cell population increases gradually, as shown in Fig. 2(b). To quantify the crowdedness of cells in the growing monolayer, we calculate the stresses in the tissue by (Ishihara and Sugimura, 2012) J 

    PJ  

1  2 AJ



i , j E J

Tij

lij lij lij

,

(17)

where  and  run along the coordinate indices x and y , and lijx and lijy represent the projection components of edge ij along the x and y directions, respectively. The

8

summation



computes over all edges E J of cell J . PJ   Ka  AJ  A0 J  denotes

i , j E J

the stress induced by area deformation, and Tij    Kc  LI  LJ  represents the edge tension of edge ij that connects cells I and J . Therefore, the mean normal stress within





J    J    yy J  , which also quantifies the crowdedness cell J can be derived as  m  1 2 xx

degree of cell J . The mean normal stress in the tumor itself is calculated as 1

c

m

       m J  AJ    AJ  , where the summation  J Ccancer   J Ccancer 



runs over all cancer cells

J Ccancer

Ccancer . Similarly, we can calculate the mean normal stress  m  in the normal cells. As the n

tumor cells proliferate, the overall size of tumor monolayer grows. During this process, the imbedded tumor gradually pushes the surrounding normal tissue, leading to a crowded state and a compressive stress field in the tumor (Fig. 2(a)). It has been reported that normal cells growing in a confined space may also engender crowding and compression (Marinari et al., 2012). The compressive stress state induced by confined cell division is in contrast to the tensile stress state in an epithelial monolayer spreading in a free space (Tambe et al., 2011; Serra-Picamal et al., 2012). The increasing compressive stress in the tumor has been shown to inhibit spheroid growth and enhance the invasive phenotypes of cancer cells (Jain et al., 2014). Furthermore, the compressive stress  m increases in both cancer cells and normal cells as tumor grows. The increase in  m is particularly marked in cancer cells (Fig. 2(c)), rendering an increase in the number of apoptotic cells (Fig. 2(b)).

3.2 Role of intercellular adhesion Cancer progression is characterized by an increasing change in the cell–cell cohesiveness that originates from such processes as the epithelial-to-mesenchymal transition (Chaffer and Weinberg, 2011). Therefore, we investigate how the adhesions between cancer cells and normal cells influence the cellular dynamics within a growing tumor. It is found that when  nc  1   nn   cc  , the dispersion of cancer cells increases; and, consequently, they may 2

invade the stroma, enhancing the tumor growth (Figs. 3(a) and (c)). In contrast, when  nc  1   nn   cc  , the cancer cells will be arrested the stroma, as shown in Figs. 3(b) and 2

(c). These different tumor growth patterns, associated with cancer–normal cell adhesion, coincide with the differential adhesion hypothesis (Foty and Steinberg, 2005) and the cell 9

sorting phenomenon observed in experiments (Hayashi and Carthew, 2004; Krieg et al., 2008). To quantify the tumor invasion, we define the degree of heterogeneity in a cancer–normal cell system as (Belmonte et al., 2008)

 t  

n  J J n   n  J

where n

J

,

(18)

J

denotes the number of the different-type neighbors of cell J (i.e. cancer–

normal) and n

J

is the number of the same-type neighbors of cell J (i.e. cancer–cancer or

normal–normal). 

J

stands for the average over all cells. Then, the invasion degree is

defined as



 t f  1 ,   t0 

(19)

where t0 denotes the time at the initial state, and t f is the ending time of the simulation. The time interval t  t f  t0 is set to be constant. Figure 3(c) indicates that the invasion degree  increases with the decrease in the cancer–normal cell interfacial tension  nc or, in other words, with the increase in the cancer–normal cell adhesion. This is because stronger cancer–normal cell adhesions will induce cancer cells to diffuse into the normal cells (Fig. 3(a)). Such a dispersive pattern will partly release the compressive stresses in the tumor and further facilitate tumor growth. The above results demonstrate that cancer–normal cell adhesions play a significant role in the growth and invasion of tumor.

4. Tumor invasion 4.1 Collective invasion We next investigate active tumor invasion while excluding the effect of cell proliferation. To characterize the intrinsic invasiveness of cancer cells, we set the initial preferred migration direction e pJ along the radial direction of the cancer cell cluster. It has been known that cancer cells exhibit a proclivity of decreased self-adhesion and enhanced adhesion to extracellular matrix as cancer deteriorates and becomes invasive (Liotta and Kohn, 2001). In the simulations, therefore, we take an increased cancer–cancer interfacial tension cc  1.5 10

