Directed and persistent movement arises from mechanochemistry of the ParA/ParB system

Significance

Cells typically use processive motor proteins or the growth/shrinkage of cytoskeletal filaments to power directed and persistent movement of cellular structures. What if there are no motor proteins or filaments? Here, we establish a third mechanism of processive motility exemplified by the ParA/ParB system, which faithfully segregates low-copy number plasmids during bacterial cell division. The DNA cargos recruit ParB, which binds to and stimulates the ATPase activity of ParA bound to the nucleoid. ATP hydrolysis dissociates ParA from the nucleoid. The transient tethering arising from the ParA–ParB bonds collectively drives forward movement of the cargo and quenches lateral diffusive motions, producing a strikingly persistent trajectory. This operational principle could be important in early evolution and conserved for many systems.

Abstract

The segregation of DNA before cell division is essential for faithful genetic inheritance. In many bacteria, segregation of low-copy number plasmids involves an active partition system composed of a nonspecific DNA-binding ATPase, ParA, and its stimulator protein ParB. The ParA/ParB system drives directed and persistent movement of DNA cargo both in vivo and in vitro. Filament-based models akin to actin/microtubule-driven motility were proposed for plasmid segregation mediated by ParA. Recent experiments challenge this view and suggest that ParA/ParB system motility is driven by a diffusion ratchet mechanism in which ParB-coated plasmid both creates and follows a ParA gradient on the nucleoid surface. However, the detailed mechanism of ParA/ParB-mediated directed and persistent movement remains unknown. Here, we develop a theoretical model describing ParA/ParB-mediated motility. We show that the ParA/ParB system can work as a Brownian ratchet, which effectively couples the ATPase-dependent cycling of ParA–nucleoid affinity to the motion of the ParB-bound cargo. Paradoxically, this resulting processive motion relies on quenching diffusive plasmid motion through a large number of transient ParA/ParB-mediated tethers to the nucleoid surface. Our work thus sheds light on an emergent phenomenon in which nonmotor proteins work collectively via mechanochemical coupling to propel cargos—an ingenious solution shaped by evolution to cope with the lack of processive motor proteins in bacteria.

Fidelity of chromosome segregation is critical for cell proliferation and survival. Unlike eukaryotic mitosis, DNA segregation in bacteria is not well understood (1–3). Diffusion-based random partitioning is sufficient for the stable inheritance of small high-copy number plasmids (4). However, to ensure faithful partitioning of low-copy number plasmids of large size, active processes need to work against diffusion (or lack of it) and orchestrate directed transport and positioning. This active process is mediated by conserved tripartite segregation machineries, whose central component is a nucleoside triphosphatase (NTPase) driving the partitioning reaction. The NTPases have been classified as actin-like (ParM), tubulin-like (TubZ), or a deviant Walker-type ATPase (ParA) (5). It is clear that systems using actin or tubulin homologs function by means of filament-based pushing or pulling mechanisms (6, 7). Most low-copy number plasmids and chromosomes are, however, segregated by the ParA ATPase family (5, 8).

A series of protein–protein and protein–DNA interactions are required for ParA-mediated DNA segregation (9): the centromere region of the plasmid, marked by parS, recruits a large number of ParB to form a partition complex, which in turn interacts with ParA that is nonspecifically bound to the nucleoid—the large compact structure that mainly consists of the bacterial chromosome and DNA-associated RNA and proteins. Plasmid-bound ParB binds ParA and stimulates its ATPase activity, which triggers ParA dissociation from the nucleoid. Once dissociated, ParA undergoes a time delay before becoming competent again for nucleoid binding, rendering this dissociation step effectively irreversible on relevant timescales. It is unclear, however, how the ParA family of ATPases mechanically harnesses these biochemical reactions to direct persistent movement of plasmids and ensure segregation fidelity of low-copy number genomic units.

In part based on in vitro observations that ParA tends to form fibrous aggregates at high concentrations, it has been proposed that ParAs form polymers that move plasmids by repeated polymerization/depolymerization cycles, similar to ParM or TubZ (10–13). However, recent experiments cast doubt on the feasibility of this proposal (14–17). A diffusion ratchet model was then put forward, which posited that a concentration gradient of ParA dimers on the nucleoid could act as the driving force for DNA segregation (18). This proposed mechanism gained direct support from a recent in vitro experiment that reconstituted the segregation system of the Escherichia coli F plasmid (14). In this experiment, the ParB protein (F SopB) specifically bound to centromere sites (F sopC) that were immobilized on a microbead. Upon contacting the ParA·ATP (F SopA·ATP), which was nonspecifically bound to a DNA carpet that mimicked the nucleoid surface, the microbead was observed to undergo directed and persistent movement over microns at a speed of ∼0.1 μm/s. Interestingly, the microbead movement left a ParA-depleted zone in its wake, consistent with the proposed diffusion ratchet model for cargo motion. Simulations of a continuum model indicate that it is possible to drive persistent ParB-bound cargo motion by maintaining a ParA concentration gradient around the cargo if the cargo diffusion constant is reduced to that observed during persistent bead motion, which is much smaller than that of the free microbead (14, 19). However, the previous study did not address the origin of the cargo diffusion suppression or its potential relationship with the persistence of the microbead motion.

Whereas the spatial ParA gradient was correlated with directed and persistent movement in the in vitro experiments (14), key questions remained: What is the causal relationship between the motion and the gradient? What is the molecular basis for the highly persistent movement? What restricts random movements orthogonal to the principal ParA spatial gradient, where nothing but diffusion is at play according to the simple diffusion ratchet model? Finally, what is the capacity of this segregation machinery to drive directed and persistent movement of plasmids?

To address these questions, we theoretically examined the underlying principle of ParA/ParB-mediated motility. We established a theoretical model based on the setup of the in vitro reconstituted microbead experiment with implications for plasmid segregation in vivo. Through stochastic simulations, we found that bead diffusion spontaneously broke symmetry and initiated movement. The subsequent movement caused dissociation of ParA–ParB bonds at the back of the bead, whereas new bonds were established at the front. Due to the slow rate of dissociated ParA resetting its DNA-binding capability, a ParA-depleted zone emerges behind the moving bead, thus rectifying the initial asymmetry. Furthermore, the model suggests that the tethers mediated by ParA–ParB bonds not only drive directed movement but also quench the diffusive motion orthogonal to the principal direction of movement. Whereas our model is a Brownian-ratchet mechanism, it highlights the importance of the mechanochemical coupling for ParA/ParB-mediated motility. In a general sense, our model distills fundamental principles of how chemical reactions can collectively mediate directed transport without conventional motor proteins or filament-based pulling and pushing forces.

Model Development

Mechanical Model.

Based on the measured diffusion constant of free microbeads near the substrate surface (D ∼ 0.1 μm²/s) and the measured microbead speed (0.1 μm/s) (14), the net driving force was estimated to be ∼4 fN from the Einstein relation between diffusion and the viscous drag coefficient (γ): $D\gamma =k_{\mathrm{B}}T$. First, let us put aside the origin of this driving force and ask the question: Can a simple mechanical model (Eq. 1), in which the microbead is propelled by this constant driving force in a fixed direction, recapitulate the observed directed and persistent movement in the presence of diffusion?

For a microscopic particle moving in a viscous medium, its Reynolds number is small, so inertial effects can be ignored. The particle motion can thus be described with the overdamped Langevin equation:

$$\mathit{\gamma }\frac{\mathit{d}\mathbf{r}\left(\mathit{t}\right)}{\mathit{dt}}=\mathbf{f}\left(\mathit{t}\right)+\mathit{\eta }\left(\mathit{t}\right),$$ [1]

where $\mathbf{r}\left(\mathit{t}\right)$ is the microbead position at time t, $\mathbf{f}\left(\mathit{t}\right)$ is the external force acting on the bead, and $\mathit{\eta }\left(\mathit{t}\right)$ represents random force resulting from thermal motion of the solvent molecules: $\langle {\mathit{\eta }}_{\mathit{i}}\left(t\right)\rangle =0$ and $\langle {\mathit{\eta }}_{\mathit{i}}\left(t\right){\mathit{\eta }}_{\mathit{j}}(t\prime )\rangle =2\mathit{\gamma }{\mathit{k}}_{\mathrm{B}}T{\mathit{\delta }}_{\mathit{ij}}\mathit{\delta }\left(\mathit{t}\prime \mathit{-t}\right)$, where ${\mathit{\delta }}_{\mathit{\iota j}}$ is Kronecker delta and $\mathit{\delta }\left(\mathit{t}\prime \mathrm{-}\mathit{t}\right)$ is the Dirac delta function.

The following equation was used to advance the microbead position $\mathbf{r}\left(\mathit{t}\right)$ over time (20):

$$\mathbf{r}\left(\mathit{t}+\mathrm{\Delta }\mathit{t}\right)=\mathbf{r}\left(\mathit{t}\right)+\frac{D\mathbf{f}}{{\mathit{k}}_{\mathrm{B}}T}\mathrm{\Delta }\mathit{t+}\sqrt{\mathrm{2}\mathit{D}\mathrm{\Delta }\mathit{t}}\mathit{\xi },$$ [2]

where the $\mathit{\xi }$ term represents a random displacement at time $\mathit{t}$, and each component of $\mathit{\xi }$ satisfies a Gaussian distribution with average value zero and unit variance.

We systematically characterized the microbead movement by two parameters: speed and persistence. We adopted a definition of persistence based on that commonly used in polymer physics; persistence is defined as the ratio between the end-to-end distance and the contour length of the movement trajectory based on the experimental bin time of 2 s. This bin time is fixed throughout the paper if not otherwise mentioned. With this definition, the maximum persistence is 1, reflecting a straight line. The persistence of the observed directed movement is >0.6 (14). With our simple mechanical model, a driving force of 4 fN could recapitulate the observed speed ∼0.1 μm/s (Fig. S1A). However, the microbead diffused freely in the orthogonal directions of the driving force, and the resulting persistence was low (∼0.2) (Fig. S1 A and B). Whereas the persistence increased with larger driving forces, the speed also increased (Fig. S1A). To obtain the observed movement persistence >0.6 (14), the microbead speed needs to be ∼0.5 μm/s (Fig. S1A), approximately five times faster than that observed (14). These significant discrepancies suggest that a simple mechanical model with a fixed driving force and intrinsic diffusion cannot explain the observed directed and persistent movement. This insight is also consistent with the conclusion from recent simulations based on a simple “chemophoresis” model (19, 21).

Fig. S1.

