PHYSICAL REVIEW E 91, 012712 (2015)
Collective cell migration induced by mechanical stress and substrate adhesiveness Michael H. Kopf* Departement de Physique, Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, France (Received 14 October 2014; published 28 January 2015) Mechanical stress normal to the boundary of a tissue sheet can arise in both constrained as well as unconstrained epithelial layers through pushing and pulling of surrounding tissue and substrate adhesiveness, respectively. A continuum model is used to investigate how such stress influences the epithelial dynamics. Four types of spreading and motility can be identified: a uniformly stretched stationary state, uniform sheet migration, active stress compensation by polarization close to the boundary, and a wormlike progression by deformation waves. Analytical and numerical solutions are presented along with bifurcation diagrams using normal stress and active force as control parameters. PACS number(s): 87.18.Hf, 05.45.—a, 87.85.gj
DOI: 10.1103/PhysRevE.91.012712 I. INTRODUCTION
Collective cell migration is ubiquitous in multicellular organisms and represents a common feature of a tumor invading surrounding healthy tissue [1], a regenerating ep ithelial layer spreading into the void left by a wound [2,3], and a cell sheet reorganizing during embryogenesis [4]. In contrast to single cell migration, collective migration maintains physical and functional connections between cells and exhibits supracellular polarity and organization of the actin cytoskeleton [5,6]. Here, we focus on thin epithelial sheets that adhere to a substrate and collectively become motile after activation by mechanical cue. Even though the tissue is assumed to polarize only in reaction to deformation and not spontaneously, it will be shown that stress on its boundaries can induce uniform polarization as well as polarization wave trains. Such a stress can either result from the substrate or from a direct pushing or pulling by surrounding tissue. We thus consider the two scenarios shown in Fig. 1: (a) a freely spreading tissue and (b) a tissue embedded in an ambient tissue or matrix. The substrate is known to significantly influence the behav ior of adherent cells and tissues. Its rigidity has been observed to determine the motility [7-10] and even the differentiation of cells [11]. Cooperative substrate deformation by many cells can serve as a long-distance coupling that controls collective migration [12]. Even a perfectly rigid substrate—and this is the case we want to focus on, here—can have a strong impact on the behavior of an adherent tissue due to another important factor: the wettability (adhesiveness) of the substrate with respect to the tissue. Similar to liquids, unconstrained cell layers are known to spread across substrates due to differences in the intercellular adhesion ycc and the adhesiveness between cells and substrate ycs. The wettability is quantified by a spreading coefficient S = ycs — ycc [13]. The importance of direct mechanical cues has been demonstrated in experiments where controlled pulling by aspiration induces a shivering motion of spherical aggregates of sarcomere cells inside a micropipette [14], Similarly, the activation of a kinase (MAPK) by stretching has been
*[email protected] 1539-3755/2015/91 (1 )/012712(6)
observed in canine kidney (MDCK) epithelia [15] and results in collective migration that often goes along with the formation of fingerlike protrusions at the leading edge [16]. Prevalent theoretical models of collective cell migration are continuum models in the spirit of active gel theory [17] or agent-based models of the Vicsek type [18,19]. In the case of tissues on substrates, the dynamics is friction dominated (the “dry” case in Ref. [20]) and has to be distinguished from the inertial dynamics of active cell suspensions (the “wet” case). Here, we employ a continuum model and describe a thin epithelial layer as a quasi-two-dimensional polarizable and active elastic material whose polarization state is described by a vector field p. We restrict our investigation to time scales, which are short in respect to the characteristic time scales of cell proliferation and intercalation, that is, to time scales on which viscoelastic effects can be neglected. The cells exert active forces that are produced in reaction to a chemical signal (like MAPK in the case of MDCK epithelia) whose concentra tion is described by a scalar field c. This signal is emitted by the cells in reaction to mechanical deformation which is described by a displacement vector field u. On a molecular level, this mechanosensitivity can be explained by the effect of deforma tion on transduction channels in the cellular membrane [21,22]. A chemomechanical feedback model of this type has been applied to describe the spreading of unconstrained epithelial layers in wound healing experiments and was implemented in a cell-based [23] as well as a continuous model [24], The substrate’s wettability is captured by a wetting force acting in normal direction to the tissue’s leading edge and enters the model as a boundary condition. It is noteworthy that within a quasi-two-dimensional model, a mechanical load arising from a surrounding tissue or matrix leads to exactly the same type of boundary condition. In the following we will therefore simply speak of a normal stress Fw irrespective of its origin. Fw represents a natural control parameter in the problem and can be fixed for example in wetting experiments by varying either ycs using a PEG-fibronectin substrate coating, or soft gels, or by varying ycc by tuning the level of cadherin expression [25]. We will first introduce the model equations, then describe the different solution types that can be found dependent on the normal stress, and finally summarize the results in bifurcation diagrams.
