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PLoS Comput Biol. Sep 2011; 7(9): e1002145.
Published online Sep 22, 2011. doi:  10.1371/journal.pcbi.1002145
PMCID: PMC3178632
Filament Depolymerization Can Explain Chromosome Pulling during Bacterial Mitosis
Edward J. Banigan,1 Michael A. Gelbart,2,3 Zemer Gitai,4 Ned S. Wingreen,4* and Andrea J. Liu1*
1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
2Graduate Program in Biophysics, Harvard University, Boston, Massachusetts, United States of America
3Department of Physics, Princeton University, Princeton, New Jersey, United States of America
4Department of Molecular Biology, Princeton University, Princeton, New Jersey, United States of America
Dennis R. Livesay, Editor
UNC Charlotte, United States of America
* E-mail: ajliu/at/physics.upenn.edu (AJL); wingreen/at/princeton.edu (NSW)
Conceived and designed the experiments: EJB ZG NSW AJL. Performed the experiments: EJB MAG. Analyzed the data: EJB. Wrote the paper: EJB NSW AJL.
Received April 23, 2011; Accepted June 20, 2011.
Chromosome segregation is fundamental to all cells, but the force-generating mechanisms underlying chromosome translocation in bacteria remain mysterious. Caulobacter crescentus utilizes a depolymerization-driven process in which a ParA protein structure elongates from the new cell pole, binds to a ParB-decorated chromosome, and then retracts via disassembly, pulling the chromosome across the cell. This poses the question of how a depolymerizing structure can robustly pull the chromosome that disassembles it. We perform Brownian dynamics simulations with a simple, physically consistent model of the ParABS system. The simulations suggest that the mechanism of translocation is “self-diffusiophoretic”: by disassembling ParA, ParB generates a ParA concentration gradient so that the ParA concentration is higher in front of the chromosome than behind it. Since the chromosome is attracted to ParA via ParB, it moves up the ParA gradient and across the cell. We find that translocation is most robust when ParB binds side-on to ParA filaments. In this case, robust translocation occurs over a wide parameter range and is controlled by a single dimensionless quantity: the product of the rate of ParA disassembly and a characteristic relaxation time of the chromosome. This time scale measures the time it takes for the chromosome to recover its average shape after it is has been pulled. Our results suggest explanations for observed phenomena such as segregation failure, filament-length-dependent translocation velocity, and chromosomal compaction.
Reliable chromosome segregation is crucial to all dividing cells. In some bacteria, segregation has been found to occur in a rather counterintuitive way: the chromosome attaches to a filament bundle and erodes it by causing depolymerization of the filaments. Moreover, unlike eukaryotic cells, bacteria do not use molecular motors and/or macromolecular tethers to position their chromosomes. This raises the general question of how depolymerizing filaments alone can continuously and robustly pull cargo as the filaments themselves are falling apart. In this work, we introduce the first quantitative physical model for depolymerization-driven translocation in a many-filament system. Our simulations of this model suggest a novel underlying mechanism for robust translocation, namely self-diffusiophoresis, motion of an object in a self-generated concentration gradient in a viscous environment. In this case, the cargo generates and sustains a concentration gradient of filaments by inducing them to depolymerize. We demonstrate that our model agrees well with existing experimental observations such as segregation failure, filament-length-dependent translocation velocity, and chromosomal compaction. In addition, we make several predictions–including predictions for the specific modes by which the chromosome binds to the filament structure and triggers its disassembly–that can be tested experimentally.
Several processes involved in DNA partitioning rely on depolymerization of filaments for translocation. In eukaryotes, depolymerizing microtubules [1] position chromosomes before cell division via macromolecular couplers and/or molecular motors bound to the microtubules [2], [3]. In prokaryotes, however, no such coupler or motor has been identified. Instead, proteins bound to the chromosome or plasmid bind directly to filaments and trigger their depolymerization [4], [5]. This poses the question of whether in the absence of a coupler, DNA can be pulled in a robust fashion, without becoming detached from the filaments as they disassemble.
Type I low-copy-number-plasmids [6], [7], chromosome I of Vibrio cholerae [8], and the chromosome of Caulobacter crescentus [9][12] all share a common segregation mechanism that relies on pulling mediated by filament depolymerization. This conserved system relies on three central components: the ATPase ParA, the DNA-binding protein ParB, and a centromere-like DNA locus. ParA is a deviant Walker-type ATPase that upon binding ATP forms dimers that can polymerize and associate with DNA [10], [13]. ParB interacts with ParA directly and stimulates ATP hydrolysis, causing ParA to dissociate into free monomers [13]. The spatial and temporal organization of ParA and the ParB-binding parS chromosomal locus can lead to robust chromosome segregation in vivo. For example, in C. crescentus, the chromosomal origin (ori) is initially localized at a single cell pole (the “stalked” pole) [14], and must translocate to the opposite “swarmer” pole before cell division. In predivisional cells, approximately one thousand ParB are bound via parS near the origin of the chromosome (ori) [9], [15]. There appear to be several distinct stages of ParB-parS-ori complex translocation [11]; our focus is on the final, most rapid stage in which the complex binds to filaments of ParA and translates from partway across the cell to the swarmer pole at a velocity of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e001.jpg [9], [11], [16], [17]. As the ParA bundle depolymerizes, presumably due to ParB-induced ATP hydrolysis or nucleotide exchange [7][11], [13], [15], [18], [19], the ParB-parS-ori complex remains localized near the edge of the ParA structure [8], [10][12].
For eukaryotic chromosome segregation driven by depolymerization of microtubules [2], [3], models generally assume the existence of a “coupler” that attaches the chromosome to the depolymerizing microtubules. This coupler moves along the microtubule ahead of the depolymerizing end, either because it slides along it diffusively [20][24], because it is pushed by conformational changes near the tip of the microtubule [23][26], or because it has a complex internal structure that causes it to process [3], [27]. Of the existing models of bacterial chromosome segregation [28][33], only a few address the question of how depolymerizing proteins can cause translocation. Typically, these models attempt to explain ParAB partitioning systems with reaction-diffusion models or general thermodynamic arguments, but do not address the conditions required for robust translocation [31], [32].
Here we ask whether depolymerization of ParA by ParB without a coupler is sufficient to explain the observed translocation in prokaryotic DNA partitioning. We performed Brownian dynamics simulations that explicitly incorporate the biochemistry of the primary constituents of the ParABS segregation system. In our simulations, a polymer representing the ParB-parS-ori complex (henceforth referred to as the “ParB polymer”), binds to a filamentous ParA bundle and initiates disassembly of ParA. We find that the ParB polymer can indeed exhibit robust, depolymerization-driven translocation via a novel mechanism (Fig. 1), provided certain conditions are met.
Figure 1
Figure 1
Schematic model for chromosome segregation and simulation snapshots.
