One-dimensional cell motility patterns

During migration, cells exhibit a rich variety of seemingly random migration patterns, which makes unraveling the underlying mechanisms that control cell migration a difficult challenge. For efficient migration, cells require a mechanism for polarization, so that traction forces are produced in the direction of motion, while adhesion is released to allow forward migration. To simplify the study of this process, cells have been studied when placed along one-dimensional tracks, where single cells exhibit both smooth and stick-slip migration modes. The stick-slip motility mode is characterized by protrusive motion at the cell front, coupled with a slow elongation of the cell, which is followed by a rapid retraction of the cell rear. In this study, we explore a minimal physical model that couples the force applied on the adhesion bonds to the length variations of the cell and to the traction forces applied by the polarized actin retrograde flow. We show that the rich spectrum of cell migration patterns emerges from this model as different deterministic dynamical phases. This result suggests a source for the large cell-to-cell variability (CCV) in cell migration patterns observed in single cells over time and within cell populations: fluctuations in the cellular components, such as adhesion strength or polymerization activity, can shift the cells from one migration mode to another, due to crossing the dynamical phase transition lines. Temporal noise is shown to drive random changes in the cellular polarization direction, which is enhanced during the stick-slip migration mode. The model contains an emergent critical length for cell polarization, whereby cells that retract below this length loose polarity, and are prone to making direction changes in migration. These results offer a new framework to explain experimental observations of migrating cells, resulting from noisy switching between underlying deterministic migration modes.

Figure 1

The simplified model. (a) Illustration of the physical model of a cell migrating along a linear track. $x_b$ and $x_f$ represent the back and front part of the cell of the cell which are connected by an effective spring with a stiffness $k$. $x_0$ is the rest length of the spring (cell) and $l$ is the length of the cell. $v$ is the velocity of the actin retrograde flow which is assumed to be constant. At the front acts a protrusion force (red arrow) and a drag force (teal arrow). At the back acts a friction force due to the slip bonds (blue arrow). (b) The physical model of the stick slip adhesion at the back $x_b$. Stochastic linkers with stiffness $\kappa$ attach with an average rate of $k_{\mathrm{on}}$ and detach with a length dependent rate of $k_{\mathrm{off}}\left(l\right)$ at the back [Eq. (8)]. The linkers stretch on average with a displacement of $\mathrm{\Delta }x$ [Eq. (10)]. $F_{\mathrm{back}}^{\pm }$ [Eqs. (5) and (6)] are the forces that act on the trailing edge of the cell. (c) The physical model of the protrusion force acting at the front $x_f$. $F_{\mathrm{front}}^{\pm }$ [Eqs. (1) and (2)] are the forces that act on/from the sliding actin filaments, which are balanced by catch adhesions (green short lines).

Figure 2

(a) $k-r$ phase diagram. Blue region correspond to the polarized stick-slip cells. White region corresponds to polarized smooth migrating cells with a constant length. Red region corresponds to bistability between polarized cells with a constant length and polarized stick slip migrating cells. Blue/Red curves are the Hopf/saddle node bifurcation transition lines. Black dashed line is the $r$ cross section for $k=0.8$. Black dots correspond to the points $(k,r)=0.8,1;(k,r)=0.8,5;$ and $(k,r)=0.8,8.4$. Red star correspond to the point $(k,r)=(0.8,7.7)$. [(b)–(d)] The dynamics at $(k,r)=0.8,1$. [(b)–(d)] The dynamics at $(k,r)=0.8,5$. [(h)–(i)] The dynamics at $(k,r)=0.8,1$. Blue/orange curves in (b), (e), and (h)correspond to the time series of the cell length and adhesion concentration at the rear. Red curve in (c), (f), and (i) represent the trajectory in $l,n$ phase space. Black curves in (d), (g), and (i) are the kymographs where the black curves are cell edges $x_b$ and $x_f$. Parameters: $f_s=5,\kappa =20,\text{and}v=10$.

Figure 3

Stability analysis along the line $k=0.8$ [vertical black dashed line in Fig. 2]. (a) The maximal and minimal length as a function of $r$. (b) The maximal and minimal force applied on a linker at the back part of the cell as a function of $r$. (c) The maximal and minimal adhesion concentration at the rear as a function of $r$. (d) The time period of the limit cycle as a function of $r$. Solid black curves indicate the stable limit cycle. Dashed black curves indicate the unstable limit cycle. Parameters: $v=10,\kappa =20,\text{and}{f}_s=5$.

