Active gel model of amoeboid cell motility

In our model, gel motion arises from active stress-induced gel phase separation and network turnover. To see how this comes about we start with a simplified description of active gel theory, in which spatial dependence occurs only along the x-direction. First, the polymer volume fraction ϕ(x,t), whose value is ϕ₀ in the homogeneous state, satisfies the continuity equation

$$\begin{equation} \partial_t\phi-U\partial_x\phi+\partial_x J_{\mathrm{p}}=-k_{\mathrm{d}}(\phi-\phi_0)\,. \end{equation} \tag{ 1 }$$

In this equation, U(t) and J_p(x,t) are, respectively, the layer velocity and polymer flux with respect to the fixed frame and k_d is the bulk filament depolymerization rate. In our description, filaments undergo uniform bulk polymerization with the rate k_d ϕ₀, assuming that diffusion of free monomers is fast enough that we may consider their concentration to be fixed at its homogeneous, unperturbed value. Our approach is different from those that restrict filament polymerization to one or more boundaries, thereby breaking the symmetry of the problem by hand. We note that the coordinate x is measured with respect to a fixed point in the frame moving with velocity U. Incompressibility of the combined polymer plus solvent system implies that the length of the layer is constant.

Flows involved in cell motility occur at low Reynolds numbers and thus the gel layer is always at mechanical equilibrium. The forces acting on the polymer component of the gel include a force due to stresses in the network; an osmotic force due to composition inhomogeneities; polymer–solvent friction; and polymer-substrate friction. The polymer–solvent friction force is much smaller than the polymer–substrate force: the ratio of the two is η_s/(ξℓ²), where η_s is the solvent viscosity, ξ is the polymer–substrate friction coefficient, and ℓ is the network mesh size. Taking η_s = 10⁻³ Pa s, ξ = 10¹⁶ Pa s m⁻² [8] (for a layer thickness of 1 μm), and ℓ = 50 nm, we find η_s/(ξℓ²) ∼ 10⁻⁴, and thus we may neglect momentum exchange between polymer and solvent. As a result, the balance of forces on the polymer component of the gel gives

$$\begin{equation} \partial_x\left(\sigma^{\rm n}-\Pi\right)=\xi J_{\mathrm{p}}, \end{equation} \tag{ 2 }$$

where σⁿ is the xx-component of the network stress and Π is the osmotic pressure. We note that in equation (2) the polymer–substrate friction force is written as ξJ_p = ξ ϕ v_p, as opposed to the usual form ξ v_p, where v_p is the polymer velocity, to account for the dependence of the friction force on polymer volume fraction.

The network stress, σⁿ, contains passive and active contributions. Here, we consider the liquid limit of the gel, valid on timescales greater than the Maxwell time τ_M = 1 − 10 s. In this limit, the passive part of the network stress, arising from polymer convection and crosslink remodeling, has the viscous form η_p∂_xJ_p, where η_p is the polymer viscosity [5, 30, 31]. However, this contribution does not qualitatively affect the flows and motile behavior of the gel layer and we do not consider it further⁵. The active part, on the other hand, is essential for motility. Since the magnitude of the active stress, arising from motor activity on filaments, increases with ϕ a simple choice for the network stress is⁶

$$\begin{equation} \sigma^{\mathrm{n}}=\zeta\phi. \end{equation} \tag{ 3 }$$

A linear dependence of the active stress on polymer concentration has also been considered in recent work on motile active droplets [22]. We consider positive values of the activity coefficient ζ, corresponding to contractile behavior of active actin networks [32, 33]. Moreover, ζ > 0 can give rise to contractile instabilities and gel phase separation [27, 32]. We also note that the active stress has been interpreted in the context of blebbing and amoeboid motility as a myosin-induced active hydrostatic pressure [18, 34]. Though this active stress results from the isotropic part of the stress tensor [5], it is physically meaningful since the polymer component of the gel is generally compressible.