Simulating ParB translocation
To understand the mechanism by which ParA translocates ParB, we performed Brownian dynamics simulations of a ParB polymer interacting with an anchored ParA filament bundle (Fig. 1c). The ParB polymer, shown in Figs. 1b–c, corresponds to the ParB-parS-ori complex. It is represented by a semi-flexible chain of monomeric subunits, typically of length 100 subunits. The center section (dark green in Fig. 1b), typically of length 50 subunits, represents the part of the chromosome that binds to ParA via ParB, while the two peripheral segments (light green in Fig. 1b only) cannot bind to ParA.
During robust translocation, the ParB polymer remains localized near the tip of the ParA bundle and moves across the cell (see snapshots in Fig. 1c and Video S1). By inducing disassembly, ParB creates a concentration gradient of ParA filaments that remains fixed with respect to the center of mass of the ParB polymer. Thus, the ParA concentration profile translocates with the ParB, and exhibits only small, short-lived fluctuations around a well-defined steady-state mean (Fig. 1c).
Translocation is most robust when ParB binds side-on to ParA
Since the precise nature of the ParB–ParA interaction is unknown, we used our simulations to identify the modes of binding and disassembly that provide robust translocation. In our model (see Methods), ParB binds to ParA subunits in the filament bundle (Fig. 1c). The ParB polymer hydrolyzes ParA subunits that it binds to; once a subunit at the tip of a ParA filament is hydrolyzed, it can depolymerize from the filament. Monomers rapidly diffuse away once they have depolymerized. Some interaction/disassembly mechanisms or parameter ranges lead to robust translocation of the ParB polymer, while others lead to failure by rapid detachment:
Tip-only binding
In this model, ParB binds only to the tips of ParA filaments (Fig. 2a). Since the number of ParA filament tips is limited, the ParB polymer is held only weakly to the ParA bundle, and small fluctuations can cause it to detach (Fig. 2a). In principle, this failure mode could be suppressed by increasing the number of ParA filaments within the bundle, but translocation is intrinsically fragile for this model.
Figure 2
Figure 2
Mean time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e002.jpg, to first detachment of ParB polymer from ParA for various failure modes.
Side-binding with filament severing
As an alternative, we allow ParB to bind to the sides of the ParA filaments (Fig. 1b). In this model, ParA filaments can disassemble by severing in addition to disassembling from the filament tips (Fig. 2b). Severing may occur at the location of any ParA subunit that has been hydrolyzed by ParB. Typically, we find that the ParB polymer binds to multiple severed ParA segments, preventing ParB from binding to the remaining filaments in the ParA bundle (Video S2). As a result, the ParB polymer rapidly detaches from the anchored bundle. ParA severing therefore does not lead to reliable ParB translocation.
Side-binding with tip-only disassembly
In this model, ParB binds side-on to ParA filaments (Figs. 1b–c) and ParA disassembles only at the filament tips. In this case, the ParB polymer translocates across the cell without detaching from the ParA bundle for a wide range of parameters. However, under certain extreme conditions, translocation fails:
Weak binding
If the ParB–ParA binding energy, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e005.jpg, is too small, ParB quickly detaches from ParA due to thermal noise and the force from the rest of the ParB polymer (Fig. 2c).
Fast hydrolysis and depolymerization
Rapid detachment occurs if the ParA hydrolysis rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e006.jpg, and ParA depolymerization rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e007.jpg, are both too large (Fig. 2d).
Our major result is that translocation is most robust in the side-binding model with disassembly only from the tips of ParA filaments. The rest of our simulations use this robust mode of disassembly and translocation, and henceforth, we refer to side-binding with tip-only disassembly as our standard model.
The rate of disassembly controls the ParB translocation velocity
To understand how ParA translocates ParB, we identified variables controlling the translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e008.jpg. In all cases, we find that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e009.jpg is given by the mean rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e010.jpg, of disassembly of a ParA filament, so that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e011.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e012.jpg is the length of a ParA subunit. In order for a subunit to disassemble from the tip of a ParA filament, the subunit must bind to ParB, its ATP must hydrolyze, and the subunit must fall off. An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e013.jpg therefore depends on the distance, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e014.jpg, that the ParB polymer typically penetrates into the ParA bundle and causes ParA-ATP hydrolysis, the rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e015.jpg, of ParA-ATP hydrolysis, and the rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e016.jpg, at which a ParA subunit depolymerizes once hydrolyzed.
In turn, the penetration length, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e017.jpg, depends on the shape of the ParB polymer. In our simulations, the freely diffusing ParB polymer adopts an isotropic, globular equilibrium shape. The maximum value, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e018.jpg, of the penetration length, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e019.jpg, is achieved if the ParB polymer is able to maintain this equilibrium shape as it is pulled by ParA. If the disassembly rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e020.jpg, is too high, the ParB polymer is pulled along so rapidly that it does not have time to relax to its equilibrium shape. In this case, the ParA bundle pulls the leading region of the ParB polymer faster than the rear of the polymer can respond to the perturbation and the ParB polymer stretches out. Because the part of ParB polymer does not keep pace with the retraction of the depolymerizing ParA bundle, the ParB polymer does penetrate as deeply into the ParA bundle, so An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e021.jpg.
We now estimate the time for the ParB polymer to relax to its equilibrium size. In our simulations, since ParB decorates the center section of the polymer and binds to ParA, the undecorated peripheral segments of the chain are the first ones to stretch out when the ParB polymer is pulled too rapidly (Video S3). The stretching of the peripheral segments is governed by the equation:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e022.jpg
(1)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e023.jpg is the ensemble-averaged An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e024.jpg-distance between the ends of a peripheral segment pulled by one end in the An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e025.jpg-direction, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e026.jpg is the diffusion coefficient of the segment, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e027.jpg is the An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e028.jpg-component of its equilibrium radius of gyration, and the relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e029.jpg, is the ratio of its internal drag, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e030.jpg, to the effective spring constant, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e031.jpg (see Text S1). Stretching is appreciable if An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e032.jpg, so for translocation in steady state (An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e033.jpg), stretching becomes appreciable for An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e034.jpg, or, equivalently, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e035.jpg (inset to Fig. 3a). The shape of the pulled ParB polymer is therefore governed by the product An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e036.jpg, where we have defined
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e037.jpg
(2)
Figure 3
Figure 3
Dependence of translocation velocity on disassembly rate and relaxation time.
The penetration length, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e081.jpg, depends directly on the shape of the ParB polymer. For large An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e082.jpg the ParB polymer is pulled rapidly and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e083.jpg is small. This is because the ParB polymer is pulled away from the ParA bundle, leading to less overlap of the volume of the ParB polymer with the volume of the ParA bundle. As a result, there is less binding between individual ParB subunits with ParA subunits. As An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e084.jpg decreases, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e085.jpg increases and saturates at An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e086.jpg for An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e087.jpg (inset to Fig. 3b). In the latter regime, the disassembly rate is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e088.jpg, where
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e089.jpg
(3)
Thus, the translocation velocity is controlled by the effective relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e090.jpg, and the maximum disassembly rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e091.jpg.