Figure 4

[(a)–(c)] Dynamics in the bistability regime [red star in Fig. 2. (a) Blue and orange curves are the time series of the cell length and adhesion concentration at the rear. Black dashed line indicates the time when noise was injected into the solution of the differential equation [Eq. (17)]. (b) $n-l$ phase diagram. Red curve is the trajectory and black curve is the separatrix between the two modes of motion. (c) The corresponding kymograph: black curves represent the edges of the cell $x_b$ and $x_f$. Dashed black line indicates the time when noise was injected. Parameters: $v=10,k=0.8,\kappa =20,\text{and}{f}_s=5$.

Figure 5

Comparison of the basic stick-slip model [Eqs. (17)and (18)] to the stick-slip dynamics observed in experiments (patient-derived Human glioma propagating cell NNI21 transfected with fluorescently tagged vinculin and seeded on laminin-coated lines of 5 $\mu \mathrm{m}$ width and imaged every 110 sec). (a) Movie snap shots, where the cells have fluorescently labeled vinculin (represents the extent of adhesion complexes). The region that defines the cell back is to the left of the vertical dashed line, i.e., behind the nucleus. (b) A kymograph of the cell migration, with the stick slip events marked. (c) The dynamics of the cell length and the total amount of vinculin signal at the cell back region, as function of time. The black line on both graphs indicates the stick-slip cycle used to plot the phase-space limit cycle shown in (d).

Figure 6

Comparison of the basic stick-slip model [Eqs. (17) and (18)] to experiments on C6 glioma cells seeded on laminin-coated lines of 5 $\mu \mathrm{m}$ width (imaged every 30 sec). Kymographs correspond to total time 2.5 hours. (a) Comparison to model results with $r=7$, then abruptly changed to $r=3$ (indicated by the horizontal dashed line). (b) As in (a), with $r=8.5$ then changing abruptly to $r=1$. Model parameters: $v=10,k=0.8,\kappa =20,\text{and}{f}_s=5$.

Figure 7

The stick-slip UCSP model. (a) Illustration of a the physical model. Unlike the simple model of Fig. 1, the treadmilling velocity is now dependent on the cell length $v\left(l\right)$, due to the advection of an inhibitory cue. The concentration of the inhibitory cue is represented by the colorbar on the right. The steady-state concentration of the inhibitory cue, $c\left(x\right)$, is shown in [(b), top], which is an exponential function due to a balance between diffusion and advection [Eq. (19)]. The effect of $c\left(x\right)$ on the local actin polymerization rate is given by a Hill function [Eq. (20)] [(b), bottom]: where the inhibitory cue concentration is low the local actin polymerization rate is large. (c) The steady-state retrograde flow speed as a function of cell length [solution of Eq. (21)]. Red solid line represent the stable solution. Red dashed line represent the unstable solution. Black dashed line notes the critical length $l_c$ of polarization.

Figure 8

(a) $k-r$ phase diagram. Striped texture regions correspond to regimes where there is no motility (stationary cell). Blue region corresponds to a bistable region of polarized stick-slip cells and stationary cells. White region corresponds to bistability between stick slip and smooth migration. Red region corresponds to tristability between stick slip, smooth migration, and stationary cells. Blue/red/brown curves are the Hopf/saddle node/Hopf bifurcation transition lines. Black dashed line is the $r$ cross section for $k=0.8$. Black dots correspond to the points $(k,r)=(0.8,0.4),(0.8,0.7),(0.8,1.65),(0.8,5),(0.8,8)$. Red star correspond to the point $(k,r)=(0.8,7.7)$. (b) Maximal and minimal amplitude of the $l$ state variable as a function of $r$ for the limit cycles in the vector field along the cross section of $k=0.8$. Dashed blue curve correspond to the unstable discontinuous limit cycle. Solid black curve corresponds to the stable limit cycle and the cell length. Black dashed curve is the unstable limit cycle. (c) The time period of the limit cycles. Blue dashed line indicate the discontinuous unstable limit cycle. Black solid line indicate the stable limit cycle (period of stick slip). Black dashed line indicates the unstable limit cycle in the tri-stable region. [(d)–(f)] The tristable region. Solution of Eqs. (23) and (24) with noise of amplitude $(\delta l,\delta v)=(0.5,0.11)$ injected at $t=3.55$ and $(\delta l,\delta n)=(0.5,0.05)$ injected at $t=16.2$, to demonstrate the transition between the phases (across the separatrix lines). Blue/orange/red curves in (d) correspond to the dynamics of length/adhesion concentration and actin retrograde flow. Gray dashed lines indicate the time point in which the noise was injected. Green curve in (e) is the trajectory in the $l\text{−}n\text{−}v$ phase space. Black solid lines are the separatrices. (f) displays the kymograph which corresponds to [(d) and (e)]. Parameters: $f_s=5, \kappa =20, \beta =11,c=3.85,\text{and}D=3.85$.