and a decreased cancer-normal interfacial tension  nc  0.5 . Our simulations suggest that initially, some cancer cells migrate individually to invade the normal region; subsequently the leader cells, in the form of cell strands or multicellular streams, emerge at the tumor margin to dominate and induce collectively directional invasion (Fig. 4). This finding agrees with experimental observations (Friedl et al., 2012). Note that in our model, all cancer cells are assumed to be polarized equally, and no cells are specified as leader cells. However, leader cells emerge spontaneously at the tumor margin through a process attributed to the feedback (i.e., Eq. (11)) and cohesiveness of the cancer cell cluster. This invasive pattern is reminiscent of the deterioration process observed in melanoma (Balois and Ben Amar, 2014). The evolution of the mean normal stress underscores the compression of cells at the leading edge of tumor invasion and tension of cells at the tailing edge (Fig. 4), which is engendered by the collectively directional invasion of the cancer cell cluster. It is found that the degree of heterogeneity  (t ) increases progressively, indicating an increase in invasiveness (see Fig. S2 in Supplementary Material). To quantify the features of tumor invasion, we calculate the mean squared displacement (MSD) of cancer cells (see Fig. S3 in Supplementary Material). The effects of cancer–normal cell adhesion and cancer cell motility on tumor invasion are examined. It is found that the former promotes tumor invasion but appears to have negligible effects on the migration modes of cancer cells, with a constant MSD exponent   1.96 (Fig. 5(a)). In contrast, the motile force intensity  has a dramatic influence on tumor invasion. Weak motile force tends to engender constrained migration, whereas sufficiently strong motile force may induce directional migration and subsequent tumor invasion (Fig. 5(b)). Besides, the cancer cell coercivity promotes tumor invasion, as shown in Fig. 5(c). In addition, the MSD exponent  increases from 0 to 2 as the coercivity p increases from 0 to 1, indicating the changes in the cancer cell migration modes. Our simulations demonstrate that strongly individualistic cancer cells ( p  1 ) orchestrate themselves into collectively directional migration, as studied and described above. However, strongly collectivist cancer cells ( p  0 ) will exhibit a dynamic behavior of collectively rotational oscillations, which will be studied in the next subsection. 4.2 Effect of cell coercivity We now explore the collectively rotational oscillations of a cancer cell cluster due to strong collectivism ( p  0 ). It is found that the cancer cells exhibit arrested behavior, accompanied 11

by a rhythmic pulsing in the normal cells. The effect of intercellular adhesion on the migration mode of cancer cells is here neglected (Fig. 5(a)), and thus, the interfacial tensions of all cancer and normal cells are assumed to be identical and set as cc   nc  1.0 . The trajectories of a cancer cell and its neighboring cells are plotted in Fig. 6(a). It reveals that cancer cells may undergo collective clockwise rotations around their equilibrium positions. The rotational direction may be right- or left-handed, depending on the initial preferred migration direction field and the system noises. The cancer cells in the rotational cluster show collective areal oscillations simultaneously (Fig. 6(b)). The kymograph of cancer cell velocity, which exhibits regular plus-to-minus alternative strips, clearly reveals the oscillatory characteristics of collective rotations (Fig. 6(c)). Spectral analysis is also performed to further elucidate the dynamic features of the collectively rotational and oscillating cells. It is found that cancer cells tend to undergo synchronous rotations with uniform amplitudes (Figs. 6(d1) and (d3)). However, the oscillation amplitudes of cell areas manifest a spatial gradient distribution: the area oscillations of marginal cancer cells are stronger than those of inner cancer cells, which can be ascribed to the confinement of normal cells (Fig. 6(d2)). In addition, a yin-yang phase pattern is observed for the area oscillations of cancer cells (Fig. 6(d4)). The frequencies of the rotations and area oscillations of cancer cells are illustrated in Fig. 6(e). It indicates a collective dynamics and coordination between the rotations and area oscillations. Moreover, the cancer cell rotation phases  X and  Y are very close to the line  Y  X  π 2 (Fig. 6(f)), confirming clockwise rotations and synchronization in the collective rotations. To evaluate the collective rotations, we define the rotation intensity as

Rm 

xJ  yJ 

,

(20)

J

where x

J

and yJ  are the oscillation amplitudes of the geometric centre

 x  , y   of J

c

J

c

cell J in the x and y directions, respectively. It is found that Rm is proportional to the motile force intensity  (Fig. 7(a)). However, the rotation frequency is independent of  , as shown in Fig. 7(a). Figures 7(b) and (c) provide two phase diagrams illustrating how cancer cell motility regulates the dynamic behavior of the cancer cell cluster confined by normal cells. It shows that under the conditions of a weak motile force intensity  , weak coercivity p , and fast memory decay rate  , the cancer cell cluster tends to undergo collectively rotational 12

oscillations. In contrast, the cluster may undergo collectively directional migration under the conditions of a strong motile force intensity  , strong coercivity p , and slow memory decay rate  .

