One dimensional cell motility patterns

Consider a cell of length $l$ that migrates along a linear track. The two ends of the cell, the front and back, denoted by $x_{f}$ and $x_{b}$, are connected by a spring (Fig. 1A). The stiffness of the spring $k$, represents the effective elasticity of the cell cytoplasm and membrane. Such a mechanical coupling between the front and rear was recently demonstrated experimentally [27].

In this part of the model, the motion of the cell is considered to be already polarized, such that the actin treadmilling from the front to the back occurs at a constant velocity $v$, and produces a constant protrusive force at the front

where $\alpha$ describes the strength of the coupling of the actin retrograde flow and the effective friction force generated by adhesions which grip the sliding filaments. This is a simplified representation of the ”clutch” mechanism [23], that converts the sliding of the actin to a protrusive force that pushes on the membrane. This coupling is dependent on the adhesion strength, and we therefore expect a term of the form: $r/(r+r_{0})$ to multiply the r.h.s. of Eq.1, where $r$ is the ratio between the binding/unbinding rates of the cell-substrate adhesion molecules $r=k_{on}/k_{off}^{0}$, and $r_{0}$ quantifies the the cell-substrate adhesion saturation. Such a term accounts for the loss of traction force when the adhesion diminishes $r\rightarrow 0$. We do not explicitly describe here the catch-bond property of these adhesions, as we are not interested in the stick-slip dynamics of the leading edge, but rather wish to focus on the stick-slip events on the whole cell scale. For the rest of this paper we effectively work in the limit of $r_{0}\ll r$ (the effects of $r_{0}$ are shown in Fig.23).

The pushing force is balanced by a drag force which is proportional to the speed of the moving front, and a restoring force due to the global cell elasticity

where $\gamma$ represents the effective resistance to the motion of the cell front due to the friction generated by the contact of adhesion molecules with the substrate. As in Eq.(1), we expect a term of the form $r/(r+r_{0})$ to multiply the first term on the r.h.s. of Eq.2, but we neglect it by choosing to work in the limit of $r_{0}\ll r$. The parameter $k$ represents the stiffness of the spring which describes the effective elasticity of the cell, and the parameter $x_{0}$ represents the rest length of the cell.

Equating (1) and (2) due to the force balance at $x_{f}$ yields

which allow to obtain the equation of motion for the moving cell front

At the back, $x_{b}$, the pulling force due to the cell elasticity

is balanced by a friction force which results from the stretching of bound linkers, which model slip bond adhesions

where $n$ is the mean number of bound linkers, $\kappa$ is the spring constant of the linkers, and $\langle\Delta x\rangle$ is the average displacement of the linkers (Fig.1C). The number of bound linkers evolves dynamically and obeys the following kinetics

where $N$ is the total number of linkers, $k_{on}$ is the basal attachment rate, and $k_{off}$ is a detachment rate which depends exponentially on the force applied on each bound linker [24, 23]

where $k_{off}^{0}$ is the basal detachment rate and $f_{s}$ represents the susceptibility of the linkers to the applied force [24]. The stretching force per bound linker is

The average stretch of each linker $\langle\Delta x\rangle$ depends on the dissociation rate of the linkers (8,7) and can be written as

By combining (8,9,10) into the force balance between (5) and (6) we obtain

where $\kappa$ is the effective spring constant of the linkers, and we substituted for the average stretching of each linker: $\langle\Delta x\rangle=\dot{x}_{b}/k_{off}$. By reorganizing (11) we obtain the equation of motion of the moving back

Combining (4) and (12), and changing coordinates to $l=x_{f}-x_{b}$ yields the following dynamical system

Next, the system (13,14) is rescaled by the time and length scales of $k_{off}^{-1}$,$x_{0}$, and by the total number of adhesion sites $N$, as well as rescaled by the parameters $\frac{v}{x_{0}k_{off}^{0}}\rightarrow\tilde{v}$, $\frac{k}{k_{off}^{0}}\rightarrow\tilde{k}$, $\frac{f_{s}N}{x_{0}k_{off}^{0}}\rightarrow\tilde{f}_{s}$, $\frac{N\kappa}{k_{off}^{0}}\rightarrow\tilde{\kappa}$ and $\frac{k_{on}^{0}}{k_{off}^{0}}\rightarrow r$ to obtain

Finally, we set $\alpha=1$ and $\gamma=1$ and remove the tilde signs in Eqs.(15,16), to obtain

Note that the dynamics is now captured by two ODEs, where the spatial component appears only as the total cell length variable. These reduced equations have four structural parameters ($r,k,\kappa,f_{s}$) and the parameter describing the strength of the actin treadmilling flow ($v$).