Figure 9

The dynamics along the line of constant $k=0.8$ [vertical dashed line in Fig. 8]. [(a)–(d)] $r=0.4$, [(e)–(h)] 0.7, [(i)–(l)] 1.65, [(m)–(p)] 5, and [(q)–(t)] $r=8$. Blue/orange/red curves in (a), (e), (i), (m), and (q) correspond to the time series of the cell length, adhesion concentration and actin retrograde flow respectively (bold/thin lines corresponds to green/purple trajectory). Green and purple lines (b), (f), (j), (n), and (r) demonstrate the trajectories in the bistable $l-n-v$ phase space regime. Black solid curves are the separatrices. (d), (g), and (i) display the corresponding kymographs. Parameters: $f_s=5,\kappa =20,\beta =11, c=3.85$, and $D=3.85$.

Figure 10

The effect of a finite actin response rate $\delta$ on the dynamics [Eq. (25)]. (a) The $k-r$ phase diagram for different values of $\delta$ [in Eq. (25)]. Dashed black lines indicate the coalescence of the discontinuous unstable and stable limit cycles, which produce the no-motility regime for different values of $\delta$. The phase transition line for $\delta \to \infty$ is the same as in Fig. 8. [(b)–(d)] Demonstration of the lagging time for values of $\delta$ at the point $(k,r)=(0.8,5)$ on the phase diagram (black dot). Blue and red curves are the time series of the cell length and actin retrograde flow speed, respectively. Black dashed line is the critical length of polarization and gray dashed line is the rest length of the cell. Blue, red, and green kymographs correspond to $\delta =625,325,$ and 25 respectively. (e) The duration of the stall as a function of $\delta$, indicating the values of $\delta =625,325,\text{and}25$ at the point $(k,r)=(0.8,5)$. Gray region indicates where there is no motility for every initial condition. (f) The stable limit cycles which correspond to (b)–(d) in the $l-n-v$ phase space. (g) Projections of the stable limit cycles which correspond to (b)–(d) in the $l-n$ and $l-v$ phase spaces. Other parameters: $f_s=5,\kappa =20,\beta =11, c=3.85$, and $D=3.85$.

Figure 11

The symmetric model. (a) Illustration of the physical model. $x_b$ and $x_f$ are the back and front part of the cell which are connected by a spring with a stiffness $k$. $x_0$ is the rest length of the spring and $l$ is the length of the cell. $v_f$ and $v_b$ are the steady-state velocities of actin retrograde flow from both ends of the cell, which are coupled to the gradient of an inhibitory cue. The concentration of the inhibitory cue is represented by the colorbar on the left. At both ends of the cell, three forces act: a protrusion force (red arrow), which is proportional to the retrograde flow speed (dark red arrow), a drag force (teal arrow) when the edge is moving forward, and a friction force due to the slip bonds (blue arrow) when the edge is moving backwards. (b) Demonstration of the friction model in Eq. (28): upper panel displays in blue the derivative of the moving rear ${\dot{x}}_b$. Below in orange is the Heaviside theta function of the time series of ${\dot{x}}_b$. Right panel displays the corresponding kymograph. (c) The steady-state retrograde flow speed as a function of cell length [Eq. (31)]. Red solid line represents the stable solution. Red dashed line represents the unstable solution. Black dashed line denotes the critical length of polarization $l_c$. Parameters: $\beta =19, c=9$, and $D=40$.