Three regimes of translocation velocity
We find that the translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e092.jpg, falls into three regimes, depending on An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e093.jpg:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e094.jpg
(4)
For An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e095.jpg (regime I), the ParB polymer retains its equilibrium shape as it is pulled across the cell at the velocity An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e096.jpg. For An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e097.jpg (regime II), the ParB polymer stretches as it is pulled and does not penetrate deeply into the ParA bundle. Since fewer ParA subunits bind to ParB, fewer are hydrolyzed and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e098.jpg drops below An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e099.jpg. For An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e100.jpg (regime III), the ParB polymer is so elongated that ParB binds to very few ParA subunits and the ParB polymer quickly detaches from the ParA bundle, leading to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e101.jpg.
This physical picture explains the results shown in Fig. 3, where we vary both the disassembly rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e102.jpg (Figs. 3a–b) and the effective relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e103.jpg (Figs. 3c–d). Specifically, Fig. 3a shows how An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e104.jpg depends on the depolymerization rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e105.jpg. For the black circles in Fig. 3a, the hydrolysis rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e106.jpg, is effectively infinite so that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e107.jpg (Eq. 3). In this case, for sufficiently small An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e108.jpg, the system is in regime I and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e109.jpg. As An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e110.jpg increases, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e111.jpg also increases; as a result, the ParB polymer stretches (inset to Fig. 3a) and the system crosses into regime II, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e112.jpg drops below An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e113.jpg. At very large An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e114.jpg, the system reaches regime III, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e115.jpg.
In contrast, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e116.jpg is small (red triangles in Fig. 3a), then An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e117.jpg cannot exceed An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e118.jpg as An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e119.jpg increases (Eq. 3). Therefore, for small An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e120.jpg, the ParB polymer remains in regime I, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e121.jpg, for all An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e122.jpg, so that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e123.jpg and translocation is robust for any An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e124.jpg. Thus, by decreasing the overall rate of disassembly by lowering An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e125.jpg, the system can achieve robust translocation, albeit at a cost to velocity.
Fig. 3b shows how An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e126.jpg varies as An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e127.jpg increases. In this case, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e128.jpg saturates to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e129.jpg at large An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e130.jpg (Eq. 3). Since An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e131.jpg is chosen to be small, we find An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e132.jpg over the entire range of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e133.jpg, meaning the system is in regime I and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e134.jpg.
The different velocity regimes can also be explored by varying An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e135.jpg instead of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e136.jpg. Fig. 3c shows that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e137.jpg is insensitive to the total drag, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e138.jpg, on the polymer when An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e139.jpg and thus An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e140.jpg are small. In this case, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e141.jpg is small, and the system is in regime I. As An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e142.jpg increases, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e143.jpg increases, causing An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e144.jpg to drop below An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e145.jpg as the system crosses into regime II.
Fig. 3d shows the effect of the total contour length, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e146.jpg, of the ParB polymer. For small An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e147.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e148.jpg is constant since the system is in regime I. As An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e149.jpg increases, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e150.jpg increases, a9nd when An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e151.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e152.jpg crosses into regime II and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e153.jpg drops below An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e154.jpg.
Dependence of the translocation velocity on binding energy, binding sites, applied load, and other physical variables
Fig. 4 shows that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e155.jpg has a threshold dependence on the ParB–ParA binding energy, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e156.jpg. As shown in Figs. 2c, ,4,4, ParB rapidly detaches from the ParA bundle if An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e157.jpg is too small. However, as long as An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e158.jpg is sufficiently large, the ParB polymer remains attached to the bundle throughout the simulation and translocates with a velocity that is insensitive to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e159.jpg and is set by An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e160.jpg (Eq. 4). We observe similar behavior as the number of binding sites on the ParB polymer is varied. If there are too few binding sites, the ParB polymer quickly detaches from ParA. Above a threshold value, however, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e161.jpg does not sensitively depend on the length of the binding strip (Fig. S1). The translocation velocity is also insensitive to the filament density within the ParA bundle, the arrangement of filaments in the bundle, and stiffness of the ParB polymer (Figs. S2, S3, S4). Finally, we have also verified that our main results hold when the form of the ParB–ParA binding potential is altered to allow binding by multiple points on ParB and/or ParA subunits.
Figure 4
Figure 4
Dependence of translocation velocity on ParB–ParA binding energy.
Detachment force for the ParB polymer
We next investigate the extent to which motility is robust to an external force on the ParB polymer that opposes translocation. The external force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e168.jpg, opposes translocation by pulling on each end of the ParB polymer. In our simulations, we find that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e169.jpg is unperturbed for An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e170.jpg (Fig. S5). For An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e171.jpg, however, the ParB polymer rapidly detaches from the ParA bundle and translocation stalls.
In order to understand this behavior, we analytically estimate the “detachment force,” An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e172.jpg, required to pull the ParB polymer off of the ParA bundle in a time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e173.jpg, that is approximately equal to the time required for the ParB polymer to translocate across the cell (see Text S1 for details).
In our simulations, we model the ParB-parS-ori complex as a polymer chain comprised of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e174.jpg monomeric subunits. Each subunit in the central strip of the ParB polymer binds with a binding energy, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e175.jpg, to a subunit in the ParA bundle. Thus, the total strength of the attraction between the ParB polymer and the ParA bundle is approximately proportional to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e176.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e177.jpg is the number of ParB subunits actually bound to ParA. Since ParB subunits lie in approximately a Gaussian distribution about the center of mass of the ParB polymer [34], An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e178.jpg, depends on the location, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e179.jpg, of the center of mass of the ParB polymer.
Now consider the effect of a force An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e180.jpg on the ParB polymer that opposes translocation in the An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e181.jpg direction. At the simplest level, based on the above analysis, the ParB polymer may be replaced by a point particle at the center of mass of the ParB polymer, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e182.jpg, in an effective potential given by
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e183.jpg
(5)
The first term is due to ParB binding to ParA and the second term is the work done by the external pulling force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e184.jpg. As An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e185.jpg increases, the minimum of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e186.jpg shifts to lower values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e187.jpg and the number of bound ParB sites decreases, eventually leading to unbinding of the ParB polymer from the ParA bundle.
The mean time for the particle to escape from the potential well (to detach from the ParA bundle) is well approximated by the Kramers escape time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e188.jpg for this potential [35], [36]:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e189.jpg
(6)
Given these expressions, we calculate the detachment force An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e190.jpg to be the force An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e191.jpg for which the escape time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e192.jpg, is equal to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e193.jpg, the time required for the ParB polymer to translocate across the cell.
In simulations with our standard model, the central binding strip has An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e194.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e195.jpg. There are An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e196.jpg ParB subunits that bind to ParA with energy, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e197.jpg, so the maximum total binding energy is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e198.jpg. The ParB polymer translocates at An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e199.jpg, so that the time to translocate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e200.jpg is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e201.jpg. With these parameters, we estimate that the detachment force is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e202.jpg. An estimate for the detachment force under more realistic conditions (in vivo) is given in the Discussion section.