A simple mechanical model of the microbead driven by a fixed driving force in the presence of noise does not recapitulate the observed directed persistent movement. (A) Dependence of persistency (defined in the main text) and velocity on the driving force. Note that, at a velocity of 0.1 μm/s, the corresponding persistency of the movement in this simple model is ∼0.2, much smaller than the persistency of >0.6 observed experimentally (14). (B) A representative simulation trajectory with a fixed driving force of 5 fN. In this case, fluctuations of the bead position are significant and the persistency of movement is much lower than with that from our mechanochemical model (Fig. 2). For all simulated trajectories, the bin time (2 s) was chosen to be the same as that of the tracking experiments.

Mechanochemical Model.

Because a constant driving force by itself cannot account for the directed and persistent movement, additional forces are needed to restrict the diffusive motion in the orthogonal directions. To address this missing component, we developed a mechanochemical model that extends the previously proposed diffusion ratchet mechanism (14). In our model, the chemical bond between ParA and ParB not only drives movement but also provides the transient tethers that quench the orthogonal excursions from the principal direction of motion. Instead of driving a power stroke as in conventional linear stepping motor proteins, ATP hydrolysis renders ParA–ParB bond dissociation events essentially irreversible, resulting in the generation and perpetuation of a spatial asymmetry in the ParA distribution around the moving cargo (i.e., a ParA concentration gradient). Thermal motion is rectified via this spatial asymmetry in binding affinity to drive directed and persistent movement. Our model essence is thus a Brownian ratchet, similar to a burnt-bridge mechanism (22, 23). However, unlike typical burnt-bridge systems in which the cargo interacts with and inactivates a single substrate site (the “bridge”) at a time, our model involves a large number of ParB bound to the cargo, interacting in parallel with a large number of surface-bound ParA. The movement reflects the collective behavior of the formation and dissociation of these bonds, which could differ substantially from conventional burnt-bridge models. For example, the driving force from individual ParA–ParB bonds does not necessarily need to overcome thermal noise and contribute directly to the persistent forward motion of a cargo, as long as the ensemble collectively produces persistent forward motion. We now set out to test the feasibility of this hypothesis.

To discern the fundamental principle, we considered the simplest scheme (Fig. 1). ParA⋅ATP and ParB are tethered through DNA to the substrate and microbead surface, respectively. For the initial exercise, we assumed that ParA⋅ATP converts to the ADP-state obligatorily and only upon binding a ParB dimer on the microbead (Fig. 1A). Because ParA·ADP rapidly dissociates from DNA, and there is a sufficiently long time delay before ParA·ADP converts back to ParA·ATP (17), we assumed that, once the ParA⋅ATP–ParB bond breaks, the site occupied by the ParA remains empty and incapable of binding ParA or ParB. Without explicit representation of ATP hydrolysis, we thus consolidated all reaction pathways pertaining to ATP hydrolysis into one irreversible step. Additionally, the ParA⋅ATP–ParB bond in the model effectively refers to the chemical bond between ParB, ParA⋅ATP, in addition to the DNA segments that connect the proteins to the microbead and the substrate (14, 17). From a mechanical viewpoint, this ParA⋅ATP–ParB bond acts as an elastic spring (Fig. 1 B and C), which implicitly includes DNA elasticity influenced by DNA crowding and ParB-mediated DNA cross-linking (Supporting Information). The deformation of this spring could generate restoring forces on the bead. Collectively across the bead surface, the vector sum of many ParA⋅ATP–ParB bonds could thus generate a net driving force that displaces the bead. Whereas the ParA⋅ATP–ParB bond has an intrinsic dissociation rate, the bond length change (hence the tensile force change) associated with the bead movement alters the ParA⋅ATP–ParB bond dissociation kinetics. Like a typical chemical bond, the ParA⋅ATP–ParB bond is weakened by force and breaks if stretched beyond a critical limit by the bead movement. This dissociation results in the depletion of ParA behind the bead. Meanwhile, ParB on the leading edge continues to establish new bonds with ParA on the unexplored regions of the substrate, where the ParA⋅ATP concentration is higher. The microbead movement therefore maintains the asymmetric biochemical environment that in turn supports further forward movement. Thus, the essential feature of this model hinges on the nature of the mechanochemical coupling.

Fig. 1.

Schematic description of mechanochemical model. (A) Simplified ParA and ParB reaction scheme. (B) Top view of the model layout. (C) Side view of a series of microbead movements arising from the coupling between elasticity, formation, and dissociation of ParA·ATP–ParB bonds. For clarity, only a handful ParA and ParB dimers are shown, whereas there are hundreds to thousands of them in the simulations. The microbead surface is grafted with DNA that binds ParB dimers (green filled circles). The substrate is coated with DNA to which ParA⋅ATP dimers are bound (magenta ovals). The ParA·ATP–ParB bond in the model effectively refers to the DNA–ParA·ATP–ParB–DNA linkage, which acts as elastic spring and generates a restoring force when stretched. Taking into account experimental conditions, the model treats the DNA in the bond as an effective linear spring that reflects the collective behavior of the DNA on the surfaces, instead of a single polymer. This is due to the dense DNA packing on the surface and the ParB-mediated DNA bridging of closely packed DNA on the bead (Supporting Information). (D) Dependence of ParA·ATP–ParB bond dynamics on the bond length. (Upper) Definitions of different lengths in the model. L_e is the equilibrium length of ParA·ATP–ParB bond. L is the instantaneous ParA·ATP–ParB bond length. L_a is the maximum length of a newly formed ParA·ATP–ParB bond. X_C is the maximum bond length extension. (Lower) Dependence of ParA·ATP–ParB bond dissociation rate on bond length extension. When it extends beyond (L_e + X_C), the ParA·ATP–ParB bond breaks instantaneously. Importantly, ParA·ATP and ParB can form a bond with an extended bond length, longer than the equilibrium length L_e (Fig. 1D). As the maximum energy penalty from this extension $\frac{1}{2}{k}_{\mathrm{s}}{\left(L_{\mathrm{a}}-L_{\mathrm{e}}\right)}^2$ is less than k_BT, newly formed bonds are typically prestretched by thermal energy. When these prestretched bonds drive the microbead to move rightward (as in Fig. 1B), the bonds near the left end of the bead dissociate. The bond dissociation is assumed to immediately convert the ParA⋅ATP to ParA⋅ADP (blue rectangles) that releases from the substrate without delay. The resulting vacancy thus remains noninteracting throughout the simulation (gray rectangle). As the bead forms more new prestretched bonds to its right than to its left, the asymmetry in the bonds drives further rightward movement. This way, the irreversible ParA·ATP–ParB bond dissociation rectifies the thermally driven movement, perpetuating the directional movement. Hence, the model essence is a Brownian ratchet that harnesses the mechanochemical feedback between chemical bond dynamics, mechanical force, and bead movement.

We quantitatively studied the dynamics of microbead movement by Brownian dynamics simulations. Because the bead does not roll during persistent movement (14), only its proximal portion contacting the substrate is relevant. The model therefore simplified the microbead as a flat circular disk. In simulations, ParA·ATP was initially distributed with 10-nm spacing uniformly over the substrate surface. A total of 5,000 ParB dimers was uniformly distributed over the surface of the disk that has a diameter of 1 μm. The positions of ParA and ParB in the model represent the anchoring points of the DNA molecules on the substrate and the cargo disk surface, respectively, to which the proteins bind. We modeled the ParA⋅ATP–ParB bond as an elastic spring. The vertical distance between the substrate and the disk was fixed at the assumed equilibrium bond length of DNA–ParA⋅ATP–ParB–DNA (L_e) throughout the simulation (Fig. 1D). The subsequent stochastic reactions between ParA and ParB were simulated with the kinetic Monte Carlo scheme. At each simulation time step, each ParB dimer could interact with the available ParA⋅ATP within a distance L_a (Fig. 1D), and bind only one ParA at a time. The probability of binding is proportional to $\exp \left(-\frac{1}{2}{k}_{\mathrm{s}}{\left(L-L_{\mathrm{e}}\right)}^2/k_{\mathrm{B}}T\right)$ for $L_{\mathrm{a}}\ge L\ge L_{\mathrm{e}}$; otherwise, it is zero. Here, k_s is the spring constant of the bond. L denotes the separation between the ParB and the ParA⋅ATP. If this bond forms, L is the instantaneous bond length, and $\frac{1}{2}{k}_{\mathrm{s}}{\left(L-L_{\mathrm{e}}\right)}^2$ represents the associated elastic energy penalty. Importantly, given the model parameters (Table S1), the maximum of this energy penalty is less than thermal energy k_BT. Consequently, thermal energy can be readily harnessed to prestretch the new bond, which in turn provides an elastic force. In the simulation, we summed the elastic forces from all of the ParA⋅ATP–ParB bonds over the disk surface. This net force coupled with diffusion drives the motion of the microbead for one time step. The model ignores the force in the z direction, which is balanced by the combination of magnetic force, gravitation, and charge repulsion between the bead and substrate in our in vitro experiments. In the next time step, the bond lengths of the ParA⋅ATP–ParB complexes were updated by this motion, from which the dissociation rates of the existing ParA⋅ATP–ParB complexes were calculated according to the following force dependence (Fig. 1D): When the bond extension (L − L_e) > X_C, the bond breaks instantaneously; otherwise, the dissociation rate is determined from $k_{\mathrm{off}}\left(f\right)=k_{\mathrm{off}}^0{e}^{f/f_{\mathrm{C}}}$, where $k_{\mathrm{off}}^0$ is the intrinsic bond dissociation rate, f is the elastic force from the bond stretching f = k_s(L − L_e), and f_c is the characteristic bond dissociation force. The dissociation reaction was next implemented in the stochastic simulation. Meanwhile, because of the bead movement in the previous time step, ParBs that were not in the ParA-bound form could now explore the new territory and form bonds with available ParAs. We then updated the net force from all of the ParA·ATP–ParB bonds, including changes in the existing and the newly formed complexes. The bead movement was then calculated as in the previous time step. We repeated these steps throughout the simulation over time.

Table S1.

Model parameter table

Parameter	Physical meaning	Value, units	Reference
ks	Elastic constant of the ParA⋅ATP–ParB bond	0.05 pN/nm*	Estimated from ref. 40
$k_{\mathrm{off}}^{\left(0\right)}$	Intrinsic dissociation rate of ParA⋅ATP–ParB bond	3/s	This article
$k_{\mathrm{on}}$	Association rate of ParA⋅ATP–ParB bond	5 ×104/s	This article
R	Bead radius	0.5 μm	Ref. 14
fc	Characteristic bond dissociation force	0.8 pN	This article
NB	Number of ParB on the bead	∼5,000	Ref. 14
[A]	ParA density	1/100 nm2	Ref. 17
D	Microbead diffusion constant	0.1 μm2/s	Ref. 14
Le	Equilibrium bond length	100 nm	This article
La	Maximum bond length for newly formed bond	103 nm†	This article
XC	Threshold bond length extension of existing bonds	10 nm	This article

^*Due to the dense packing and cross-linking, the DNA in the model represents an effective elastic element instead of a single chain. Please see Supporting Information for detailed estimation.

