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:

The rate of disassembly controls the ParB translocation velocity

To understand how ParA translocates ParB, we identified variables controlling the translocation velocity, . In all cases, we find that is given by the mean rate, , of disassembly of a ParA filament, so that , where 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. therefore depends on the distance, , that the ParB polymer typically penetrates into the ParA bundle and causes ParA-ATP hydrolysis, the rate, , of ParA-ATP hydrolysis, and the rate, , at which a ParA subunit depolymerizes once hydrolyzed.

In turn, the penetration length, , 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, , of the penetration length, , is achieved if the ParB polymer is able to maintain this equilibrium shape as it is pulled by ParA. If the disassembly rate, , 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 .

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:(1)where is the ensemble-averaged -distance between the ends of a peripheral segment pulled by one end in the -direction, is the diffusion coefficient of the segment, is the -component of its equilibrium radius of gyration, and the relaxation time, , is the ratio of its internal drag, , to the effective spring constant, (see Text S1). Stretching is appreciable if , so for translocation in steady state (), stretching becomes appreciable for , or, equivalently, (inset to Fig. 3a). The shape of the pulled ParB polymer is therefore governed by the product , where we have defined(2)

The penetration length, , depends directly on the shape of the ParB polymer. For large the ParB polymer is pulled rapidly and 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 decreases, increases and saturates at for (inset to Fig. 3b). In the latter regime, the disassembly rate is , where(3)Thus, the translocation velocity is controlled by the effective relaxation time, , and the maximum disassembly rate .

Three regimes of translocation velocity

We find that the translocation velocity, , falls into three regimes, depending on :(4)For (regime I), the ParB polymer retains its equilibrium shape as it is pulled across the cell at the velocity . For (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 drops below . For (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 .

This physical picture explains the results shown in Fig. 3, where we vary both the disassembly rate, (Figs. 3a–b) and the effective relaxation time, (Figs. 3c–d). Specifically, Fig. 3a shows how depends on the depolymerization rate, . For the black circles in Fig. 3a, the hydrolysis rate, , is effectively infinite so that (Eq. 3). In this case, for sufficiently small , the system is in regime I and . As increases, also increases; as a result, the ParB polymer stretches (inset to Fig. 3a) and the system crosses into regime II, where drops below . At very large , the system reaches regime III, and .

In contrast, if is small (red triangles in Fig. 3a), then cannot exceed as increases (Eq. 3). Therefore, for small , the ParB polymer remains in regime I, , for all , so that and translocation is robust for any . Thus, by decreasing the overall rate of disassembly by lowering , the system can achieve robust translocation, albeit at a cost to velocity.

Fig. 3b shows how varies as increases. In this case, saturates to at large (Eq. 3). Since is chosen to be small, we find over the entire range of , meaning the system is in regime I and .

The different velocity regimes can also be explored by varying instead of . Fig. 3c shows that is insensitive to the total drag, , on the polymer when and thus are small. In this case, is small, and the system is in regime I. As increases, increases, causing to drop below as the system crosses into regime II.

Fig. 3d shows the effect of the total contour length, , of the ParB polymer. For small , is constant since the system is in regime I. As increases, increases, a9nd when , crosses into regime II and drops below .

Dependence of the translocation velocity on binding energy, binding sites, applied load, and other physical variables

Fig. 4 shows that has a threshold dependence on the ParB–ParA binding energy, . As shown in Figs. 2c, 4, ParB rapidly detaches from the ParA bundle if is too small. However, as long as is sufficiently large, the ParB polymer remains attached to the bundle throughout the simulation and translocates with a velocity that is insensitive to and is set by (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, 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.

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, , opposes translocation by pulling on each end of the ParB polymer. In our simulations, we find that is unperturbed for (Fig. S5). For , 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,” , required to pull the ParB polymer off of the ParA bundle in a time, , 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 monomeric subunits. Each subunit in the central strip of the ParB polymer binds with a binding energy, , 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 , where 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], , depends on the location, , of the center of mass of the ParB polymer.

Now consider the effect of a force on the ParB polymer that opposes translocation in the 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, , in an effective potential given by(5)The first term is due to ParB binding to ParA and the second term is the work done by the external pulling force, . As increases, the minimum of shifts to lower values of 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, for this potential [35], [36]:(6)Given these expressions, we calculate the detachment force to be the force for which the escape time, , is equal to , the time required for the ParB polymer to translocate across the cell.