012712-1
©2015 American Physical Society
MICHAEL H. KOPF
Fw
c e lls
PHYSICAL REVIEW E 91,012712 (2015)
Fw
fw
s u b s tr a te
c e lls
JdL
s u b s tr a te
FIG. 1. (Color online) Two scenarios of stressed tissues. If un constrained (left), a tissue can be under stress arising from wettability. If the tissue is embedded in another tissue (right), pushing and pulling from the surrounding cells leads to stress. II. MODEL EQUATIONS To capture the essential dynam ics we consider a sim plified one-dim ensional version o f the m odel used in Ref. [24]. Since we w ant to focus on the effect o f norm al stress at the boundary and the onset o f the spreading dynam ics, w e neglect both cell proliferation and nonlinear elastic effects, w hich becom e effective only on larger tim e scales. The model equations for the signal concentration c, the polarization p, and the displacem ent field u take the follow ing nondim ensional shape (see A ppendix A for inform ation about the used scaling):
c = c" — c + fiu',
(1)
P = p" - a 2p ( l + p 2) + cp,
(2)
u — u" + qcp.
(3)
FIG. 2. (Color online) Type-III solutions at Fw = 0 .0 1 2 and (a) q = 0.01 and (b) q = 0.045. The tissue actively pulls inwards to compensate the external stress. The smaller the active force q , the stronger the tissue has to polarize to achieve an equilibrium.
activation fi < 0. A uniform ly polarized state is possible, w here the w hole tissue m oves w ith constant velocity v and under constant stretch: Type II:
Here, w e use the notation / = d f/d t and / ' = d f / d x as shorthand for tem poral and spatial derivatives, respectively. D eform ation is quantified by u! w hile fi determ ines how m uch o f the signal is generated or annihilated in reaction to deform ation. In the absence o f the chem ical signal, the m aterial rem ains quiescent and the dam ping force is quantified by the positive param eter a 2 > 0. T his choice o f signs distinguishes the m aterial from a spontaneously polarizing tissue for which the linear term a 2p w ould have positive sign. Finally, q denotes the strength o f the active force. It follow s from Eq. (3) that the center o f m ass o f the bounded system can translate and does so w ith the propagation velocity
q fL v = — / dxcp. L Jo
(4)
In the presence o f a norm al stress F w,defined positive if pulling outw ard, we em ploy the elastic boundary conditions u \ 0) = u'(L) = Fw and no-flux conditions for p and c, that is, c'(0) = c’{L) = p \ 0) - p \ L ) = 0. III. SOLUTIONS AND BIFURCATION ANALYSIS We w ill now discuss the possible types o f stationary and m otile solutions. T he sim plest case is a fam ily o f uniform ly stretched states w ith a hom ogeneous concentration c. We will refer to this fam ily as type-I solutions: Type I:
(c,p,u) - 08F w,0 ,F wx + n 0),
(c,p,u) — (ySFw, ± p n , u 0 + Fwx ± vt),
(6)
w ith p u = y7(fiFvj/a1) — 1 and v = qfiF^Pii- O f course, the problem is sym m etric, and positive and negative velocities are equivalent. If the system starts from random initial conditions and finally settles on a type-II solution, the direction o f m otion depends on the initial conditions. Two m ore fam ilies o f nonuniform states can be calculated num erically. Solutions o f type III correspond to nonuniform ly stretched tissues w ith nonzero polarization close to the boundaries, w here the cells either pull o r push to balance the external stress Fw. Typical solutions o f this type are show n in Fig. 2. Finally, solutions o f type IV exhibit a kind o f m otility very different from the uniform m otion o f type-II solutions: a