^†The maximum bond length of newly formed bond corresponds to the scenario that the microbead-bound DNA–ParB complex can bind to the substrate-bound ParA within the lateral range of ∼50 nm.

We stress that the model essence remains robust over a broad range of model parameter (Fig. S2) and does not depend on the specific force dependence of the dissociation rate, as other force-dependent bond dissociation models work equally well (Fig. S2). The model parameters were estimated from existing experimental measurements wherever possible (see Table S1 and the parameter derivations in Supporting Information, Model Parameter Consideration or Estimation). Unless otherwise noted, the Brownian dynamics time step $\mathrm{\Delta }\mathit{t}$ was set to 10 μs in the simulations, which is small enough to account for the fastest reaction/diffusion process in the system. Choosing smaller time steps did not affect the model results (Fig. S3). Below, we first present typical model results.

Fig. S2.

Directed and persistent movement is preserved across a broad range of model parameters. A–D show the dependence of bead velocity on (A) bead diffusion constant, (B) ParA⋅ATP–ParB bond formation rate (k_on), (C) the limit of bond length extension in X–Y plane ($\sqrt{{\left(L_{\mathrm{e}}+X_{\mathrm{C}}\right)}^2-L_{\mathrm{e}}^2}$), and (D) the effective range of new bond association in X–Y plane ($\sqrt{L_{\mathrm{a}}^2-L_{\mathrm{e}}^2}$). (E) Catch-slip–type bond behavior enhances the persistency and increases the velocity of directed and persistent movement slightly. For catch-slip–type behavior, the bonds are stabilized when stretched due to a decrease in dissociation rate compared with the nonstretched state.

Fig. S3.

Robustness of model results to variation of simulation time step. Note that the typical timescale for protein conformation change is on the order of 10⁻⁶ s. As the model does not resolve any internal dynamics of protein conformation changes, the microsecond-timescale sets the lower bound of the model temporal resolution. On the other hand, the time step cannot be too large. In Brownian dynamics simulation, the displacement from diffusive motion in one time step should not be much larger than the unit size of the lattice; otherwise, reaction events become too rare. Given that the unit size of the lattice L₀ is 10 nm in the simulation and the intrinsic diffusion constant of the microbead D is ∼0.1 μm²/s, the upper limit of the simulation time step is ∼(L₀)²/D ∼ 10⁻³ s, well above the simulation time step of 10⁻⁵ s used for all of the results.

Results

Directed and Persistent Movement Emerges from a Mechanochemical Model of ParA–ParB Interactions.

Fig. 2A shows an example of directed and persistent movement of a microbead generated by our mechanochemical model. According to the model, stochastic fluctuations triggered the initial symmetry-breaking event. Snapshots of the chemical bond distributions on the microbead revealed drastic changes during symmetry breaking (Fig. 2B). At the beginning, the nonuniform distribution of forces from the stochastically formed chemical bonds together with the diffusive motion of the microbead drove its initial movement, which subsequently induced the dissociation of bonds at the trailing end. Bond dissociation is coupled to the formation of ParA-ADP, which dissociates from the substrate. As the bead moved forward, it established new ParA⋅ATP–ParB bonds in the new territory, which provided the net force that drove the bead forward. Consequently, the initial asymmetry was perpetuated and the ensuing bead movement was directed and persistent. There was no preference in the direction of persistent movement (Fig. S4), in line with experimental observations (14).

Fig. 2.

Persistent movement of microbead arises from the mechanochemical model. (A) A typical trajectory (gray path) of directed and persistent movement of a microbead. (B) The top views of the model simulation snapshots showing the spatial-temporal evolution of ParA binding states during the directed and persistent microbead movement as in A. Here, ParA·ATP that does not bind to ParB is in magenta, the ParA depletion zone is in gray, and the ParA⋅ATP that forms ParA·ATP–ParB bond is in blue. The green dashed line marks the boundary of the microbead. For clarity, only 25% of the ParA·ATP–ParB bonds are shown in the figure. (C) Time evolution of the ParA·ATP–ParB bond number as in A. (D) A typical energy landscape of the microbead movement on the substrate. Here, we plotted the energy penalty incurred by virtual displacements (δx and δy) from the instantaneous position of the microbead (marked by the pink circular disk). These virtual displacements do not involve bond formation or dissociation. The green star marks the lowest energy displacement, which has a negative energy barrier. That is, there exists a driving force propelling the microbead toward the location of this lowest energy point, consistent with the overall persistent direction of movement (marked by the arrow).

Fig. S4.

Histogram of the initial motion direction shows that there is no directional bias in the symmetry breaking of bead movement.

Furthermore, the model predicted the differential spatial pattern of ParA around the microbead (Fig. 2 A and B). Whereas the leading edge was enriched in ParA⋅ATP–ParB bonds, the microbead left a path of ParA depletion in its wake. The microbead movement essentially burnt its own bridge and there was no way back. This self-generated polarity provided an overall directed driving force that was in the persistent direction. Note that, if we incorporated the diffusion of ParA molecules, ParA·ATP rebinding to substrate, and an explicit reverse reaction in the initial binding step, the sharp boundary of the ParA depletion zone behind the microbead was smoothed out (Fig. S5), remarkably resembling experimental observations (14). Importantly, including these additional factors did not qualitatively affect the directed and persistent movement (Fig. S5). We therefore ignored these more detailed processes for the remainder of the results, keeping the model minimal.

Fig. S5.

Directed and persistent movement is preserved in more detailed ParA/ParB reaction schemes. (A) Reaction scheme. This version of the model includes the following: (i) the reverse reaction that dissociates ParA·ATP–ParB into ParA·ATP on the substrate and ParB on the microbead without the conversion of ATP to ADP in ParA; (ii) ParA·ATP diffusion on the substrate; (iii) exchange of ParA·ATP between the substrate and the cytoplasm; (iv) dissociation of ParA·ADP on the substrate; and (v) nucleotide exchange between ParA·ATP and ParA·ADP in solution. Step v is typically very slow (∼10 min) and hence does not appreciably affect the final outcome. For B–D, the additional parameter values in the reaction scheme were as follows: k_a = 0.01/s, k_d = 1/s, k_on = 5 × 10⁴/s, k_off = 2.7/s, k_r = 0.3/s, and ParA diffusion constant along the substrate was 10³ nm²/s reflecting transient ParA–DNA substrate interactions. (B) Simulation snapshot showing the spatial distribution of ParA·ATP (blue), ParA⋅ATP–ParB bonds (magenta), ParA·ADP (light green), and the ParA depletion zone (i.e., nonspecific DNA, white) around the microbead during directed and persistent movement. (C) The trajectory of the directed and persistent movement of the microbead corresponding to B. In this plot, the microbead moves processively for 20 s. (D) A typical spatial profile of ParA (ParA·ATP and ParA·ADP) along the orthogonal direction of the microbead movement corresponding to B. Note that, with the more complicated reaction-diffusion scheme, the concentration gradient of ParA is smoother while directed and persistent movement is preserved.

The above results are consistent with the proposed diffusion ratchet mechanism (14, 18). However, the important question left unanswered by the previous proposal is: What restricts microbead deviation orthogonal to the principal movement direction? The answer lies in the tethering of the microbead to the substrate by the ParA⋅ATP–ParB bonds. Fig. 2C shows the temporal evolution of the bond number during microbead movement. At steady state, there were ∼250 bonds for this set of model parameters, analogous to 250 springs tethering the microbead to the substrate. From the viewpoint of the energy landscape, these tethers imposed an energy barrier quenching diffusion (Fig. 2D). To calculate the tethering effects on the motion, we chose a time point at which the microbead was undergoing steady-state movement, and we virtually moved it through various displacements without changing the status of existing bonds. We then calculated the resulting changes in the elastic energy from these chemical bonds. As shown in Fig. 2D, an orthogonal deviation of 5 nm incurred an energy penalty of ∼2.5 k_BT. These bonds collectively defined a potential that trapped the microbead and suppressed lateral deviations. Importantly, the force involved at each time step was in the piconewton range as indicated by the energy landscape (energy/displacement) (Fig. 2D). However, because the microbead movement driven by chemical bond-mediated forces was gated by the dissociation of these bonds, this effective internal resistance greatly slowed the speed. Consequently, the net driving force in the femtonewton range equaled the difference between the instantaneous driving force and the chemical bond-mediated internal resistance. That is, these chemical bonds not only provided the driving force for microbead movement but also suppressed deviations in the orthogonal direction and slowed directed motion. This finding resolves a point of confusion concerning plasmid diffusion constants in vivo. The measured apparent diffusion constant of plasmid movement during directed movement in vivo is small (24), which has been interpreted as an indication that diffusion ratchet mechanisms are not capable of generating directed plasmid movement (24). Our results demonstrate that the slow apparent diffusion is a natural consequence of free diffusion of the plasmid constrained by the ParA⋅ATP–ParB bonds, similar to our microbead experiments (14). Our model thus predicts that, instead of diffusion-limited motion, the mechanochemical coupling ensures directed and persistent movement.

The Number of ParA⋅ATP–ParB Bonds and Cargo Size Influence Movement Persistence and Velocity.

We next investigated how the interplay between the number of ParA⋅ATP–ParB bonds and the intrinsic mobility of the cargo influence cargo movement. Multiple contributing factors determine the bond number, e.g., the on and off rates, the densities of ParA and ParB on the substrate and on the microbead, respectively. We focused on the effect of ParB density and bead size on the directed and persistent movement.