In simulations with our standard model, the central binding strip has and . There are ParB subunits that bind to ParA with energy, , so the maximum total binding energy is . The ParB polymer translocates at , so that the time to translocate is . With these parameters, we estimate that the detachment force is . 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, (Fig. 3a), large drag coefficients, (Fig. 3c), and large external pulling forces, (Fig. S5). In the first case, the mean time to first detachment is shorter than the translocation time for ; this suggests that the force, , required for rapid detachment is . Similarly, we find that the ParB polymer fails to translocate for , giving a detachment force of . In addition, we have conducted simulations in which we apply an external force, , to each of the ends of the polymer. For these simulations, we find robust translocation up to a detachment force of .

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, , 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:(7)

where and are the drag coefficients of the ParB polymer and ParA filament bundle, respectively.

In the case of a long, unanchored ParA bundle, 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), is small, so the large ParB polymer remains relatively stationary while pulling the smaller, disassembling ParA bundle towards mid-cell (Fig. 5b).

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 , 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 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, , 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, , provided disassembly is slow enough to guarantee that (Eq. 4). From the observed ori translocation velocity, [9], [11], [16], [17], we estimate the in vivo ParA disassembly rate to be , 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 , because we used high disassembly rates. Translocation is robust in our simulations at these high values of because the effective relaxation time, , 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 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 . To estimate the detachment force, , under realistic conditions, we first estimate , the maximum possible binding energy , the extent of the chromosome , and the diffusion coefficient of the chromosome . We first estimate by assuming that ParB decorates the approximately kilobase segment of the chromosome that was found to be the site of force exertion during translocation in [9]. For we therefore obtain a maximum binding energy of . For ideal polymer chains [34], and . Thus, we estimate and . This crude estimate of actually agrees well with experimental snapshots of C. crescentus during chromosome segregation [10], [11]. The estimate of falls within the range , which is measured in E. coli for DNA segments of varying sizes [45], [46]. We note that is insensitive to , and varies by less than over that range.

According to experiments [9], [11], [16], [17], the ParB-parS-ori complex translocates across the cell in about minutes. Using Eq. 6, we find that the detachment force is . This value is of the same order of magnitude as the 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, (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 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 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 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.

Biochemistry

The process of ParA disassembly begins when a ParB subunit binds to a ParA-ATP subunit. If the interaction energy, , exceeds a certain threshold, , the ParA-ATP hydrolyzes at rate . Once the ParA subunit hydrolyzes, it may detach from the ParA filament by depolymerization at rate (after which it continues to interact with other subunits by the interaction ). 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 . The typical subunit diffusion coefficient is taken to be , as measured in [50], and the diffusion coefficient for a particular subunit is (typically or , see below), giving a cell viscosity and a characteristic time scale . Typical runs are approximately and simulation steps are .

Interactions

Several interactions are included in the model; their specific forms are given below. All subunits are spheres with diameter that repel each other if they overlap:(8)where is the center-to-center distance between subunits and and . Within a ParA or ParB polymer chain, neighboring subunits are held together through an attractive harmonic potential:(9)with . 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:(10)where is the initial spacing of filaments in the ParA bundle and . ParA filaments are stiffened by a bending potential [51]:(11)where is the angle between the bond vector, , between ParA subunits and , and the bond vector, , between subunits and . Thus, , where . We take and . Similarly, the stiffness of the ParB polymer can be controlled by an interaction potential of form of Eq. 11 (however, in our standard model, 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 has a unit polarization vector, , 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 to the bond vectors connecting adjacent subunits:(12)We choose so that tends to be perpendicular to the bond vectors, and fix for ParA filaments and 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:(13)where and sets the pitch of the helix. Here, for ParA and for ParB. The second potential has the same form,(14)but constrains polarization vectors on the next-nearest-neighbor subunits with and . Here, in ParA and 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:(15)where is the vector distance between the ParA and ParB subunits and is the binding energy. In our standard model, . The normalization factor ensures that is the relevant energy scale for binding. The distance sets 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 . In Eq. 15, , , and . Binding is strongest when the two polarization vectors point towards each other and along .

We have also studied several variations of these models. For example, in a separate set of simulations, we set and 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 , where for tip binding and 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 , where for both tip-binding and side-binding. Binding only weakly depended on the orientation of the ParA-ParB bond through a modified version of , which we denote and for tip-binding and side-binding, respectively. For tip-binding without ParB polarization vectors:(16)For side-binding without ParB polarization vectors::(17)where , , , and 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, from the surrounding liquid medium:(18)(19)and(20)(21)The subunit friction constant is , where is the viscosity, and is a constant that determines the relative magnitude of the drag on subunit . Typically, for ParA and normal ParB subunits, and for ParB subunits that cannot bind to ParA. is the rotational friction coefficient.