It is convenient to plot the resulting dynamics on the $k-r$ phase diagram shown in Fig.2A. We find that above a critical value of the cell stiffness $k$ (to the right of the blue solid line), the spring is too stiff to allow for large length changes that enables the spring to store and release large forces. In this regime we get a single, stable fixed point that corresponds to smooth cell motion, and no stick-slip behavior (for the phase-diagram as function of the adhesion saturation parameter $r_{0}$ see Appendix A).

Below the critical stiffness, we find that for small values of $r$ the system corresponds to smooth motion (stable fixed point), of a short and fast-moving cell (Fig.2H-J).

Above a transition line (solid blue line in Fig.2A), the fixed point undergoes a Hopf bifurcation, which marks the transition from smooth motion to stick-slip motion (limit cycle dynamics)(Fig.2E-G). In this regime there are large length oscillations, as well as large oscillations in the number of bound adhesion linkers at the cell back (Fig.3A,B). These oscillations in $n$ occur due to the large values of the force per linker that is reached (Fig.3C), which leads to catastrophic, avalanche-like detachment events at the back. Within the stick-slip regime, we find that the duration of the limit-cycle increases with increasing $r$, i.e. the dynamics slow down with increasing substrate adhesiveness (Fig.3D).

As $r$ increases there is a second Hopf bifurcation, whereby the fixed point is stable again. However, there is a narrow region of bistability, where the phase-space is separated by a separatrix (solid black line in Fig.3B), such that the stable fixed point coexists with the limit cycle. Within this regime, noise can induce transitions from stick-slip (limit cycle) to smooth motion (stable fixed point), as demonstrated in Fig.4.

For increasing $r$ the separatrix grows until it meets the limit cycle, which marks the transition to flows that all lead to the single stable fixed point. The dynamics in this regime correspond to a smooth motion of a slow-moving and elongated cell (Fig.2B-D,Fig.3).

We now compare in detail the model’s stick-slip limit-cycle dynamics (Fig.2E-G) to experiments (see Appendix D for experimental methods). In Fig.5 we analyze the stick-slip migration mode of a C6 glioma cell (seeded on laminin-coated lines of 5$\mu$m width), by following the cell length and the intensity of membrane-bound vinculin, using total internal reflection fluorescence (TIRF) microscopy. As a component of the adhesion complex, we denote the total intensity of vinculin in the rear of the nucleus as a measure of the adhesion strength at the cell back in the model ($n$). We find that the experimentally observed dynamics during a stick-slip cycle give rise to a limit-cycle in $n,l$-phase space that has the same qualitative features as the model predicts (Fig.2F).

The robust features of the migration modes predicted by the model, can be observed in experiments (Fig.6). In Fig.6a we show a kymograph of a motile cell moving along a one-dimensional stripe, exhibiting stick-slip behavior. At a certain point along the trajectory the cell increased both its rate of stick-slip events, and its overall migration velocity. In addition, the amplitude of the stick-slip events decreased in size. All of these features are captured by the model, if the average adhesion strength between the cell and the substrate ($r$) decreases abruptly along the trajectory, and the cell moves from high to low $r$ within the stick-slip regime (Fig.2A). The cell-substrate adhesion strength is sensitive to the surface properties along the stripe, which may vary along its length. The backwards growth of the lamellipodia at the cell back, during each stick-slip cycle, are described by the more elaborate model below.

In Fig.6b we show a cell migrating smoothly, and then abruptly decreasing in length and increasing in speed. This behavior is captured by the model assuming that the adhesion strength $r$ decreases such that the cell jumps from high to low $r$ as shown in (Fig.2A,D,J).

Note that we can not exclude that the abrupt change in behavior observed for the cells in Fig.6 originates from a change in some other internal parameter of the cell. In addition, we observe highly dynamic stick-slip behavior at the leading edge of the cell, which we do not describe by our current model [23].

We now complement the length-adhesion model described above, by incorporating a model for the spontaneous self-polarization of the cell. We use a modified version of the scheme developed in [25] (Fig.7A), whereby in addition to the advection of proteins that enhance the actin treadmilling, such as myosin-II, we also consider an inhibitor of actin polymerization. The inhibitor protein is free to diffuse in the cytoplasm, and is also advected by the actin treadmilling flow. An example for an inhibitor of local actin polymerization, that is advected by the actin flow, is Arpin [28, 25]. We furthermore assume that the time-scale of the redistribution of this inhibitor across the cell is much shorter than the timescale of changes in the actin treadmilling speed. This assumption is corroborated by recent measurements of myosin-II redistribution time when advected by actin ($\sim 10$sec) [27], while the stick-slip dynamics of the cell are over timescales of tens of minutes [9].