train o f polarization waves travels across the tissue and due to the active force, these polarization waves lead to deform ation waves, rem iniscent of a w orm that locally contracts and expands in peristalsis (see m ovies in the Supplem ental M aterial for illustration [26]). W ith each wave the tissue propagates a certain distance, with a velocity that can be calculated using Eq. (4). Like type-II solutions, m otion to the left and to the right are equivalent and this sym m etry is broken by the initial conditions. Solutions o f type IV are found betw een two critical Fw values that depend on the strength o f the active force q. In order to better understand the solution structure o f the problem , we obtain a bifurcation diagram using num erical continuation w ith the softw are AUTO [27] that em ploys the pseudoarclength algorithm [28], To this end, the dom ain is discretized using N — 256 grid points and second-order finite differences. As solutions o f the m odel equations are then vectors in a 3 x 256-dim ensional space, w e define the norm
(5)
w ith integration constant u q . Type-I solutions are not polarized and thus exhibit no active forces. If the norm al stress satisfies /3FW > a 2, up to three m ore solution fam ilies, types II—IV, can be identified. N ote that since a 2 is a positive constant, they can be found for Fw > 0 only in the case o f expansive activation P > 0, and for Fw < 0 only in the case o f compressive
^Sp^ = \ ¥ l J 0 /
f
C*T)2 d x d rj
(7)
as a solution m easure, w here L is the dom ain size and T the tem poral period for tim e-periodic solutions. For steady states the tim e average (1 / T ) j d t . . . is om itted. Bifurcation diagram s for three different values o f the active force strength
012712-2
PHYSICAL REVIEW E 91, 012712 (2015)
COLLECTIVE CELL MIGRATION INDUCED BY . . .
FIG. 3. (Color online) Bifurcation diagram for q = 0.01. Stable and unstable branches are shown as solid and dashed lines, respec tively. Red dots mark the Hopf bifurcations at the ends of the type-IV branch.
q are shown in Figs. 3-5. The other parameters are set to P = \ ,ce2 = 10-2, and L = 103. From the discussion of the solution types we already know that a bifurcation occurs at Fw = a 2/ ft. With smaller Fw there are only stable type-I solutions. For higher values of Fw, type-I solutions become unstable while stable solutions of types II and III appear. If the strength of the active force q is higher than a critical value, a pair of Hopf bifurcations (HB) appears on the branch of type-II solutions. These HBs mark the end points of a branch of type-IV solutions. Thus nonuniform motion occurs only in a limited range of normal stresses Fw, similar to the observation in Ref. [14]. As the diagrams show, new HB pairs and secondary branches of type-IV solutions appear with increasing q. Also, the primary type-IV branch splits into two branches at lower Fw values between a pair of period-doubling (PD) bifurcations. However, the solutions on the split branches behave very similar [26]. While the wave train in the primary branch spans the whole tissue [see Fig. 6(a)], the wave trains
FIG. 5. (Color online) Bifurcation diagram for q = 0.045 with symbols as in Fig. 3. Up to three type-IV solutions coexist for the same Fw. Blue squares mark the solutions at Fw = 1.025 x 10~2 that are shown as (a), (b), and (c) (from top to bottom) in Fig. 6. The numbers 1-6 indicate the Hopf bifurcations corresponding to the critical modes shown in the movies ev_crit_q045_HB 1.mp4 to ev_crit_1045_HB6.mp4 in the Supplemental Material [26].
