On a poroviscoelastic model for cell crawling

Abstract

In this paper a minimal, one-dimensional, two-phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two-phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper-convected Maxwell model and demonstrate that even the simplest of two-phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill–posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling-wave solution in which the crawling velocity has a bell-shaped dependence on adhesion strength, in agreement with biological observation.

Appendices

Appendix A: System reduction by potential hodograph transformation

Potential hodograph transformations have proved useful and informative in understanding this type of system (King and Oliver 2005; Kimpton et al. 2012). We perform a similar analysis here, seeking insight into the nature of the viscoelastic governing Eqs. (39)–(41). We introduce an integrated mass variable,

$$\begin{aligned} s&= \int \limits _0^{\chi (s,t)} \theta (\xi ,t)\,\mathrm {d}\xi ,\end{aligned}$$

(128)

$$\begin{aligned} \chi (0,t)&= 0, \end{aligned}$$

(129)

and transform all the dependent variables

$$\begin{aligned} \theta (\chi (s,t),t)&= \Theta (s,t),\end{aligned}$$

(130)

$$\begin{aligned} u(\chi (s,t),t)&= U(s,t),\end{aligned}$$

(131)

$$\begin{aligned} \sigma (\chi (s,t),t)&= \Lambda (s,t), \end{aligned}$$

(132)

and introduce $\phi$ and $I$ defined

$$\begin{aligned} \phi (s,t)&= \frac{1}{\Theta (s,t)},\end{aligned}$$

(133)

$$\begin{aligned} I(s,t)&= - \phi (s,t)\int \limits _0^s \phi (\eta ,t)J(1/\phi (\eta ,t))\,\text{ d }\eta , \end{aligned}$$

(134)

to obtain, after some manipulation

$$\begin{aligned} \frac{\partial U}{\partial s}&= \frac{\partial \phi }{\partial t} + \frac{\partial I}{\partial s},\end{aligned}$$

(135)

$$\begin{aligned} U&= A(\phi )\frac{\partial \phi }{\partial s} + B(\phi )\frac{\partial }{\partial s}\left( \frac{\Lambda }{\phi }\right) - C(\phi )V,\end{aligned}$$

(136)

$$\begin{aligned} \frac{1}{D\phi }\frac{\partial U}{\partial s}&= \frac{\partial \Lambda }{\partial t} + \frac{I}{\phi }\frac{\partial \Lambda }{\partial s} + \left( \frac{1}{D}-\frac{2}{\phi }\frac{\partial U}{\partial s} \right) \Lambda , \end{aligned}$$

(137)

where $A,\,B$ and $C$ are given by

$$\begin{aligned} A(\phi )&= \frac{\varPsi '(1/\phi )}{\phi ^3\left( \alpha (1/\phi ) + \beta (1/\phi )\right) },\end{aligned}$$

(138)

$$\begin{aligned} B(\phi )&= \frac{1}{\phi \left( \alpha (1/\phi ) + \beta (1/\phi )\right) },\end{aligned}$$

(139)

$$\begin{aligned} C(\phi )&= \frac{\beta (1/\phi )}{\left( \alpha (1/\phi ) + \beta (1/\phi )\right) }. \end{aligned}$$

(140)

It is not clear how (135)–(137) can be combined to eliminate $U$ and $\Lambda$, so we simplify the problem by considering the no polymerization/depolymerization case where $J=0$. This means that $I=0$ and $\chi$ becomes a Lagrangian variable. The transformed viscoelastic stress balance becomes

$$\begin{aligned} \frac{\partial \Lambda }{\partial t} + E\Lambda = F, \end{aligned}$$

(141)

where

$$\begin{aligned} E(s,t)&= \frac{1}{D} - \frac{2}{\phi }\frac{\partial \phi }{\partial t},\end{aligned}$$

(142)

$$\begin{aligned} F(s,t)&= \frac{1}{D\phi }\frac{\partial \phi }{\partial t}, \end{aligned}$$

(143)

which has the solution

$$\begin{aligned} \Lambda = \phi ^2e^{-t/D} \left( \frac{\Lambda ^*}{\phi ^{*2}} + \frac{1}{D}\int \limits _0^t \frac{e^{\xi /D}}{\phi ^3}\frac{\partial \phi }{\partial \xi } \,\mathrm {d}\xi \right) , \end{aligned}$$

(144)

where $\Lambda ^*(x)=\Lambda (x,0)$ is the initial condition for $\Lambda$ obtained from $\sigma ^*$ and $\phi ^*(s)=\phi (s,0)$ is obtained from $\theta ^*$, the initial condition for the network volume fraction. Now combining (135) and (136), along with $I=0$, we see that

$$\begin{aligned} \frac{\partial \phi }{\partial t} = \frac{\partial }{\partial s}\left( A(\phi )\frac{\partial \phi }{\partial s} + B(\phi )\frac{\partial }{\partial s}\left( \frac{\Lambda }{\phi }\right) - C(\phi )V\right) . \end{aligned}$$