5. Conclusion The collective cellular dynamics of epithelial tumors has been investigated by using the dynamic vertex model. A feedback mechanism that integrates the influences of cell persistence and local coordination on cell migration is proposed to account for tumor invasiveness. Our results reveal that an increase in the intercellular adhesion between cancer cells and normal cells promotes both the growth and invasion of tumors. The cancer cell coercivity significantly affects the mode of tumor invasion. Cancer cells can orchestrate into collectively rotational oscillation and directional migration, in response to weak and strong cell coercivity, respectively. This study helps understand the role of mechanical factors in the growth, invasion, and metastasis of epithelial tumors.

Conflict of interest statement Authors have no financial and personal relationships that could inappropriately influence or bias this work.

Acknowledgments Supports from the National Natural Science Foundation of China (Grant Nos. 11620101001, 11672161, and 11432008), Tsinghua University (20121087991 and 20151080441), and the Thousand Young Talents Program of China are acknowledged.

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Figure captions Fig. 1. Biomechanical model of a tumor. (a) A 2D epithelial cell monolayer consisting of a disc-like cancer cell cluster surrounded by normal cells. (b) Forces at a vertex, including friction forces Fir , active forces Fia , and passive forces Fie . Fig. 2. Simulation results of epithelial tumor growth. (a) Time series of cell morphology and mean normal stress. (b) Evolution of the cancer cell and apoptotic cell populations. (c) c Mean normal stress in cancer cells and normal cells. In the simulations, we set Ka   10.0 ,

Ka n  10.0 , Kc  0.5 , cc  1.0 ,  nc  1.0 , Tc  10.0 ,  R  0.002 , g  0.15 , and

kg dc  0.5 , where d c is the mean distance between neighboring cells. Fig. 3. Effects of cancer–normal cell interfacial tension on tumor growth. Growth patterns corresponding to (a) decreased cancer–normal cell interfacial tension and (b) increased cancer–normal cell interfacial tension. (c) Increment of cancer cell population δN c and tumor invasion degree  with respect to cancer–normal cell interfacial tension. The data are mean  standard error. Here, δN c and  are computed at the same time interval

t  t f  t0 for different simulations. In the simulations, we set Ka n  10.0 , Ka c  10.0 ,

Kc  0.5 , cc  1.0 , Tc  10.0 ,  R  0.002 , g  0.15 , and kg dc  0.5 . Fig. 4. Time series of cell morphology and mean normal stress. In the second row, black c n arrows denote cell velocity. In the simulation, we set Ka   10.0 , Ka   10.0 , Kc  0.5 ,

cc  1.5 ,  nc  0.5 ,   1.0 ,   0.1 , p  1.0 , and  R  0.002 . Fig. 5. Effects of mechanical cues on tumor invasion. Influences of (a) the cancer–normal cell interfacial tension  nc , (b) the cancer cell motile force intensity  , and (c) the cancer cell coercivity p . The data are mean  standard error. In the simulations, we set

Ka c  10.0 , Ka n  10.0 , Kc  0.5 , cc  1.5 ,   0.1 and  R  0.002 ;   1.0 and p  1.0 for (a),  nc  0.5 and p  1.0 for (b), and  nc  0.5 and   1.0 for (c).

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Fig. 6. Collectively rotational oscillations of cancer cells. (a) Trajectories of geometric centres and (b) area oscillations of a cancer cell and its neighbors, where C0 stands for the selected cancer cell and C1–C6 represent its neighboring cells. (c) Kymographs of the horizontal velocities of cancer cells. (d) Spatial distribution of the relative oscillation amplitudes and phase patterns for cell geometric centre xc and cell area A . The relative

oscillation amplitude is defined as wJ  wJ wm , where wm  max J  wJ  and wJ is the oscillation amplitude of geometric centre xc or area A of cell J . (e) Histogram of the oscillation frequencies of geometric centre xc and area A of cancer cells. (f) Scatter diagram of oscillation phases of cancer cells corresponding to geometric centres xc and c n yc . In the simulations, we set Ka   10.0 , Ka   10.0 , Kc  0.5 , cc  1.0 ,  nc  1.0 ,

  1.0 ,   1.0 , p  0.0 , and  R  0.002 .

Fig. 7. Effect of cell motility on the collective behavior of cancer invasion. (a) Influence of motile force intensity on the rotation frequency and intensity. In the simulations, we set Ka c  10.0 , Ka n  10.0 , Kc  0.5 , cc  1.0 ,  nc  1.0 ,   1.0 , p  0.0 and

 R  0.002 . Phase diagrams regulated by (b) motile force intensity and cell coercivity, and by (c) cell memory decay rate and cell coercivity. In the simulations, we set Ka   10.0 , c

Ka n  10.0 , Kc  0.5 , cc  1.0 ,  nc  1.0 , and  R  0.002 . Besides,   1.0 for (b) and   1.0 for (c).

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