of the secondary branches start closer to either boundary [see Figs. 6(b) and 6(c)]. The different number of maxima in the wave trains is visible in the spectrum of the linear problem close to the HBs (see Appendix B). Figure 7 summarizes the results of 150 continuation runs at different q values in a codimension-2 bifurcation diagram. It shows the Fw values corresponding to the end points of the type-IV branches (HBs) for each value of q and illustrates the appearance of branches with increasing q. We see that while the left HB of each pair moves closer to the critical value Fw = a 2/ p , the Fw value corresponding to the rightmost HB increases with q, so that the range of normal stresses sustaining nonuniform motion grows with the strength of the active
4
2
0 -2 -4 (a)
(b)
0.01
0.01004
0.01008
0.01012
0.01016
Fw
(c) FIG. 4. (Color online) Bifurcation diagram for q = 0.01375 with symbols as in Fig. 3. Two branches of type-IV solutions coexist between two pairs of Hopf bifurcations. The inset shows a region where the primary type-IV branch splits between two period doubling bifurcations (shown as blue diamonds).
0
200
400
600
800
FIG. 6. (Color online) Snapshots of the coexisting type-IV solu tions at q = 0.045 and Fw = 1.025 x 10-2, corresponding to the blue squares in Fig. 5. The wave trains emerge in the bulk and propagate to the right.
012712-3
MICHAEL H. KOPF
PHYSICAL REVIEW E 91, 012712 (2015)
case is conceptionally straightforward and could be used to find more complicated solutions. ACKNOWLEDGMENTS
The author thanks Cristina Marchetti for helpful discus sion. This work was supported by the French National Re search Agency (ANR), grant ENS-ICFP: ANR-10-238LABX0010/ANR-10-IDEX-0001-02 PSL*. APPENDIX A: SCALING
-------- 1-------- 1-------- 1________ i_______ i 0.01
0.0102
0.0104
0.0106
0.0108
0.011
Fw FIG. 7. (Color online) Codimension-2 bifurcation diagram showing the loci of Hopf bifurcations on the branch of type-II solutions dependent on the strength of the active force q and the normal stress Fw. Intersections of the shown curves with a horizontal at a given q value correspond to Hopf bifurcations, as is indicated by the horizontal lines for the cases of Figs. 3-5. The branches are all of the same type; different line styles are only used to distinguish the lines. Hopf bifurcations on the same curve are connected by a branch of time-periodic type-IV solutions.
force q. A similar codimension-2 bifurcation cascade was recently observed in a model for the deposition of surfactant monolayers (see [29] and Figs. 10 and 11 therein). There, just as in the present model, solutions with differently sized wave trains (see Fig. 6) coexist. IV. SUMMARY AND CONCLUSIONS
We can summarize that normal boundary stress can induce motility in spreading cell layers with a simple chemomechanical feedback. The boundary stress has been found to be a necessary precondition for the observed motility. In particular, in the unstressed case Fw = 0, the trivial homogeneous solution (c , p , u ) = (0,0,0) is linearly stable for all finite parameters. Even at small stretch y3Fw < or2, the external mechanical cue cannot be bypassed by increasing the initial concentration c: the system relaxes back to its linearly stable homogeneous state. The collective cell migration is either uniform, like in collective gliding, or occurs by deformation waves. The efficiency of similar motility modes has been investigated in a purely mechanistic model with prescribed activity [30], A switching in motility modes dependent on either adhesiveness (our Fw) or active force (our q) has been observed in single keratocytes [31]. The effect of proliferation and apoptosis has not been considered in the current investigation but might be relevant as it leads to intercellular rearrangement that effectively fluidizes the tissue [32], Furthermore, stresses arising due to volumetric growth might be an additional trigger of mechanosensitive activity. Cell division and death can be readily incorporated into the description of the elastic layer by introduction of a growth tensor [33], as was done in Ref. [24], or by an additional pressure and volume source [34], Finally, an extension of the presented analysis to the two-dimensional
The model equations (l)-(3) are based on the model presented in Ref. [24] but are simplified by considering only one spatial dimension and linear elasticity. Additionally, we chose a slightly different scaling. This physically equivalent approach allows for a simpler bifurcation analysis, as the codimension-2 diagram in Fig. 7 in the main text would be come largely unclear in the original scaling. The dimensionful governing equations for c, p, and u are c ~ Dc" —kdc + fin', FpP = K2p" - (1 + K0p 2)A2p +
Collective cell migration induced by mechanical stress and substrate adhesiveness Michael H. Kopf* Departement de Physique, Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, France (Received 14 October 2014; published 28 January 2015) Mechanical stress normal to the boundary of a tissue sheet can arise in both constrained as well as unconstrained epithelial layers through pushing and pulling of surrounding tissue and substrate adhesiveness, respectively. A continuum model is used to investigate how such stress influences the epithelial dynamics. Four types of spreading and motility can be identified: a uniformly stretched stationary state, uniform sheet migration, active stress compensation by polarization close to the boundary, and a wormlike progression by deformation waves. Analytical and numerical solutions are presented along with bifurcation diagrams using normal stress and active force as control parameters. PACS number(s): 87.18.Hf, 05.45.—a, 87.85.gj