Figure 12

The polarization length. (a) The critical length $l_c$ (blue) and polarization length $l_p$ (orange) as a function of the coupling coefficient $\beta$. Gray dashed lines indicate the sample points of $\beta =4,8,\text{and}11$. (b) The steady-state velocity profile as a function of length for $\beta =4,8,\text{and}11$ (purple, teal, and yellow curves, respectively). (c) The critical coupling coefficient as a function of cell elasticity $k$. The purple, teal, and yellow dashed curves denote the critical values of $k$ for the values of $\beta$ used in (b) and (d)–(l). The black dashed line denotes the critical value of $\beta$ for $k=0.8$ [which is the value of $k$ used in (d)–(l)]. [(d)–(f)] Time series, kymograph, and phase space for $\beta =4$. [(g)–(i)] Time series, kymograph, and phase space for $\beta =8$. [(j)–(l)] Time series, kymograph, and phase space for $\beta =11$. In (d), (g), and (i), the upper blue curve is the time series of the cell length. Dashed black/gray lines are the critical/polarization lengths. Lower red/blue curves represent the time series of the actin retrograde flow speed at the front/back respectively. In (e), (h), (k), the purple/teal/yellow colors represent $\beta =4,8,\text{and}11$. In (f), (i), and (l), the red/blue curves represent the trajectories at the $l\text{−}{n}_f\text{−}{v}_f/l\text{−}{n}_b\text{−}{v}_b$ phase spaces (upper/lower part of the surface). Parameters: $\kappa =20,f_s=5,r=5,k=0.8,c=3.85,\delta =250,\text{and}D=3.85$.

Figure 13

Time series of the cell length (dark top panels) and the retrograde flow speed (bottom) of the symmetric system for different values of the parameter $c$: (a) $c=3.85$, (b) 7.7, and (c) 11.55. Black dashed line in the upper panels is the critical length of polarization $l_c$. Red/blue curves in the lower panels represent the actin flow at the front/back. Other parameters: $\beta =11,D=3.85,fs=5,\kappa =20,r=5,k=0.8,\text{and}\delta =250$.

Figure 14

The effect of $l_c$ on the overall velocity of migration during stick slip: [(a)–(c)] Above the critical length. $c=3.95,r=3,5,\text{and}7$. (Left) kymograph, (right top) cell length and speed time series. (Bottom) phase space. Parameters: $\beta =11,D=3.95,k=0.8,f_s=5,\kappa =20,\delta =210$. [(d)–(e)] Below the critical length. $c=11.85,r=3,5,\text{and}7$. (Left) kymograph, (right top) cell length and speed time series. (Bottom) Phase space. Parameters: $\beta =11,D=3.95,k=0.8,f_s=5,\kappa =20,\text{and}\delta =210$.

Figure 15

Effect of $l_c$ on the overall migration velocity during stick-clip motion. [(a)–(c)] Cell length is above the critical length. $c=5,r=2,4,\text{and}6$. (Left) Kymograph, (right top) cell length and speed time series. (Bottom) phase space. Parameters: $\beta =11,D=3.95,k=0.8,f_s=5,\kappa =20,\text{and}\delta =210$. [(d)–(e)] Below the critical length. $c=6,r=2,4,\text{and}6$. (Left) Kymograph, (right top) cell length and speed time series. (Bottom) phase space. Parameters: $\beta =14,D=20,k=1,f_s=5,\kappa =20,\text{and}\delta =120$.

Figure 16

(a) The probability to change direction as a function of the damping rate for different levels of noise in the actin velocity. (b) Demonstration of a stick slip event at which the noise level is low ($\mathrm{\Delta }v={10}^{-3}$), such that $v_f$ (blue) and $v_b$ (red) do not cross when $l<l_c$, and the cell remains polarized. (c) Demonstration of a stick slip event at which the noise amplitude is sufficiently high ($\mathrm{\Delta }v={10}^{-2}$), such that $v_f$ (blue) and $v_b$ (red) cross when $l<l_c$, which enable the polarization mechanism to switch directions. Parameters: $\kappa =20,f_s=5,r=5,k=0.8,c=11.55,D=3.85,\beta =11,\text{and}\delta =300$.

Figure 17

Demonstration of direction changes events for different values of $\delta$ with noise amplitude of $\mathrm{\Delta }v={10}^{-3}$ [see Fig. 16]. Upper/middle/lower panels correspond to the time series of the cell length/actin flow/adhesion concentration for $\delta =300,325,\text{and}350$, (a)–(c) from left to right respectively. At the bottom, there are the corresponding kymographs. Parameters: $\kappa =20,f_s=5,r=5,k=0.8,c=11.55,\beta =11,\text{and}D=3.85$.