This order of magnitude estimate agrees with our simulations at high depolymerization rates, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e203.jpg (Fig. 3a), large drag coefficients, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e204.jpg (Fig. 3c), and large external pulling forces, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e205.jpg (Fig. S5). In the first case, the mean time to first detachment is shorter than the translocation time for An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e206.jpg; this suggests that the force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e207.jpg, required for rapid detachment is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e208.jpg. Similarly, we find that the ParB polymer fails to translocate for An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e209.jpg, giving a detachment force of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e210.jpg. In addition, we have conducted simulations in which we apply an external force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e211.jpg, to each of the ends of the polymer. For these simulations, we find robust translocation up to a detachment force of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e212.jpg.
The ParB polymer translocates even when the ParA bundle is not anchored
So far, we have assumed that the ParA bundle is anchored to the pole. Recent reports suggest that in C. crescentus, ParA is localized to the swarmer pole by TipN [10], [12], but it is unclear if TipN actually anchors ParA. We therefore examined whether ParB translocation could occur if the ParA bundle is localized but not anchored.
Fig. 5 shows that the ParB polymer translocates even when the ParA bundle is unanchored. We understand this through Newton's third law, which dictates that the force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e213.jpg, that pulls ParB to ParA is equal in magnitude but opposite in direction to the force on ParA. Thus ParB is pulled towards the swarmer pole while ParA is simultaneously pulled away from it:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e214.jpg
(7)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e218.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e219.jpg are the drag coefficients of the ParB polymer and ParA filament bundle, respectively.
Figure 5
Figure 5
The ParB polymer translocates even when the ParA bundle is unanchored.
In the case of a long, unanchored ParA bundle, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e220.jpg and the ParB polymer translocates across the cell while the ParA bundle remains relatively stationary (Fig. 5b). However, if the ParA bundle is sufficiently small (e.g., when the ParB has nearly reached the swarmer pole), An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e221.jpg is small, so the large ParB polymer remains relatively stationary while pulling the smaller, disassembling ParA bundle towards mid-cell (Fig. 5b).
Based on recent experimental observations [7][12], [15], we have tested several simulation models (Fig. 2) and discovered a robust mechanism for chromosome segregation in C. crescentus via the ParABS system.
Self-diffusiophoresis can explain ParA pulling
Our simulations point to a specific physical mechanism underlying translocation in the ParABS system. We find that disassembly of ParA generates a steady-state ParA filament concentration gradient that remains fixed in the center-of-mass frame of the translocating ParB polymer (Fig. 1c). In other words, disassembly of ParA allows the ParA filament concentration gradient to translocate with the particle across the cell so that at all times the ParB polymer is moving up the concentration gradient of ParA to satisfy its attraction to ParA. Our simulations do not include fluid flow, but it is known that external concentration gradients can also drive motion of a particle in a fluid environment; the latter phenomenon is known as “diffusiophoresis.” If the particle (in this case, the ParB-parS-ori complex) is attracted to the solute (the ParA filament bundle), it will translocate up the concentration gradient towards high solute concentrations [37]. In “self-diffusiophoresis,” the particle itself (the ParB-parS-ori complex) generates and sustains the solute concentration gradient [38], [39] via disassembly of ParA. We emphasize that ParB-induced depolymerization (particle-induced destruction of solute) is central to this process. Without depolymerization, the ParA bundle would remain intact and the concentration of ParA filaments would not change with time. As a result, the ParA concentration profile would not be able to move with the particle and translocation would not occur.
This intrinsically many-body mechanism is distinct from biased diffusion. In contrast to biased diffusion mechanisms which apply to a coupler that attaches a load to a single filament or fiber [20][22], [26], self-diffusiophoretic translocation can occur even if the ParB polymer does not diffuse, as long as the ParB-ParA interaction range is finite. In self-diffusiophoresis, “diffusio” refers not to diffusion of a coupler, but to the key role of the solute gradient, just as the prefix in “electrophoresis” refers to an electric potential gradient [37]. The self-diffusiophoretic mechanism also differs from ones involving motion of a coupler [3], [20][27]; in our case, the load is not attached to a coupler that cannot detach from the depolymerizing filaments. Instead, the load is attached directly to the depolymerizing filaments via many non-permanent bonds.
It has been suggested that polymerization-driven motility, as in the case of F-actin in the lamellipodium of eukaryotic cells, also constitutes an example of self-diffusiophoretic motility [40], [41]. In that case, the object to be moved (e.g., the cell membrane) is repelled by the structure (the branched actin network) that it builds in order to move. In depolymerization-driven translocation, on the other hand, the object to be moved (the ParB-DNA complex) is attracted to the structure (ParA) that it destroys in order to move.
The self-diffusiophoretic mechanism suggests modes of failure for translocation. For example, overexpression of ParA leads to segregation defects, and it has been suggested that these defects arise due to the increase in the quantity of delocalized ParA [12], [15]. This effect may be analogous to what we observe in our simulations with severing (Video S2), where instead of binding to the ParA bundle, ParB can bind to severed ParA filaments. This disrupts the steady-state generation of a translating ParA concentration gradient so that it does not support steady-state ParB polymer translocation. Similarly, when ParA is overexpressed, extra ParA monomers or protofilaments may diminish or erase the ParA concentration gradient created by depolymerization. Alternatively, the extra ParA could saturate ParB, preventing translation of the ParA gradient.
Translocation is most robust for side-binding of ParB to ParA with disassembly only from the tip
We observe robust translocation over a wide range of physical parameters only if ParB binds to the sides of ParA filaments, triggering disassembly only from the tips of filaments (Fig. 1b–c). If ParB binds only to the tips of filaments, translocation is far less robust for two reasons. First, there are many fewer ParA subunits to which ParB can bind so the overall attraction between ParB and ParA is weaker. Second, the ParB polymer is localized near the tip of the bundle, at the very edge of the concentration gradient of ParA that drives translocation. In contrast, in the side-binding model, the ParB polymer penetrates further into the bundle so that it is localized near the steepest, central section of the concentration gradient (Fig. S6). Thus, in the tip-binding-only model, the ParB polymer is much more likely to detach from the ParA bundle due to thermal noise (Fig. 2a). This failure mode can only be averted by greatly increasing the binding energy or the number of filaments, and thus tips, in the ParA bundle.
We also find that ParA disassembly via severing does not provide robust translocation (Fig. 2b) because severed protofilaments can bind to ParB, reducing the attraction between the ParB polymer and the main ParA bundle, leading to detachment.
We therefore predict that ParB binds to the sides of ParA filaments and ParA filaments disassemble primarily from the tip. This prediction can be tested with in vitro experiments.
