Actin filaments growing against an elastic membrane: Effect of membrane tension

We study the force generation by a set of parallel actin filaments growing against an elastic membrane. The elastic membrane tries to stay flat and any deformation from this flat state, either caused by thermal fluctuations or due to protrusive polymerization force exerted by the filaments, costs energy. We study two lattice models to describe the membrane dynamics. In one case, the energy cost is assumed to be proportional to the absolute magnitude of the height gradient (gradient model) and in the other case it is proportional to the square of the height gradient (Gaussian model). For the gradient model we find that the membrane velocity is a nonmonotonic function of the elastic constant $\mu$ and reaches a peak at $\mu ={\mu }^{*}$. For $\mu <{\mu }^{*}$ the system fails to reach a steady state and the membrane energy keeps increasing with time. For the Gaussian model, the system always reaches a steady state and the membrane velocity decreases monotonically with the elastic constant $\nu$ for all nonzero values of $\nu$. Multiple filaments give rise to protrusions at different regions of the membrane and the elasticity of the membrane induces an effective attraction between the two protrusions in the Gaussian model which causes the protrusions to merge and a single wide protrusion is present in the system. In both the models, the relative time scale between the membrane and filament dynamics plays an important role in deciding whether the shape of elasticity–velocity curve is concave or convex. Our numerical simulations agree reasonably well with our analytical calculations.

Figure 1

Schematic representation of our model. The square blocks show actin monomers, which join together to form rodlike filaments. The thick solid line represents the shape of the elastic membrane. (a) A bulk site of the membrane thermally fluctuates and changes its height by an amount $d$, which in turn changes the membrane contour length by $2d$. The forward process increases the energy and occur with rate $R_{+}$ while the reverse process decreases the energy and happens with rate $R_{-}$. (b) A bulk site of the membrane changes its height by an amount $d$ but the energy remains same and thus the movement happens with rate unity. (c) A bound filament pushes the binding site by an amount $d$ that costs energy and occurs with rate $U_0{R}_{+}$. (d) A free filament polymerizes (depolymerizes) with rate $U_0$ ($W_0$).

Figure 2

$\mu –V$ curve for a single filament. (a) The average velocity of the membrane as a function of $\mu$ shows a peak at ${\mu }^{*}$. While $V\sim 1/L$, the peak position ${\mu }^{*}$ does not depend on $L$. The solid triangle on the $y$ axis marks $V_0=\frac{d}{L}(U_0-W_0)$, which is the expected value of $V$ at $\mu =0$. Inset: For large $\mu$, the membrane velocity decreases exponentially with a decay constant $\simeq 0.67$, which is close to the value of $\beta d$. Here, we have used $L=64$. (b) The variation of average velocity of the binding site ($V_{\mathrm{bind}}$) and the average velocity of a bulk site ($V_{\mathrm{bulk}}$) with $\mu$ shows that $V_{\mathrm{bind}}$ decreases monotonically with $\mu$ while $V_{\mathrm{bulk}}$ increases with $\mu$ for $\mu <{\mu }^{*}$. For $\mu >{\mu }^{*}$, these two velocities are equal. Here we have used $L=16$. For all the above plots, we have used $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.

Figure 3

The membrane contour length keeps growing for $\mu <{\mu }^{*}$ and stabilizes for $\mu >{\mu }^{*}$. (a) The ratio $\lambda =1$ for $\mu <{\mu }^{*}$ and decreases sharply at ${\mu }^{*}$ to its asymptotic value $2/L$ for large $\mu$. Inset: The saturation of $\lambda$ at large $\mu$ happens exponentially. (b) The quantity $\frac{\langle \mathcal{C}\left(t\right)-\mathcal{C}\left(0\right)\rangle }{t}$, for large $t$, decreases with $\mu$ and becomes zero above ${\mu }^{*}$. Inset: Variation of contour length with time at ${\mu }^{*}=1.748pN$. We see that after large enough time, $\mathcal{C}\left(t\right)$ grows with time as $t^{0.5}$ with a diffusion constant $D=37.41\pm 7.19 {\mathrm{nm}}^2$/s. The analytically calculated value of $D=41.352 {\mathrm{nm}}^2$/s which is close to the numerical value. Here we use $L=64$. For both panels $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.