The model predicted that, for a given bead size and hence a fixed intrinsic diffusion constant, there existed a threshold ParB density on the bead below which persistent movement could not be maintained (Fig. 3A). For a cargo of 0.5-µm radius (diffusion constant of ∼0.1 µm²/s as in ref. 14), the minimal density of ParB is ∼1,000/µm², consistent with estimates of ParB density on P1 plasmid of approximately one dimer per 30 bp (25). This points to the critical role of the “ParB-spreading” process in which ParB binding to parS on the plasmid nucleates higher-order complex assembly involving a large number of ParBs to promote plasmid segregation (26–30). Conversely, if the density of ParB was too high, too many ParA–ParB bonds formed resulting in the decrease of both the persistence and the speed (Fig. 3 A–C). Similar results hold for ParA density. Generally speaking, too many bonds effectively trapped the cargo. Our model thus provides an explanation for the frequent loss of plasmids during cell division when either ParA or ParB are in excess (31). According to our model, these phenotypes could arise from the loss of directed and persistent movement by the ParA/ParB-mediated transport machinery as plasmids become stuck to nucleoid by too many ParA⋅ATP–ParB bonds. Last, there appeared to be a critical cargo size below which diffusive movement persisted. These findings are consistent with observations that, although in principle the ParA/B system could mediate transport of diverse cargos, these cargos tend to be massive (32). The directed and persistent movement was favorable for beads of intermediate size, as the speed and persistence fell off relatively slowly as the bead size increased (Fig. 3B).

Fig. 3.

Microbead movement depends on the ParB density on the cargo and the size of the cargo. (A) Phase diagram shows that the distinct interplay of the ParB density on the cargo and the size of the cargo can give rise to different modes of microbead movement. Here, we define the “directed and persistent movement” when the persistency of the trajectory is ≥0.7. Note that the viscous drag coefficient in the current simulation corresponds to that of water. In vivo, the cytoplasm is much more viscous, and thus the diffusion of the same cargo is much slower. This will shift the minimal cargo size for directed and persistent movement to a smaller size. The yellow star represents the nominal set of model parameters over which either bead size or ParB density is varied alone in B and C, respectively. (B) Biphasic dependence of the speed of the directed and persistent movement on cargo size. (C) Biphasic dependence of the speed of the directed and persistent movement on the ParB density on the cargo.

Mechanochemistry of the ParA⋅ATP–ParB Complex Governs Microbead Motility.

We next studied how the mechanochemical coupling of the ParA⋅ATP–ParB bond governed microbead motion. Fig. 4A depicts the calculated phase diagram of the dependence of bead movement on the elasticity and lifetime of the ParA⋅ATP–ParB bond (∼1/k_off). When the spring constant of the bond was reduced, the bead underwent more diffusive motion, as the energy barrier preventing the bead deviation was insufficient to quench thermal fluctuations. Interestingly, a very large spring constant also resulted in diffusive motion. This was because larger forces were generated by these stiffer bonds and, subsequently, moved the microbead over a larger distance at each time step. When this longer “stride” stretched the chemical bond beyond its bond length limit, it caused ParA⋅ATP–ParB bond dissociation. Consequently, the microbead was essentially set free, as it was no longer tethered to the surface. Our model thus predicted that persistent movement could be maintained only for an intermediate range of bond elasticity. Within this persistent movement regime, the speed increased marginally with the bond spring constant (Fig. 4B). On the other hand, the bond lifetime critically dictated the mode of microbead movement. For a given spring constant, decreasing the bond lifetime resulted in less persistent movement. In the limit of decreasing lifetimes, the ParA⋅ATP–ParB bonds would break even before the mechanical forces from these bonds could appreciably move the microbead (Fig. 4 A and C). That is, the mechanical action was decoupled from the chemical reaction. Conversely, a longer bond lifetime prolonged the mechanical coupling and improved directional persistence until too many tethers blocked bead motion.

Fig. 4.

Mechanochemical coupling controls ParA/ParB-mediated microbead movement. (A) Phase diagram shows that directed and persistent microbead movement depends on the stiffness (k_s) and the intrinsic dissociation rate ($k_{\mathrm{off}}^0$) of ParA⋅ATP–ParB bonds. As above, directed and persistent movement corresponds to trajectory persistence of ≥0.7. The yellow star represents the nominal set of model parameters over which either the bond stiffness or the bond dissociation rate is varied alone in B and C, respectively. (B) The speed and the persistence of directed and persistent movement is insensitive to the ParA⋅ATP–ParB bonds stiffness. (C) The speed and the persistency critically depend on the dissociation rate of ParA⋅ATP–ParB bonds.

Our model further predicted that, with a combination of a smaller bond spring constant and a relatively longer bond lifetime, the microbead displayed saltatory movement instead of smooth translocation. Fig. 5 A and B show typical trajectories of such saltatory movement and the corresponding temporal evolution of the number of ParA⋅ATP–ParB bonds, respectively. The microbead spent periods of time restricted within a small area undergoing tethered Brownian motion, and occasionally took rapid jumps, after which it was again confined. During this process, the number of ParA⋅ATP–ParB bonds oscillated, with more bonds corresponding to confined diffusion and fewer bonds corresponding to jumps (Fig. 5 A and B). The model suggested that this jerky movement occurred due to the asynchrony between the mechanical action and the chemical reaction of the ParA⋅ATP–ParB bonds. Whereas the mechanical forces were sufficient to drive the microbead forward, the chemical bonds persisted and kept the bead tethered. With the rupture of a sufficient number of bonds, the bead could undergo sizable movement (a jump) driven by the mechanical force. Once a significant fraction of bonds broke, the remaining bonds were insufficient to quench diffusion. Subsequent diffusion not only introduced randomness in the movement direction but also prevented tethering and drove larger strides until a sufficient number of ParA⋅ATP–ParB bonds were reestablished to quench diffusion. Consequently, the number of ParA⋅ATP–ParB bonds negatively correlated with the step size during saltatory movement (Fig. 5 A and B). This saltatory movement thus manifested another facet of the underlying mechanochemical coupling mechanism and constituted a unique prediction of the model, which was absent from previous simple diffusion ratchet models (14, 19, 21).

Fig. 5.

Saltatory movement reflects competition between ParA⋅ATP–ParB bond chemistry and mechanics. (A) A representative trajectory of saltatory movement from simulations. (B) The number of ParA⋅ATP–ParB bonds during the saltatory movement in A. (C) A representative saltatory trajectory of a microbead from in vitro experiments. (D) The instantaneous microbead step size (red line) and the instantaneous fluorescence intensity (green line) of the microbead, which due to the TIRF illumination is inversely proportional to the distance of the microbead from the substrate, during the saltatory movement in C. For A and C, points on the x–y plot correspond to the position of the center of the 1-µm microbead at 2-s intervals.

To test this prediction, we analyzed the trajectories of microbead movements in reconstitution experiments previously described (14). The modes of microbead movement were diverse, likely reflecting heterogeneity with respect to the surface densities of SopA, SopB, and DNA, and the variation in the surface confinement force. Specifically, whereas some beads underwent directed persistent movement and some were stuck to the substrate, there was a sizable fraction of microbeads that displayed saltatory movement, alternating between confined diffusive motion and long strides (Fig. 5C). We also observed that the proximity of the SopB-coated microbead to the DNA-coated surface, as inferred from the microbead fluorescence intensity in the exponentially decaying evanescent field generated by total internal reflection, was anticorrelated with the step size of the microbead movement with average correlation coefficient of −7.1 ± 4.1 (n = 12). Namely, the microbead fluorescence intensity decreased during long strides in the saltatory movement (Fig. 5D). The microbead fluorescence intensity decrease in the total internal reflection fluorescence (TIRF) field reflects an increase in the microbead–substrate distance and vice versa. We interpreted the transient “hopping” of the microbead away from the substrate as evidence for loss of these tethering bonds.

ParA/ParB-mediated plasmid motility in vivo has been shown to vary. “Smooth” translocation, where the plasmid continuously moves close to the receding edge of a ParA (or SopA) depletion zone on the nucleoid, has been observed in vivo for pB171 plasmid and F plasmid (33). More recently, both plasmids have also been shown to support saltatory movements, whereby the plasmids are for the most part immobile and uniformly distributed at relatively fixed positions on the nucleoid (34). For P1 plasmid, ParA distributes uniformly over the nucleoid and also forms colocalized foci with immobile plasmids (16). During plasmid movement, these colocalized ParA foci disappear and only reappear once the plasmids have been repositioned. These “stick-and-move” dynamics are consistent with the “saltatory” regime of the diffusion ratchet model described here. We conclude that diverse modes of ParA/ParB-mediated motility likely use the same diffusion ratchet mechanism, but subtle differences, some of which are emphasized here, could produce transitions between smooth and saltatory modes of plasmid movement observed in vivo.

Our combined work thus suggests that directed and persistent movement requires a sufficient number of ParA⋅ATP–ParB bonds to transiently tether the microbead and quench diffusion while still being able to dissociate rapidly enough to drive microbead movement. The proper coordination between the timescales of the mechanical action and the chemistry of the ParA⋅ATP–ParB bonds orchestrates the synchrony between bond tethering and dissociation in driving persistent cargo movement.

Response of ParA/ParB-Mediated Motility to an Opposing Force.

Last, we investigated how ParA/ParB-mediated motility responds to opposing forces. Fig. 6A shows representative simulation trajectories of the microbead in which a force was applied in the opposite direction of the bead movement after the bead began directed and persistent movement. At a small opposing force (∼0.03 pN), the microbead simply diverted its path (Fig. 6A). A larger force of 1 pN or more caused the microbead to turn sharply and move backward (Fig. 6A). Regardless of the magnitude of the opposing force, ParA/ParB-mediated motility never stalled. The model predicted that, whereas an opposing force could readily alter the direction of the ParA/ParB-mediated motility over time, it did not change the speed very much until the opposing force significantly exceeded 1 pN (Fig. 6B). In the latter case, the large external force broke the bonds and drove the microbead directly, which did not reflect the inherent ParA/ParB-mediated motility.

Fig. 6.

Response of ParA/ParB-mediated directed movement to opposing force. (A) Model predictions for diversions of ParA/ParB-driven movement of microbead by an opposing force with constant direction and magnitude. Example trajectories for the cases of small (F = 30 fN) and large (F = 1 pN) opposing forces are shown. For comparison, the corresponding trajectory when there is no external force (F = 0) is shown (light blue line); the original direction of this trajectory is marked by the blue arrow (“V”). The gold dot on the trajectory marks when the external force is switched on in the simulation, and trajectories of the subsequent movement are shown in different colors (dark blue for F = 30 fN, and green for F = 1 pN). (B) Dependence of microbead speed on opposing force. The blue-shaded zone marks the regime in which the external force breaks the existing bonds and drives the microbead movement in a purely mechanical manner. The resulting speed of the microbead is thus independent of ParA/ParB system. (C) Experimental observation of moving microbeads diverted by repulsive force from stuck magnetic beads. Representative trajectories of five SopA/SopB driven beads (colored heavy lines) moving on the surface of the flow cell interacting with a stuck bead (red dot in middle of white circle). Due to the external magnetic field, the magnetic moments of the beads are aligned perpendicular to the plane of motion, which results in a repulsive interaction between beads that falls off as the inverse fourth power of their separation. (D) Simulated trajectories of the microbead movement in the presence of the same magnetic dipole–dipole interaction as in C. The position of the virtual microbead is indicated by the red dot, and five independent trajectories are shown. Note that C and D share the same force scale: The log of the repulsive force in the vicinity of the stuck beads is represented by the contour lines (spaced every 0.5 log units from the maximum of 1 pN at the purple contour) and the white shading (force scale on right side of Fig. 6C).

