Langevin computer simulations of bacterial protein filaments and the force-generating mechanism during cell division

FtsZ is a bacterial protein that forms filaments that play an essential role in midcell constriction during the process of cell division. The shape of individual filaments of different lengths imaged with atomic force microscopy was modeled considering the protein monomers as beads in a chain and a few parameters to represent their effective interactions. The flexural rigidity and persistence length of the filaments were estimated. This latter value was comparable to the filament length, implying that these biological polymers are halfway between the perfectly stiff linear aggregate whose shapes are fully controlled by the angle between the monomers and highly flexible polymers whose shapes follow a random walk model. The lateral interactions between adjacent filaments, also estimated in the modeling, were found to play an essential role in determining the final shape and kinetics of the coiled structures found in longer polymers. The estimated parameters were used to model the behavior of the polymers also on a cylindrical surface. This analysis points to the formation of helical structures that suggest a mechanism for force generation and amplification through the development of FtsZ spirals at the midcell division point.

Figure 1

AFM images of FtsZ filaments formed in the presence of GDP-${\text{AlF}}_3$. (a) Image of filaments formed shortly after incorporation of the nucleotide. (b) Same filaments imaged 50 min later showing the shape evolution due to filament growth. Scale bar is 600 nm. The filaments marked with the arrows in (a) are those selected to follow their time evolution in Fig. 7.

Figure 2

Time evolution of the FtsZ filaments. The left panel is a portion of the AFM image in Fig. 1a. The central and right panels show the same surface 7 and 13 min later, respectively.

Figure 3

Example of the data process to get bond-bond angle distribution from the AFM images explained in the text. The panel on the left shows a single FtsZ filament from the AFM image in Fig. 1a. The black squares are the AFM pixel positions with height over 3 nm, from the flat mica background. The smooth curve in the central and left panels represents the best fit to a polynomial form in polar coordinates, with an optimized choice of the origin (cross). The monomers are assumed to be at regular distances $r_0=4.5\text{nm}$ along that curve.

Figure 4

Distribution of the bond angles in the open FtsZ filaments in Fig. 1 and its Gaussian fit with the parameter in Table .

Figure 5

Filament coil structure along the LDS, described through its mean radius, for our model chain with the parameters in Table . The two different initial configurations, (a) and (b), evolve to the equilibrium structure (c), which corresponds to small thermal fluctuations around the minimum of the potential energy $U$, as sketched in the inset.

Figure 6

Typical equilibrium configurations of coils formed in our LDS for two values of the lateral attraction parameter: (a) $\beta ϵ=1$; (b) $\beta ϵ=3$. The inner and outer radii are shown in (a).

Figure 7

AFM images of the FtsZ filaments indicated by an arrow in Fig. 1a. Time zero is the first image taken after addition of GDP-${\text{AlF}}_3$. Scale bar is 100 nm.

Figure 8

LDS time evolution of the mean radius for filaments of $N$ monomers, from initially open configurations. The sudden decrease of $⟨R⟩$ observed for $N\ge 120$ corresponds to the transition to coiled structures, as indicated by the sketched forms over the $N=140$ curve. The typical time for that conformational change depends on the filament length, with a minimum value of $t\approx 400t_0$, in terms of the LDS time scale, obtained for $N\approx 140$. For $N\le 80$ the filaments remain in the open configuration for all the sampled time. Notice that the time scale is different in the left- and right-hand columns.

Figure 9

A helical filament formed by $N=3000$ monomers on a cylindrical surface of radius $R_c=470\text{nm}$, with the energy parameters in Table . A detailed view of the region within the dashed line is shown on the right-hand side for a typical equilibrium configuration in the Langevin dynamics simulation. The attractive interactions along the filament would produce a radial force $F_r$ on the cylindrical surface as explained in the text.

Figure 10

Mean energy of helical filaments with $N=3000$ (circles) and $N=4000$ (diamonds) monomers on a cylindrical surface of radius $R_c$, averaged along LD simulations with the energy parameters in Table .