DOI: 10.1103/PhysRevE.91.012712 I. INTRODUCTION
Collective cell migration is ubiquitous in multicellular organisms and represents a common feature of a tumor invading surrounding healthy tissue [1], a regenerating ep ithelial layer spreading into the void left by a wound [2,3], and a cell sheet reorganizing during embryogenesis [4]. In contrast to single cell migration, collective migration maintains physical and functional connections between cells and exhibits supracellular polarity and organization of the actin cytoskeleton [5,6]. Here, we focus on thin epithelial sheets that adhere to a substrate and collectively become motile after activation by mechanical cue. Even though the tissue is assumed to polarize only in reaction to deformation and not spontaneously, it will be shown that stress on its boundaries can induce uniform polarization as well as polarization wave trains. Such a stress can either result from the substrate or from a direct pushing or pulling by surrounding tissue. We thus consider the two scenarios shown in Fig. 1: (a) a freely spreading tissue and (b) a tissue embedded in an ambient tissue or matrix. The substrate is known to significantly influence the behav ior of adherent cells and tissues. Its rigidity has been observed to determine the motility [7-10] and even the differentiation of cells [11]. Cooperative substrate deformation by many cells can serve as a long-distance coupling that controls collective migration [12]. Even a perfectly rigid substrate—and this is the case we want to focus on, here—can have a strong impact on the behavior of an adherent tissue due to another important factor: the wettability (adhesiveness) of the substrate with respect to the tissue. Similar to liquids, unconstrained cell layers are known to spread across substrates due to differences in the intercellular adhesion ycc and the adhesiveness between cells and substrate ycs. The wettability is quantified by a spreading coefficient S = ycs — ycc [13]. The importance of direct mechanical cues has been demonstrated in experiments where controlled pulling by aspiration induces a shivering motion of spherical aggregates of sarcomere cells inside a micropipette [14], Similarly, the activation of a kinase (MAPK) by stretching has been
*[email protected] 1539-3755/2015/91 (1 )/012712(6)
observed in canine kidney (MDCK) epithelia [15] and results in collective migration that often goes along with the formation of fingerlike protrusions at the leading edge [16]. Prevalent theoretical models of collective cell migration are continuum models in the spirit of active gel theory [17] or agent-based models of the Vicsek type [18,19]. In the case of tissues on substrates, the dynamics is friction dominated (the “dry” case in Ref. [20]) and has to be distinguished from the inertial dynamics of active cell suspensions (the “wet” case). Here, we employ a continuum model and describe a thin epithelial layer as a quasi-two-dimensional polarizable and active elastic material whose polarization state is described by a vector field p. We restrict our investigation to time scales, which are short in respect to the characteristic time scales of cell proliferation and intercalation, that is, to time scales on which viscoelastic effects can be neglected. The cells exert active forces that are produced in reaction to a chemical signal (like MAPK in the case of MDCK epithelia) whose concentra tion is described by a scalar field c. This signal is emitted by the cells in reaction to mechanical deformation which is described by a displacement vector field u. On a molecular level, this mechanosensitivity can be explained by the effect of deforma tion on transduction channels in the cellular membrane [21,22]. A chemomechanical feedback model of this type has been applied to describe the spreading of unconstrained epithelial layers in wound healing experiments and was implemented in a cell-based [23] as well as a continuous model [24], The substrate’s wettability is captured by a wetting force acting in normal direction to the tissue’s leading edge and enters the model as a boundary condition. It is noteworthy that within a quasi-two-dimensional model, a mechanical load arising from a surrounding tissue or matrix leads to exactly the same type of boundary condition. In the following we will therefore simply speak of a normal stress Fw irrespective of its origin. Fw represents a natural control parameter in the problem and can be fixed for example in wetting experiments by varying either ycs using a PEG-fibronectin substrate coating, or soft gels, or by varying ycc by tuning the level of cadherin expression [25]. We will first introduce the model equations, then describe the different solution types that can be found dependent on the normal stress, and finally summarize the results in bifurcation diagrams.