We can therefore use the steady-state distribution of the concentration of this inhibitor $c(x)$, which is given by an exponential function (Fig.7B)

where $v$ is the effective instantaneous actin treadmilling velocity, $D$ the effective diffusion coefficient of the inhibitor and $c_{tot}$ is the total amount of inhibitor molecules in the cytoplasm (see Appendix B).

To complete the model [25], the net treadmilling flow is given by the difference between the actin polymerization flows created at the two ends of the cell, which are inhibited by the local concentration of the inhibitor at the front and rear

where $\beta$ gives the scale of the actin flow in the cell, and the polymerization activity at each end is diminished by the inhibitor concentration at each end, given by a simple Hill function: $\tilde{c}(x_{b,f})=\frac{c_{s}}{c_{s}+c(x_{b,f})}$ (Fig.7B), where $c_{s}$ is the concentration at which the effect of the inhibitor saturates.

After rescaling (see Appendix C), using Eqs.19,20, we obtain the following implicit equation for the net actin treadmilling flow

In the persistent regime (sufficiently large $c_{s}$), we find that as $l$ increases above a critical value Eq.(21) undergoes a pitchfork bifurcation, where spontaneous actin treadmilling appears, corresponding to a polarized and motile cell (Fig.7C). The critical length is given by

which provides a lower bound for the coupling strength $\beta>D/c$, below which there is no spontaneous motility. A lower critical length for polarization was also obtained in [29].

The equations of motion are now extended beyond the basic model (Eqs.17,18) to include the dynamics of the actin treadmilling flow

where $v_{ss}(l)$ is the steady-state solution to Eq.21, which is the length-dependent actin flow speed, and $\delta$ is the rate at which the actin flow adjusts to length changes, i.e. the rate at which $v$ approaches $v_{ss}(l)$.

For a cell that has a rest-length that is longer than the critical length for polarization, $l_{c}<1$, the resting state of the cell is polarized. Since the velocity saturates to a constant value for $l>l_{c}$ (Fig.7C), the results are qualitatively similar to those we obtained for a constant, length-independent, $v$ (Fig.3).

We will therefore focus on the more interesting case where $l_{c}>1$, and the rest-length of the cell is smaller than the critical length for polarization. In this case we find another fixed point that corresponds to $v=0$

where the cell is at its rest length, and stationary.

In Fig.8A we plot the $k-r$ phase diagram for the simpler case of $\delta\rightarrow\infty$, such that the actin flow speed is given directly by the solution $v_{ss}(l)$ (Eq.20, Fig.7C). In comparison with the case of fixed actin flow (Fig.2A), there are several new phases, despite the overall similar shape of the phase diagram.

For very low values of $r$, we find that there is a transition line below which the cell is non-motile (Brown region in Fig.8A), which corresponds to all the flows in the $n-l$ phase-space leading to the single fixed point at $v=0$ (Eq.26). Above this transition line, with increasing $r$, there are phases of coexistence between smooth motion or stick-slip, and non-motile behavior. We plot in Fig.9 the dynamics for different points of increasing value of $r$, for fixed $k=0.8$ (vertical dashed line in Fig.8A). Two flows are demonstrated for each case (green and purple paths and corresponding kymographs in Fig.9), exposing the coexistence of either smooth motion (E-H, Q-T) or stick-slip (I-L) with the non-motile phase. The bifurcations of the stable and unstable solutions are denoted by their cell length (and the limit cycle amplitude), as well as the limit cycle periods, in Fig.8B,C (along the same $k=0.8$ line).

There is even a thin region of tri-stability (Fig.8A), which we demonstrate in Fig.8D-F, by inducing a transition between the three phases by adding noise at specific times. For a fuller exploration of the $k-r$ phase diagram see Appendix E.

Next, we explore the effect of a finite value of $\delta$, where the treadmilling velocity of the actin does not instantaneously adjust to its length-dependent value $v_{s}s(l)$ (Eq.25).

In Fig.10A we plot the shift in the stick-slip transition line to lower values of $k$, for decreasing values of $\delta$. The region of no-motility is now pushed to lower values of $k$. The reason for this is shown in Fig.10B-D: when the length drops below $l_{c}$ after the slip event, the velocity of the actin does not drop to zero instantaneously (as in the case when $\delta\rightarrow\infty$). There is therefore a region of the phase-diagram where the length recovers and increases above $l_{c}$, allowing the stick-slip limit-cycle to survive. The ”stall duration”, during which $l<l_{c}$ and the cell is almost stalled, increases with increasing value of $\delta$ (Fig.10E).

