Article
Scattering of Cell Clusters in Confinement Amit Pathak1,* 1
Department of Mechanical Engineering and Materials Science, Washington University, St. Louis, Missouri
ABSTRACT Epithelial-to-mesenchymal transition (EMT) enables scattering of cell clusters and disseminates motile cells to distant locations in vivo during embryonic development and cancer metastasis. Both stiffness and topography of the extracellular matrix (ECM) have been shown to influence EMT. In this work, we examine how the integrity of epithelial cell clusters is regulated by subcellular forces, protrusions, and adhesions for varying ECM inputs, such as stiffness, topography, and dimensionality. Our model simulates multicell networks of defined sizes and shapes in ECMs of varied stiffness and geometry. The integrity of cell clusters is dictated by cell-cell junctions, which depend on subcellular forces and adhesion dynamics within each cell of the cluster. Our simulations demonstrate an enhanced dissociation of cell-cell junctions in stiffer and more confined three-dimensional (3D) environments, consistent with experimental findings. In narrow channels, the cell edges parallel to the axis of channels lose their cell-cell junctions more readily than those oriented in the perpendicular direction. The inhibition of protrusive activity and cell polarity disables confinement-dependent cell scattering. Here, cell adhesion and spreading along channel walls is found to be essential for scattering. The model also predicts that two-dimensional (2D) confinement of clusters restricts cell spreading and simultaneously blunts the confinement-sensitive cell scattering. This new, to our knowledge, multiscale model integrates molecular adhesion dynamics, subcellular forces, cellular deformation, and macroscale mechanical properties of the ECM to predict the state of cell clusters of defined shapes and sizes. The predictions made by our model not only match experimental findings from a number of experimental setups, but also provide a new conceptual framework for understanding mechanosensitive cell scattering and EMT.
INTRODUCTION During both embryonic development and cancer metastasis, the dissemination of motile cells to distant locations is initiated by a cellular program called epithelial-to-mesenchymal transition (EMT). The coordination of actomyosin forces, polarization of cell shape, and loss of cellular contacts with the basement membrane cumulatively enable the detachment and migration of cells. An erratic sequence of these steps is implicated in triggering the onset of tumor invasion and metastasis. The cells at the periphery of tumor microenvironment interact with mechanically heterogeneous extracellular matrices (ECMs), undergo EMT-based scattering, escape from the primary tumor, migrate to distant locations, and subsequently repopulate secondary tumors. Thus, the physical interaction of individual cells with their ECM can play a crucial role in determining the structural integrity of multicell clusters. Over the years, several studies have established that stiffer ECMs cause greater subcellular forces and protrusions, stronger cell-ECM adhesions, and more polarized morphology of individual cells, which in
Submitted March 11, 2016, and accepted for publication August 31, 2016. *Correspondence: [email protected] Editor: Sean Sun http://dx.doi.org/10.1016/j.bpj.2016.08.034 Ó 2016 Biophysical Society.
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turn lead to the rupture of cell-cell junctions followed by EMT (1–4). Since cell polarization is a key precursor to the stiffness-induced cell scattering in these studies, we asked whether the cells trapped inside confined environments could undergo a similar shape polarization, as we and others have shown earlier (5–8), and commence dissociation from their native cluster. This is an important open question given that tissue environments often vary in their topography and dimensionality without significant stiffness variations. Our recent experiments revealed that epithelial clusters disintegrate more readily in confined environments even in soft ECMs (9). When the cells were rendered unable to extend protrusions or polarize, by inhibiting the function of microtubules, this confinement-dependent EMT vanished. The cell-ECM adhesions were found to be essential for stiffness-dependent EMT. These experimental findings point toward a complex machinery of intercoupled cellular, subcellular, and extracellular features that operate at varying time and length scales to trigger varying phenotypes of cell scattering. The molecular level mechanosensing of ECM stiffness regulates nanometer- and submicron-scale changes in cytoskeletal structure and cell-ECM adhesions, which lead to micron-scale changes in single-cell morphology,
Cell Scattering in Confinement
detachment of cells, and a large-scale destabilization of cell colonies and tissues. Similarly, micron-scale complexities in ECM geometry can affect single-cell morphology and cause the scattering of large multicell clusters. To deconstruct the relative contributions of these cellular and extracellular inputs on the integrity of cell clusters, we need a biophysical model that combines mechanosensitive subcellular mechanisms of forces, adhesions, and protrusions to simulate the deformation of single cells within the multicell networks adhered to ECMs of varying stiffness, confinement, and dimensionality. Such a computational model can serve as a powerful tool to interrogate specific influences of extracellular and subcellular inputs in regulating the integrity of cell clusters. The existing computational models have studied cell behavior through varied approaches—from continuum mechanics–based methods for single-cell behavior (10) to network models with discrete cell elements for the dynamics of cell clusters. The discrete cell models based on cellular-automata and agent-based methods (11,12) have informed the roles of intercellular signaling pathways in regulating the behavior of large cell populations. However, these models do not explicitly account for subcellular mechanical features, such as actomyosin forces and ECM adhesions, which enable individual cells in the population to deform and mechanically interact with their neighbors. These missing mechanical features in discrete cell models may explain important aspects of mechanosensing in cell populations. In this work, we present a new model, to our knowledge, for cell clustering scattering that combines the following subcellular and cellular processes: 1) formation and rupture dynamics of integrin-based cell-ECM adhesions, 2) cellular polarization and frontward protrusions, and 3) generation of actomyosin forces. These subcellular processes collectively dictate the state of cell-cell junctions in each cell of a given epithelial cluster. The model also accounts for stiffness and geometry of the ECM surrounding the multicell network. We test the model for varying ECM stiffness (0.1–1000 kPa) and confinement due to channel widths between 20 and 80 mm, emulating several previous experiments (1–4,9,13). Consistent with the known ability of cells to actively respond to ECM stiffness (3,4,14), our model predicts that cell clusters scatter more readily on stiffer ECMs. Surprisingly, the cells trapped within 2D islands do not undergo scattering (13) as they do inside 3D channels with vertical walls (9). After the inhibition of microtubules-based protrusions and cell polarization, cell clusters exhibit a confinement-insensitive behavior. Our calculations corresponding to the inhibition of cell-ECM adhesions predict a stiffness-insensitive behavior of cell clusters. To our knowledge, this is the first model to account for subcellular forces and single-cell morphology in dictating the dynamical state of cell-cell junctions and thus predicting both the stiffness-, confinement-, and dimensionality-dependent scattering of cell clusters. Taken together, these calcu-
lations reveal relative contributions of physical properties (such as ECM stiffness, topography, and dimensionality) and biochemical parameters (such as actomyosin forces, microtubules-based protrusions and polarization, and integrin/ cadherin-based adhesion dynamics) in regulating the integrity of epithelial cell clusters. MATERIALS AND METHODS Overview of the model The cluster of epithelial cells is modeled as a network of square-shaped cells interconnected with one another through cell-cell junctions. In this multicell network, each cell is comprised of adhesion nodes at the cell periphery, a deformable nucleus in the middle of the cell, and cytoskeletal elements connecting adhesion and nucleus nodes (Fig. 1). The cellular nodes at the periphery have two components: cell-cell (CC) junctions, and cellECM (CE) adhesions. Within individual cells, a net direction of frontrear polarity is calculated according to a probability of adhesion formation and the protrusive forces are accordingly applied on cellular nodes in the front region of the cell. The actin-myosin contractile forces depend on the length of cytoskeletal elements and signaling due to the formation of CE adhesions. The movement of nodes due to various cellular forces dictates morphological changes in individual cells, which continually reorganize the entire cell cluster. Various simulations are carried out for multicell clusters of different sizes in contact with ECMs of varying stiffness and geometry.