The predicted trajectory diversion and resilience of speed in the regime of directed and persistent movements are natural consequences of the diffusion ratchet mechanism. The resisting force presents a bias to the stochastic fluctuations that gradually remodel the distribution of ParA⋅ATP–ParB bonds over time, and ultimately drive the microbead in the direction of the force. Such behavior is in contrast to conventional motor proteins such as kinesin, whose motility and hence stepping uses the chemical energy in a more deterministic manner. In other words, the Brownian-ratchet nature of ParA/ParB-mediated motility has the capacity to avoid obstacles, which may be important for transporting large cargos inside bacteria.

To experimentally examine this prediction, we exploited an experimental configuration in which some magnetic beads stuck on the substrate (probably due to a higher than average density of sopC-SopB on the bead), whereas others underwent directed and persistent movement (Fig. 6C). The external magnetic field aligns the magnetic moments of the beads perpendicular to the x–y plane, which results in a repulsive dipole–dipole interaction between beads that falls off as the inverse fourth power of their separation. The fixed beads therefore produced a distance-dependent opposing force against the moving beads, the magnitude of which can be determined from the known magnetic properties of the beads and the applied magnetic field (14). We observed that the microbeads moving in head-on paths toward a fixed bead got diverted more than those moving marginally (Fig. 6C). The speed changes somewhat after the diversion (27 ± 40%, n = 6). Our model simulation with the same distance-dependent opposing force shows the same behavior (Fig. 6D). Our experimental observation thus confirms this diversion behavior, which highlights the underlying mechanism as Brownian ratchet rather than a more deterministic mechanism.

Discussion and Conclusions

In this study, we developed a theoretical model for ParA/ParB-mediated directed and persistent cargo motility observed both in vitro and in vivo. Our model adds to the growing body of data refuting the filament-based models for ParA-mediated plasmid motion.

Combining the mechanistic insights from the model with experimental results, we arrived at the following picture of the ParA/B-mediated directed and persistent motion (also see our analytic results in Supporting Information and Fig. S6). Initially, with the depletion of the ParA⋅ATPs directly underneath the microbead (Fig. 2 and Fig. S6A), thermal fluctuations displace the microbead in a random direction without causing significant restoring forces, thus breaking symmetry. This displacement makes more ParA⋅ATP available for ParB binding on one side of the microbead, subsequently referred to as the front. Because the energy penalty associated with the bond extension is small [e.g., $\frac{1}{2}{k}_{\mathrm{s}}{(L_{\mathrm{a}}-L_{\mathrm{e}})}^2<<k_{\mathrm{B}}T$], newly formed ParA⋅ATP–ParB bonds are typically prestretched by thermal fluctuations (i.e., L_a > L_e; Fig. 1D and Fig. S6A). The asymmetric ParA⋅ATP distribution around the microbead thus creates conditions favoring the formation of more forward-stretched new bonds at the front than the backward-stretched bonds at the rear (Fig. S6A). This results in a net forward bias of the stretched bond orientation, and further drives the microbead movement in the direction of the displacement. The resulting asymmetric chemical environment (i.e., binding site distribution) serves to rectify thermal fluctuations of the cargo and perpetuate the directed and persistent movement. This is characteristic of a Brownian ratchet.

Fig. S6.

Model schematics model for analytical study of directed and persistent movement. (A) Quasi-1D representation of the system for linear stability analysis. The ParA-covered substrate and the ParB-covered microbead are represented by the gray or blue lines and the green line, respectively, whose spatial coordinates are represented by x₁ and x₂. The gray line represents the ParA depletion zone, which also corresponds to the original location of the microbead before the perturbed displacement. The right boundary of the ParA depletion zone marks the origin for both the x₁ and x₂ axes. From the steady state, the microbead is perturbed by a displacement $\delta l$ in the positive direction of x₂. The maximum bond length of newly formed bonds is L_a, and the equilibrium bond length is L_e. This way, the size of the available binding zone on the right is $\mathrm{\Delta }L+\delta l$, whereas that on the left of the bead is $\mathrm{\Delta }L-\delta l$. It is this asymmetric binding environment that sets the stage of perpetuating the directed movement. (B) Schematic of directed and persistent movement of the microbead in a mean-field approximation. Here, all of the bonds at the leading edge are assumed to behave the same way. The lower panel shows an expanded region of the bond dynamics of a single ParB. Just before the bond breaks (the dashed line), the bond extension is L_break. The origin for the x₁ axis is marked at the right leading edge of the microbead. The new bond forms in the region $L_{\mathrm{break}}<x_1<\mathrm{\Delta }L$, and is represented by the thick blue line.

At steady state, we derived an analytic expression for the ParA/ParB-mediated motility speed based on a mean-field approximation of the model (Supporting Information). It suggests that the speed is proportional to the maximum lateral distance between the ParB on the microbead and the ParA⋅ATP on the substrate for which binding can occur (i.e.,$\sqrt{L_{\mathrm{a}}^2-L_{\mathrm{e}}^2}$; Fig. 1D). This maximum lateral distance is analogous to the step size of a linear stepper motor protein. Additionally, the speed is inversely proportional to the effective timescale arising from the interplay between the mechanical response of the system and the chemical reactions of ParA⋅ATP–ParB bond formation and turnover. This effective timescale in ParA/ParB-mediated motility is thus similar to the duration of ATP hydrolysis cycles in conventional motor proteins.

However, we emphasize that our mechanism fundamentally differs from the stepping mechanism of conventional molecular motors. Although ATP hydrolysis is not explicitly described in the model, it is involved in the irreversible ParB-stimulated dissociation of ParA from the DNA carpet. Consequently, instead of directly driving a power stroke as in conventional motors, ATP hydrolysis is primarily involved in generating the ParA depletion zone behind the moving cargo. Whereas the spatial ParA concentration gradient of the ParA-depleted zone drives forward movement, it does not impose a priori constraint on lateral diffusive movement. Persistent movement requires a sufficient number of ParA⋅ATP–ParB bonds to tether the cargo and quench diffusive motion in the lateral directions while dissociating frequently enough to allow forward motion (Figs. 2–5). We note that this tethering effect opposes lateral excursions with a shallow energy barrier of approximately a few k_BT (Fig. 2D); escaping from the potential well becomes probabilistic. Indeed, significant lateral excursions were predicted to occur occasionally (Fig. S7), consistent with the in vitro experiments (14). Together, the mechanochemical coupling of the ParA⋅ATP–ParB chemical bonds drives the forward cargo motion, imposes resistance to the forward motion, and constrains diffusive course deviation ensuring persistent movement (Figs. 2–5). The transient tethering aspect of the model provides an explanation for the apparent reduced diffusion constants measured in both in vitro and in vivo experiments (14, 24, 35).

Fig. S7.

An example trajectory showing directed but not persistent movement. The bead moved tens of microns before making wide turns.

Recently, a “DNA relay” model proposed that, for the Caulobacter crescentus ParABS system, DNA-bound ParA-ATP dimers tether cargo to the bacterial chromosome but require chromosomal elastic dynamics to generate the translocation force that “relays” cargo from one DNA region to another (35). This mechanical contribution from the chromosome was evoked because simulations of a simple “diffusion-binding” model indicated that the cargo diffused too slowly to produce directed cargo movement (35). However, the diffusion constant of the cargo was likely significantly underestimated in this simulation. The modeling was performed with the measured diffusion constant (35), rather than an effective free diffusion constant chosen to produce simulation results that recapitulate the experimental data, including the measured diffusion constant. As a result, the decrease in diffusion due to tethering was effectively doubled because the experimentally determined diffusion constant, which necessarily includes the hindering effects of the tethering process, was input into the model that further reduced the effective diffusion due to tethering of the cargo in the Brownian simulations (35). Furthermore, the DNA relay model requires a ParA/ParB bond lifetime that far exceeds measured values of the bond lifetime for related Par and Sop systems (17). Refinements in modeling require further experimental delineation of mechanistic details of the apparently different behaviors displayed by different ParABS systems.

In our model, the ParA/ParB-mediated Brownian-ratchet motility is mechanically driven by the collective dynamics of an ensemble of bonds. We note that the Brownian-ratchet mechanism does not strictly depend on the direct mechanical pulling force of individual bonds, as it solely hinges on how the asymmetry of the system rectifies thermal fluctuations. The cargo could therefore undergo directed motion through a simple burnt-bridge Brownian-ratchet mechanism without chemical bond-mediated mechanical forces. However, in the absence of mechanical force, the cargo is not tethered. Consequently, movement persistency cannot be readily maintained, as nothing prevents diffusive excursions in the orthogonal directions of the directed motion, similar to the predicted case with a very small bond spring constant (Fig. 4).

The mechanochemical coupling mechanism examined here does not constrain certain aspects of the biochemical parameters, e.g., the temporal sequence of ParA⋅ATP–ParB bond dissociation and ATP hydrolysis. The model assumes a cutoff bond length beyond which the ParA⋅ATP–ParB bond dissociates immediately. The model essence is preserved when this cutoff length varies (Fig. S2C). Moreover, most of the bonds spontaneously dissociate at steady state in the model rather than being stretched to the breaking point. Similar dissociation kinetics could arise from more elaborate biochemical dissociation pathways in reality including multiple substeps or nonlinear terms. Quantitative biochemical characterization of ParB-stimulated ATP hydrolysis of DNA-bound ParA⋅ATP will assist further refinement of the model in the future.