In the model we developed so far, we did not allow for the actin polymerization to produce local traction and protrusive forces at both ends, but rather assumed that the competition between the two ends gives rise to a single leading edge (or to an unpolarized cell) (Fig.7A).

We now extend the model to describe the local protrusive forces that are produced by actin polymerization at the two ends of the cell (Fig.11A).

The dynamics of the moving front and back depend on their respective direction of motion: when they protrude there is a drag force that is given by a constant drag coefficient, while when they retract the friction depend on the slip-bond activity of the adhesion (Fig.11B). We therefore have the following equations of motion for the two ends of the cell

where the friction coefficients are given by

where $\Theta$ is the theta Heaviside function. This way the friction coefficient depends on the local motion of each end. This is illustrated in Fig.11B, where the friction at the back is due to the slip-bonds when the back end of the cell is retracting $\dot{x}_{b}>0$ and $\Theta\left(\dot{x}_{b}\right)=1$.

Note that the force that pulls on the adhesions during retraction (in the exponential function in Eq.28) is not only given by the elastic spring force (as in the previous models presented above), but is countered by the local force of actin polymerization that is pushing the membrane

Similarly, the dynamics of the adhesions are given by

The dynamics of the polymerization/tread-milling of the actin is now calculated separately on both sides of the cell, given by

where

The distribution of the inhibitor along the cell is affected by the competition between the treadmilling actin from both ends, so its given by Eq.19 with: $v=v_{f}-v_{b}$.

The solution of Eq.31 is shown in Fig.11C. The solution shows that the cell remains unpolarized for $l<l_{c}$, and the actin tredmilling flows are equal at the two ends of the cell, given by $v_{f/b}\beta\left(\frac{l}{c+l}\right)$. A polarized cell for $l>l_{c}$ has a higher treadmilling flow from the front, and in the limit of large $l$ the treadmilling velocity approaches these limiting values: $v_{f}^{*}=\beta$ and $v_{b}^{*}=d/c$.

We find that there a new length scale in the system, which determines the ability of the cell to elongate and reach polarization. For the the unpolarized cell we equate the protrusive forces at both ends to the restoring force of the spring

which yields the polarization length

By equating the critical length $l_{c}$ (22) to the polarization length $l_{p}$ (33) we obtain a critical value of $\beta$, above which the cell polarizes (Fig.12A,B)

As for the previous model, the interesting behavior is for a cell that has a rest-length that is smaller than $l_{c}>1$. For such a system we demonstrate the dynamics for different values of $\beta$ (above and below $\beta_{c}$), in Fig.12D-L. For $\beta<\beta_{c}$ the cell elongates symmetrically (Fig.12D-F), but does not polarize (no spontaneous symmetry breaking). For $\beta>\beta_{c}$ we find regimes of smooth motion ($\beta=8$, Fig.12G-I) and stick-slip ($\beta=11$, Fig.12J-L) for parameters that correspond to these phases in the phase diagram of the stationary treadmilling system (Fig.2).

When the cell is within the stick-slip regime, it may end up after each slip event with a cell length that is shorter than $l_{c}$. Since in the symmetric model the protrusive force acts locally at both ends, the cell elongates symmetrically until $l>l_{c}$ and it repolarizes. This is demonstrated in Fig.11B. In Fig.20 we show the effect of the parameter $c$ on the stick-slip migration, where increasing $c$ decreases the retrograde flow and increases the time it takes the cell to repolarize following each slip event.

Furthermore, since the cell is effectively stalled when its length is shorter than $l_{c}$, the time it takes for the cell to expand back to $l_{c}$ after each stick-slip event determines the overall migration velocity. This is demonstrated in Figs.21,22.

During each stick-slip event, since the cell loses polarity, it may repolarize in a new direction. This occurs when we add random noise to the equations of motion: the system is solved as a stochastic differential equation where additive noise is added to the actin flow $v_{f}$ and $v_{b}$. For different amplitudes of the noise term we plot in Fig. 13A the probability $p$ that the cell changes its direction of motion per stick-slip event. We find that for each level of noise there is a critical value of the actin response rate $\delta$ (Fig.30) that sets a threshold between $p=0$ and $p=0.5$. During each stick-slip event, when $l<l_{c}$, the actin flow velocities on both sides approach each other exponentially with a time-scale of $\delta^{-1}$. As $v_{f}/b$ approach within the noise amplitude, they cross each other (Fig. 13B,C), and the side that has the higher flow when $l=l_{c}$ determines the new direction of cell motion. The dynamics around the critical $\delta$ where the turning probability increases strongly, are demonstrated in Fig.14.