Cell-ECM adhesion dynamics As cells in the cluster interact with one another, their physical interaction with the ECM is also maintained through CE focal adhesions. It has been shown that forces exerted on focal adhesions stretch the receptor-ligand bonds and aid the growth of adhesions (15). The softer ECMs undergo larger deformation and thus lead to greater distances between receptors and ligands, which in turn is thought to impair adhesion bond formation (16). Conversely, the stiffer ECMs are expected to deform minimally, keep receptors and ligands in closer proximity, and enhance focal adhesion formation. We implement this stiffness-dependent adhesion dynamics in terms of bond displacement d ¼ Fam =akaE , which depends on actin-myosin force Fam . Here, the spring constant kaE of a CE bond accounts for both the stiffness of the receptor-ligand bond (ka ¼ 1 nN/mm) and the Young’s modulus of the ECM (E) according to (17): kaE ¼ ðao Eka Þ=ðao E þ 3ka Þ, where ao ¼ 0:006 mm is the perimeter of the circular area occupied by an average adhesion. In multicell clusters, the integrin-cadherin crosstalk (18,19) dictates that greater number of CC junctions (b) slows down CE bond formation. We combine these two influences of cell-ECM and cellcell interactions to describe the rate of formation of CE bonds for each node (a) as follows:
a_ þ ¼ raþ ða aÞð1 b=bo Þexpðd=do Þ;
(1a)
(henceforth, an over-dot denotes first time-derivative) with raþ ¼ 0:05 h-1 b Þ=N is the maximum number of as a rate constant. a ¼ ð1 þ zÞðb ao a b o ¼ 104 is the total numCE bonds in a given node at current time, where a b is the number of bonded and free receptors per cell at any time (20,21), a bo a b is the ber of bonded receptors in a cell at a given time, and thus a number of free receptors in the whole cell, with N ¼ 36 nodes per cell. The vertically distributed adhesion nodes on channel walls, with height z > 0, are represented by combining (1 þ z) number of nodes into one node. The force-based rupture rate of CE bond dissociation rate according to Bell’s law (20,22) is written as koff ¼ ra expðFam D=kB TÞ, where kB is the Boltzmann constant, T is the absolute temperature, ra ¼ 0:001 h-1 is
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Forces in cytoskeletal elements The cytoskeletal elements connected to CE nodes generate contractile forces proportional to the number of actin-myosin compartments in a given element and mechnotransductive signaling due to CE adhesions in the entire cell ðb a Þ as follows:
b a b o; Fam ¼ fam ða þ bÞlam a
(2)
where fam ¼ 12 pN is the constant force generated per actin-myosin compartment (23) and the size of the connected adhesion node (a þ b) represents the number of actin-myosin compartments per unit length of the element ðlam Þ. The cytoskeletal elements are assumed to deform in a linear elastic manner, such that the passive force generated within an element is written as follows:
Fcy ¼ kcy ða þ bÞðlam lo Þ;
(3)
where kcy ¼ 0:1 nN/mm is a spring constant and lo ¼ 3 mm is a reference length for all elements despite their different initial lengths depicted in Fig. 1 a. The number of springs in each element is proportional to the number of actomyosin compartments (proportional to the adhesion size a þ b).
Protrusion forces and cell polarization The initiation of polarity and extension of protrusions (24,25) are dynamic processes that occur within individual cells in the epithelial cluster. In this model, the direction of polarization is evaluated from a set of 12 possible angles between 0 and 2p, equally discretized at an interval of p/6, for each cell every 2 h. The probability of choosing a given angle of polarity is weighted by the total number of CE adhesions in the front region (defined by the direction of polarity under consideration) of the cell. Thus, a polarization direction corresponding to larger adhesions in the front of the cells is more likely to be picked. This selection criterion for protrusion direction is based on the cell’s ability to dynamically polarize in directions that are most suitable for forming stable adhesions. The magnitude of the protrusion force on a given cellular node comprises an active outward force proportional to the size of the adhesion and a uniform expansion of the cell body, written as follows: FIGURE 1 (a) Schematic of a cell cluster confined within channels of defined width modeled as a network of multiple cells connected together through cell-cell junctions. (b) Each cell is composed of adhesion nodes and cytoskeletal elements (gray) that support protrusive (Fpr) and actomyosin (Fam) forces. (c) The adhesion nodes (green) combine both cell-cell junctions and cell-ECM adhesion and cytoskeletal element combines active (actomyosin) and passive (elastic) forces. To see this figure in color, go online. a rate constant, Fam is the acto-myosin force on the bond, and D is the bond deformation for potential-energy minimum. For an adhesion node with a bonds and ka as the stiffness per bond, the bond deformation is D ¼ Fam =aka . Thus, we write the bond dissociation rate for a given node as follows:
Fpr ¼
fpr a þ fex p;
(4)
where fpr ¼ 0:4 nN is force per CE adhesion. An adhesion-independent spreading force fex ¼ ðfex =NÞð2Ai AÞ=A is also applied on each node to mimic cell spreading such that the cell area A can reach up to twice the initial area Ai ¼ 100 mm2, with fex ¼ 300 nN (estimated relative to other force constants used in these simulations). The direction of the protrusion force on each node is outward from the cell nucleus, i.e., opposite of the actomyosin force (Fig. 1 b). Based on the chosen protrusion direction, the polarity parameter p ¼ maxðd=dmax ; 0Þ defines nodes in the front of the cell, with d as nodal distance from the nucleus along the direction of polarization and dmax as the distance of the node that is farthest from the nucleus.
a_ ¼ ra a exp
F2am : aka kB T
(1b)
Nodal displacement and cell movement
Taken together, the dynamics of CE adhesion bonds (a) combines the rates for formation (Eq. 1a) and rupture (Eq. 1b) rates as follows:
The force equilibrium at each node is ensured at every time step and the displacement of the adhesion nodes is calculated according to a vector addition of all forces as follows:
a_ ¼ a_ þ a_ :
0 ¼~ Fpr þ ~ Fam þ ~ Fcy þ ~ Fmc þ h~ x_ c ;
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(1c)
(5a)
Cell Scattering in Confinement where h ¼ 5 nNmm-1 s is a constant drag coefficient estimated for these simulations, and ~ xc is the position vector of adhesion nodes. Here, Fmc is a linear elastic force proportional to the deformation of spring-like cell membrane elements and a spring constant of kmc ¼ 10 nN=mm. Similarly, the displacement of the nuclear nodes is also calculated according to the following force equilibrium:
0 ¼ ~ Fam ~ Fcy þ ~ Fmn þ ~ Fnu þ h~ x_ n ;
(5b)
where Fmn is an elastic force due to deformation of the nucleus membrane element, Fnu is an elastic force due to deformation of the nucleus body, and xn is the position vector of nuclear nodes. All nodes on the nucleus body are connected to each other through nuclear elements, which deform linearly based on a spring constant knu ¼ ð75=NÞ nN=mm per node. The membrane elements on the periphery the nucleus deform linearly according to spring constant kmn ¼ 40 nN=mm.
Dynamics of cell-cell junctions The CC junctions dissociate in a force-dependent manner (13,20,26,27) according to a relationship that accounts for different bond rupture rates in two force regimes, similar to Evans and Ritchie’s equations (22,28):
( b_ ¼
fb < fb rb bfb fb ; 2 ; rb b fb fb ; fb Rfb
(6a)
where rb ¼ 0:02 h-1 is a rate constant, fb is the net force on the node due to cytoskeletal and membrane elements of the two neighboring cells, and fb ¼ 0:4 nN is a force constant. The cytoplasmic E-cadherin binds to b-catenin and translocates to the cell membrane to form new cell-cell bonds (29). Thus, in the model, the ruptured CC complexes are allowed to form new CC junctions as follows:
b =b a o Þ; b_ þ ¼ rb ðbo bÞð1 a
(6b)
where bo ¼ 5000=N is the maximum number of CC junctions per node. According to the known integrin-cadherin crosstalk (18,19), the cell-ECM adb ) oppose the formation of CC junctions. Thus, the net rate hesions (greater a of change of the number of CC junctions at any node is as follows:
b_ ¼ b_ þ þ b_ :
(6c)
Simulation of cell clusters in defined ECMs To gain quantitative insights into how the complex interplay of subcellular and extracellular cues might lead to cell scattering, as evident by our recent experimental findings (9), we simulate the interaction of cell clusters with ECMs of varying stiffness and confinement. We integrate the rate-dependent models presented above (Eqs. 1–6) to simulate the physical interaction of cell clusters with ECMs of varied stiffness and geometry by ensuring force balance at each node at all times. Initially, cell dimensions are chosen as 10 10 mm, with 4 mm nucleus diameter. Each cell is discretized into N ¼ 36 adhesion nodes and the same number of cytoskeletal elements. Initially, CC junctions are at their maximum value bo , with no CE adhesions. The system of ordinary differential equations (ODEs) developed above (Eqs. 1, 5, and 6), corresponding to CE adhesions a, CC junctions b, cellular adhesion node position xc , and nuclear node position xn , is solved using Runge-Kutta scheme from time t ¼ 0–120 h, consistent with experimental culture period (1–4,9). As the CE adhesions grow (Eq. 1), the protrusion and actomyosin forces rise (Eqs. 2–4), which displaces the leading edge in the outward direction and polarizes the cell (Eq. 5). Thus, due to the
surge of forces and nodal displacements in each cell, the CC junctions start to decay (Eq. 6).