Last, because bacteria do not have typical linear stepper motors, the operational principle of ParA/ParB system perhaps played important roles in a variety of biological systems since early evolution, and could thus be conserved in many systems. In fact, this driving–quenching mechanism is similar to the “Pac-Man”–type motility evidenced for the DAM1 ring (36, 37). The DAM1 ring directly connects chromosomes “end-on” to microtubules in yeast mitosis. Whereas the Dam1 ring is highly diffusive (36), its end-on binding to microtubule suppresses its diffusive excursions and allows it to steadily track the depolymerizing microtubule end, driving chromosome movement with high fidelity in mitosis (38). Our study distills a general physical principle for collective behavior from properly coordinated chemical reactions. Unlike classical molecular motors, the ParA/ParB system does not have an inherent directionality. Directionality arises from symmetry breaking and self-organization without the guide of a well-defined track. The subsequent motion diverts its direction upon encountering an opposing force, instead of stalling (Fig. 6). Such flexibility reflects the fact that ParA/ParB-mediated motility is driven by the collective behavior of many weak Brownian ratchet-type interactions. Because the cargo motion depends on the ParA depletion zone, multiple cargos in proximity mutually influence their motion through interaction of the depletion zones. Thus, the ParA-based cargo transport systems have been proposed to be able to self-organize equidistant distribution of multiple cargos for partition in vivo. Dissecting these additional features in bacterial cell division will be our future topic of research.

Methods

Proteins.

Protein expression, purification, and labeling were performed as previously described in detail (39).

DNA-Carpeted Flow Cell.

Flow cell assembly, pacification, and carpeting with sonicated salmon sperm DNA was done as previously described in detail (39).

DNA–Bead Coupling.

A biotinylated and Alexa 647-labeled DNA fragment (3.36 kb) containing a sopC centromere site was PCR amplified and coupled to MyOne Streptavidin C1 Dynal beads (Invitrogen) as previously described in detail (39).

Imaging and Analysis.

The illumination, microscope, and camera settings were previously described (14).

Model Parameter Consideration or Estimation

Diffusion Constant of the Microbead (D).

The diffusion constant D ∼ 0.1 μm²/s (14) corresponds to the measured value obtained from the mean square displacement of the microbeads confined near the DNA-coated surface by a magnetic field. The experimental conditions were identical to those in which directed bead movement was measured, except that ParB, ParA, or ATP was not present (figure S2 in ref. 14). If not otherwise mentioned, this diffusion constant is used throughout the paper.

ParA⋅ATP–ParB Equilibrium Bond Length (L_e) and Critical Bond Length Extension (X_C).

In our in vitro reconstitution experiments (14, 17), ParA⋅ATP and ParB are tethered through a segment of DNA to the substrate and microbead surface, respectively. The ParA⋅ATP–ParB bond in the model is an effective entity encompassing the series of linkages between the bead and the surface: DNA–ParA⋅ATP–ParB–DNA. The substrate-bound DNA has an average length of ∼500 bp, but it is affixed to the surface at both ends (17). The microbead-bound DNA is ∼3,300 bp with 12 repeats of the 43-bp sopC binding sequence to which 12 or more SopB dimers bind at ∼1,700 bp from the end tethered to the bead, which is followed by an additional 1,100 bp of DNA (14). In total, there are ∼1,000 DNA segments per bead, each with up to 12 or more SopB dimers bound roughly at the middle of each DNA molecule (14). As a result, the bead-bound DNA in the model represents an effective elastic element with elastic parameters influenced by the close packing of the DNA segments, instead of a single isolated DNA chain. The multiple ParB dimers on the same DNA segment could in principle contribute two effects. First, they could cross-link adjacent DNA segments (27), which will increase the effective elastic constant of the DNA segment in the model (see below). Second, the corresponding contour length of the DNA segment per ParB in the model is significantly reduced to approximately tens of nanometers or shorter. Given that both ParA⋅ATP and ParB are small proteins with typical dimensions of a few nanometers, the majority of the ParA⋅ATP–ParB bond length in the model stems from the DNA segments. With the above considerations, we assumed that the equilibrium length, L_e, of the ParA⋅ATP–ParB bond in the model is 100 nm, which is comparable to but shorter than the contour length of the DNA segments. We further assumed that this bond could be stretched by 10% beyond its equilibrium length before rupturing (X_C = 10 nm). The qualitative results of the model were not sensitive to the values of ParA⋅ATP–ParB equilibrium bond length (L_e) or the threshold bond extension (X_C).

ParA⋅ATP–ParB Bond Stiffness (k_s).

As the DNA segments make up the majority of the DNA–ParA⋅ATP–ParB–DNA bond length, we assumed that the effective bond stiffness in the model arises from the DNA subjected to several considerations. First, on the single-molecule level, the stiffness of DNA resulting from the entropic elasticity depends on the contour length of the DNA chain, l (40): $k_{\mathrm{s}}=3k_{\mathrm{B}}T/2Pl=\left(0.12/l\right)\mathrm{pN}$, where $P=50\hspace{0.17em}\mathrm{nm}$ is the persistence length of DNA. Because the substrate-bound DNA is affixed at both ends, it is not in subject to deformation very much. We thus neglect the contribution of substrate-bound DNA in the bond spring constant. The microbead-bound DNA is 3,300 bp, and the ParB/SopB dimers bind to the ParS sites covering the DNA at ∼1,700–2,200 bp from the end tethered to the bead. The effective DNA segment in ParA–ParB bond is thus 1,700–2,200 bp, which has a contour length 578–748 nm. The spring constant of a single microbead-bound DNA is thus (2.1–1.6) × 10⁻⁴ pN/nm. Second, the ParB-mediated cross-links between neighboring DNA segments could further stiffen the DNA elasticity. As a rough estimate, one can consider the 1,000 DNA segments on the bead as 1,000 springs working in parallel, which results in an effective spring constant of (2.1–1.6) × 10⁻¹ pN/nm. In reality, however, it may not be that all of the DNA segments are cross-linked together. Suppose only 10% DNA is cross-linked, the spring constant is ∼10⁻² pN/nm. Third, ParA and ParB are densely grafted to the surfaces of the substrate and the microbead, respectively. For instance, ∼320 DNA/μm² renders the average distance between neighboring DNA tethers ∼60 nm, which is much smaller than the expected radius of gyration of a 3.3-kbp DNA segment. This presents a crowded environment. As bond extension in the model does not change the bead–substrate distance vertically, it has to tilt and “push” its neighbors in the crowded environment, which incurs repulsions from the neighboring DNA segments. Together, the spring constants of the DNAs that connect ParA to the substrate and ParB to the bead thus reflect the collective property of many interacting DNA segments, instead of that of a single polymer. Whereas the qualitative aspects of the model were not sensitive to the values of k_s within a reasonable range, we chose the value of the bond stiffness as k_s = 5 × 10⁻² pN/nm in our simulation, if not otherwise mentioned. Importantly, with k_s ∼10⁻² pN/nm, our model can recapitulate not only the observed speed and persistency of the bead movement but also the effective quenched diffusion constant evidenced in experiment (14).

ParA⋅ATP Density on Substrate.

The maximum density of ParA on the DNA carpet was estimated to be ∼6/(100 nm²) (17). We used a ParA density of 1/(100 nm²) in our simulations.

ParA⋅ATP–ParB Bond Association Rate (k_on).

Given the density of ParA ∼1/(100 nm²) and the vertical distance between the microbead and the substrate is ∼100 nm, the effective concentration of ParA is on the order of 10⁻³ M. The diffusion-limited reaction rate constant is ∼10⁹/(M⋅s) based on the classical Smoluchowski expression (41), which sets the upper limit of the reaction rate. In our case, this upper limit is ∼10⁶/s. On the other hand, typical protein–protein association rate is ∼10⁶/(M⋅s) (41). Assuming an effective ParA concentration of 10⁻³ M, we obtain an association rate of ∼10³/s. We considered this value as the lower limit of the association rate in the model. The typical ParA⋅ATP–ParB bond association rate was chosen to be k_on = 5 × 10⁴/s in our simulations, if not otherwise mentioned. The essence of model results was not sensitive to k_on in the range between 10³/s and 10⁶/s.

ParA⋅ATP–ParB Bond Dissociation Rate (k_off⁰).

Our model assumes that ParA⋅ATP–ParB bond dissociation immediately and irreversibly triggers the conversion of ParA⋅ATP to ParA⋅ADP, which then releases from the DNA carpet. To estimate this dissociation rate, we took advantage of the experimental measurement of the SopA/SopB system (17), which is homologous to the ParA/ParB system and is believed to have similar biochemical properties. The rate of SopA dissociation from DNA carpet in the presence of DNA and SopB was measured to be ∼0.13/s (17). Because SopA can rebind to DNA carpet in this experiment, the measured rate sets a lower limit for the dissociation rate specified in our model. Using the SopA system as a guide, we typically set k_off⁰ = 3/s in our simulation. Although movement persistency was not sensitive to the k_off values as low as ∼0.1/s, the velocity decreased with reductions in k_off.

Characteristic Force for ParA⋅ATP–ParB Bond Dissociation (f_c).

We assumed a weak force dependence of the ParA⋅ATP–ParB bond dissociation rate as $k_{\mathrm{off}}=k_{\mathrm{off}}^0\exp \left(f/f_{\mathrm{C}}\right)$, where f is the instantaneous force experienced by the bond, and we set f_c = 0.8 pN. The qualitative results of the model were not sensitive to the specific forms of bond dissociation rate or the precise value of f_c.

Timescale of Force Balance.

We can derive the characteristic mechanical response time of the system, at which the force balance can be established. With the diffusion constant of the bead D = 0.1 μm²/s, its effective viscous drag coefficient is γ = k_BT/D = 4 × 10⁻⁵ pN⋅s/nm. With ∼250 bonds at steady state as shown in our typical simulation result (Fig. 2) and the typical spring constant of individual bond chosen to be k_s = 0.05 pN/nm in the model, the effective elastic spring constant of the system is ∼12.5 pN/nm. The viscous drag coefficient divided by this elastic spring constant yields a characteristic timescale ∼3.2 × 10⁻⁶ s, which is the force balance timescale. Our typical simulation time step is 10⁻⁵ s, over which force balance can thus be established. We further note that the force dependence of the bond dissociation rate implicitly assumes that the timescale for establishing force balance for individual bonds (γ/k_s = 8 × 10⁻⁴ s) is much shorter than that of bond dissociation. Only in this limit can the force faithfully be transduced to the bond. As shown in our simulation results, the proposed mechanochemical coupling is effective only when the ParA⋅ATP–ParB bond dissociation rate is relatively slow. The above assumption of timescale separation is thus self-consistent with our model results.

Analytical Treatment

In this section, we describe the mathematical model in more details and elucidate the reason for symmetry breaking and determinants of bead velocity.

Symmetry Breaking.