012712-1
©2015 American Physical Society
MICHAEL H. KOPF
Fw
c e lls
PHYSICAL REVIEW E 91,012712 (2015)
Fw
fw
s u b s tr a te
c e lls
JdL
s u b s tr a te
FIG. 1. (Color online) Two scenarios of stressed tissues. If un constrained (left), a tissue can be under stress arising from wettability. If the tissue is embedded in another tissue (right), pushing and pulling from the surrounding cells leads to stress. II. MODEL EQUATIONS To capture the essential dynam ics we consider a sim plified one-dim ensional version o f the m odel used in Ref. [24]. Since we w ant to focus on the effect o f norm al stress at the boundary and the onset o f the spreading dynam ics, w e neglect both cell proliferation and nonlinear elastic effects, w hich becom e effective only on larger tim e scales. The model equations for the signal concentration c, the polarization p, and the displacem ent field u take the follow ing nondim ensional shape (see A ppendix A for inform ation about the used scaling):
c = c" — c + fiu',
(1)
P = p" - a 2p ( l + p 2) + cp,
(2)
u — u" + qcp.
(3)
FIG. 2. (Color online) Type-III solutions at Fw = 0 .0 1 2 and (a) q = 0.01 and (b) q = 0.045. The tissue actively pulls inwards to compensate the external stress. The smaller the active force q , the stronger the tissue has to polarize to achieve an equilibrium.
activation fi < 0. A uniform ly polarized state is possible, w here the w hole tissue m oves w ith constant velocity v and under constant stretch: Type II:
Here, w e use the notation / = d f/d t and / ' = d f / d x as shorthand for tem poral and spatial derivatives, respectively. D eform ation is quantified by u! w hile fi determ ines how m uch o f the signal is generated or annihilated in reaction to deform ation. In the absence o f the chem ical signal, the m aterial rem ains quiescent and the dam ping force is quantified by the positive param eter a 2 > 0. T his choice o f signs distinguishes the m aterial from a spontaneously polarizing tissue for which the linear term a 2p w ould have positive sign. Finally, q denotes the strength o f the active force. It follow s from Eq. (3) that the center o f m ass o f the bounded system can translate and does so w ith the propagation velocity
q fL v = — / dxcp. L Jo
(4)
In the presence o f a norm al stress F w,defined positive if pulling outw ard, we em ploy the elastic boundary conditions u \ 0) = u'(L) = Fw and no-flux conditions for p and c, that is, c'(0) = c’{L) = p \ 0) - p \ L ) = 0. III. SOLUTIONS AND BIFURCATION ANALYSIS We w ill now discuss the possible types o f stationary and m otile solutions. T he sim plest case is a fam ily o f uniform ly stretched states w ith a hom ogeneous concentration c. We will refer to this fam ily as type-I solutions: Type I:
(c,p,u) - 08F w,0 ,F wx + n 0),
(c,p,u) — (ySFw, ± p n , u 0 + Fwx ± vt),
(6)
w ith p u = y7(fiFvj/a1) — 1 and v = qfiF^Pii- O f course, the problem is sym m etric, and positive and negative velocities are equivalent. If the system starts from random initial conditions and finally settles on a type-II solution, the direction o f m otion depends on the initial conditions. Two m ore fam ilies o f nonuniform states can be calculated num erically. Solutions o f type III correspond to nonuniform ly stretched tissues w ith nonzero polarization close to the boundaries, w here the cells either pull o r push to balance the external stress Fw. Typical solutions o f this type are show n in Fig. 2. Finally, solutions o f type IV exhibit a kind o f m otility very different from the uniform m otion o f type-II solutions: a