Figure 4

Peak position and peak height of $\mu –V$ curve depends on $\mathcal{S}/L$. (a) Scaled $V$ as a function of $\mu$ for different $\mathcal{S}/L$ values. The scaling factor $V_{\max }$ has been used such that the data for different $\mathcal{S}/L$ values can be compared. In this plot the symbols $\circ$ correspond to $\mathcal{S}/L=2^{-2}$ ($V_{\max }\simeq 5.84\times {10}^{-2}$ nm/s), the $\square$ correspond to $\mathcal{S}/L=2^{-1}$ ($V_{\max }\simeq 5.84\times {10}^{-2}$ nm/s), the $▵$ correspond to $\mathcal{S}/L=1$ ($V_{\max }\simeq 7.89\times {10}^{-2}$ nm/s) and the $\nabla$ correspond to $\mathcal{S}/L=2^6$ ($V_{\max }\simeq 2.23$ nm/s). For smaller values of $\mathcal{S}/L$, the curve is monotonic as the velocity is mostly determined by the binding site. As $\mathcal{S}/L$ increases, the $\mu –V$ curve develops a peak at ${\mu }^{*}$. Inset: ${\mu }^{*}$ decreases with $\mathcal{S}/L$. The discrete points from simulations match well with continuous line from Eq. (6). We use $L=64$ here. (b) The membrane velocity at ${\mu }^{*}$ plotted against $\mathcal{S}/L$ shows a power-law increase. The solid line represents a function $\sim x^{0.87}$ and goes parallel to our numerical data points. Here we have used $L=64$. Other simulation parameters are as in Table 1.

Figure 5

Variation of membrane velocity and contour length against $\mu$ for different values of $\mathcal{S}/L$. (a) The $\mu –V$ curve changes from convex to concave as $\mathcal{S}/L$ is increased. For $\mathcal{S}/L\ll 1$, the curve is convex while for $\mathcal{S}/L\gg 1$, it becomes concave. Here we have scaled $V$ by $V_{\max }$ such that we can compare them in the same scale. The values of $V_{\max }$ are $4.67\times {10}^{-1}$ nm/s, $4.67\times {10}^{-1}$ nm/s, 3.73 nm/s, and 3.73 nm/s, respectively, in the increasing order of $\mathcal{S}/L$. Here we have used $L=8$. (b) Average contour length of the membrane scaled by $Ld$ as a function of $\mu$ for different values of $\mathcal{S}/L$. The continuous line is from analytical calculation in the equilibrium limit, which matches with simulation for very high $\mathcal{S}/L$. Other simulation parameters are as in Table 1.

Figure 6

For $\mu \gg 1$, the membrane behaves as a rigid obstacle. (a) The contact probability $p_0$ for different values of $\mathcal{S}/L$. The continuous lines are from the analytical predictions. For large $\mu$, the value of $p_0$ saturates to $1/2$, which is expected for a rigid barrier. As $\mathcal{S}/L$ decreases, the rigid barrier behavior sets in for smaller $\mu$ values. The continuous lines show analytical calculation in $\mu <{\mu }^{*}$ regime. (b) The variation of $\lambda$ for different values of $\mathcal{S}/L$. For small $\mu$, the value of $\lambda$ is unity, which means that the contour length of the membrane diverges with time. For high-enough value of $\mu , \lambda$ saturates to $2/L$. We use $L=64$ for both the plots. Other simulation parameters are as in Table 1.

Figure 7

$\mu –V$ curve for multiple filaments with $\rho \ll 1$. (a) The velocity scales with $\rho$ similar to the single filament case. It also shows a peak at $\mu ={\mu }^{*}$ which is independent of $\rho$. We have used $\mathcal{S}/L=1$ here. (b) Scaled velocity as a function of $\mu$ for different time scales which show that the curve changes from convex to concave similar to the single filament case. For $\mathcal{S}/L\ll 1$, the curve is convex while for $\mathcal{S}/L\gg 1$, the curve becomes concave. The values of $V_{\max }$ are 0.467 nm/s, 3.71 nm/s, 3.74 nm/s, and 3.75 nm/s for $\mathcal{S}/L=2^{-2},2^5,2^7$, and $2^{10}$, respectively. Here we have used $N=4$ and $L=32$. Other simulation parameters are as in Table 1.