We describe the model in the continuous limit with Eqs. S1–S3 (Fig. S6):

$$\frac{\partial B_1(\overrightarrow{r},t)}{\partial t}=k_{\mathrm{on}}\left(S\right)\left(B_{\mathrm{T}}-B_1\left(\overrightarrow{r},t\right)\right)A_0\left(\overrightarrow{r},t\right)-k_{\mathrm{off}}^0\exp \left(\frac{L\left(\overrightarrow{r},t\right)-L_{\mathrm{e}}}{L_{\mathrm{C}}}\right)B_1\left(\overrightarrow{r},t\right),$$ [S1]

$$\frac{\partial A_0(\overrightarrow{r},t)}{\partial t}=-k_{\mathrm{on}}\left(B_0-B_1\left(\overrightarrow{r},t\right)\right)A_0\left(\overrightarrow{r},t\right),$$ [S2]

$$\gamma \overrightarrow{V}=k_{\mathrm{s}}{\displaystyle \int d\overrightarrow{r}}{B}_1\left(\overrightarrow{r},t\right)\cdot {\overrightarrow{L}}_{x-y}\left(\overrightarrow{r},t\right)+\zeta \left(\overrightarrow{r},t\right).$$ [S3]

Here, B_T is the total ParB density and is a constant. B₁ represents the local density of ParB on the microbead surface that is bound to the ParA⋅ATP on the substrate, i.e., the local density of ParA⋅ATP–ParB bonds. A₀ is the local density of the free ParA⋅ATP on the substrate. $\overrightarrow{V}$ is the microbead velocity in the X–Y plane. We consider the initial condition that ParA⋅ATP fully covers the entire substrate, the microbead is stationary, and that ParB on the microbead has not established any bonds with ParA⋅ATP on the substrate. Following the simplification adopted in the main text, the microbead surface is modeled as a circular flat disk, which we still refer as the microbead, if not otherwise mentioned. $\overrightarrow{r}$ is the spatial coordinate both for the microbead surface and the substrate area directly underneath it. L is the average length of the existing local ParA⋅ATP–ParB bonds, whereas L_e is the equilibrium bond length, which is assumed to be the vertical distance between the substrate and the microbead surface. The association rate of ParA⋅ATP–ParB bond, $k_{\mathrm{on}}\left(S\right)=k_{\mathrm{on}}^{0}p\left(S\right)$ has a spatial dependence: the free ParB on the microbead can bind to the available ParA⋅ATP on the substrate as long as the separation between them, S, satisfies L_a ≥ S ≥ L_e, where L_a is the maximum binding range for ParA⋅ATP–ParB bond association (Fig. S6A). In the model, the probability density of binding, p(S), is a function of the separation S as $p\left(S\right)=p_0\exp \left(-\frac{1}{2}{k}_{\mathrm{s}}{\left(S-L_{\mathrm{e}}\right)}^2/k_{\mathrm{B}}T\right)$ for L_a ≥ S ≥ L_e; otherwise, p(S) = 0. p₀ is a constant normalization factor. The dissociation rate of the ParA⋅ATP–ParB is modulated by the bond length extension $L\left(\overrightarrow{r},t\right)-L_{\mathrm{e}}$, in addition to the intrinsic rate $k_{\mathrm{off}}^0$. In particular, the ParA⋅ATP–ParB bond is assumed to dissociate instantaneously if the bond length is beyond the threshold bond length (L_e + X_C), where (L_e + X_C) > L_a > L_e. ${\overrightarrow{L}}_{x-y}\left(\overrightarrow{r},t\right)$ represents the ParA⋅ATP–ParB bond extension in the X–Y plane. The model ignores elastic force in the Z direction, as it is balanced by the magnetic force in our in vitro experiments. k_s is the spring constant of the ParA⋅ATP–ParB bond. γ is the viscous drag coefficient of the microbead, and ζ is the random force arising from thermal noise.

We now set out to study the symmetry-breaking event. To this end, we seek the steady-state solution of Eqs. S1–S3 in the absence of noise, and investigate the stability of this solution. Combining Eqs. S1 and S2 shows that the total ParA on the substrate [i.e., free and ParB-bound ParA (A₀ + B₁)] is monotonically decreasing. The only steady state of the system corresponds to the scenario in which the velocity is zero ($\overrightarrow{V}=0$), none of the ParBs on the microbead surface are in a ParA-bound state [$B_1\left(\overrightarrow{r},t\right)=A_1\left(\overrightarrow{r},t\right)$= 0], the local density of ParA⋅ATP directly underneath the microbead is zero [$A_0\left(\overrightarrow{r},t\right)$= 0], and ParA⋅ATP fully covers the rest of the substrate. Essentially, this case can be realized by holding the microbead for a long enough time so that all of the ParA⋅ATPs directly underneath the microbead are depleted. This event is similar to the full model simulation result on the bond dynamics before symmetry breaking (Fig. 2B).

Now consider a perturbation of this steady state that displaces the microbead by a very small distance$\mathrm{\Delta }\overrightarrow{l}$ in the x–y plane. We determine whether this $\mathrm{\Delta }\overrightarrow{l}$ increases (unstable and symmetry breaking) or decreases (stable) with time. For simplicity, we consider a quasi-1D version of the problem (Fig. S6A), in which the perturbed displacement of the microbead ($\mathrm{\Delta }\overrightarrow{l}$) is toward the right in the X direction. The conclusion from the quasi-1D case can readily be generalized to the 2D case as shown subsequently.

First, we focus on the dynamics near the front/right end of the microbead in Fig. S6A. This boundary of the ParB-covered domain is at the location $x_2=\mathrm{\Delta }l$. The furthest location on the substrate that the ParB on the microbead can bind is thus at $x_1=\mathrm{\Delta }L+\mathrm{\Delta }l$, where x₁ is the spatial coordinate on the substrate. $\mathrm{\Delta }L=\sqrt{L_{\mathrm{a}}^2-L_{\mathrm{e}}^2}\ge 0$. For each ParB on the microbead, $\mathrm{\Delta }L$ defines the effective range of its binding zone along the substrate, and $\mathrm{\Delta }L=0$ means that the ParB can only bind to the ParA directly underneath at the distance L_e. ParA⋅ATP within the range of $0\le x_1\le \mathrm{\Delta }L+\mathrm{\Delta }l$ can bind ParB at the rate of k_on, where the available ParB should be in the range of $-\mathrm{\Delta }L+x_1\le x_2\le \mathrm{\Delta }l$. For simplicity, we assume that all binding events within this range have equal probability, i.e., $p\left(S\right)=1/\left(\mathrm{\Delta }L+\mathrm{\Delta }l-x_1\right)$. This approximation is justified for the following reason: In the full simulation, the maximum length of a new bond L_a is 103 nm, the equilibrium bond length L_e is 100 nm, and the bond spring constant k_s = 0.05 pN/nm. The maximum elastic energy incurred during bond formation is thus $\frac{1}{2}{k}_{\mathrm{s}}{\left(L_{\mathrm{a}}-L_{\mathrm{e}}\right)}^2$ ∼ 0.1$k_{\mathrm{B}}T$ << $k_{\mathrm{B}}T$, $\exp \left(-\frac{1}{2}{k}_{\mathrm{s}}{\left(S-L_{\mathrm{e}}\right)}^2/k_{\mathrm{B}}T\right)$ is very close to 1, and $p\left(S\right)=p_0$. As such, upon the ParA⋅ATP at $x_1$ establishing the ParA⋅ATP–ParB bond, the average bond extension in the x direction will be ${\displaystyle {\int }_{-\mathrm{\Delta }L+x_1}^{\mathrm{\Delta }l}d{x}_2\left(x_1-x_2\right)\left(1/\left(\mathrm{\Delta }L+\mathrm{\Delta }l-x_1\right)\right)}$. Integrating the average bond extension over all ParA⋅ATPs within the range of $0\le x_1\le \mathrm{\Delta }L+\mathrm{\Delta }l$ over a short time duration δt yields the net extension of all of the new bonds near the right end of the microbead: $\left(k_{\mathrm{on}}\delta t\right)\left(1/L_0\right){\displaystyle {\int }_0^{\mathrm{\Delta }L+\mathrm{\Delta }l}d{x}_1}{\displaystyle {\int }_{-\mathrm{\Delta }L+x_1}^{\mathrm{\Delta }l}d{x}_2\left(x_1-x_2\right)}\left(1/\left(\mathrm{\Delta }L+\mathrm{\Delta }l-x_1\right)\right)$, where $1/L_0$ is the ParA⋅ATP density on the substrate, and k_on has already absorbed the total density of ParB (i.e., B_T) for simplicity. We ignore the volume exclusion effect that is expected to introduce a minor correction as long as the density of bonds is sparse. This latter condition is valid, as the δt is very small so that not many bonds have yet formed from the steady state with zero bonds.

Similarly, we obtain the corresponding net bond extension in the x direction near the left end of the microbead ∼$\left(k_{\mathrm{on}}\delta t\right)\left(1/L_0\right){\displaystyle {\int }_{-\left(\mathrm{\Delta }L-\mathrm{\Delta }l\right)}^{0}d{x}_1}{\displaystyle {\int }_{\mathrm{\Delta }l}^{\mathrm{\Delta }L+x_1}d{x}_2\left(x_1-x_2\right)}\left(1/\left(\mathrm{\Delta }L-\mathrm{\Delta }l+x_1\right)\right)$. Here, the spatial coordinates are shifted to discern the difference in the ParA⋅ATP–ParB bond formation between the front and the rear of the microbead.

Summing up the two integrals, we have the average bond length extension of the new ParA⋅ATP–ParB bonds formed from the steady state during the time interval δt:

$$\begin{array}{l}\frac{\left(k_{\mathrm{on}}\delta t\right)}{L_0}{\displaystyle \underset{0}{\overset{\mathrm{\Delta }L+\mathrm{\Delta }l}{\int }}d{x}_1}{\displaystyle \underset{-\mathrm{\Delta }L+x_1}{\overset{\mathrm{\Delta }l}{\int }}d{x}_2}\left(x_1-x_2\right)\frac{1}{\left(\mathrm{\Delta }L+\mathrm{\Delta }l-x_1\right)}+\frac{\left(k_{\mathrm{on}}\delta t\right)}{L_0}{\displaystyle \underset{-\left(\mathrm{\Delta }L-\mathrm{\Delta }l\right)}{\overset{0}{\int }}d{x}_1}{\displaystyle \underset{\mathrm{\Delta }l}{\overset{\mathrm{\Delta }L+x_1}{\int }}d{x}_2}\left(x_1-x_2\right)\frac{1}{\left(\mathrm{\Delta }L-\mathrm{\Delta }l+x_1\right)}\\ \\ =\frac{\mathrm{\Delta }L}{L_0}\mathrm{\Delta }l\left(k_{\mathrm{on}}\delta t\right).\end{array}$$

Importantly, with the extended binding range ($\mathrm{\Delta }L$ > 0), the new bonds pull the microbead in the direction of the perturbed displacement. Consequently, the net force arising from the perturbed displacement $\mathrm{\Delta }l$ is ∼$k_{\mathrm{s}}\left(\mathrm{\Delta }L/L_0\right)\mathrm{\Delta }l\left(k_{\mathrm{on}}\delta t\right)$, which causes a positive change in velocity as $dV/dt=\left(k_{\mathrm{s}}/\lambda \right)\left(\mathrm{\Delta }L/L_0\right)k_{\mathrm{on}}\mathrm{\Delta }l$ and thus breaks symmetry. A similar argument involving integration over x and y dimensions holds for the 2D case, in which case the corresponding net force is ∼$k_{\mathrm{s}}{\left(\mathrm{\Delta }L/L_0\right)}^2\mathrm{\Delta }\overrightarrow{l}\left(k_{\mathrm{on}}\delta t\right)$ and thus results in a positive change in the velocity.