(145)

Finally we can substitute for $\Lambda$ using (144) to obtain an integro-partial-differential equation for $\phi$

$$\begin{aligned} \frac{\partial \phi }{\partial t} \!=\! \frac{\partial }{\partial s}\left( A(\phi )\frac{\partial \phi }{\partial s} \!+\! B(\phi )\frac{\partial }{\partial s}\left( \phi e^{-t/D} \left( \frac{\Lambda ^*}{\phi ^{*2}} \!+\! \frac{1}{D}\int \limits _0^t \frac{e^{\xi /D}}{\phi ^3}\frac{\partial \phi }{\partial \xi } \,\mathrm {d}\xi \right) \right) \!-\! C(\phi )V\right) ,\nonumber \\ \end{aligned}$$

(146)

which depends on one extra unknown, $V(t)$, which is determined by a transformed version of the integral constraint (44).

When the network volume fraction is in the contractile regime $A<0$, conversely $A>0$ when the network is in the swelling regime. Therefore, the potential hodograph transform is informative in identifying that contraction of the network can be viewed as a destabilising, backwards diffusion term which drives the cell motion. We see explicitly that the viscoelastic network rheology introduces a term which is nonlocal in time. We also see that the timescale over which the network’s flow history effects the current network configuration decreases as $D\rightarrow 0$.

Appendix B: Convergence graphs

In order to ensure that our code converges as the spatial and temporal discretization is refined we solve the initial value problem (39)–(50) on a sequence of grids. We first fix the time step at $10^{-3}$ and double the number of space elements successively from 200 to 64,000. The maximum absolute difference between the crawling velocity of the strip, $V$, calculated on successive grids is plotted against the size of the spatial element $\varDelta x$ in Fig. 10a. We see that the difference decreases as the spatial grid is refined, so the code converges as the number of spatial elements increases. The spatial convergence of the numerical approximation to $V$ given by the code is order 3/2. Temporal convergence is similarly demonstrated in Fig. 10b by plotting the difference in crawling velocity of the strip as a function of step size. The temporal convergence of the numerical approximation to $V$ given by the code was shown to be first order by solving the same problem on a grid with $800$ space elements for a range of time steps doubling from $1.25^{-4}$ to $4\times 10^{-3}$. Full details of the initial conditions and parameters used are given in the caption.

Fig. 10

a The rate of spatial convergence for initial conditions $\theta ^* = \theta _E + 10^{-4}\cos (\pi x),\,\sigma ^* = 10^{-4}\cos (\pi x)$ and the parameter values $\theta _L = 0.01,\,\theta _E = 0.02,\,\theta _R = 0.03,\,\varPsi ^* = 15{,}000,\,\alpha ^* = 0.4,\,\beta ^* = 3$ and $D = 0.1$. b The rate of temporal convergence for the viscoelastic code with the same initial conditions and parameters as in a except that here $\varPsi ^*=12{,}000$

Appendix C: Myosin driven network contraction

Some parameter choices in the viscoelastic formulation lead to an ill-posed system. A key feature of cell motility that our model currently neglects is the distribution and transport of myosin. Myosin can bind to and move along actin filaments, it can also detach from the network and be transported through the cytosol. A number of modellers have produced models for cell motility that track the distribution and binding of myosin, see e.g. (Rubinstein et al. 2009) who implement an advection–reaction equation for bound myosin density and an advection–reaction–diffusion equation for unbound myosin density. The unbound, free myosin is assumed to be spatially uniform and a reaction–advection equation is used for the bound myosin concentration in Larripa and Mogilner (2006), Mogilner et al. (2001) and Wolgemuth et al. (2011). Alt et al. (2010) present equations to model the transport of bound myosin with the actin network, advection and diffusion of free myosin and binding/unbinding kinetics within the two-phase flow framework. Their calculations are done in the limit where their is no advection of bound myosin and both diffusion of free myosin and binding/unbinding kinetics are sufficiently fast that free myosin is spatially uniform and the binding dynamics are at equilibrium.

