Feedback mechanisms in a mechanical model of cell polarization

Abstract

Directed cell migration requires a spatially polarized distribution of polymerized actin. We develop and treat a mechanical model of cell polarization based on polymerization and depolymerization of actin filaments at the two ends of a cell, modulated by forces at either end that are coupled by the cell membrane. We solve this model using both a simulation approach that treats filament nucleation, polymerization, and depolymerization stochastically, and a rate-equation approach based on key properties such as the number of filaments N and the number of polymerized subunits F at either end of the cell. The rate-equation approach agrees closely with the stochastic approach at steady state and, when appropriately generalized, also predicts the dynamic behavior accurately. The calculated transitions from symmetric to polarized states show that polarization is enhanced by a high free-actin concentration, a large pointed-end off-rate, a small barbed-end off-rate, and a small spontaneous nucleation rate. The rate-equation approach allows us to perform a linear-stability analysis to pin down the key interactions that drive the polarization. The polarization is driven by a positive-feedback loop having two interactions. First, an increase in F at one side of the cell lengthens the filaments and thus reduces the decay rate of N (increasing N); second, increasing N enhances F because the force per growing filament tip is reduced. We find that the transitions induced by changing system properties result from supercritical pitchfork bifurcations. The filament lifetime depends strongly on the average filament length, and this effect is crucial for obtaining polarization correctly.

Figure A1

Schematic of two possible filament-length distributions, with the blue line having a larger standard deviation.

Figure A2

Approach to polarized steady state for $k_P^{-}=1.5k_P^{-c}$ (see Table 2). Other parameters have the values given in Tables 1 and 2. Frames a) and b) are for small l₀, while frames c) and d) are for large l₀.

Figure A3

Approach to polarized steady state for $k_B^{-}=0.5k_B^{-c}$ (see Table 2). Other parameters have the values given in Tables 1 and 2. Frames a) and b) are for small l₀; frames c) and d) are for large l₀. We do not plot the rate-equation result for the large l₀ value, because in this case the average filament length in the rate equations becomes less than 0.5l₀ at one end of the cell, so Eq. 22 for the decay rate breaks down.

Figure A4

Approach to polarized steady state for $k_r=0.5k_r^c$ (see Table 2). Other parameters have the values given in Tables 1 and 2. Frames a) and b) are for small l₀, while frames c) and d) are for large l₀.

Figure 1

Schematic of model. Actin filaments are represented by red lines and the two nucleation regions by blue dashed lines. Springs mimicking the mechanical function of the cell membrane along the sides and top of the cell are shown in purple. The schematic is compared with a polarized neutrophil, where red labels polymerized actin and green labels tubulin [29] (https://www.london-nano.com/cleanroom-and-facilities/facilities/confocal-microscopes).

Figure 2

Filament distribution from simulations, for parameters giving a symmetric steady state (frame a), and an polarized steady state (frame b). Color usage is as in Fig. 1.

Figure 3

Dynamics of F and N before reaching symmetric steady states, for $k_P^{-}=0.5k_P^{-c}$ (see Table 2). Other parameters are as in Tables 1 and 2. Frames a) and b) are for small l₀, while frames c) and d) are for large l₀.

Figure 4

Dynamics of F and N before reaching asymmetric steady states, for $G_0=1.5G_0^c$ (see Table 2). Other parameters have the values given in Tables 1 and 2. Frames a and b are for small l₀ and frames c and c are for large l₀.

Figure 5

Dynamics of cell polarization, for different treatments of the filament decay rate k_d. Parameters: $G_0^c=1.5G_0^c$ (see table 2). Other parameters have the values given in Tables 1 and 2. Initial conditions are N₁ = 2, N₂ = 1.

Figure 6

Bifurcation diagram of polarization as function of G₀, $k_P^{-},k_B^{-}$, k_r, using the small l₀ value. The parameters are varied from $0.5G_0^c,0.5k_P^{-c},0.5k_B^{-c},0.5k_r^c$ to $1.5G_0^c,1.5k_P^{-c},1.5k_B^{-c},1.5k_r^c$. In the simulation, each parameter is varied by 5% from dot to dot. In the rate equations, each parameter is varied by 1%, forming a smooth curve.

Figure 7

Schematic of essential feedback loop that leads to polarization.