RESULTS AND DISCUSSION Enhanced scattering of cell clusters confined in 3D channels In these simulations, cell cluster length is kept constant at 80 mm and cluster width equals the given channel width. First, we simulated the response of a cluster confined within a wide channel, width w ¼ 60 mm, made of soft ECM (E ¼ 1 kPa). The movement of adhesion nodes in contact with channel walls was restricted in the outward y-direction (Fig. 1 a) to keep the cells within the confines of the channels, with a maximum permitted height of 10 mm in the z-direction. As shown in Fig. 2 a, each cell in the cluster interacts with the ECM differently, deforms due to intracellular forces, and loses some CC junctions by day 3. Here, green nodes denote CC junctions and blue cytoskeletal elements depict the front of the cell at a given time. By day 5, cells neighboring the channel walls lost more CC junctions than those in the interior, but the cell cluster did not disintegrate (see cellular deformations and cell cluster rearrangement over time in Movie S1 included in the Supporting Material). We repeated simulations in a narrow channel (w ¼ 20 mm) and found that cells adhered and elongated along the channel walls, which led to a loss of CC junctions more rapidly than in wide channels. By the end of day 5, the cell cluster scattered with no remaining CC junctions (Movie S2). These simulations are consistent with experimental findings (9), as illustrated in Fig. 2. Next, we repeated simulations in a stiff ECM (E ¼ 100 kPa) inside both wide and narrow channels. On stiff ECM, cells formed more CE adhesions and generated higher forces than on the soft ECM, which led to a loss of CC junctions in both wide and narrow channels, with even more rapid declustering in the narrow channel (compare days 3 and 5 in Fig. 2, i and j, and m and n; see Movies S3 and S4). These results match experimental findings, in which stiffer ECMs caused EMT in both narrow and wide channels. To understand cell scattering in a broader range of ECM stiffness and confinement, we repeated simulations for channel widths w ¼ 20, 40, 60, and 80 mm and ECM stiffness E ¼ 0.1, 1, 10, 100, and 1000 kPa. In each condition, we computed an average number of CC junctions after day 5, normalized by their initial value (at time t ¼ 0), and plotted against ECM stiffness (Fig. 3 a). In the widest channel (unconfined ECM), higher ECM stiffness led to a greater loss of CC junctions and caused more cell scattering. This direct relationship between CC junctions and ECM stiffness vanished in the narrowest channels—compare the flat red line for 20 mm channel with higher slopes for wider channels in Fig. 3 a. Thus, the stiffness matters less in narrow channels, which is consistent with our experimental findings (9). The ECM confinement matters more in softer environments, as evidenced by the
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FIGURE 2 Simulated configurations of cell clusters confined in wide channels of 60 mm width (a, b, i, and j) and narrow channels of 20 mm width (e, f, m, and n) made of soft (1 kPa) and stiff (100 kPa) ECMs. Number of CC junctions and CE adhesions on any node are denoted by green and black circles, respectively, of proportional size. Cytoskeletal elements marked in blue denote the front of the cell, i.e., those with nonzero protrusive forces. Simulations are compared with representative immunofluorescence images (c and d, g and h, k and l, o and p), obtained from our previous study (9), of E-cadherin (green) expression, with DAPI (blue), in MCF10A cell clusters cultured for either 3 or 5 days inside channels of varying stiffness and confinement. The scale bar represents 50 mm. To see this figure in color, go online.
diverging trend-lines toward the lower ECM stiffness regime in Fig. 3 a. The CC junctions decayed more rapidly in narrower and stiffer channels (Fig. 3, b and c). We also found that the cells formed more CE adhesions in stiffer ECMs and narrower channels (Fig. 3 d). Thus, the CE adhesions and the CC junctions have an antagonistic relationship, consistent with previous experimental findings of integrincadherin crosstalk (19,30). The reduced stabilization of CEs in softer ECMs caused low protrusion and actomyosin forces, which in turn disabled the dissociation rate of CC junctions. In contrast, on stiff ECMs, the model allowed greater stabilization of mechanosensitive CEs, which led to higher forces and faster dissociation of CC junctions (Fig. 3, b and c). The CE adhesions reach a saturation level beyond the ECM stiffness of 10 kPa in all channel widths (see the overlapping lines corresponding to E ¼ 10, 100, and 1000 kPa in Fig. 3, e and f). Faster rupture of cell-cell junctions parallel to the axis of narrow confinement In our model setup, the CC junctions can form on cell edges that are oriented either orthogonal or parallel to the direction
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of the channel walls, as illustrated in Fig. 4. We repeated the simulations performed above for cell clusters in narrow or wide channels made of soft or stiff ECMs and separately plotted the time evolution of the number of CC junctions (normalized by their maximum value) on cell edges parallel and orthogonal to the channel walls. In narrow channels made of soft ECM, the CC junctions parallel to the channel walls (red) broke before the CC junctions perpendicular to the channel walls (blue), with the largest difference between the two configurations after day 3, as shown in Fig. 4 a. It is likely that as cells climb the channel walls and move apart, they generate higher forces and enable the rupture of CC bonds in the middle of channel. This predicted difference between the dissociation rates of parallel and orthogonal CC junctions indicates that cell scattering in confinement does not occur homogeneously. Instead, the probability of rupture of a given CC junction depends on the orientation of the cell edge relative to the confinement geometry. We also found slower dissociation of CC junctions in cell edges parallel to the narrow channels made of stiff ECM (Fig. 4 b). However, the preferential dissociation of CC junctions parallel to narrow channel walls (Fig. 4, a and b) did not occur in wide channels made of either soft
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FIGURE 3 Number of CC junctions (a) and CE adhesions (d) at time t ¼ 120 h, normalized by their maximum values, plotted against ECM stiffness for different channel widths. Time variations of CC junctions and CE adhesions for the cases of narrow (b and e) and wide (c and f) channels are plotted for different ECM stiffness values. To see this figure in color, go online.
or stiff ECMs (Fig. 4, c and d). Thus, the interaction of cells with the walls of narrow channels is necessary for the initial separation of cell edges that are oriented parallel to the axis of confinement. Inhibition of cell-ECM adhesions blunts stiffnesssensitive cell scattering Since higher CE adhesions on stiffer ECMs correlated with a greater loss of CC junctions and cell scattering (Fig. 3, a and d), we hypothesized that inhibiting the formation of CE adhesions might blunt the cell’s ability to sense ECM stiffness. Indeed, our recent experiments showed that FAK inhibition led to stiffness-insensitive EMT (9). We simu-
lated this adhesion inhibition by reducing the forward rate of CE formation by 1/10th, i.e., multiplying a_ þ (Eq. 1a) by 0.1, and repeated the calculations for all ECM conditions (stiffness and channel width) noted above. As expected, the dependence of CE adhesions on ECM stiffness was dramatically reduced for all channel widths (see flat lines in Fig. 5 f). Simultaneously, the CC junctions were rendered stiffness-independent (see flat lines in Fig. 5 e and cluster configurations in Fig. 5, a–d) compared with the control case (varying slopes in Fig. 3 a). While this CE-inhibition blunts the ECM stiffness-dependent decay of CC junctions, active protrusions are still able to spread along channel walls, polarize cells (Fig. 5, a–d), and enable confinement-dependent dissociation of CC junctions, as evidenced
FIGURE 4 Time variation of the number of CC junctions, normalized by their maximum quantity, oriented parallel and orthogonal to channel walls for narrow (a and b) and wide (c and d) channels made of soft (a and c) or stiff (b and d) ECMs. To see this figure in color, go online.
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FIGURE 5 Simulated configurations of cell clusters after inhibition of CE adhesions in wide (a and c) and narrow (b and d) channels made of soft (a and b) and stiff (c and d) ECMs. Normalized values of CC junctions (e), CE adhesions (f), and protrusion force (g) at time t ¼ 120 h plotted against ECM stiffness for different channel widths after inhibition of CE adhesions by slowing the forward rate of CE adhesion formation, by multiplying a_ þ (Eq. 1a) by 0.1. To see this figure in color, go online.
by the channel width-dependent differences in the protrusion forces (Fig. 5 g) and the CC junctions (separated trend-lines in Fig. 5 e). Thus, the inhibition of CE adhesions blunts stiffness-dependent cell scattering but retains the confinement-sensitive dynamics of CC junctions through protrusive forces in cells (see Figs. 5, a–e, and S1 for time variation of CC junctions and CE adhesions for each condition), which is consistent with experimental findings (9). Disruption of protrusions and polarity abrogates the effect of 3D confinement Microtubules enable front-to-rear polarization and stabilize frontward protrusions within the cells. In our recent experiments, the inhibition of microtubules reduced morphological polarization and protrusive activity in cells, which led to a confinement-insensitive EMT. We simulated a similar microtubule-inhibition by prescribing randomized (adhesion-independent) protrusion directions at 2 h time interval and reducing the protrusion force constant to 1/10th of its original value ðfpr ¼ 0:04 nNÞ. Together, these changes disrupt the adhesion-dependent front-to-rear polarity and reduce protrusion forces. Consistent with the experimental findings (9), our simulations predict a dramatically reduced dependence of CC junctions on confinement, as evidenced by the overlapping trend-lines for varying channel widths in Fig. 6 e. Here, the reduction in protrusions dramatically reduced cellular
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deformation and polarization of cell shape (see cluster configurations in Fig. 6, a–d), which meant that wide and narrow channels had similar effects on the cell cluster. Note that the confinement dependence of the CE adhesions was more pronounced after microtubule-inhibition than after adhesion-inhibition—compare Figs. 5 e and 6 e. As expected, the protrusion force after microtubule-inhibition remained low regardless of channel width (Fig. 6 g). Taken together, these results suggest that ECM stiffness- and confinement-sensitive EMT follow two different pathways: 1) CE adhesions regulate stiffnesssensitive EMT and adhesion-inhibition does not turn off confinement-sensitive EMT, and 2) protrusive forces and cell polarization regulate confinement-sensitive EMT but the stiffness-sensitive effects persist even after their inhibition. Random cellular polarization reduces declustering but preserves confinement-sensitivity Our simulations and validation with experimental findings presented above establish that adhesion-dependent cellular polarity and protrusions are necessary for confinement-sensitive scattering of cell clusters (Fig. 6). To further delineate the relative importance of the adhesions-dependent cellular polarity and the magnitude of protrusion forces, we performed simulations with randomly assigned directions of polarity without reducing the magnitude of the protrusion force. In these simulations, we randomly picked a direction
Cell Scattering in Confinement
FIGURE 6 Simulated configurations of cell clusters after inhibition of microtubules in wide (a and c) and narrow (b and d) channels made of soft (a and b) and stiff (c and d) ECMs. Normalized values of CC junctions (e), CE adhesions (f), and protrusion force (g) at time t ¼ 120 h plotted against ECM stiffness for different channel widths after inhibition of microtubule-based protrusions and polarization. To see this figure in color, go online.
