Confinement-dependent localization of diffusing aggregates in cellular geometries

Confinement has a strong influence on diffusing nano-sized clusters. In particular, biomolecular aggregates within the shell-like confining space of a bacterial cell have been shown to display a variety of localization patterns, from being midcell to the poles. How does the confining space determine where the aggregate will localize? Here, using Monte Carlo simulations we have calculated the equilibrium spatial distribution of fixed-sized clusters diffusing in spherocylindrical shells. We find that localization to the poles depends strongly on shell thickness and the size of the cluster. Compared to being at midcell, polar clusters can be more bent and hence have higher energy, but they also can have a greater number of defects and hence have more entropy. Under certain conditions this can lead to polar clusters having a lower free energy than at midcell, favoring localization to the poles. Our findings suggest possible localization selection mechanisms within shell-like geometries that can arise purely from cluster confinement.

Figure 1

(a) Schematic of a cluster of $N$ attractive particles diffusing in thin shell with thickness $h$. The shell geometry is a cylindrical shell capped with two hemispheres. The cylindrical region has a length $L$, the inner boundary has radius $R_I$, and the outer one radius $R$. The thickness of the shell is given by $h=R-R_I$. In all simulations we consider the aspect ratio to be $(2R+L)/2R=2$. Shown is a cluster positioned at two locations: the midcell region and at one of the poles (for $N=20$ from a 2D simulation). (b) A cluster with $N=80$ at midcell for $h/\sigma =1.25$ showing a highly regular hexagonal close packed structure. (c) A polar cluster with $N=80$ for the same shell thickness $h/\sigma =1.25$ showing the existence of defects (open circles) that reduce the number of nearest neighbors to fewer than six.

Figure 2

Calculated fractional residency as a function of shell thickness, $h/\sigma$. (a) Dependence of fractional residency on the cluster size, $N$, for a 2D shell geometry with $R=8\sigma$ and $L=16\sigma$. (b) Dependence on cluster size $N$ for a 3D shell, with $R=8\sigma$ and $L=16\sigma$. (c) Effect of shell curvature (changing the outer radius) on fractional residency for a fixed-size cluster with $N=80$ in 3D shells. (Error bars were estimated by halving the number of samples and calculating the change in $f_p/f_m$.)

Figure 3

Dependence of average energy difference (in units of $k_{B}T$) between polar and midcell clusters, $E_p-E_m$, on (a) cluster size for 2D shells with $R=8\sigma$, and for 3D shells where (b) $N$ is varied with shell curvature fixed or (c) curvature is varied with $N$ fixed. The number of particles or the size of the outer radius is shown in the legends. (Error bars were estimated by halving the data and calculating the fluctuation in $E_p-E_m$.)

Figure 4

Energetics (all energy is in units of $k_{B}T$) of polar and midcell clusters with $N=90$. (a) Energy per particle of polar and midcell clusters versus shell thickness in a shell with $R=8\sigma$. (b) Fitted bending energy cost, $A_B$ (in units of $k_{B}T/{\sigma }^2$) for polar and midcell clusters versus shell thickness. (c) Difference in fitted cluster energy per particle, ${\epsilon }_C$, between pole and midcell versus shell thickness. Error bars on fitted parameters shown in (b) and (c) were determined using the bootstrap method. (d) The dependence of bending energy per particle, ${\epsilon }_B$ as a function of $1/R^2$ (in units of $1/{\sigma }^2$) at different shell thicknesses.

Figure 5

(a) Ratio of the number of cluster configurations at the pole to midcell $w_p/w_m$ as a function of shell thickness, $h/\sigma$. (b) The average number of particles in a cluster, $N_{\mu }(r/\sigma )$ within a fixed-size sphere with radius $r/\sigma$ at different shell thicknesses (here $\mu =m$, midcell). (c) The average difference in the number of nearest neighbors $\Delta N$ between midcell and polar clusters. Positive $\Delta N$ correspond to midcell clusters having on average a higher number of nearest neighbors and is a measure of the number of defects that exist within the polar cluster. In all figures, the cluster size was $N=80$ and $R=8\sigma$ unless otherwise specified.