train o f polarization waves travels across the tissue and due to the active force, these polarization waves lead to deform ation waves, rem iniscent of a w orm that locally contracts and expands in peristalsis (see m ovies in the Supplem ental M aterial for illustration [26]). W ith each wave the tissue propagates a certain distance, with a velocity that can be calculated using Eq. (4). Like type-II solutions, m otion to the left and to the right are equivalent and this sym m etry is broken by the initial conditions. Solutions o f type IV are found betw een two critical Fw values that depend on the strength o f the active force q. In order to better understand the solution structure o f the problem , we obtain a bifurcation diagram using num erical continuation w ith the softw are AUTO [27] that em ploys the pseudoarclength algorithm [28], To this end, the dom ain is discretized using N — 256 grid points and second-order finite differences. As solutions o f the m odel equations are then vectors in a 3 x 256-dim ensional space, w e define the norm
(5)
w ith integration constant u q . Type-I solutions are not polarized and thus exhibit no active forces. If the norm al stress satisfies /3FW > a 2, up to three m ore solution fam ilies, types II—IV, can be identified. N ote that since a 2 is a positive constant, they can be found for Fw > 0 only in the case o f expansive activation P > 0, and for Fw < 0 only in the case o f compressive
^Sp^ = \ ¥ l J 0 /
f
C*T)2 d x d rj
(7)
as a solution m easure, w here L is the dom ain size and T the tem poral period for tim e-periodic solutions. For steady states the tim e average (1 / T ) j d t . . . is om itted. Bifurcation diagram s for three different values o f the active force strength
012712-2
PHYSICAL REVIEW E 91, 012712 (2015)
COLLECTIVE CELL MIGRATION INDUCED BY . . .
FIG. 3. (Color online) Bifurcation diagram for q = 0.01. Stable and unstable branches are shown as solid and dashed lines, respec tively. Red dots mark the Hopf bifurcations at the ends of the type-IV branch.
q are shown in Figs. 3-5. The other parameters are set to P = \ ,ce2 = 10-2, and L = 103. From the discussion of the solution types we already know that a bifurcation occurs at Fw = a 2/ ft. With smaller Fw there are only stable type-I solutions. For higher values of Fw, type-I solutions become unstable while stable solutions of types II and III appear. If the strength of the active force q is higher than a critical value, a pair of Hopf bifurcations (HB) appears on the branch of type-II solutions. These HBs mark the end points of a branch of type-IV solutions. Thus nonuniform motion occurs only in a limited range of normal stresses Fw, similar to the observation in Ref. [14]. As the diagrams show, new HB pairs and secondary branches of type-IV solutions appear with increasing q. Also, the primary type-IV branch splits into two branches at lower Fw values between a pair of period-doubling (PD) bifurcations. However, the solutions on the split branches behave very similar [26]. While the wave train in the primary branch spans the whole tissue [see Fig. 6(a)], the wave trains
FIG. 5. (Color online) Bifurcation diagram for q = 0.045 with symbols as in Fig. 3. Up to three type-IV solutions coexist for the same Fw. Blue squares mark the solutions at Fw = 1.025 x 10~2 that are shown as (a), (b), and (c) (from top to bottom) in Fig. 6. The numbers 1-6 indicate the Hopf bifurcations corresponding to the critical modes shown in the movies ev_crit_q045_HB 1.mp4 to ev_crit_1045_HB6.mp4 in the Supplemental Material [26].