of polarity for each cell in the cluster every 2 h without concern for adhesions in the front or rear of the cell. In narrow-soft, wide-soft, and wide-stiff channels, the randomly polarized cells had a higher number of CC junctions compared with the control case (Fig. 7 a). Thus, we predict that cells without the ability to polarize in an adhesiondependent manner lose their CC junctions less efficiently as compared with the control case. We also predict that randomly polarized cells form a lesser number of CE adhesions (Fig. 7 b) and generate smaller protrusion forces (Fig. 7 c). Strikingly, even without the stable polarization, cells were able to distinguish between narrow and wide channels in both soft and stiff ECMs, as evidenced by the difference in red and blue bars in Fig. 7. Nonadhesive channel walls disable confinementdependent cell scattering Through simulations and validation with experiments, we have shown that CE adhesions and protrusions are necessary for stiffness-sensitive and confinement-sensitive cell scattering, respectively (Figs. 5 and 6). According to our predictions (Fig. 7, a–c), whereas the adhesion-independent cellular polarization reduces CC dissociation, it preserves
confinement-sensitivity through interaction with channel walls. Next, we sought to isolate the role of CE adhesions along channel walls in the ability of cells to sense confined environments. To this end, we performed simulations in which adhesion nodes in contact with the channel edges were not allowed to grow on the channel walls. In this case, we predict that the cells in narrow channels made of soft ECM are able to preserve their CC junctions much more efficiently compared with the control case (red bar in Fig. 7 d). As a result, the number of CC junctions in both narrow and wide channels made of soft ECM end up at similar levels. Thus, without a robust interaction with the channel walls, cells lose their ability to undergo confinement-sensitive scattering. We also predict that the number of CE adhesions (Fig. 7 e) and the protrusions forces (Fig. 7 f) remain confinement-independent when the cells are rendered unable to form new adhesions with the channel walls. Scattering of cell clusters in 2D confinement depends on spreading area In a recent study, protrusions have been shown to regulate forces at the cell-cell interface and the cell scattering
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FIGURE 7 After implementing randomized cell polarity, the calculated number of CC junctions (a), number of CE adhesions (b), and protrusion force (c) at time t ¼ 120 h, inside narrow (20 mm) or wide (60 mm) channels of varying stiffness. In case of nonadhesive channel walls, predicted CC junctions (d), number of CE adhesions (e), and protrusion force (f) at time t ¼ 120 h for varying stiffness and channel width. To see this figure in color, go online.
stopped when cell clusters were trapped within a 2D adhesive island, despite the addition of a mechanosensitive growth factor (13). Thus, protrusive activity and cell spreading are essential for cell scattering. To examine if our model can explain this behavior, we performed simulations where a cluster of four cells was allowed to spread on a 2D adhesive island of defined area. Here, cells were forced to stay within a boundary by assigning zero protrusion forces on nodes that lie outside the defined adhesive area. When the area of the 2D island (gray region in Fig. 8) was twice the initial area occupied by the cluster ðA=Ao ¼ 2Þ, similar to the experiments (13), cells had ample room to extend protrusions and spread, which led to a loss of CC junctions even on soft ECM of E ¼ 1 kPa (Fig. 8 a; Movie S5). The cell scattering increased on stiff ECM of E ¼ 1000 kPa (Fig. 8 d; Movie S6). On smaller adhesive islands ðA=Ao ¼ 1:5Þ, cell spreading was rendered relatively more constrained and the decay of CC junctions was significantly reduced in both soft and stiff ECMs (Fig. 8, b and e). When the adhesive island completely trapped the cell cluster, i.e., A=Ao ¼ 1, the CC junctions remained stable even on the stiff ECM (Fig. 8, c and f; Movies S7 and S8). The upregulation of mechanosensitive subcellular pathways on stiffer ECMs in our calculations can be considered analogous to the growth factor (HGF-based) stimulation of cells in the experiments. Thus, our predictions match the experimental findings (13) in which cell scattering on adhesive islands stopped even after growth factor stimulation.
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CONCLUSIONS Over the years, numerous experimental studies have revealed new phenotypes of mechanosensitive and ECMdependent cell scattering and EMT (1–4,9,13,19,31,32). However, a clear conceptual framework of how cellular mechanisms of forces and adhesions physically interact with the ECM and enable EMT was missing. In this study, we have presented a new model, to our knowledge, that calculates the dynamics of CC junctions and simulates cell scattering by combining protrusive and contractile forces, CE adhesions, and morphological polarization of cells, all of which operate in an ECM-dependent manner. Our simulations of cell scattering in variegated ECMs and subcellular perturbations demonstrate that the predictive capabilities of this model are not limited to one controlled experiment. Instead, we have employed the model to simulate a variety of cellular and extracellular perturbations— including cell clusters in channels of varied stiffness and confinement, on 2D adhesive patterns of varying area, channels with nonadhesive walls, and the inhibition of focal adhesions and microtubules. By isolating various cellular and extracellular inputs and comparing the simulations with corresponding experimental findings, our model highlights the importance of the following cellular mechanisms in regulating stiffness- and confinement-sensitive cell scattering: 1. By simulating the cell clusters inside the channels of varied stiffness and confinement, we find that stiffer ECMs enable a rise in CE adhesions, accompanied by the
Cell Scattering in Confinement
action with the channel walls may be an integral part of confinement-sensitive cell scattering. 3. The global inhibition of CE adhesions disables the ability of cells to sense ECM stiffness, which leads to similar forces and CC dissociation in both soft and stiff ECM. However, because the cells are still able to interact with the channel walls through active protrusions, the confinement-sensitive dissociation of CC junctions and cell scattering persists. 4. Indeed, after the abrogation of cellular polarization and protrusions, the cells are unable to probe the topography of their environment and thus distinguish between narrow and wide channels, leading to a confinementinsensitive cell scattering. In case of randomized (adhesion-independent) cell polarization, cells lose their ability to extend sustained protrusions in the directions that are most favorable for forming CE adhesions, which slows force generation and the related dissociation of CC junctions. 5. Finally, our simulations of cell clusters cultured on adhesive islands of a defined area reveal that a 2D confinement of cells constricts their spreading and thus preserves the integrity of cell clusters, which is consistent with experimental results (13). Thus, 2D and 3D ‘‘confinements’’ are not alike. Instead, the available surface area for cell spreading dictates the fate of CC junctions regardless of dimensionality of the environment. FIGURE 8 Simulated configurations of cell clusters, at time t ¼ 120 h, confined to 2D adhesive islands (denoted by the gray regions in each panel) of area A that is twice (a and d), 1.5 times (b and e), or equal to (c and f) the initial cell cluster area Ao. Simulations are repeated on ECM stiffness of E ¼ 1 kPa (a–c) and 1000 kPa (d–f). To see this figure in color, go online.
dissociation of CC junctions and enhanced cell scattering, which match several recent experiments (1–4). The model also captures our recent finding of enhanced EMT in narrow 3D channels (9), even in case of a soft ECM. We predict that CC junctions arranged parallel to the axis of confinement (channels in this case) dissociate more readily than those that lie orthogonally. This result indicates that the ability of cells to interact with the opposing channel walls may be crucial in triggering the preferential dissociation of CC junctions parallel to the channel walls. 2. The spreading of individual cells along channel walls through stronger CE adhesions pulls them away from the cell cluster, which may be responsible for the enhanced decay of CC junctions in confined environments. Indeed, our simulations for cell clusters confined within channels with nonadhesive walls predict that the cells are unable to undergo a confinement-dependent dissociation of CC junction. Thus, a robust cellular inter-
In summary, our simulations of cell clusters in variegated matrix environments present a novel framework, to our knowledge, for stiffness- and confinement-sensitive scattering of cells. According to our model, the stiffness-sensitive cell scattering critically depends on CE adhesions. Importantly, our model helps elucidate several cellular and extracellular features regulating the confinement-sensitive dissociation of CC junctions—adhesive channel walls enable cellular movement away from the cluster and aid de-clustering; adhesion-dependent cell polarization allows the cells to efficiently morph themselves according to the ECM confinement; microtubules-based cell polarity and protrusions are critical for probing heterogeneous ECM topography; larger adhesive area in 2D leads to greater cell spreading, longer stress fibers, higher forces, and enhanced dissociation of CC junctions. Given that our simulations recapitulate a range of experimental frameworks, the predictive capability of this model may expedite experimental studies by providing preliminary assessments of EMT markers for an even wider array of cellular and extracellular perturbations.
SUPPORTING MATERIAL Two figures and eight movies are available at http://www.biophysj.org/ biophysj/supplemental/S0006-3495(16)30756-1.
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ACKNOWLEDGMENTS This work was in part supported by grants to A.P. from the National Science Foundation (CAREER Award 1454016) and the Edward Mallinckrodt, Jr. Foundation (New Investigator Award).