The osmotic pressure, Π, is formally given by Π = −F/V +ϕ δF/δϕ, where F(ϕ,∂_xϕ) is the ϕ-dependent part of the gel free energy and V is the gel volume [31]. Physically, Π acts to saturate the linear instability causing gel phase separation and to smooth the interface between polymer-rich and polymer-poor regions. Thus, a simple, phenomenological form for Π is

$$\begin{equation} \Pi=\alpha (\phi-\phi_0)^3-\gamma \partial_x^2\phi, \end{equation} \tag{ 4 }$$

where α and γ are positive coefficients⁷. A term linear in ϕ in Π, describing filament diffusion, has been omitted since it can be absorbed into σⁿ. Finally, combining equations (1)–(4) we obtain the equation of motion for ϕ

$$\begin{equation} \partial_t\phi-U\partial_x\phi+\frac{1}{\xi}\partial_x^2(\zeta\phi-\alpha(\phi-\phi_0)^3+\gamma\partial_x^2\phi)=-k_{\mathrm{d}}(\phi-\phi_0). \end{equation} \tag{ 5 }$$

Equation (5) has, minus the convective term −U∂_xϕ, the form of a Cahn–Hilliard equation including a chemical reaction term [35]. This model has, in particular, been used to study spinodal decomposition and steady states of binary solutions with chemical exchange between the components [36, 37]. By including the convective term, we will show how equation (5) predicts inhomogeneous, moving steady-states.

The solution of equation (5) requires the specification of four integration constants, determined from boundary conditions at the layer ends, and the unknown layer velocity U(t). No-flux boundary conditions on the polymer are expressed as J_p(0,t) = Uϕ(0,t) and J_p(L,t) = Uϕ(L,t). These equations imply no-flux of the solvent at the boundaries, as well. In addition, assuming that there are no specific surface energies or dissipation at x = 0 and L implies that ∂_xϕ(0,t) = 0 and ∂_xϕ(L,t) = 0 [38].⁸ Finally, U(t) can be found by requiring that the gel layer plus the end pistons are at mechanical equilibrium, namely

$$\begin{equation} \xi_{\mathrm{l}} U+\xi\int_0^L J_{\mathrm{p}}\,\mathrm{d}x=0, \end{equation} \tag{ 6 }$$

where ξ_l is the friction coefficient between the piston loads and the surface.

In this section, we demonstrate that an instability of the homogeneous, stationary gel layer gives rise to gel phase separation and layer motion. We will focus on the onset of unstable states, depending on the activity parameter ζ, the depolymerization rate k_d, and the drag coefficient ξ_l. First, a straightforward linear stability analysis can be performed in the limit ξ_l → ∞, i.e. for fixed pistons, in which case the eigenmodes ϕ(x,t) − ϕ₀ ∼ e^{λ∞ _nt}cos(nπx/L), n = 1,2,3,... , substituted into equation (5), yield the dispersion relation

$$\begin{equation} \lambda^{\infty}_n=-k_{\mathrm{d}}+\frac{1}{\xi}\left(\frac{n\pi}{L}\right)^2\left(\zeta-\left(\frac{n\pi}{L}\right)^2\gamma\right)\,. \end{equation} \tag{ 7 }$$

Equation (7) reveals that activity drives the linear instability of the gel; the mixing effect of filament depolymerization suppresses long wavelength modes; and the smoothing term −γ ∂²_xϕ cuts off short wavelength modes.

The stability analysis for the case of moving pistons is more involved, as the third spatial derivative of the eigenmodes no longer vanishes at the layer ends. To highlight the role of ζ, k_d, and ξ_l, we introduce the dimensionless quantities $\overline {\zeta }=\zeta L^2/\gamma$, $\overline {k}_{\mathrm {d}}=k_{\mathrm {d}}\xi L^4/\gamma$, $\overline {\xi }_{\mathrm {l}}=\xi _{\mathrm {l}}/(\xi L)$, $\overline {\alpha }=\alpha L^2/\gamma$, $\overline {J}_{\mathrm {p}}=J_{\mathrm {p}}\xi L^3 /\gamma$, and $\overline {U}=U \xi L^3/\gamma$. Furthermore, we non-dimensionalize x by L and t by ξL⁴/γ, keeping the old variable names for the new ones. As a result, the dimensionless equation of motion for ϕ is

$$\begin{equation} \partial_t\phi-\overline{U}\partial_x\phi+\partial_x^2(\overline{\zeta}\phi-\overline{\alpha}(\phi-\phi_0)^3+\partial_x^2\phi)=-\overline{k}_{\mathrm{d}}(\phi-\phi_0)\,. \end{equation} \tag{ 8 }$$