Figure 18

Demonstrating the different migration modes as a function of the strength of the actin flow ($\beta$), in comparison to observed cell migrations (C6 glioma cells seeded on laminin-coated lines of 5 $\mu \mathrm{m}$ width, imaged every 30 sec). Kymographs correspond to total time of 3 hours: (a) a cell spreading symmetrically and not polarizing: $\beta =8.5$ (in our model $\beta <{\beta }_c$, Fig. 12). (b) Smooth migration: $\beta =9$. Note that the length variations at the front are associated with actin waves which propagate in the cells and are not part of this model [15]. (c) Stick-slip migration: $\beta =10.5$. Other model parameters: $\kappa =20,f_s=5,r=3,k=0.75,c=7.12,\text{and}D=15.4$. The white line in the model kymographs denotes the trajectory of the cell's geometric center.

Figure 19

Stick-slip migration analysis in experiments and the model. [(a)–(c)] Experiment: C6 glioma cells seeded on laminin-coated lines of 5 $\mu \mathrm{m}$ width (imaged every 30 sec). Kymograph correspond to total time of 3 hours. (a) Kymograph of a cell with stick-slip migration. (b) Upper panel is the time-dependent length of the cell. Lower panel is the cell speed (from the trajectory of the geometric center). (c) Phase-space trajectory of the cell. [(d)–(f)] Model results for a cell in the stick-slip regime: (a) kymograph, (b) length and velocity time series, and (c) phase space trajectory. parameters: $\beta =14,c=6,D=20,r=4,k=0.8,f_s=5,\kappa =20$, and $\delta =120$.

Figure 20

Stick-slip migration comparison to experiment of C6 glioma cells migrating on laminin-coated lines of 5 $\mu \mathrm{m}$ width (frames are 30 sec apart). [(a)–(c)] Experiment. [(d)–(f)] Model calculation. For both we plot the kymograph, cell length and speed time series, and phase space trajectories. Model parameters: $\beta =12,c=8.37,D=10.86,k=0.9,f_s=5,r=6,\kappa =16,\text{and}\delta =150$.

Figure 21

As in Fig. 20. [(a)–(c)] Experiment. [(d)–(f)] Model calculation. Model parameters: $\beta =19,c=9,D=40,k=1.1,f_s=5,r=0.4,\kappa =20,\text{and}\delta =120$.

Figure 22

Comparison of different migration patterns between experiments and model. [(a), (c), and (d)] Kymographs of C6 glioma cells migrating on laminin-coated lines of 5 $\mu \mathrm{m}$ width (frames are 30 sec apart). (a) Stick-slip motion with frequent direction changes. Model parameters: $\beta =18,c=9,D=30,k=1.05,f_s=5,r=0.5,\kappa =20,\text{and}\delta =220$. (b) Kymograph of 3T3 cells seeded on a nanofiber coated with fibronectin (image extracted from Ref. [15]). The cell initially spreads by developing lamellipodia in both directions, and eventually breaks symmetry and migrates persistently with an approximately constant length. Model parameters: $\beta =8,c=3.85,d=3.85,k=2,f_s=5,r=1,\kappa =20,\text{and}\delta =220$. (c) Persistent stick-slip motion, with large lamellipodia extensions at the rear of the cell (image extracted from Ref. [9]). Model parameters: $\beta =40,c=9,D=40,k=1.1,f_s=5,r=0.4 \kappa =20$, and $\delta =60$. (d) A cell performing stick-slip, goes into mitosis (nonmotile), divides and generates two daughter cells migrating in opposite directions. The transitions between the different stages are indicated by the horizontal dashed lines. Model parameters: (i) $\beta =40,r=10;$ (ii) $\beta =15,r=10;$ (iiia) $\beta =19,r=12;\text{and}$ (iiib) $\beta =19,r=20$. Rest of the parameters $c=9,D=40,k=1.1,f_s=5,\kappa =20,\text{and}\delta =300$.

Figure 23

The effects of the adhesion saturation parameter $r_0$ on the cellular dynamics. (a) $k-r$ phase diagram for different values of $r_0$. Blue/red/blue curves represent the transition between migration in constant length and stick slip for. [(a)–(d)] Kymographs of the model for $k=0.8$ and $r=2,5,\text{and}8.25$. Blue/Red/Green colors represent $r_0\to 0/r_0=0.2/r_0=2$. Other parameters: $f_s=5\text{and}\kappa =20$.