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Scattering of Cell Clusters in Confinement Amit Pathak1,* 1
Department of Mechanical Engineering and Materials Science, Washington University, St. Louis, Missouri
ABSTRACT Epithelial-to-mesenchymal transition (EMT) enables scattering of cell clusters and disseminates motile cells to distant locations in vivo during embryonic development and cancer metastasis. Both stiffness and topography of the extracellular matrix (ECM) have been shown to influence EMT. In this work, we examine how the integrity of epithelial cell clusters is regulated by subcellular forces, protrusions, and adhesions for varying ECM inputs, such as stiffness, topography, and dimensionality. Our model simulates multicell networks of defined sizes and shapes in ECMs of varied stiffness and geometry. The integrity of cell clusters is dictated by cell-cell junctions, which depend on subcellular forces and adhesion dynamics within each cell of the cluster. Our simulations demonstrate an enhanced dissociation of cell-cell junctions in stiffer and more confined three-dimensional (3D) environments, consistent with experimental findings. In narrow channels, the cell edges parallel to the axis of channels lose their cell-cell junctions more readily than those oriented in the perpendicular direction. The inhibition of protrusive activity and cell polarity disables confinement-dependent cell scattering. Here, cell adhesion and spreading along channel walls is found to be essential for scattering. The model also predicts that two-dimensional (2D) confinement of clusters restricts cell spreading and simultaneously blunts the confinement-sensitive cell scattering. This new, to our knowledge, multiscale model integrates molecular adhesion dynamics, subcellular forces, cellular deformation, and macroscale mechanical properties of the ECM to predict the state of cell clusters of defined shapes and sizes. The predictions made by our model not only match experimental findings from a number of experimental setups, but also provide a new conceptual framework for understanding mechanosensitive cell scattering and EMT.
INTRODUCTION During both embryonic development and cancer metastasis, the dissemination of motile cells to distant locations is initiated by a cellular program called epithelial-to-mesenchymal transition (EMT). The coordination of actomyosin forces, polarization of cell shape, and loss of cellular contacts with the basement membrane cumulatively enable the detachment and migration of cells. An erratic sequence of these steps is implicated in triggering the onset of tumor invasion and metastasis. The cells at the periphery of tumor microenvironment interact with mechanically heterogeneous extracellular matrices (ECMs), undergo EMT-based scattering, escape from the primary tumor, migrate to distant locations, and subsequently repopulate secondary tumors. Thus, the physical interaction of individual cells with their ECM can play a crucial role in determining the structural integrity of multicell clusters. Over the years, several studies have established that stiffer ECMs cause greater subcellular forces and protrusions, stronger cell-ECM adhesions, and more polarized morphology of individual cells, which in
Submitted March 11, 2016, and accepted for publication August 31, 2016. *Correspondence: [email protected] Editor: Sean Sun http://dx.doi.org/10.1016/j.bpj.2016.08.034 Ó 2016 Biophysical Society.
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turn lead to the rupture of cell-cell junctions followed by EMT (1–4). Since cell polarization is a key precursor to the stiffness-induced cell scattering in these studies, we asked whether the cells trapped inside confined environments could undergo a similar shape polarization, as we and others have shown earlier (5–8), and commence dissociation from their native cluster. This is an important open question given that tissue environments often vary in their topography and dimensionality without significant stiffness variations. Our recent experiments revealed that epithelial clusters disintegrate more readily in confined environments even in soft ECMs (9). When the cells were rendered unable to extend protrusions or polarize, by inhibiting the function of microtubules, this confinement-dependent EMT vanished. The cell-ECM adhesions were found to be essential for stiffness-dependent EMT. These experimental findings point toward a complex machinery of intercoupled cellular, subcellular, and extracellular features that operate at varying time and length scales to trigger varying phenotypes of cell scattering. The molecular level mechanosensing of ECM stiffness regulates nanometer- and submicron-scale changes in cytoskeletal structure and cell-ECM adhesions, which lead to micron-scale changes in single-cell morphology,
Cell Scattering in Confinement
detachment of cells, and a large-scale destabilization of cell colonies and tissues. Similarly, micron-scale complexities in ECM geometry can affect single-cell morphology and cause the scattering of large multicell clusters. To deconstruct the relative contributions of these cellular and extracellular inputs on the integrity of cell clusters, we need a biophysical model that combines mechanosensitive subcellular mechanisms of forces, adhesions, and protrusions to simulate the deformation of single cells within the multicell networks adhered to ECMs of varying stiffness, confinement, and dimensionality. Such a computational model can serve as a powerful tool to interrogate specific influences of extracellular and subcellular inputs in regulating the integrity of cell clusters. The existing computational models have studied cell behavior through varied approaches—from continuum mechanics–based methods for single-cell behavior (10) to network models with discrete cell elements for the dynamics of cell clusters. The discrete cell models based on cellular-automata and agent-based methods (11,12) have informed the roles of intercellular signaling pathways in regulating the behavior of large cell populations. However, these models do not explicitly account for subcellular mechanical features, such as actomyosin forces and ECM adhesions, which enable individual cells in the population to deform and mechanically interact with their neighbors. These missing mechanical features in discrete cell models may explain important aspects of mechanosensing in cell populations. In this work, we present a new model, to our knowledge, for cell clustering scattering that combines the following subcellular and cellular processes: 1) formation and rupture dynamics of integrin-based cell-ECM adhesions, 2) cellular polarization and frontward protrusions, and 3) generation of actomyosin forces. These subcellular processes collectively dictate the state of cell-cell junctions in each cell of a given epithelial cluster. The model also accounts for stiffness and geometry of the ECM surrounding the multicell network. We test the model for varying ECM stiffness (0.1–1000 kPa) and confinement due to channel widths between 20 and 80 mm, emulating several previous experiments (1–4,9,13). Consistent with the known ability of cells to actively respond to ECM stiffness (3,4,14), our model predicts that cell clusters scatter more readily on stiffer ECMs. Surprisingly, the cells trapped within 2D islands do not undergo scattering (13) as they do inside 3D channels with vertical walls (9). After the inhibition of microtubules-based protrusions and cell polarization, cell clusters exhibit a confinement-insensitive behavior. Our calculations corresponding to the inhibition of cell-ECM adhesions predict a stiffness-insensitive behavior of cell clusters. To our knowledge, this is the first model to account for subcellular forces and single-cell morphology in dictating the dynamical state of cell-cell junctions and thus predicting both the stiffness-, confinement-, and dimensionality-dependent scattering of cell clusters. Taken together, these calcu-
lations reveal relative contributions of physical properties (such as ECM stiffness, topography, and dimensionality) and biochemical parameters (such as actomyosin forces, microtubules-based protrusions and polarization, and integrin/ cadherin-based adhesion dynamics) in regulating the integrity of epithelial cell clusters. MATERIALS AND METHODS Overview of the model The cluster of epithelial cells is modeled as a network of square-shaped cells interconnected with one another through cell-cell junctions. In this multicell network, each cell is comprised of adhesion nodes at the cell periphery, a deformable nucleus in the middle of the cell, and cytoskeletal elements connecting adhesion and nucleus nodes (Fig. 1). The cellular nodes at the periphery have two components: cell-cell (CC) junctions, and cellECM (CE) adhesions. Within individual cells, a net direction of frontrear polarity is calculated according to a probability of adhesion formation and the protrusive forces are accordingly applied on cellular nodes in the front region of the cell. The actin-myosin contractile forces depend on the length of cytoskeletal elements and signaling due to the formation of CE adhesions. The movement of nodes due to various cellular forces dictates morphological changes in individual cells, which continually reorganize the entire cell cluster. Various simulations are carried out for multicell clusters of different sizes in contact with ECMs of varying stiffness and geometry.