Writing $\phi (x,t)-\phi _0=\mathrm {e}^{\overline {\lambda }_n t}\phi _n(x)$, inserting this ansatz into equation (8), and keeping only linear terms in ϕ_n yields

$$\begin{equation} \phi_n''''(x)+\overline{\zeta}\phi_n''(x)=-(\overline{k}_{\mathrm{d}}+\overline{\lambda}_n)\phi_n(x)\,, \end{equation} \tag{ 9 }$$

where primes (') denote differentiation with respect to x. Writing the layer velocity as $\overline {U}(t)=\tilde {U}_n \mathrm {e}^{\overline {\lambda }_n t}$, the boundary conditions on the third derivatives of ϕ and the flux $\overline {J}_{\mathrm {p}}$ at x = 0 and 1 read

$$\begin{equation} \phi_n'(0)=0,\quad \phi_n'(1)=0, \end{equation} \tag{ 10a }$$

$$\begin{equation} \phi_n'''(0)=\tilde{U}_n\phi_0,\quad \phi_n'''(1)=\tilde{U}_n\phi_0, \end{equation} \tag{ 10b }$$

$$\begin{equation} \overline{\xi}_{\mathrm{l}} \tilde{U}_n+\overline{\zeta} \phi_n\Big|_0^1+\phi_n''\Big|_0^1=0. \end{equation} \tag{ 11 }$$

The modes with non-zero $\tilde {U}_n$ are antisymmetric about x = 1/2. We thus obtain

$$\begin{equation} \phi_n(x)=A_{n+}\sin{\left[k_{n+}(x-1/2)\right]}+A_{n-}\sin{\left[k_{n-}(x-1/2)\right]}\,, \end{equation} \noindent \tag{ 12 }$$

where A_n± are constants and

$$\begin{equation} k_{n\pm}=\sqrt{\frac{\overline{\zeta}}{2}\pm \frac{1}{2}\sqrt{\overline{\zeta}^2-4(\overline{k}_{\mathrm{d}}+\overline{\lambda}_n)}}\,. \end{equation} \tag{ 13 }$$

The condition for non-trivial values of A_n± is obtained by setting the determinant of the two-by-two matrix obtained from the boundary and force balance conditions, equations (10)–(11), to zero. This results in the characteristic equation

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle k_{n+} \cos{(k_{n+}/2)} \left(\frac{k_{n-}^3\overline{\xi}_{\mathrm{l}}}{\phi_0}\cos{(k_{n-}/2)}-2(\overline{\zeta}^2-k_{n-}^2)\sin{(k_{n-}/2)} \right) \\ \displaystyle - k_{n-} \cos{(k_{n-}/2)} \left(\frac{k_{n+}^3\overline{\xi}_{\mathrm{l}}}{\phi_0}\cos{(k_{n+}/2)}-2(\overline{\zeta}^2-k_{n+}^2)\sin{(k_{n+}/2)} \right) = 0, \end{array} \end{eqnarray} \tag{ 14 }$$

which can be solved numerically for the growth rate $\overline {\lambda }_n$ for given $\overline {\zeta }$, $\overline {k}_{\mathrm {d}}$, and $\overline {\xi }_{\mathrm {l}}$. In the following, we will focus on values of $\overline {\zeta }$ for which n = 1 is the most unstable mode, corresponding, approximately, to $\pi ^2<\overline {\zeta }<9\pi ^2$. However, in principle higher, odd modes are excited for larger $\overline {\zeta }$, though the lowest odd mode is likely the most relevant for cell motility. We note that a finite drag coefficient ξ_l affects the dispersion relation in a non-trivial way, as can be seen from equation (14). Qualitatively, we expect that the smaller the piston drag, the more unstable the gel layer is. Indeed, figure 2 reveals that the growth rate of the first mode, $\overline {\lambda }_1$, decreases as $\overline {\xi }_{\mathrm {l}}$ increases. Interesting, if, in the polymer force balance, the polymer–solvent drag were to dominate the polymer friction with the substrate, i.e. J_p − ϕ U ∼ ∂_x(σⁿ − Π), then $\overline {\lambda }$ would no longer depend on $\overline {\xi }_{\mathrm {l}}$, since no-flux at x = 0,1 would mean that the eigenmodes for ϕ(x,t) − ϕ₀ are always of the form cos(nπx).

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In the regime $\overline {\lambda }_1>0$, a contractile instability of the gel layer appears, which is triggered by active stress, but is counteracted by filament turnover. However, filament turnover is essential for moving steady-states of the gel to exist. Indeed, if $\overline {k}_{\mathrm {d}}=0$, polymer conservation implies that at steady-state $\overline {J}_{\mathrm {p}}(x)=\overline {U}\phi (x)$; substituting this into the global force balance, equation (6), and solving for $\overline {U}$, it follows that the layer speed is zero.