Here we consider a simple scenario in which myosin is advected with the cell’s actin network. Our governing equations are given in (39)–(44) along with

$$\begin{aligned} \frac{\partial }{\partial t}(m\theta ) + \frac{\partial }{\partial x}(m\theta u)=0, \end{aligned}$$

where $m$ is the non-dimensional concentration of bound myosin per unit volume of actin network. An initial condition is required: $m(x,0)=m^*(x)$, for $0<x<1$. The aim of this section is not to develop a sophisticated model for myosin transport; that will be the topic of a future paper. Instead we demonstrate that allowing myosin concentration as opposed to network volume fraction drive the network contraction is enough to eliminate the ill-posedness seen in some parameter regimes. We replace (48) with a constitutive law that depends on both the network volume fraction and myosin concentration, a candidate constitutive law taken from Alt et al. (2010) would be

$$\begin{aligned} \varPsi = -S^*ln(1-\theta )-C^*\theta m, \end{aligned}$$

(147)

where $S^*$ is a nondimensional swelling strength and $C^*$ is a nondimensional contractile strength, chosen so that $\varPsi _{\theta }(\theta _E,M)>0$ and $\varPsi _{m}(\theta _E,M)<0$, where $M$ is the spatially uniform equilibrium concentration of myosin.

To see that coupling our viscoelastic model to this simple model for myosin transport eliminates the ill-posed behaviour seen in some parameter regimes for our viscoelastic model, we consider a linear stability analysis on an unbounded domain. We consider the perturbations $\theta =\theta _E+\varepsilon \theta _1+\cdots ,\,\sigma =\varepsilon \sigma _1+\cdots ,\,m=M+\varepsilon m_1+\cdots ,\,u=\varepsilon u_1+\cdots$ and $V=\varepsilon V_1+\cdots$ and obtain at leading order

$$\begin{aligned} \frac{\partial \theta _1}{\partial t}+\theta _E\frac{\partial u_1}{\partial x}&= -\theta _1,\\ -\varPsi _{\theta E}\frac{\partial \theta _1}{\partial x}-\varPsi _{mE}\frac{\partial m_1}{\partial x}+\theta _E\frac{\partial \sigma _1}{\partial x}&= \alpha _E u_1 + \beta _E(u_1+V_1),\\ D\frac{\partial \sigma _1}{\partial t}+\sigma _1&= \frac{\partial u_1}{\partial x},\\ M\frac{\partial \theta _1}{\partial t}+\theta _E\frac{\partial m_1}{\partial t} + \theta _EM\frac{\partial u_1}{\partial x}&= 0, \end{aligned}$$

where a subscript $E$ indicates a function is evaluated at the steady state. After some manipulation we can eliminate the variables in favour of $\theta _1$, which is governed by

$$\begin{aligned}&\left( D\frac{\partial }{\partial t} +1 \right) \left( (\alpha _E+\beta _E)\left( \frac{\partial ^2\theta _1}{\partial t^2}+\frac{\partial \theta _1}{\partial t}\right) - \varPsi _{\theta E}\theta _E\frac{\partial ^3\theta _1}{\partial x^2t} - \varPsi _{mE}M\frac{\partial ^2\theta _1}{\partial x^2} \right) \\&\qquad \qquad \qquad \qquad =\theta _E\frac{\partial ^2}{\partial x^2} \left( \frac{\partial ^2\theta _1}{\partial t^2}+\frac{\partial \theta _1}{\partial t}\right) . \end{aligned}$$

Now we consider a perturbation to the steady state, $\theta _1\propto \exp (\omega t + ikx)$ and obtain the following cubic equation for the growth rate of a perturbation as a function of the wavenumber of the perturbation

$$\begin{aligned}&D(\alpha _E+\beta _E)\omega ^3 + \left[ (\alpha _E+\beta _E)(1+D)+(1+D\varPsi _{\theta E}\theta _E)k^2\right] \omega ^2\\&\qquad + \left[ \alpha _E+\beta _E+(1+DM\varPsi _{mE}+\varPsi _{\theta E}\theta _E)k^2\right] \omega + M\varPsi _{mE}k^2 =0. \end{aligned}$$

Seeking dominant balances in the large $k$ limit reveals that one root decays, with

$$\begin{aligned} \omega \sim -\frac{1+D\varPsi _{\theta E}\theta _E}{D(\alpha _E+\beta _E)}k^2, \end{aligned}$$

while the two remaining roots tend to constants, $\omega _{\infty }$, satisfying

$$\begin{aligned} (1+D\varPsi _{\theta E}\theta _E)\omega _{\infty }^2 + (1+DM\varPsi _{mE} +\varPsi _{\theta E}\theta _E)\omega _{\infty } + M\varPsi _{mE} = 0. \end{aligned}$$

Labelling the coefficients according to $a\omega ^2+b\omega +c=0$, we note that $a>0$ and $c<0$ so the root of the discriminant, $b^2-4ac$, is bigger than $b$ in modulus and regardless of the sign on $b$ there is always one positive real root and one negative real root. Therefore, coupling a simple advection model for myosin to our viscoelastic system allows us to drive network contraction through a slightly different physical mechanism with $\varPsi _{\theta E}>0,\,\varPsi _{mE}<0$. This analysis demonstrates that on an unbounded domain the resulting system of equations will have a bounded largest growth rate that plateaus to a constant for large wavenumber perturbations, so that the governing equations are always well-posed.