Cell-ECM adhesion dynamics As cells in the cluster interact with one another, their physical interaction with the ECM is also maintained through CE focal adhesions. It has been shown that forces exerted on focal adhesions stretch the receptor-ligand bonds and aid the growth of adhesions (15). The softer ECMs undergo larger deformation and thus lead to greater distances between receptors and ligands, which in turn is thought to impair adhesion bond formation (16). Conversely, the stiffer ECMs are expected to deform minimally, keep receptors and ligands in closer proximity, and enhance focal adhesion formation. We implement this stiffness-dependent adhesion dynamics in terms of bond displacement d ¼ Fam =akaE , which depends on actin-myosin force Fam . Here, the spring constant kaE of a CE bond accounts for both the stiffness of the receptor-ligand bond (ka ¼ 1 nN/mm) and the Young’s modulus of the ECM (E) according to (17): kaE ¼ ðao Eka Þ=ðao E þ 3ka Þ, where ao ¼ 0:006 mm is the perimeter of the circular area occupied by an average adhesion. In multicell clusters, the integrin-cadherin crosstalk (18,19) dictates that greater number of CC junctions (b) slows down CE bond formation. We combine these two influences of cell-ECM and cellcell interactions to describe the rate of formation of CE bonds for each node (a) as follows:
a_ þ ¼ raþ ða aÞð1 b=bo Þexpðd=do Þ;
(1a)
(henceforth, an over-dot denotes first time-derivative) with raþ ¼ 0:05 h-1 b Þ=N is the maximum number of as a rate constant. a ¼ ð1 þ zÞðb ao a b o ¼ 104 is the total numCE bonds in a given node at current time, where a b is the number of bonded and free receptors per cell at any time (20,21), a bo a b is the ber of bonded receptors in a cell at a given time, and thus a number of free receptors in the whole cell, with N ¼ 36 nodes per cell. The vertically distributed adhesion nodes on channel walls, with height z > 0, are represented by combining (1 þ z) number of nodes into one node. The force-based rupture rate of CE bond dissociation rate according to Bell’s law (20,22) is written as koff ¼ ra expðFam D=kB TÞ, where kB is the Boltzmann constant, T is the absolute temperature, ra ¼ 0:001 h-1 is
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Forces in cytoskeletal elements The cytoskeletal elements connected to CE nodes generate contractile forces proportional to the number of actin-myosin compartments in a given element and mechnotransductive signaling due to CE adhesions in the entire cell ðb a Þ as follows:
b a b o; Fam ¼ fam ða þ bÞlam a
(2)
where fam ¼ 12 pN is the constant force generated per actin-myosin compartment (23) and the size of the connected adhesion node (a þ b) represents the number of actin-myosin compartments per unit length of the element ðlam Þ. The cytoskeletal elements are assumed to deform in a linear elastic manner, such that the passive force generated within an element is written as follows:
Fcy ¼ kcy ða þ bÞðlam lo Þ;
(3)
where kcy ¼ 0:1 nN/mm is a spring constant and lo ¼ 3 mm is a reference length for all elements despite their different initial lengths depicted in Fig. 1 a. The number of springs in each element is proportional to the number of actomyosin compartments (proportional to the adhesion size a þ b).
Protrusion forces and cell polarization The initiation of polarity and extension of protrusions (24,25) are dynamic processes that occur within individual cells in the epithelial cluster. In this model, the direction of polarization is evaluated from a set of 12 possible angles between 0 and 2p, equally discretized at an interval of p/6, for each cell every 2 h. The probability of choosing a given angle of polarity is weighted by the total number of CE adhesions in the front region (defined by the direction of polarity under consideration) of the cell. Thus, a polarization direction corresponding to larger adhesions in the front of the cells is more likely to be picked. This selection criterion for protrusion direction is based on the cell’s ability to dynamically polarize in directions that are most suitable for forming stable adhesions. The magnitude of the protrusion force on a given cellular node comprises an active outward force proportional to the size of the adhesion and a uniform expansion of the cell body, written as follows: FIGURE 1 (a) Schematic of a cell cluster confined within channels of defined width modeled as a network of multiple cells connected together through cell-cell junctions. (b) Each cell is composed of adhesion nodes and cytoskeletal elements (gray) that support protrusive (Fpr) and actomyosin (Fam) forces. (c) The adhesion nodes (green) combine both cell-cell junctions and cell-ECM adhesion and cytoskeletal element combines active (actomyosin) and passive (elastic) forces. To see this figure in color, go online. a rate constant, Fam is the acto-myosin force on the bond, and D is the bond deformation for potential-energy minimum. For an adhesion node with a bonds and ka as the stiffness per bond, the bond deformation is D ¼ Fam =aka . Thus, we write the bond dissociation rate for a given node as follows:
Fpr ¼
fpr a þ fex p;
(4)
where fpr ¼ 0:4 nN is force per CE adhesion. An adhesion-independent spreading force fex ¼ ðfex =NÞð2Ai AÞ=A is also applied on each node to mimic cell spreading such that the cell area A can reach up to twice the initial area Ai ¼ 100 mm2, with fex ¼ 300 nN (estimated relative to other force constants used in these simulations). The direction of the protrusion force on each node is outward from the cell nucleus, i.e., opposite of the actomyosin force (Fig. 1 b). Based on the chosen protrusion direction, the polarity parameter p ¼ maxðd=dmax ; 0Þ defines nodes in the front of the cell, with d as nodal distance from the nucleus along the direction of polarization and dmax as the distance of the node that is farthest from the nucleus.
a_ ¼ ra a exp
F2am : aka kB T
(1b)
Nodal displacement and cell movement
Taken together, the dynamics of CE adhesion bonds (a) combines the rates for formation (Eq. 1a) and rupture (Eq. 1b) rates as follows:
The force equilibrium at each node is ensured at every time step and the displacement of the adhesion nodes is calculated according to a vector addition of all forces as follows:
a_ ¼ a_ þ a_ :
0 ¼~ Fpr þ ~ Fam þ ~ Fcy þ ~ Fmc þ h~ x_ c ;
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(1c)
(5a)
Cell Scattering in Confinement where h ¼ 5 nNmm-1 s is a constant drag coefficient estimated for these simulations, and ~ xc is the position vector of adhesion nodes. Here, Fmc is a linear elastic force proportional to the deformation of spring-like cell membrane elements and a spring constant of kmc ¼ 10 nN=mm. Similarly, the displacement of the nuclear nodes is also calculated according to the following force equilibrium:
0 ¼ ~ Fam ~ Fcy þ ~ Fmn þ ~ Fnu þ h~ x_ n ;
(5b)
where Fmn is an elastic force due to deformation of the nucleus membrane element, Fnu is an elastic force due to deformation of the nucleus body, and xn is the position vector of nuclear nodes. All nodes on the nucleus body are connected to each other through nuclear elements, which deform linearly based on a spring constant knu ¼ ð75=NÞ nN=mm per node. The membrane elements on the periphery the nucleus deform linearly according to spring constant kmn ¼ 40 nN=mm.
Dynamics of cell-cell junctions The CC junctions dissociate in a force-dependent manner (13,20,26,27) according to a relationship that accounts for different bond rupture rates in two force regimes, similar to Evans and Ritchie’s equations (22,28):
( b_ ¼
fb < fb rb bfb fb ; 2 ; rb b fb fb ; fb Rfb
(6a)
where rb ¼ 0:02 h-1 is a rate constant, fb is the net force on the node due to cytoskeletal and membrane elements of the two neighboring cells, and fb ¼ 0:4 nN is a force constant. The cytoplasmic E-cadherin binds to b-catenin and translocates to the cell membrane to form new cell-cell bonds (29). Thus, in the model, the ruptured CC complexes are allowed to form new CC junctions as follows:
b =b a o Þ; b_ þ ¼ rb ðbo bÞð1 a
(6b)
where bo ¼ 5000=N is the maximum number of CC junctions per node. According to the known integrin-cadherin crosstalk (18,19), the cell-ECM adb ) oppose the formation of CC junctions. Thus, the net rate hesions (greater a of change of the number of CC junctions at any node is as follows:
b_ ¼ b_ þ þ b_ :
(6c)
Simulation of cell clusters in defined ECMs To gain quantitative insights into how the complex interplay of subcellular and extracellular cues might lead to cell scattering, as evident by our recent experimental findings (9), we simulate the interaction of cell clusters with ECMs of varying stiffness and confinement. We integrate the rate-dependent models presented above (Eqs. 1–6) to simulate the physical interaction of cell clusters with ECMs of varied stiffness and geometry by ensuring force balance at each node at all times. Initially, cell dimensions are chosen as 10 10 mm, with 4 mm nucleus diameter. Each cell is discretized into N ¼ 36 adhesion nodes and the same number of cytoskeletal elements. Initially, CC junctions are at their maximum value bo , with no CE adhesions. The system of ordinary differential equations (ODEs) developed above (Eqs. 1, 5, and 6), corresponding to CE adhesions a, CC junctions b, cellular adhesion node position xc , and nuclear node position xn , is solved using Runge-Kutta scheme from time t ¼ 0–120 h, consistent with experimental culture period (1–4,9). As the CE adhesions grow (Eq. 1), the protrusion and actomyosin forces rise (Eqs. 2–4), which displaces the leading edge in the outward direction and polarizes the cell (Eq. 5). Thus, due to the
surge of forces and nodal displacements in each cell, the CC junctions start to decay (Eq. 6).