Comparisons with experiments on Par-mediated chromosome pulling
Our model is sufficiently versatile to account for a range of experimental observations. For example, by varying the initial density and cross-linking of the ParA filament bundle in our simulations, we find cases in which some ParA filaments remain partially assembled even though the ParB polymer has translocated across the cell (Fig. S7). This is in agreement with the observations of Ptacin et al. [10], who found that in some cases, a fiber of ParA extended across the predivisional cell after ori had translocated.
We find that the robustness of translocation is primarily controlled by the quantity An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e222.jpg, the product of an effective relaxation time (Eq. 2) and the maximum rate of disassembly of ParA (Eq. 3). The underlying details of the ParB polymer are only important insofar as they affect quantitative results such as the precise value of the relaxation time; they do not affect the qualitative physical principles described above.
Specifically, if An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e223.jpg is too high, the ParB polymer stretches out and can detach from the ParA bundle. This finding suggests a possible role for chromosome organizing factors such as the SMC protein [14], [42]. In order to translocate reliably and efficiently, the chromosome of four million base pairs [14], [16] must be organized such that it does not overload the pulling mechanism. We propose that one important physical function of chromosomal organization and condensation is to minimize the effective relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e224.jpg, so that the chromosome can keep up with the retracting ParA bundle, to ensure robust translocation.
In addition, we find that the velocity is simply the product of the ParA subunit length and the maximum disassembly rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e225.jpg, provided disassembly is slow enough to guarantee that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e226.jpg (Eq. 4). From the observed ori translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e227.jpg [9], [11], [16], [17], we estimate the in vivo ParA disassembly rate to be An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e228.jpg, which is slower than the measured disassembly rate of dynamically unstable ParM filaments [43], but comparable to the disassembly rate of actin filaments [44].
The translocation velocity in our simulations is considerably higher, typically several An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e229.jpg, because we used high disassembly rates. Translocation is robust in our simulations at these high values of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e230.jpg because the effective relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e231.jpg, of our ParB polymer is fairly short. In the real system, where the effective relaxation time of the chromosome is likely to be considerably longer, it could be a biological necessity that both ParA disassembly and ori translocation proceed at slower than the simulated rates.
Likewise, in our simulations the ParB polymer detaches when it is pulled with a force An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e232.jpg of order tens of pN, but this detachment force is likely to be much higher in the real system. The most important difference between our simulations and the actual bacterium lies in the number of ParB binding sites An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e233.jpg. To estimate the detachment force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e234.jpg, under realistic conditions, we first estimate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e235.jpg, the maximum possible binding energy An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e236.jpg, the extent of the chromosome An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e237.jpg, and the diffusion coefficient of the chromosome An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e238.jpg. We first estimate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e239.jpg by assuming that ParB decorates the approximately An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e240.jpg kilobase segment of the chromosome that was found to be the site of force exertion during translocation in [9]. For An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e241.jpg we therefore obtain a maximum binding energy of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e242.jpg. For ideal polymer chains [34], An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e243.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e244.jpg. Thus, we estimate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e245.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e246.jpg. This crude estimate of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e247.jpg actually agrees well with experimental snapshots of C. crescentus during chromosome segregation [10], [11]. The estimate of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e248.jpg falls within the range An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e249.jpg, which is measured in E. coli for DNA segments of varying sizes [45], [46]. We note that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e250.jpg is insensitive to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e251.jpg, and varies by less than An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e252.jpg over that range.
According to experiments [9], [11], [16], [17], the ParB-parS-ori complex translocates across the cell in about An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e253.jpg minutes. Using Eq. 6, we find that the detachment force is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e254.jpg. This value is of the same order of magnitude as the An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e255.jpg stall force for chromosome segregation along kinetochore fibers in eukaryotes [47], [48]. Thus, this estimate suggests that the mechanism we have proposed is both physically reasonable and biologically relevant.
Implications for other phenomena
Insights from our results may extend to plasmid segregation by ParAB. In Escherichia coli, the ParA concentration profile is known to oscillate as plasmid pB171 is partitioned [6], [19], [49]. This dynamic behavior appears to be required for proper plasmid partitioning [7], [19]. We suggest that ParB creates a moving ParA filament concentration gradient that pulls the plasmid along as ParA disassembles.
In addition, our findings suggest an alternative explanation for observations that the distance that plasmid pB171 translocates in a given time interval increases approximately linearly with the initial ParA filament length [7]. Ringgaard et al. [7] suggest that this effect arises from a ParA filament-length-dependent plasmid detachment rate. However, we have shown that the relative velocities of the ParB polymer and the ParA bundle depend on the ratio of the viscous drags on ParA and ParB, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e256.jpg (Fig. 5). Thus, the observed dependence of plasmid translocation distances and velocities on ParA filament length may simply be a result of Newton's third law, due to the variation of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e257.jpg with ParA filament length.
Our simulations with unanchored ParA filaments suggest a new possibility for the mechanism of terminus segregation in C. crescentus. As translocation begins, the ParA filaments are long, so An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e258.jpg and the ParB polymer is pulled rapidly towards the swarmer pole. However, as the ParB polymer nears the swarmer pole the ParA filaments are much shorter and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e259.jpg may be satisfied, so that the ParA bundle is pulled toward mid-cell. Experiments have indicated that ParA binds non-specifically to DNA [7], [10], [18]. Thus, we propose that DNA near the terminus is non-specifically bound to ParA and translocates away from the swarmer pole as ParA filaments are pulled toward mid-cell by the ParB-parS-ori complex. In contrast to previously suggested passive mechanisms [16], [30], [33], this is an active process, directly linked to ori translocation.
Our results provide a new paradigm for understanding depolymerization-driven translocation in prokaryotic DNA segregation systems. Since self-assembly and disassembly are ubiquitous in cellular systems, the creation of concentration gradients by these processes provides a general and robust mechanism for translocation.
At the start of each simulation, ParA monomeric subunits form a cross-linked bundle of filaments. The ParB-decorated chromosome is represented by a semi-flexible chain of monomeric subunits, typically of length 100 subunits, divided into three sections. The center section, typically of length 50 subunits, represents the part of the chromosome bound to ParB; these subunits can bind specifically to ParA subunits. The two end sections of the ParB polymer flanking the ParB section do not bind to ParA.
Biochemistry
The process of ParA disassembly begins when a ParB subunit binds to a ParA-ATP subunit. If the interaction energy, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e260.jpg, exceeds a certain threshold, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e261.jpg, the ParA-ATP hydrolyzes at rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e262.jpg. Once the ParA subunit hydrolyzes, it may detach from the ParA filament by depolymerization at rate An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e263.jpg (after which it continues to interact with other subunits by the interaction An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e264.jpg). In our standard model, ParB binds to the sides of ParA filaments, and a hydrolyzed ParA subunit can only depolymerize if it is located at the tip of a ParA filament.