Figure 8

Results for multiple filaments with high filament density. In the main plot, we show $\mu –V$ curve for multiple filaments for $\rho \sim 1$. We note that $V$ does not scale with $\rho$. The peak position ${\mu }^{*}$ decreases with $\rho$. The inset shows the variation of ${\mu }^{*}$ with $\rho$. We note that for $\rho \ll 1$, the value of ${\mu }^{*}$ remains constant and then decreases with $\rho$ for high value of $\rho$. We use $\mathcal{S}/L=1$ here. Other simulation parameters are as in Table 1.

Figure 9

Variation of velocity and contour length with $\nu$ for Gaussian model. (a) The $\nu –V$ curve for single filament shows $1/L$ scaling. The curve is monotonic for all $\nu \ne 0$. Top inset: The velocity falls exponentially with decay constant $\simeq 3.13$, which is close to $2\beta d^2$. Bottom inset: In the limit of very small $\nu$, the velocity saturates to a value $\sim 0.074$ nm/s, that is distinctly different from $V_0=0.058$ nm/s at $\nu =0$. We use $L=64$ for the insets and $\mathcal{S}/L=1$ for all the plots. (b) Average contour length of the membrane scaled by $d$ as a function of $\nu$. For very high value of $\nu , \langle \mathcal{C}\rangle$ becomes equal to the system size $Ld$. Top inset: Contour length decreases exponentially for large $\mu$ with a decay constant $\simeq 3.2$ which is again close to $2\beta d^2$. Bottom inset: For very small $\nu$, contour length falls off with $\nu$ as a power law with an exponent $\simeq 0.9$. For all the above plots, we use $L=64$ and $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.

Figure 10

Contact probability and velocity as a function of $\nu$ in a Gaussian model for different values of $\mathcal{S}/L$. (a) For $\mathcal{S}/L\ll 1, p_0$ does not show much variation and its value remains close to $1/2$. For $\mathcal{S}/L\sim 1$, in the small $\nu$ region, the value of $p_0$ is higher than $1/2$ which then saturates to its rigid barrier limit for high $\nu$. For $\mathcal{S}/L\gg 1, p_0$ starts with very small value and then increases with $\nu$, then saturates to $1/2$. Higher the value of $\mathcal{S}/L$, slower the tendency to reach $1/2$. We use $L=8$ here. (b) The $\nu –V$ curve changes from convex to concave as $\mathcal{S}/L$ is increased. The values of $V_{\max }$ are 0.475 nm/s, 3.50 nm/s, 3.72 nm/s, and 3.73 nm/s for $\mathcal{S}/L=1,2^3,2^9$, and $2^{12}$ respectively. Other simulation parameters are as in Table 1.

Figure 11

Velocity of the barrier as a function of $\nu$ for multiple filaments. There is no qualitative difference between the results of a single filament and the multiple filaments. For large value of $\nu$, the velocity falls exponentially with a decay constant $\simeq 3.18$, which is close to the single filament case. For the inset we use $N=8;L=128$ and for all the plots, we use $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.

Figure 12

Merging of protrusions. We plot the quantity $\frac{(l-k)}{k}$, for the membrane with only two binding sites at a distance $k$ apart, with $l$ being the contour length between the two binding sites. For a given $k$, this quantity decreases with $\nu$ exponentially with a decay constant $\simeq 1.44$, which indicates merging of two protrusions for large $\nu$. Here we use $L=1024$ and $\mathcal{S}/L=1$. Inset: Two protrusions merge as we increase $\nu$. The average height profile of the membrane is shown for different values of $\nu$. Note that the height of the protrusions also become shorter as $\nu$ increases. The $\nu$ values appearing in the legends are in the unit of pN/nm. These data are for $k=256$. We use $L=1024$ and $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.

Figure 13

Height gradient around the binding site. (a) The probability that $z\ge 0$ as a function of $\mu$. We see that the probability is exactly one for $\mu <{\mu }^{*}$ and even for $\mu >{\mu }^{*}$, it is very close to one. (b) Here we show the probability that $h_{b\pm 1}\ge h_{b\pm 2}$ as a function of $\mu$. The red triangles are for $prob(h_{b+1}\ge h_{b+2})$ while the blue circles are for $prob(h_{b-1}\ge h_{b-2})$. These probabilities are also close to one for entire range of $\mu$. Here, $L=64$ and $\mathcal{S}/L=1$. Other simulation parameters are as in Table 1.