RESULTS AND DISCUSSION Enhanced scattering of cell clusters confined in 3D channels In these simulations, cell cluster length is kept constant at 80 mm and cluster width equals the given channel width. First, we simulated the response of a cluster confined within a wide channel, width w ¼ 60 mm, made of soft ECM (E ¼ 1 kPa). The movement of adhesion nodes in contact with channel walls was restricted in the outward y-direction (Fig. 1 a) to keep the cells within the confines of the channels, with a maximum permitted height of 10 mm in the z-direction. As shown in Fig. 2 a, each cell in the cluster interacts with the ECM differently, deforms due to intracellular forces, and loses some CC junctions by day 3. Here, green nodes denote CC junctions and blue cytoskeletal elements depict the front of the cell at a given time. By day 5, cells neighboring the channel walls lost more CC junctions than those in the interior, but the cell cluster did not disintegrate (see cellular deformations and cell cluster rearrangement over time in Movie S1 included in the Supporting Material). We repeated simulations in a narrow channel (w ¼ 20 mm) and found that cells adhered and elongated along the channel walls, which led to a loss of CC junctions more rapidly than in wide channels. By the end of day 5, the cell cluster scattered with no remaining CC junctions (Movie S2). These simulations are consistent with experimental findings (9), as illustrated in Fig. 2. Next, we repeated simulations in a stiff ECM (E ¼ 100 kPa) inside both wide and narrow channels. On stiff ECM, cells formed more CE adhesions and generated higher forces than on the soft ECM, which led to a loss of CC junctions in both wide and narrow channels, with even more rapid declustering in the narrow channel (compare days 3 and 5 in Fig. 2, i and j, and m and n; see Movies S3 and S4). These results match experimental findings, in which stiffer ECMs caused EMT in both narrow and wide channels. To understand cell scattering in a broader range of ECM stiffness and confinement, we repeated simulations for channel widths w ¼ 20, 40, 60, and 80 mm and ECM stiffness E ¼ 0.1, 1, 10, 100, and 1000 kPa. In each condition, we computed an average number of CC junctions after day 5, normalized by their initial value (at time t ¼ 0), and plotted against ECM stiffness (Fig. 3 a). In the widest channel (unconfined ECM), higher ECM stiffness led to a greater loss of CC junctions and caused more cell scattering. This direct relationship between CC junctions and ECM stiffness vanished in the narrowest channels—compare the flat red line for 20 mm channel with higher slopes for wider channels in Fig. 3 a. Thus, the stiffness matters less in narrow channels, which is consistent with our experimental findings (9). The ECM confinement matters more in softer environments, as evidenced by the
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FIGURE 2 Simulated configurations of cell clusters confined in wide channels of 60 mm width (a, b, i, and j) and narrow channels of 20 mm width (e, f, m, and n) made of soft (1 kPa) and stiff (100 kPa) ECMs. Number of CC junctions and CE adhesions on any node are denoted by green and black circles, respectively, of proportional size. Cytoskeletal elements marked in blue denote the front of the cell, i.e., those with nonzero protrusive forces. Simulations are compared with representative immunofluorescence images (c and d, g and h, k and l, o and p), obtained from our previous study (9), of E-cadherin (green) expression, with DAPI (blue), in MCF10A cell clusters cultured for either 3 or 5 days inside channels of varying stiffness and confinement. The scale bar represents 50 mm. To see this figure in color, go online.
diverging trend-lines toward the lower ECM stiffness regime in Fig. 3 a. The CC junctions decayed more rapidly in narrower and stiffer channels (Fig. 3, b and c). We also found that the cells formed more CE adhesions in stiffer ECMs and narrower channels (Fig. 3 d). Thus, the CE adhesions and the CC junctions have an antagonistic relationship, consistent with previous experimental findings of integrincadherin crosstalk (19,30). The reduced stabilization of CEs in softer ECMs caused low protrusion and actomyosin forces, which in turn disabled the dissociation rate of CC junctions. In contrast, on stiff ECMs, the model allowed greater stabilization of mechanosensitive CEs, which led to higher forces and faster dissociation of CC junctions (Fig. 3, b and c). The CE adhesions reach a saturation level beyond the ECM stiffness of 10 kPa in all channel widths (see the overlapping lines corresponding to E ¼ 10, 100, and 1000 kPa in Fig. 3, e and f). Faster rupture of cell-cell junctions parallel to the axis of narrow confinement In our model setup, the CC junctions can form on cell edges that are oriented either orthogonal or parallel to the direction
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of the channel walls, as illustrated in Fig. 4. We repeated the simulations performed above for cell clusters in narrow or wide channels made of soft or stiff ECMs and separately plotted the time evolution of the number of CC junctions (normalized by their maximum value) on cell edges parallel and orthogonal to the channel walls. In narrow channels made of soft ECM, the CC junctions parallel to the channel walls (red) broke before the CC junctions perpendicular to the channel walls (blue), with the largest difference between the two configurations after day 3, as shown in Fig. 4 a. It is likely that as cells climb the channel walls and move apart, they generate higher forces and enable the rupture of CC bonds in the middle of channel. This predicted difference between the dissociation rates of parallel and orthogonal CC junctions indicates that cell scattering in confinement does not occur homogeneously. Instead, the probability of rupture of a given CC junction depends on the orientation of the cell edge relative to the confinement geometry. We also found slower dissociation of CC junctions in cell edges parallel to the narrow channels made of stiff ECM (Fig. 4 b). However, the preferential dissociation of CC junctions parallel to narrow channel walls (Fig. 4, a and b) did not occur in wide channels made of either soft
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FIGURE 3 Number of CC junctions (a) and CE adhesions (d) at time t ¼ 120 h, normalized by their maximum values, plotted against ECM stiffness for different channel widths. Time variations of CC junctions and CE adhesions for the cases of narrow (b and e) and wide (c and f) channels are plotted for different ECM stiffness values. To see this figure in color, go online.
or stiff ECMs (Fig. 4, c and d). Thus, the interaction of cells with the walls of narrow channels is necessary for the initial separation of cell edges that are oriented parallel to the axis of confinement. Inhibition of cell-ECM adhesions blunts stiffnesssensitive cell scattering Since higher CE adhesions on stiffer ECMs correlated with a greater loss of CC junctions and cell scattering (Fig. 3, a and d), we hypothesized that inhibiting the formation of CE adhesions might blunt the cell’s ability to sense ECM stiffness. Indeed, our recent experiments showed that FAK inhibition led to stiffness-insensitive EMT (9). We simu-
lated this adhesion inhibition by reducing the forward rate of CE formation by 1/10th, i.e., multiplying a_ þ (Eq. 1a) by 0.1, and repeated the calculations for all ECM conditions (stiffness and channel width) noted above. As expected, the dependence of CE adhesions on ECM stiffness was dramatically reduced for all channel widths (see flat lines in Fig. 5 f). Simultaneously, the CC junctions were rendered stiffness-independent (see flat lines in Fig. 5 e and cluster configurations in Fig. 5, a–d) compared with the control case (varying slopes in Fig. 3 a). While this CE-inhibition blunts the ECM stiffness-dependent decay of CC junctions, active protrusions are still able to spread along channel walls, polarize cells (Fig. 5, a–d), and enable confinement-dependent dissociation of CC junctions, as evidenced
FIGURE 4 Time variation of the number of CC junctions, normalized by their maximum quantity, oriented parallel and orthogonal to channel walls for narrow (a and b) and wide (c and d) channels made of soft (a and c) or stiff (b and d) ECMs. To see this figure in color, go online.
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FIGURE 5 Simulated configurations of cell clusters after inhibition of CE adhesions in wide (a and c) and narrow (b and d) channels made of soft (a and b) and stiff (c and d) ECMs. Normalized values of CC junctions (e), CE adhesions (f), and protrusion force (g) at time t ¼ 120 h plotted against ECM stiffness for different channel widths after inhibition of CE adhesions by slowing the forward rate of CE adhesion formation, by multiplying a_ þ (Eq. 1a) by 0.1. To see this figure in color, go online.
by the channel width-dependent differences in the protrusion forces (Fig. 5 g) and the CC junctions (separated trend-lines in Fig. 5 e). Thus, the inhibition of CE adhesions blunts stiffness-dependent cell scattering but retains the confinement-sensitive dynamics of CC junctions through protrusive forces in cells (see Figs. 5, a–e, and S1 for time variation of CC junctions and CE adhesions for each condition), which is consistent with experimental findings (9). Disruption of protrusions and polarity abrogates the effect of 3D confinement Microtubules enable front-to-rear polarization and stabilize frontward protrusions within the cells. In our recent experiments, the inhibition of microtubules reduced morphological polarization and protrusive activity in cells, which led to a confinement-insensitive EMT. We simulated a similar microtubule-inhibition by prescribing randomized (adhesion-independent) protrusion directions at 2 h time interval and reducing the protrusion force constant to 1/10th of its original value ðfpr ¼ 0:04 nNÞ. Together, these changes disrupt the adhesion-dependent front-to-rear polarity and reduce protrusion forces. Consistent with the experimental findings (9), our simulations predict a dramatically reduced dependence of CC junctions on confinement, as evidenced by the overlapping trend-lines for varying channel widths in Fig. 6 e. Here, the reduction in protrusions dramatically reduced cellular
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deformation and polarization of cell shape (see cluster configurations in Fig. 6, a–d), which meant that wide and narrow channels had similar effects on the cell cluster. Note that the confinement dependence of the CE adhesions was more pronounced after microtubule-inhibition than after adhesion-inhibition—compare Figs. 5 e and 6 e. As expected, the protrusion force after microtubule-inhibition remained low regardless of channel width (Fig. 6 g). Taken together, these results suggest that ECM stiffness- and confinement-sensitive EMT follow two different pathways: 1) CE adhesions regulate stiffnesssensitive EMT and adhesion-inhibition does not turn off confinement-sensitive EMT, and 2) protrusive forces and cell polarization regulate confinement-sensitive EMT but the stiffness-sensitive effects persist even after their inhibition. Random cellular polarization reduces declustering but preserves confinement-sensitivity Our simulations and validation with experimental findings presented above establish that adhesion-dependent cellular polarity and protrusions are necessary for confinement-sensitive scattering of cell clusters (Fig. 6). To further delineate the relative importance of the adhesions-dependent cellular polarity and the magnitude of protrusion forces, we performed simulations with randomly assigned directions of polarity without reducing the magnitude of the protrusion force. In these simulations, we randomly picked a direction
Cell Scattering in Confinement
FIGURE 6 Simulated configurations of cell clusters after inhibition of microtubules in wide (a and c) and narrow (b and d) channels made of soft (a and b) and stiff (c and d) ECMs. Normalized values of CC junctions (e), CE adhesions (f), and protrusion force (g) at time t ¼ 120 h plotted against ECM stiffness for different channel widths after inhibition of microtubule-based protrusions and polarization. To see this figure in color, go online.