An inhomogeneous polymer density profile and filament turnover give rise to moving states in the regime $\overline {\lambda }_1>0$. The state diagram in figure 3 indicates the region in the $(\overline {\zeta },\,\overline {k}_{\mathrm {d}},\,\overline {\xi }_{\mathrm {l}})$ parameter space in which such moving steady-states exist, bound by the surfaces $\overline {k}_{\mathrm {d}}=0$ and $\overline {\lambda }_1=0$. The moving states are characterized by an inhomogeneous profile ϕ(x) and a non-zero steady state velocity $\overline {U}_{\infty }$, which are obtained by numerically solving the nonlinear equation (8), completed by boundary conditions and global force balance. Figure 1 shows an example of a moving steady-state for non-zero $\overline {k}_{\mathrm {d}}$ that develops after the initial instability of the gel, and in which the layer is enriched in polymer at its rear. The polymer volume fraction, ϕ, is indicated by the blue curve in figure 1. For non-zero $\overline {k}_{\mathrm {d}}$, net filament depolymerization occurs in parts of the layer where ϕ > ϕ₀, whereas net polymerization occurs in regions where ϕ < ϕ₀. As a result, polymer flux is directed from right to left in figure 1; exchange of momentum between the gel and the substrate provides a rightward force and drives motion of the layer to the right.

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The moving states can be characterized by the load force–velocity relationship for the moving layer, described here by the dependence of $\overline {\xi }_{\mathrm {l}}$ on $\overline {U}_{\infty }$; see figure 4. Figures 4(a) and (b) show that $\overline {\xi }_{\mathrm {l}}$ decreases with increasing $\overline {U}_{\infty }$ for, respectively, fixed $\overline {\zeta }$ and varying $\overline {k}_{\mathrm {d}}$ and fixed $\overline {k}_{\mathrm {d}}$ and varying $\overline {\zeta }$. We note that if the dispersion relation is such that $\overline {\lambda }^{\infty }_1>0$ (obtained for $\overline {\xi }_{\mathrm {l}}\to \infty$), then one must have $\overline {\xi }_{\mathrm {l}}\to \infty$ as $\overline {U}_{\infty }\to 0$, since only rigidly fixed pistons can prevent a linearly unstable gel from being set into motion. On the other hand, if $\overline {\lambda }^{\infty }_1<0$ (obtained for $\overline {\xi }_{\mathrm {l}}\to \infty$), moving steady-state exist for $\overline {\xi }_{\mathrm {l}}<\overline {\xi }_{\mathrm {l}}^*$, where $\overline {\xi }_{\mathrm {l}}^*$ depends on $\overline {\zeta }$ and $\overline {k}_{\mathrm {d}}$. In fact, $\overline {\xi }_{\mathrm {l}}^*$ is the load friction for which the slab is marginally stable: $\overline {\lambda }_1(\overline {\zeta },\overline {k}_{\mathrm {d}},\overline {\xi }_{\mathrm {l}}^*)=0$. The state diagram in figure 3 shows that if the depolymerization rate is $\overline {k}_{\mathrm {d}}$ is large enough, the mixing effect of filament turnover suppresses gel phase separation and self-sustained moving states no longer exist. This illustrates the subtle role played by filament turnover in our model: it is necessary for moving states to be sustained, yet if it is too high, cell polarity, needed for motion, is abolished.

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Motion of the gel layer requires a transfer of momentum from the substrate to the gel. The dependence of the steady-state layer speed, U_∞, on the polymer–substrate friction coefficient, ξ, allows a comparison between our model and cell motility experiments that probe the dependence of migration speed on surface adhesion. Figure 5 reveals that there is an optimum value of ξ, which maximizes U_∞. This prediction agrees with biphasic behavior of migration speed as a function of surface adhesion observed in lamellipodium-driven crawling cell motility on flat surfaces [39, 40] and in amoeboid motility of cells confined in microchannels [23]. Our model predicts that the optimum value of ξ shifts to higher values as the activity ζ increases, in agreement with the work of [40]. In the model presented here, the layer speed is small for low ξ because the polymers do not get enough traction from the substrate, and is small again for high ξ, since, according to equation (5), the amplitude of gel phase separation, and hence the layer speed, decreases with decreasing ζ/ξ.

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