The above linear stability analysis suggests symmetry breaking requires the satisfaction of two conditions in our system. The first specifies the steady-state condition that the ParA⋅ATP directly underneath the microbead needs to be depleted. Under this steady-state condition, a forward-moving microbead will have more ParA⋅ATP available for ParB binding in the front than in the rear. The second condition for symmetry breaking is the extended binding zone ($\mathrm{\Delta }L$ > 0). Otherwise, the ParB can only bind the ParA⋅ATP directly underneath ($\mathrm{\Delta }L$ = 0) and there is no net force in the horizontal direction. In a sense, this extended binding range and the associated probabilistic binding events reflect the effect of thermal fluctuations. Through this process, thermal fluctuations are harnessed through the asymmetric biochemical environment surrounding the forward-moving microbead, perpetuating the movement. Therefore, our underlying mechanism is a Brownian-ratchet mechanism. As such, more ParA⋅ATP–ParB bonds will be established that drag the microbead forward, breaking symmetry and increasing velocity in the perturbed direction. As the microbead moves after symmetry breaking, less ParA⋅ATP is available for the ParB binding at the back than at the front of the microbead. Gradually, only the ParB near the front of the microbead can establish ParA⋅ATP–ParB bonds. This simplified sequence of events is consistent with our full simulation shown in Fig. 2B. That is, the number of ParA⋅ATP–ParB bonds decreases significantly before the microbead breaks symmetry. Furthermore, the distribution of ParA⋅ATP–ParB bonds is positively correlated with the direction of the persistent movement.

In the above quasi-1D case, the direction of persistent movement is fixed after symmetry breaking. However, in two dimensions, there exists a degree of freedom orthogonal to the chosen primary movement direction after symmetry breaking. The question is: What happens if there is a perturbation in this orthogonal direction? The above analysis demonstrates that symmetry breaking or instability entails the depletion of ParA⋅ATP underneath the microbead. Conversely, the existing bonds will contribute to a net force approximately $-k_{\mathrm{s}}N\mathrm{\Delta }\overrightarrow{l}$ in the linear regime, where N is the bond number. This force is negative and opposes the perturbation in the orthogonal direction, thus maintaining the persistent directional movement. This insight is consistent with our full simulation results, which reveal an energy barrier established by preexisting bonds that inhibits lateral excursions of the microbead (Fig. 2D).

Speed at Steady State.

We next examined what controls the speed of the directed and persistent movement at steady state. We took a mean-field approach, which is valid in the limit of a very large number of existing ParA⋅ATP–ParB bonds at any time. This approximation ignores stochastic fluctuations and assumes that all of the existing ParA⋅ATP–ParB bonds behave in the same way. That is, all of the bonds are in synchrony as one unit. Under this mean-field approximation, we focused on the behavior of a single ParB when the microbead moves at an average speed $\overline{V}$ (Fig. S6B).

In our simulations, the ParA⋅ATP–ParB association rate is much faster than the dissociation rate. Consequently, nearly all of the ParBs at the microbead front end are in the ParA⋅ATP-bound state. Because ParB can only bind to one ParA⋅ATP at any time, the formation of a new bond from the ParB must wait until the old one dissociates. This imposes an ordered sequence on the chemical bond dynamics, leading to a simplified picture: Bonds form and dissociate in cycles that couple with microbead movement. The duration of such a cycle is $\left(1/k_{\mathrm{off}}^0+1/k_{\mathrm{on}}\right)$, where we ignored the load dependence of the off rate as a further simplification. Suppose that, just before the ParA⋅ATP–ParB bond dissociates, the bond length L is$\sqrt{L_{\mathrm{e}}^2+L_{\mathrm{break}}^2}$. Note that L_break must be positive in the mean-field picture to maintain forward movement. Given the equal probability of binding events for $\mathrm{\Delta }L\ge x_1\ge L_{\mathrm{break}}$, the average location on the substrate at which the new bond forms is at $x_1=\left(1/2\right)\left(L_{\mathrm{break}}+\mathrm{\Delta }L\right)$ with the average bond length $\overline{L}=\sqrt{L_{\mathrm{e}}^2+\left(1/4\right){\left(L_{\mathrm{break}}+\mathrm{\Delta }L\right)}^2}$. The average traveling distance during one cycle is $x_1-L_{\mathrm{break}}=\left(1/2\right)\left(\mathrm{\Delta }L-L_{\mathrm{break}}\right)$, which should equal to the constant speed times the duration of chemical cycle, i.e., $\left(1/2\right)\left(\mathrm{\Delta }L-L_{\mathrm{break}}\right)=\overline{V}\left(1/k_{\mathrm{off}}^0+1/k_{\mathrm{on}}\right)$. Based on this self-consistent scheme, the constant speed of the microbead is determined by the bond extension-mediated elastic force $\overline{V}=\left(k_{\mathrm{s}}/\gamma \right)N\left(\overline{L}-L_{\mathrm{e}}\right)\left(\left(1/2\right)\left(L_{\mathrm{break}}+\mathrm{\Delta }L\right)/\overline{L}\right)$, where N is the average number of existing ParA⋅ATP–ParB bonds, and the effective viscous drag coefficient γ relates to the microbead diffusion constant as $\gamma \sim k_{\mathrm{B}}T/D$. Combining this expression of speed with that of traveled distance in one bond association–dissociation cycle, it leads to a cubic equation for the velocity $\overline{V}$, the roots of which can be solved exactly but has a complicated formula. For model parameters near the nominal case, $\left(\gamma /k_{\mathrm{s}}N\right)\left(k_{\mathrm{off}}^0{k}_{\mathrm{on}}/\left(k_{\mathrm{off}}^0+k_{\mathrm{on}}\right)\right)<<1$. This reduces the root formula to Eq. S4, where we kept the first-order correction $O\left(\left(\gamma /k_{\mathrm{s}}N\right)\left(k_{\mathrm{off}}^0{k}_{\mathrm{on}}/\left(k_{\mathrm{off}}^0+k_{\mathrm{on}}\right)\right)\right)$ in the solution so that it exposes the underlying physics more explicitly:

$$\overline{V}=\left(\mathrm{\Delta }L-{\left(\frac{\gamma }{k_{\mathrm{s}}N}\frac{k_{\mathrm{off}}^0{k}_{\mathrm{on}}}{k_{\mathrm{off}}^0+k_{\mathrm{on}}}\right)}^{\frac{1}{3}}{\left(2L_{\mathrm{e}}^2\mathrm{\Delta }L\right)}^{\frac{1}{3}}\right)\frac{k_{\mathrm{off}}^0{k}_{\mathrm{on}}}{k_{\mathrm{off}}^0+k_{\mathrm{on}}}.$$ [S4]

For our nominal case, $k_{\mathrm{off}}^0$ ∼ 3/s, k_on ∼ 5 × 10⁴/s, k_s ∼ 0.05 pN/nm, $\gamma \sim k_{\mathrm{B}}T/D$ with D ∼ 0.1 μm²/s, L_e is 100 nm, $\mathrm{\Delta }L=\sqrt{L_{\mathrm{a}}^2-L_{\mathrm{e}}^2}$ ∼ 25 nm, and N ∼ 200 (from our full simulation result). With these parameters, Eq. S4 yields a speed of ∼0.07 μm/s, consistent with the full model simulation result of ∼0.09 μm/s. In addition, Eq. S4 predicts the following features:

• i)$\mathrm{\Delta }L$ could be regarded as the effective step size. We emphasize that our underlying mechanism here is a Brownian ratchet, and this step size ($\mathrm{\Delta }L$) reflects the effect of thermal fluctuations. The microbead speed $\overline{V}$ is proportional to $\mathrm{\Delta }L$ and modulated by the interplay between the three intrinsic timescales of the system: the timescale for bond formation (∼$1/k_{\mathrm{on}}$), the lifetime of the bond ∼$1/k_{\mathrm{off}}$, and the mechanical response time of the microbead ∼$\gamma /k_{\mathrm{s}}=k_{\mathrm{B}}T/D{k}_{\mathrm{s}}$.
• ii)In the parameter regime close to our nominal case, $\overline{V}$ is independent of the diffusion of the microbead D, the ParA⋅ATP–ParB bond association rate k_on, and the spring constant of the bond $k_s$. Instead, the speed is proportional to the off rate $k_{\mathrm{off}}^0$. This is because $k_{\mathrm{s}}N/\gamma \sim \left(k_{\mathrm{s}}N/k_{\mathrm{B}}T\right)D>>\left(k_{\mathrm{off}}^0{k}_{\mathrm{on}}/\left(k_{\mathrm{off}}^0+k_{\mathrm{on}}\right)\right)$ (i.e., the mechanical timescale of the system is much faster than that of chemical reaction), and $k_{\mathrm{on}}>>k_{\mathrm{off}}^0$ in our nominal case. In this limit, Eq. S4 reduces to the following: $\overline{V}=\mathrm{\Delta }L{k}_{\mathrm{off}}^0$. This means that the microbead movement in the nominal case is gated by the off rate of the chemical bond. It is not limited by microbead diffusion, but controlled by chemical reactions.
• iii)The speed will drop off when the on rate and/or the diffusion constant are significantly reduced.

All of these predictions are qualitatively consistent with results of our full model simulations (Fig. 4 B and C and Fig. S2 A–D). However, the mean-field approximation has limitations. It cannot account for saltatory movement (Fig. 5) nor can it determine the number of bonds independently. Nonetheless, our analytic result clarifies the underlying physics, and is consistent with, and corroborated by, the full model.