Units
Simulation units are converted into physical units by taking the subunit length to be An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e265.jpg. The typical subunit diffusion coefficient is taken to be An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e266.jpg, as measured in [50], and the diffusion coefficient for a particular subunit is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e267.jpg (typically An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e268.jpg or An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e269.jpg, see below), giving a cell viscosity An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e270.jpg and a characteristic time scale An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e271.jpg. Typical runs are approximately An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e272.jpg and simulation steps are An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e273.jpg.
Interactions
Several interactions are included in the model; their specific forms are given below. All subunits are spheres with diameter An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e274.jpg that repel each other if they overlap:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e275.jpg
(8)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e276.jpg is the center-to-center distance between subunits An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e277.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e278.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e279.jpg. Within a ParA or ParB polymer chain, neighboring subunits are held together through an attractive harmonic potential:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e280.jpg
(9)
with An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e281.jpg. In order to hold the ParA bundle together, we typically take 40% of ParA subunits to be cross-linked to a subunit in a nearby filament through an attractive potential:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e282.jpg
(10)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e283.jpg is the initial spacing of filaments in the ParA bundle and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e284.jpg. ParA filaments are stiffened by a bending potential [51]:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e285.jpg
(11)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e286.jpg is the angle between the bond vector, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e287.jpg, between ParA subunits An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e288.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e289.jpg, and the bond vector, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e290.jpg, between subunits An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e291.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e292.jpg. Thus, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e293.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e294.jpg. We take An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e295.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e296.jpg. Similarly, the stiffness of the ParB polymer can be controlled by an interaction potential of form of Eq. 11 (however, in our standard model, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e297.jpg in the ParB polymer).
In addition, we introduce interactions so that binding between ParA and ParB occurs in specific spatial locations on the spheres representing the subunits. Each subunit An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e298.jpg has a unit polarization vector, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e299.jpg, that determines the location of the binding site for the ParB–ParA interaction, and the following interaction potential aligns it to be at an angle An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e300.jpg to the bond vectors An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e301.jpg connecting adjacent subunits:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e302.jpg
(12)
We choose An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e303.jpg so that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e304.jpg tends to be perpendicular to the bond vectors, and fix An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e305.jpg for ParA filaments and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e306.jpg in the ParB polymer, which is relatively more flexible. Binding sites are arranged helically on the ParA filaments and the ParB polymer due to two additional interaction potentials. The first constrains polarization vectors on nearest-neighbor subunits on a given chain:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e307.jpg
(13)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e308.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e309.jpg sets the pitch of the helix. Here, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e310.jpg for ParA and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e311.jpg for ParB. The second potential has the same form,
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e312.jpg
(14)
but constrains polarization vectors on the next-nearest-neighbor subunits with An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e313.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e314.jpg. Here, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e315.jpg in ParA and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e316.jpg for ParB. Note that in addition to regulating the locations of the binding sites, Eqs. 13 and 14 implicitly regulate torsion within the ParB polymer.
Finally, ParB binds to ParA with a site-specific, short-ranged interaction potential:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e317.jpg
(15)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e318.jpg is the vector distance between the ParA and ParB subunits and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e319.jpg is the binding energy. In our standard model, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e320.jpg. The normalization factor An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e321.jpg ensures that An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e322.jpg is the relevant energy scale for binding. The distance An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e323.jpg sets An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e324.jpg as the minimum of the binding potential. Binding site specificity is implemented through regulation of the angles between the polarization vectors on the ParA and ParB subunits as well as An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e325.jpg. In Eq. 15, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e326.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e327.jpg, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e328.jpg. Binding is strongest when the two polarization vectors point towards each other and along An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e329.jpg.
We have also studied several variations of these models. For example, in a separate set of simulations, we set An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e330.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e331.jpg for both ParA and ParB, so that the binding sites were not arranged helically on the ParA filaments and ParB polymer. The orientation of the polarization vectors was set by An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e332.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e333.jpg for tip binding and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e334.jpg for side-on binding. We also studied cases in which monomeric ParB subunits did not possess specific orientations (polarization vectors). In these cases, ParA polarization vectors were set by An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e335.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e336.jpg for both tip-binding and side-binding. Binding only weakly depended on the orientation of the ParA-ParB bond through a modified version of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e337.jpg, which we denote An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e338.jpg and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e339.jpg for tip-binding and side-binding, respectively. For tip-binding without ParB polarization vectors:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e340.jpg
(16)
For side-binding without ParB polarization vectors::
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e341.jpg
(17)
where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e342.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e343.jpg, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e344.jpg, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e345.jpg are as defined above.
Equations of motion
All subunits in the system translate and rotate according to Brownian dynamics [52]. Thus, we solve a system of coupled Langevin equation where the velocity of each subunit is governed by the forces exerted by other subunits in the system as well as thermal forces, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e346.jpg from the surrounding liquid medium:
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e347.jpg
(18)
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e348.jpg
(19)
and
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e349.jpg
(20)
A mathematical equation, expression, or formula.
 Object name is pcbi.1002145.e350.jpg
(21)
The subunit friction constant is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e351.jpg, where An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e352.jpg is the viscosity, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e353.jpg is a constant that determines the relative magnitude of the drag on subunit An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e354.jpg. Typically, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e355.jpg for ParA and normal ParB subunits, and An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e356.jpg for ParB subunits that cannot bind to ParA. An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e357.jpg is the rotational friction coefficient.
Figure S1
Behavior of the ParB polymer as a function of the length of the central ParB strip that binds to ParA. If too few of the ParB can bind to ParA, the ParB polymer detaches in an observably finite average time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e358.jpg (open symbols). When the percentage of binding sites is above threshold, the translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e359.jpg, is non-zero. If there are enough binding sites to cause disassembly at all of the ParA filament tips simultaneously, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e360.jpg is insensitive to the number of ParB that can bind ParA. The dashed line separates the regimes of detachment and translocation.
(PDF)
Figure S2
Dependence of translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e361.jpg, on the density of ParA filaments within the ParA bundle. For ParA bundles of equal diameter, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e362.jpg, but different numbers of ParA filaments, the translocation velocities are approximately equal. Thus, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e363.jpg is insensitive to the density of filaments in the ParA bundle.
(PDF)
Figure S3
Snapshots of a simulation with a “ParA tube”. The ParA filaments in the ParA bundle are arranged cylindrically. The snapshots are slightly rotated into the page and the thin black circle indicates the base of the cylinder. Translocation of the ParB polymer is insensitive to whether the ParA filaments are arranged as a tube or as a bundle. Depolymerized ParA monomers are not shown.
(PDF)
Figure S4
Dependence of translocation velocity, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e364.jpg, on the stiffness of the ParB polymer. In our standard model, the ParB polymer is flexible, and the bending stiffness is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e365.jpg. In order to simulate a stiff ParB polymer, we apply the bending potential in Eq. 11 to the ParB polymer. An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e366.jpg is insensitive to the bending stiffness over the observed range of An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e367.jpg.
(PDF)
Figure S5
Force-velocity relation for ParB polymer translocation in our simulations. In these simulations, an external force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e368.jpg, pulls on each of the two ends of the ParB polymer, thus opposing depolymerization-driven translocation. Translocation of the ParB polymer is unperturbed when subjected to external pulling forces up to An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e369.jpg.