Figure 24

Example of patterned laminin-coated linear tracks used in this study. The coating has been done with a mixture 1 to 5 with 2 g/ml of alexa 647 BSA and 10 g/ml of laminin. (a) High resolution stitched field of view the robust patterning with occasional existence of several microns-scale gaps in coating (asterisk). (b) The zoomed image of the region marked by the dashed rectangle in (a), highlights the local submicrometer coating fluctuations, with variability in protein deposition highlighted by the color spectrum and quantified along a zone of $3\mu \mathrm{m}$ in width, a size chosen to be in the range of usual focal adhesion size. (c) The fluctuations along the zone marked by the dashed rectangle in (b) are presented in the graph, illustrating local variation of intensity in protein concentration.

Figure 25

(a) $k-r$ phase diagram as presented in Fig. 9. Black dashed lines correspond to the $r$ cross section for $k=0.5,0.8,\text{and}1.1$ and $k$ cross sections of $r=0.5,1.5,3.5,\text{and}5,8$. Red dots correspond to intersections (dynamics at the intersections are shown in Figs. 25, 26, 27, 28). [(b)–(d)] The time period of the limit cycles in the model along the cross sections of $k=1.1$/0.8/0.5 respectively. Black dashed line indicate the unstable limit cycles. Black solid lines indicate the stable limit cycles. [(e)–(g)] Maximal amplitude of the $n$ state variable as a function of $r$ for the limit cycles in the vector field along the cross section of $k=1.1$/0.8/0.5, respectively. Black dashed curves correspond to the unstable limit cycles. Solid black curves correspond to the stable limit cycles. Dashed blue curve corresponds to the $n$ value of the fixed point in the region where $l<x_0$ ($l<1$ in the normalized units). Black dot corresponds to the transition between a finite and infinite period of the discontinuous unstable limit cycle (the $n$ cross section in which the unstable limit cycle crosses the section where $n=\frac{r}{1+r}$. Parameters: $f_s=5, \kappa =20, \beta =11,c=3.85,\text{and}D=3.85$.

Figure 26

The dynamics along the line of constant $k=1.1$ [vertical dashed line in Fig. 25]. [(a)–(d)] $r=0.5$, [(e)–(h)] 1.5, [(i)–(l)] 3.5, [(m)–(p)] 5, and [(q)–(t)] 8. Blue/orange/red curves in (a), (e), (i), (m), and (q) correspond to the time series of the cell length, adhesion concentration, and actin retrograde flow, respectively (bold/thin lines corresponds to green/purple trajectory). Green and purple curves in (b), (f), (j), (n), and (r) demonstrate the trajectories in the $l-n-v$ phase space. Black solid curves are the separatrices. (d), (g), and (i) display the corresponding kymographs. Parameters: $f_s=5,\kappa =20,\beta =11, c=3.85$, and $D=3.85$.

Figure 27

The dynamics along the line of constant $k=0.8$ [vertical dashed line in Fig. 25]. [(a)–(d)] $r=0.5$, [(e)–(h)] 1.5, [(i)–(l)] 3.5, [(m)–(p)] 5, and [(q)–(t)] 8. Blue/orange/red curves in (a), (e), (i), (m), and (q) correspond to the time series of the cell length, adhesion concentration and actin retrograde flow respectively (bold/thin lines corresponds to green/purple trajectory). Green and purple curves in (b), (f), (j), (n), and (r) demonstrate the trajectories in the $l-n-v$ phase space. Black solid curves are the separatrices. (d), (g), and (i) display the corresponding kymographs. Parameters: $f_s=5,\kappa =20,\beta =11, c=3.85$, and $D=3.85$.

Figure 28

The dynamics along the line of constant $k=0.5$ [vertical dashed line in Fig. 25]. [(a)–(d)] $r=0.5$, [(e)–(h)] 1.5, [(i)–(l)] 3.5, [(m)–(p)] 5, and [(q)–(t)] 8. Blue/orange/red curves in (a), (e), (i), (m), and (q) correspond to the time series of the cell length, adhesion concentration and actin retrograde flow respectively (bold/thin lines corresponds to green/purple trajectory). Green and purple curves in (b), (f), (j), (n), and (r) demonstrate the trajectories in the $l-n-v$ phase space. Black solid curves are the separatrices. (d), (g), and (i) display the corresponding kymographs. Parameters: $f_s=5,\kappa =20,\beta =11, c=3.85$, and $D=3.85$.