Rebuilding cytoskeleton roads: Active-transport-induced polarization of cells

Many cellular processes require a polarization axis which generally initially emerges as an inhomogeneous distribution of molecular markers in the cell. We present a simple analytical model of a general mechanism of cell polarization taking into account the positive feedback due to the coupled dynamics of molecular markers and cytoskeleton filaments. We find that the geometry of the organization of cytoskeleton filaments, nucleated on the membrane (e.g., cortical actin) or from a center in the cytoplasm (e.g., microtubule asters), dictates whether the system is capable of spontaneous polarization or polarizes only in response to external asymmetric signals. Our model also captures the main features of recent experiments of cell polarization in two considerably different biological systems, namely, mating budding yeast and neuron growth cones.

Figure 1

(Color online) Model geometry. Filaments nucleated at the (a) cell center and (b) membrane. Arrows indicate the velocity of directed transport. Concentration of molecular markers $\mu$ on the membrane and $c$ in the bulk.

Figure 2

(Color online) Solution $s\left(k\right)$ of Eq. (8). $k$ and $\lambda$ are plotted in dimensionless units normalized by the inverse of the cell radius, $R^{-1}=2\pi /L$. Parameter values used are $k^{\text{on}}=1\mu \text{m}{\text{s}}^{-1}$, $k^{\text{off}}=0.1{\text{s}}^{-1}$, $D_b=0.1\mu {\text{m}}^2{\text{s}}^{-1}$, $D_m=0.01\mu {\text{m}}^2{\text{s}}^{-1}$, and $R=10\mu \text{m}$. The dotted line shows the small $k$ asymptote $s=\left(D_b{k}^{\text{off}}/k^{\text{on}}\right)k$.

Figure 3

(Color online) (a) The most unstable mode $k_{\text{max}}$ as a function of $\lambda =\alpha {\mu }^0\pi /D_b$ (both normalized by $R^{-1}$ as in Fig. 2). Cartoons of the distribution of markers are shown above the curve. (b) $\tau =1/s\left(k_{\text{max}}\right)$ as a function $\lambda$. The calculated solid line is compared to data (points) from [9] for the measured polarization time $t$ against pheromone concentration $c_{\text{ph}}$. We used a scaling of $\lambda =2c_{\text{ph}}$ and $\tau =0.14t$ to obtain the observed fit. Other parameter values used are as in Fig. 2.

Figure 4

(Color online) Density of polarity markers $c\left(x,z\right)$ for driven polarization with $\alpha \left(x\right)$ (a) plotted in the $\left\{x,z\right\}$ model plane with parameters as in Fig. 2 and $\lambda =6$, $k=1$, $M=1000$, ${\alpha }_k/{\alpha }_0=0.2$ and (b) mapped onto a cartoon circular cell. Darker shading represents higher density.