(PDF)
Figure S6
Steady-state ParA concentration profiles for tip-binding-only and side-binding models. Steady-state ParA concentration is plotted versus position relative to the center of mass of the ParB polymer, which is located at An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e370.jpg and indicated by the dotted green line. When ParB binds only to the tips of ParA filaments, the center of mass of the ParB polymer (dotted green line) localizes near the edge of the ParA filament concentration gradient (dashed black curve). This enables the ParB polymer to easily escape the ParA concentration gradient and detach from the ParA bundle due to thermal noise. However, when ParB can bind to the sides of ParA filaments, the ParB polymer penetrates further into the ParA bundle, and thus the center of mass (green) of the ParB polymer is localizes near the center of the ParA concentration gradient (dashed red curve). Thus, the ParB polymer is not susceptible to falling out of the ParA gradient and detaching from the ParA bundle due to thermal noise.
(PDF)
Figure S7
Snapshots of a simulation in which several ParA filaments remain after the ParB polymer has translocated. If the initial spacing, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e371.jpg, of the ParA filaments in the bundle is large, the ParB polymer may translocate by disassembling some, but not all, of the ParA filaments. In the snapshots shown, the initial ParA filament spacing is An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e372.jpg, four times greater than the initial spacing, used in our standard simulations. This simulation demonstrates the versatility of our model by replicating one of the observations of Ptacin et al. (2010) [10]. This result can also be obtained with closely packed (e.g., An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e373.jpg) ParA filaments if the filament bundle contains a large number of filaments.
(PDF)
Text S1
Polymer relaxation time and estimated detachment force for the ParB polymer. This text explains how to calculate the characteristic polymer relaxation time, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e374.jpg, and the peripheral segment diffusion coefficient, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e375.jpg. In addition, we provide details for the estimation of the detachment force, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e376.jpg.
(PDF)
Video S1
A movie of translocation of the ParB polymer in our standard simulation conditions. The ParB polymer remains localized near the tip of the ParA bundle and translocates as the ParA bundle disassembles. Depolymerized ParA monomers are not shown.
(MOV)
Video S2
A movie of a simulation run for the model in which ParB binds to the sides of ParA filaments and severs them. The ParB polymer translocates briefly until severed ParA protofilaments bind to the ParB polymer and disrupt its binding to the main ParA filament bundle.
(MOV)
Video S3
Stretching of the ParB polymer. When the maximum ParA disassembly rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e377.jpg, is sufficiently large, the ParB polymer does not have time to relax to its equilibrium shape as it is pulled, and therefore stretches out. The ParB polymer consists of three segments; the two peripheral segments (light green), which cannot bind to ParA, and the central segment (dark green), which can bind to ParA. Note that the peripheral segments of the ParB polymer stretch, while the central segment of the ParB polymer is initially bound to the ParA filament bundle. When the peripheral segments stretch too far, they start to stretch the central segment, thus decreasing the length, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e378.jpg, that the ParB polymer penetrates into the ParA bundle. This can lead to detachment for sufficiently high An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e379.jpg. In this movie, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e380.jpg, five times greater than the ParA disassembly rate, An external file that holds a picture, illustration, etc.
Object name is pcbi.1002145.e381.jpg, in our standard simulations (see Fig. 3 caption).
(MOV)
Acknowledgments
We thank Rob Phillips for helpful discussions.
Footnotes
The authors have declared that no competing interests exist.
We gratefully acknowledge the support of the NSF (http://www.nsf.gov) through DMR-0605044 (EJB and AJL), DMR-0520020 (AJL), and PHY-0957573 (NSW). ZG is supported by a HFSP (http://www.hfsp.org/) Young Investigator Award and NIH (http://www.nih.gov/) grant 1DP2OD004389. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
1. Mitchison T, Kirschner M. Dynamic instability of microtubule growth. Nature. 1984;312:237–242. [PubMed]
2. Walczak CE, Cai S, Khodjakov A. Mechanisms of chromosome behaviour during mitosis. Nat Rev Mol Cell Biol. 2010;11:91–102. [PMC free article] [PubMed]
3. McIntosh JR, Volkov V, Ataullakhanov FI, Grishchuk EL. Tubulin depolymerization may be an ancient biological motor. J Cell Sci. 2010;123:3425–3434. [PubMed]
4. Gerdes K, Howard M, Szardenings F. Pushing and pulling in prokaryotic DNA segregation. Cell. 2010;141:927–942. [PubMed]
5. Kirkpatrick CL, Viollier PH. A polarity factor takes the lead in chromosome segregation. EMBO J. 2010;29:3035–3036. [PubMed]
6. Ebersbach G, Gerdes K. he double par locus of virulence factor pB171: DNA segregation is correlated with oscillation of ParA. Proc Natl Acad Sci USA. 2001;98:15078–15083. [PubMed]
7. Ringgaard S, van Zon J, Howard M, Gerdes K. Movement and equipositioning of plasmids by ParA filament disassembly. Proc Natl Acad Sci USA. 2009;106:19369–19374. [PubMed]
8. Fogel MA, Waldor MK. A dynamic, mitotic-like mechanism for bacterial chromosome segregation. Genes Dev. 2006;20:3269–3282. [PubMed]
9. Toro E, Hong SH, McAdams HH, Shapiro L. Caulobacter requires a dedicated mechanism to initiate chromosome segregation. Proc Natl Acad Sci USA. 2008;105:15435–15440. [PubMed]
10. Ptacin JL, Lee SF, Garner EC, Toro E, Eckart M, et al. A spindle-like apparatus guides bacterial chromosome segregation. Nat Cell Biol. 2010;12:791–798. [PMC free article] [PubMed]
11. Shebelut CW, Guberman JM, van Teeffelen S, Yakhnina AA, Gitai Z. Caulobacter chromosome segregation is an ordered multistep process. Proc Natl Acad Sci USA. 2010;107:14194–14198. [PubMed]
12. Schofield WB, Lim HC, Jacobs-Wagner C. Cell cycle coordination and regulation of bacterial chromosome segregation dynamics by polarly localized proteins. EMBO J. 2010;29:3068–3081. [PubMed]