of polarity for each cell in the cluster every 2 h without concern for adhesions in the front or rear of the cell. In narrow-soft, wide-soft, and wide-stiff channels, the randomly polarized cells had a higher number of CC junctions compared with the control case (Fig. 7 a). Thus, we predict that cells without the ability to polarize in an adhesiondependent manner lose their CC junctions less efficiently as compared with the control case. We also predict that randomly polarized cells form a lesser number of CE adhesions (Fig. 7 b) and generate smaller protrusion forces (Fig. 7 c). Strikingly, even without the stable polarization, cells were able to distinguish between narrow and wide channels in both soft and stiff ECMs, as evidenced by the difference in red and blue bars in Fig. 7. Nonadhesive channel walls disable confinementdependent cell scattering Through simulations and validation with experiments, we have shown that CE adhesions and protrusions are necessary for stiffness-sensitive and confinement-sensitive cell scattering, respectively (Figs. 5 and 6). According to our predictions (Fig. 7, a–c), whereas the adhesion-independent cellular polarization reduces CC dissociation, it preserves
confinement-sensitivity through interaction with channel walls. Next, we sought to isolate the role of CE adhesions along channel walls in the ability of cells to sense confined environments. To this end, we performed simulations in which adhesion nodes in contact with the channel edges were not allowed to grow on the channel walls. In this case, we predict that the cells in narrow channels made of soft ECM are able to preserve their CC junctions much more efficiently compared with the control case (red bar in Fig. 7 d). As a result, the number of CC junctions in both narrow and wide channels made of soft ECM end up at similar levels. Thus, without a robust interaction with the channel walls, cells lose their ability to undergo confinement-sensitive scattering. We also predict that the number of CE adhesions (Fig. 7 e) and the protrusions forces (Fig. 7 f) remain confinement-independent when the cells are rendered unable to form new adhesions with the channel walls. Scattering of cell clusters in 2D confinement depends on spreading area In a recent study, protrusions have been shown to regulate forces at the cell-cell interface and the cell scattering
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FIGURE 7 After implementing randomized cell polarity, the calculated number of CC junctions (a), number of CE adhesions (b), and protrusion force (c) at time t ¼ 120 h, inside narrow (20 mm) or wide (60 mm) channels of varying stiffness. In case of nonadhesive channel walls, predicted CC junctions (d), number of CE adhesions (e), and protrusion force (f) at time t ¼ 120 h for varying stiffness and channel width. To see this figure in color, go online.
stopped when cell clusters were trapped within a 2D adhesive island, despite the addition of a mechanosensitive growth factor (13). Thus, protrusive activity and cell spreading are essential for cell scattering. To examine if our model can explain this behavior, we performed simulations where a cluster of four cells was allowed to spread on a 2D adhesive island of defined area. Here, cells were forced to stay within a boundary by assigning zero protrusion forces on nodes that lie outside the defined adhesive area. When the area of the 2D island (gray region in Fig. 8) was twice the initial area occupied by the cluster ðA=Ao ¼ 2Þ, similar to the experiments (13), cells had ample room to extend protrusions and spread, which led to a loss of CC junctions even on soft ECM of E ¼ 1 kPa (Fig. 8 a; Movie S5). The cell scattering increased on stiff ECM of E ¼ 1000 kPa (Fig. 8 d; Movie S6). On smaller adhesive islands ðA=Ao ¼ 1:5Þ, cell spreading was rendered relatively more constrained and the decay of CC junctions was significantly reduced in both soft and stiff ECMs (Fig. 8, b and e). When the adhesive island completely trapped the cell cluster, i.e., A=Ao ¼ 1, the CC junctions remained stable even on the stiff ECM (Fig. 8, c and f; Movies S7 and S8). The upregulation of mechanosensitive subcellular pathways on stiffer ECMs in our calculations can be considered analogous to the growth factor (HGF-based) stimulation of cells in the experiments. Thus, our predictions match the experimental findings (13) in which cell scattering on adhesive islands stopped even after growth factor stimulation.
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CONCLUSIONS Over the years, numerous experimental studies have revealed new phenotypes of mechanosensitive and ECMdependent cell scattering and EMT (1–4,9,13,19,31,32). However, a clear conceptual framework of how cellular mechanisms of forces and adhesions physically interact with the ECM and enable EMT was missing. In this study, we have presented a new model, to our knowledge, that calculates the dynamics of CC junctions and simulates cell scattering by combining protrusive and contractile forces, CE adhesions, and morphological polarization of cells, all of which operate in an ECM-dependent manner. Our simulations of cell scattering in variegated ECMs and subcellular perturbations demonstrate that the predictive capabilities of this model are not limited to one controlled experiment. Instead, we have employed the model to simulate a variety of cellular and extracellular perturbations— including cell clusters in channels of varied stiffness and confinement, on 2D adhesive patterns of varying area, channels with nonadhesive walls, and the inhibition of focal adhesions and microtubules. By isolating various cellular and extracellular inputs and comparing the simulations with corresponding experimental findings, our model highlights the importance of the following cellular mechanisms in regulating stiffness- and confinement-sensitive cell scattering: 1. By simulating the cell clusters inside the channels of varied stiffness and confinement, we find that stiffer ECMs enable a rise in CE adhesions, accompanied by the
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action with the channel walls may be an integral part of confinement-sensitive cell scattering. 3. The global inhibition of CE adhesions disables the ability of cells to sense ECM stiffness, which leads to similar forces and CC dissociation in both soft and stiff ECM. However, because the cells are still able to interact with the channel walls through active protrusions, the confinement-sensitive dissociation of CC junctions and cell scattering persists. 4. Indeed, after the abrogation of cellular polarization and protrusions, the cells are unable to probe the topography of their environment and thus distinguish between narrow and wide channels, leading to a confinementinsensitive cell scattering. In case of randomized (adhesion-independent) cell polarization, cells lose their ability to extend sustained protrusions in the directions that are most favorable for forming CE adhesions, which slows force generation and the related dissociation of CC junctions. 5. Finally, our simulations of cell clusters cultured on adhesive islands of a defined area reveal that a 2D confinement of cells constricts their spreading and thus preserves the integrity of cell clusters, which is consistent with experimental results (13). Thus, 2D and 3D ‘‘confinements’’ are not alike. Instead, the available surface area for cell spreading dictates the fate of CC junctions regardless of dimensionality of the environment. FIGURE 8 Simulated configurations of cell clusters, at time t ¼ 120 h, confined to 2D adhesive islands (denoted by the gray regions in each panel) of area A that is twice (a and d), 1.5 times (b and e), or equal to (c and f) the initial cell cluster area Ao. Simulations are repeated on ECM stiffness of E ¼ 1 kPa (a–c) and 1000 kPa (d–f). To see this figure in color, go online.
dissociation of CC junctions and enhanced cell scattering, which match several recent experiments (1–4). The model also captures our recent finding of enhanced EMT in narrow 3D channels (9), even in case of a soft ECM. We predict that CC junctions arranged parallel to the axis of confinement (channels in this case) dissociate more readily than those that lie orthogonally. This result indicates that the ability of cells to interact with the opposing channel walls may be crucial in triggering the preferential dissociation of CC junctions parallel to the channel walls. 2. The spreading of individual cells along channel walls through stronger CE adhesions pulls them away from the cell cluster, which may be responsible for the enhanced decay of CC junctions in confined environments. Indeed, our simulations for cell clusters confined within channels with nonadhesive walls predict that the cells are unable to undergo a confinement-dependent dissociation of CC junction. Thus, a robust cellular inter-
In summary, our simulations of cell clusters in variegated matrix environments present a novel framework, to our knowledge, for stiffness- and confinement-sensitive scattering of cells. According to our model, the stiffness-sensitive cell scattering critically depends on CE adhesions. Importantly, our model helps elucidate several cellular and extracellular features regulating the confinement-sensitive dissociation of CC junctions—adhesive channel walls enable cellular movement away from the cluster and aid de-clustering; adhesion-dependent cell polarization allows the cells to efficiently morph themselves according to the ECM confinement; microtubules-based cell polarity and protrusions are critical for probing heterogeneous ECM topography; larger adhesive area in 2D leads to greater cell spreading, longer stress fibers, higher forces, and enhanced dissociation of CC junctions. Given that our simulations recapitulate a range of experimental frameworks, the predictive capability of this model may expedite experimental studies by providing preliminary assessments of EMT markers for an even wider array of cellular and extracellular perturbations.
SUPPORTING MATERIAL Two figures and eight movies are available at http://www.biophysj.org/ biophysj/supplemental/S0006-3495(16)30756-1.
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ACKNOWLEDGMENTS This work was in part supported by grants to A.P. from the National Science Foundation (CAREER Award 1454016) and the Edward Mallinckrodt, Jr. Foundation (New Investigator Award).
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