of the secondary branches start closer to either boundary [see Figs. 6(b) and 6(c)]. The different number of maxima in the wave trains is visible in the spectrum of the linear problem close to the HBs (see Appendix B). Figure 7 summarizes the results of 150 continuation runs at different q values in a codimension-2 bifurcation diagram. It shows the Fw values corresponding to the end points of the type-IV branches (HBs) for each value of q and illustrates the appearance of branches with increasing q. We see that while the left HB of each pair moves closer to the critical value Fw = a 2/ p , the Fw value corresponding to the rightmost HB increases with q, so that the range of normal stresses sustaining nonuniform motion grows with the strength of the active
4
2
0 -2 -4 (a)
(b)
0.01
0.01004
0.01008
0.01012
0.01016
Fw
(c) FIG. 4. (Color online) Bifurcation diagram for q = 0.01375 with symbols as in Fig. 3. Two branches of type-IV solutions coexist between two pairs of Hopf bifurcations. The inset shows a region where the primary type-IV branch splits between two period doubling bifurcations (shown as blue diamonds).
0
200
400
600
800
FIG. 6. (Color online) Snapshots of the coexisting type-IV solu tions at q = 0.045 and Fw = 1.025 x 10-2, corresponding to the blue squares in Fig. 5. The wave trains emerge in the bulk and propagate to the right.
012712-3
MICHAEL H. KOPF
PHYSICAL REVIEW E 91, 012712 (2015)
case is conceptionally straightforward and could be used to find more complicated solutions. ACKNOWLEDGMENTS
The author thanks Cristina Marchetti for helpful discus sion. This work was supported by the French National Re search Agency (ANR), grant ENS-ICFP: ANR-10-238LABX0010/ANR-10-IDEX-0001-02 PSL*. APPENDIX A: SCALING
-------- 1-------- 1-------- 1________ i_______ i 0.01
0.0102
0.0104
0.0106
0.0108
0.011
Fw FIG. 7. (Color online) Codimension-2 bifurcation diagram showing the loci of Hopf bifurcations on the branch of type-II solutions dependent on the strength of the active force q and the normal stress Fw. Intersections of the shown curves with a horizontal at a given q value correspond to Hopf bifurcations, as is indicated by the horizontal lines for the cases of Figs. 3-5. The branches are all of the same type; different line styles are only used to distinguish the lines. Hopf bifurcations on the same curve are connected by a branch of time-periodic type-IV solutions.
force q. A similar codimension-2 bifurcation cascade was recently observed in a model for the deposition of surfactant monolayers (see [29] and Figs. 10 and 11 therein). There, just as in the present model, solutions with differently sized wave trains (see Fig. 6) coexist. IV. SUMMARY AND CONCLUSIONS
We can summarize that normal boundary stress can induce motility in spreading cell layers with a simple chemomechanical feedback. The boundary stress has been found to be a necessary precondition for the observed motility. In particular, in the unstressed case Fw = 0, the trivial homogeneous solution (c , p , u ) = (0,0,0) is linearly stable for all finite parameters. Even at small stretch y3Fw < or2, the external mechanical cue cannot be bypassed by increasing the initial concentration c: the system relaxes back to its linearly stable homogeneous state. The collective cell migration is either uniform, like in collective gliding, or occurs by deformation waves. The efficiency of similar motility modes has been investigated in a purely mechanistic model with prescribed activity [30], A switching in motility modes dependent on either adhesiveness (our Fw) or active force (our q) has been observed in single keratocytes [31]. The effect of proliferation and apoptosis has not been considered in the current investigation but might be relevant as it leads to intercellular rearrangement that effectively fluidizes the tissue [32], Furthermore, stresses arising due to volumetric growth might be an additional trigger of mechanosensitive activity. Cell division and death can be readily incorporated into the description of the elastic layer by introduction of a growth tensor [33], as was done in Ref. [24], or by an additional pressure and volume source [34], Finally, an extension of the presented analysis to the two-dimensional
The model equations (l)-(3) are based on the model presented in Ref. [24] but are simplified by considering only one spatial dimension and linear elasticity. Additionally, we chose a slightly different scaling. This physically equivalent approach allows for a simpler bifurcation analysis, as the codimension-2 diagram in Fig. 7 in the main text would be come largely unclear in the original scaling. The dimensionful governing equations for c, p, and u are c ~ Dc" —kdc + fin', FpP = K2p" - (1 + K0p 2)A2p +