13. Leonard TA, Butler PJ, Löwe J. Bacterial chromosome segregation: Structure and DNA binding of the Soj dimer – a conserved biological switch. EMBO J. 2005;24:270–282. [PubMed]
14. Jensen RB, Shapiro L. The Caulobacter crescentus smc gene is required for cell cycle progression and chromosome segregation. Proc Natl Acad Sci USA. 1999;96:10661–10666. [PubMed]
15. Mohl DA, Gober JW. Cell cycle-dependent polar localization of chromosome partitioning proteins in Caulobacter crescentus. Cell. 1997;88:675–684. [PubMed]
16. Viollier PH, Thanbichler M, McGrath PT, West L, Meewan M, et al. Rapid and sequential movement of individual chromosomal loci to specific subcellular locations during bacterial DNA replication. Proc Natl Acad Sci USA. 2004;101:9257–9262. [PubMed]
17. Jensen RB. Coordination between chromosome replication, segregation, and cell division in Caulobacter crescentus. J Bacteriol. 2006;188:2244–2253. [PMC free article] [PubMed]
18. Easter J, Jr, Gober JW. ParB-stimulated nucleotide exchange regulates a switch in functionally distinct ParA activities. Mol Cell. 2002;10:427–434. [PubMed]
19. Ebersbach G, Ringgaard S, Möller-Jensen J, Wang Q, Sherratt DJ, et al. Regular cellular distribution of plasmids by oscillating and filament-forming ParA ATPase of plasmid pB171. Mol Microbiol. 2006;61:1428–1442. [PubMed]
20. Hill TL. Theoretical problems related to the attachment of microtubules to kinetochores. Proc Natl Acad Sci USA. 1985;82:4404–4408. [PubMed]
21. Peskin CS, Oster GF. Force production by depolymerizing microtubules: Load-velocity curves and run-pause statistics. Biophys J. 1995;69:2268–2276. [PubMed]
22. Raj A, Peskin CS. The inuence of chromosome exibility on chromosome transport during anaphase A. Proc Natl Acad Sci USA. 2006;103:5349–5354. [PubMed]
23. Efremov A, Grishchuk EL, McIntosh JR, Ataullakhanov FI. In search of an optimal ring to couple microtubule depolymerization to processive chromosome motions. Proc Natl Acad Sci USA. 2007;104:19017–19022. [PubMed]
24. Armond JW, Turner MS. Force transduction by the microtubule-bound Dam1 ring. Biophys J. 2010;98:1598–1607. [PubMed]
25. Molodtsov MI, Grishchuk EL, Efremov AK, McIntosh JR, Ataullakhanov FI. Force production by depolymerizing microtubules: A theoretical study. Proc Natl Acad Sci USA. 2005;102:4353–4358. [PubMed]
26. Liu J, Onuchic JN. A driving and coupling “Pac-Man” mechanism for chromosome poleward translocation in anaphase A. Proc Natl Acad Sci USA. 2006;103:18432–18437. [PubMed]
27. McIntosh JR, Grishchuk EL, Morphew MK, Efremov AK, Zhudenkov K, et al. Fibrils connect microtubule tips with kintetochores: A mechanism to couple tubulin dynamics to chromosome motion. Cell. 2008;135:322–333. [PMC free article] [PubMed]
28. Jacob F, Brenner S, Cuzin F. On regulation of DNA replication in bacteria. Cold Spring Harb Symp Quant Biol. 1963;28:329–348.
29. Norris V. Hypothesis: Chromosome separation Escherichia coli involves autocatalytic gene expression, transertion and membrane-domain formation. Mol Microbiol. 1995;16:1051–1057. [PubMed]
30. Lemon KP, Grossman AD. Localization of bacterial DNA polymerase: Evidence for a factory model of replication. Science. 1998;282:1516–1519. [PubMed]
31. Hunding A, Ebersbach G, Gerdes K. A mechanism for ParB-dependent waves of ParA, a protein related to DNA segregation during cell division in prokaryotes. J Mol Biol. 2003;329:35–43. [PubMed]
32. Adachi S, Hori K, Hiraga S. Subcellular positioning of F plasmid mediated by dynamic localization of SopA and SopB. J Mol Biol. 2006;356:850–863. [PubMed]
33. Jun S, Mulder B. Entropy-driven spatial organization of highly confined polymers: Lessons for the bacterial chromosome. Proc Natl Acad Sci USA. 2006;103:12388–12393. [PubMed]
34. Doi M, Edwards SF. Oxford:: Clarendon Press; 1986. The theory of polymer dynamics. pp. 14–16, 22–23, 95.
35. Kramers HA. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica (Utrecht) 1940;7:284–304.
36. Risken H. Berlin: Springer-Verlag; 1996. The Fokker-Planck quation. pp. 96–99, 122–125.
37. Anderson JL. Colloid transport by interfacial forces. Annu Rev Fluid Mech. 1989;21:61–99.
38. Golestanian R, Liverpool TB, Ajdari A. Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys Rev Lett. 2005;94:220801. [PubMed]
39. Golestanian R, Liverpool TB, Ajdari A. Designing phoretic micro- and nano-swimmers. New J Phys. 2007;9:126–133.
40. Lee KC, Liu AJ. New proposed mechanism of actin-polymerization-driven motility. Biophys J. 2008;95:4529–4539. [PubMed]
41. Lee KC, Liu AJ. Force-velocity relation for actin-polymerization-driven motility from Brownian dynamics simulations. Biophys J. 2009;97:1295–1304. [PubMed]
42. Sullivan NL, Marquis KA, Rudner DZ. Recruitment of SMC by ParB-parS organizes the origin region and promotes efficient chromosome segregation. Cell. 2009;137:697–707. [PMC free article] [PubMed]
43. Campbell CS, Mullins RD. In vivo visualization of type II plasmid segregation: Bacterial actin filaments pushing plasmids. J Cell Biol. 2007;179:1059–1066. [PMC free article] [PubMed]
44. Pollard TD, Blanchoin L, Mullins RD. Molecular mechanims controlling actin filament dynamics in nonmuscle cells. Annu Rev Biomol Struct. 2000;29:545–576. [PubMed]
45. Cunha S, Woldringh CL, Odijk T. Restricted diffusion of DNA segments within the isolated Escherichia coli nucleoid. J Struct Biol. 2005;150:226–232. [PubMed]
46. Elmore S, Müller M, Vischer N, Odijk T, Woldringh CL. Single-particle tracking of oriC - GFP uorescent spots during chromosome segregation in escherichia coli. J Struct Biol. 2005;151:275–287. [PubMed]
47. Nicklas RB. Measurements of the force produced by the mitotic spindle in anaphase. J Cell Biol. 1983;97:542–548. [PMC free article] [PubMed]
48. Westermann S, Drubin DG, Barnes G. Structures and functions of yeast kinetochore complexes. Annu Rev Biohcem. 2007;76:563–591. [PubMed]
49. Ebersbach G, Gerdes K. Bacterial mitosis: Partitioning protein ParA oscillates in spiralshaped structures and positions plasmids at mid-cell. Mol Microbiol. 2004;52:385–398. [PubMed]
50. Elowitz MB, Surette MG, Wolf PE, Stock JB, Leibler S. Protein mobility in the cytoplasm of Escherichia coli. J Bacteriol. 1999;181:197–203. [PMC free article] [PubMed]
51. Rapaport DC. Cambridge: Cambridge University Press; 1995. The art of molecular dynamics simulations.
52. Allen MP, Tildesley DJ. Oxford University Press; 1989. Computer simulation of liquids. pp. 259–264.
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