Entropy-driven spatial organization of highly confined polymers: lessons for the bacterial chromosome

Abstract

Despite recent progress in visualization experiments, the mechanism underlying chromosome segregation in bacteria still remains elusive. Here we address a basic physical issue associated with bacterial chromosome segregation, namely the spatial organization of highly confined, self-avoiding polymers (of nontrivial topology) in a rod-shaped cell-like geometry. Through computer simulations, we present evidence that, under strong confinement conditions, topologically distinct domains of a polymer complex effectively repel each other to maximize their conformational entropy, suggesting that duplicated circular chromosomes could partition spontaneously. This mechanism not only is able to account for the spatial separation per se but also captures the major features of the spatiotemporal organization of the duplicating chromosomes observed in Escherichia coli and Caulobacter crescentus.

Fig. 1.

Blob approach to segregation of two partly intermingled chains in a cylindrical pore. R_‖ ∼ ND^1−(1/ν) is the longitudinal size of a single chain. The free energy of overlap F_blob scales as k_BTR_overlap/D.

Fig. 2.

Segregation of two chains of various topology in a rod-shaped box of length L_tot (normalized center-to-center distance between two chains vs. Monte Carlo sweeps). (A) Linear. (B) Ring. (C) Branched. In all three cases, we used a box of the same dimensions and repeated each simulation for 10 different initial configurations. The aspect ratio of each graph has been chosen such that the “timing” of the segregation can be compared.

Fig. 3.

Chains of nontrivial topology and their typical conformations in strong confinement. (A) A schematic representation of the slowly evolving topology of circular chromosome during replication. (Note: not to be confused with actual polymer conformations.) (B) Results of two stages of the replication process, indicated by the vertical gray bar in A, namely, the asymmetric and symmetric figure-θ conformations, respectively. For each case, a snapshot of a typical conformation is shown (top of each image) and the spatial distribution of the midpoints of the colored segments (blue, red, and gray) around their average position (bottom of each image). (C) The conformation of a figure-eight chain consisting of a total of 2,047 beads (>120 times the width D of the box) in a rod-shaped box of aspect ratio 6. Note the mirror-like, linear ordering of the chain within the rod-shaped geometry, although each bead shows a positional fluctuation comparable with the size of the confinement width D.

Fig. 4.

The concentric-shell model of replicating nucleoid, which we employ in the simulation of DNA replication.

Fig. 5.

Chromosome segregation in E. coli, comparing simulation vs. experiment. (Left) A series of typical conformations of a replicating circular chain (mother strand in gray, two daughter strands in red and blue). We also present two sets of segregation pathways (ori-ter trajectories during replication) in the presence (Left) and absence (Right) of a replication factory. See the schematic diagrams of the chain topology; the black pentagons represent the replisomes. The dotted lines show the results of 10 individual simulations, and the solid lines show average trajectories. (Center) We juxtapose the simulations with the published data in Bates and Kleckner (7) in an attempt to capture the main features of the experimental observations (note that, to show the average cell growth, we have scaled back the normalized cell lengths presented in ref. , using their data). For comparison, we used the fraction-replicated f as our “universal clock”.

Fig. 6.

Chromosome segregation of C. crescentus, comparing simulation (Left) vs. experiment (3) (Right). The simulated trajectories are the average of 26 individual simulation runs (or “cells”); we show the trajectories of nine representative loci (including ori and ter) on the right-arc of a circular chromosome for the entire duration of replication (up to 99.9%), whereas experimental data are only available for trajectories up to 50% of replication. For clarity, we only show the trajectories from the onset of replication of each locus. A full trajectory of ter is shown, however, to emphasize its slow drift from the cell pole to the cell center during replication, in contrast to the fast, directed diffusion of ori2 in the nucleoid periphery (on the other hand, we kept ori1 in the volume near the stalked pole until 10–20% of the chain has been replicated). The final spatial organization after the completion of replication bears an interesting resemblance to the mirror-like symmetry of the figure-eight chain (see Fig. 3).