Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ dimers to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and PomZ proteins. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter’s midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ on the nucleoid is slow relative to the cluster’s velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.
The formation of protein patterns and the intracellular positioning of proteins is a major prerequisite for many important processes in bacterial cells, such as cell division. In order to maintain the genetic content of the bacterial cell, the chromosome (nucleoid) is duplicated during the cell cycle and must be segregated into the two cell halves prior to cell division. The future division site is defined by the FtsZ ring, which forms at midcell and recruits the cytokinetic machinery. Interestingly, FtsZ is highly conserved in bacteria, while the protein systems responsible for the positioning of the FtsZ ring, and with it the cell division site, are not [1–3]. Recently, Schumacher et al. identified a set of proteins, called PomX, PomY and PomZ, in Myxococcus xanthus cells that are important for midcell localization and formation of the FtsZ ring [4–6]. PomZ is an ATPase, which belongs to the family of ParA / MinD ATPases [3]. It binds non-specifically to DNA in its dimeric, ATP-bound state, and its activity is stimulated by interactions with PomX, PomY and DNA. PomX and PomY form a single cluster, which is tethered to the nucleoid via PomZ dimers bound to the chromosome. Starting from an off-center position near one nucleoid pole, the cluster moves towards midnucleoid, coinciding with midcell [5]. When the cluster has reached midcell, the FtsZ ring forms there and the cell divides. During cell division, the cluster splits into two halves, such that each half is located at one pole of the nucleoids in the daughter cells, and the same cycle repeats. Notably, the Pom proteins localize to midcell before FtsZ and also in the absence of FtsZ [4,5]. The mechanism underlying midcell localization of the FtsZ ring is well understood in Escherichia coli cells [7–21]. Here, Min proteins (MinC, MinD and MinE) guide the formation of the FtsZ ring at midcell. Importantly, the system for midcell localization in E. coli and M. xanthus differ substantially, even though both systems contain an ATPase (PomZ and MinD, respectively) and perform the same task in the cell, i. e. midcell sensing. First, the scaffold to which the ATP-bound ATPase binds is different: MinD binds to the cell membrane and PomZ to the bacterial nucleoid in the cytoplasm. Second, MinD-bound MinC inhibits [22], whereas the Pom cluster promotes FtsZ ring formation at midcell [5]. Finally, the observed protein patterns differ: the Pom proteins colocalize in a cluster that moves towards midcell, while the Min proteins, which do not form a cluster, oscillate from pole to pole [22,23]. Conversely, the Pom system is mechanistically more similar to plasmid and chromosome segregation systems that involve a ParABS system. Like the Pom system, plasmid and chromosome segregation systems make use of an ATPase that shuttles one or several cargoes (such as a plasmid, a partition complex or a protein cluster) along the nucleoid. To ensure the equal distribution of low-copy number plasmids to the daughter cells, they are tethered to the nucleoid and positioned at equal distances along the nucleoid by ParABS systems [24–26]. A ParABS system consists of the proteins ParA and ParB, and a DNA sequence, parS. ParA proteins are ATPases, which bind non-specifically to DNA as ATP-bound dimers [27–29]. Their ATPase activity is stimulated in the presence of ParB [30–32], which binds to the parS sequence on the chromosome (to form the partition complex) or on the plasmid [3]. Besides plasmid and chromosome segregation [33], ParABS systems are also important for the positioning of cellular components (e. g. chemotactic clusters or carboxysomes) [34,35]. Several different cargo dynamics involving ParABS systems have been observed. For one cargo these localization patterns include, among others, midcell localization [26,36,37], oscillatory movement of ParA and its cargo [24,25,38] as well as movement from one cell pole to the other [39]. Multiple cargoes are found to equidistantly position along the nucleoid [24–26,36,37]. To account for the dynamics observed in Par systems, various mechanisms have been proposed. Some models rely on ParA filament formation [24,30,33,40–43], others challenge this assumption in in vivo systems [39,44,45]. A diffusion-ratchet mechanism for the movement of ParB-coated beads in vitro and DNA segregation in vivo has been introduced [32,45–49]. Based on the observation that DNA has elastic properties [39,50], a DNA-relay mechanism for the movement of the partition complex was proposed [39,51]. Here, the force exerted on the cargo is attributed to the elastic properties of the chromosome. Ietswaart et al. observed that if a plasmid is located off-center on the nucleoid, the ParA flux from the left and right sides of the plasmid differ [36]. Based on this idea, they proposed a model that produces equal plasmid spacing over the nucleoid as long as the plasmid moves in the direction of the higher ParA concentration [36]. Additionally, models based on reaction-diffusion equations for Par protein dynamics, have been introduced [52–55]. In order to account for the experimental observations in M. xanthus cells, we have proposed a model for midcell localization [5] that is inspired by, but also differs from previous models for Par systems (see S1 Text). The key experimental observations for the Pom system are: First, PomZ accumulates at the cluster consisting of PomX and PomY proteins, which is in contrast to a low ParA density at plasmids / the partition complex [24,47]. Conversely, there are also positioning systems that show an accumulation of the ATPase at the cargo [34,56], which resembles the observations for the Pom system. Second, the cluster is relatively large (0. 7 μm in length, [5]) compared to plasmids / partition complexes (about 0. 1 μm in length, [39]). Third, the PomZ proteins diffuse quickly on the nucleoid compared to a slowly moving Pom cluster [5]. Finally, fluorescence micrographs of M. xanthus cells do not show a clear depletion zone in PomZ in the wake of the cluster, in contrast to observations for Par systems [24,38,47]. The latter observation can be explained by the fast PomZ dynamics on the nucleoid. Our model suggests a positioning mechanism that relies on the biasing of fluxes of PomZ dimers on the nucleoid, similar to the equipositioning mechanism proposed by Ietswaart et al. [36]. With this model we were able to reproduce midnucleoid localization with physiologically relevant parameters [5], but it remained unclear how the movement of the cluster changes when the rates of the key biological processes involved are varied. Here, we investigate the robustness of Pom cluster dynamics in our model with respect to different parameters, so that our model can be tested by experimentally examining this robustness. Interestingly, we observe that there exists an intermediate ATP hydrolysis rate that minimizes the time the clusters need to reach midnucleoid. Furthermore, we find that fast diffusion of PomZ dimers on the cluster accelerates the movement of the cluster towards midnucleoid. To gain a better understanding of the cluster dynamics observed in the in silico parameter sweeps, we investigate how PomZ dimers generate a net force on the cluster in our model. For the case where the PomZ gradient builds up faster than the velocity of cluster movement, we derive a semi-analytical approximation for the average cluster trajectory, which dissects the generation of a net force into two parts: the difference between the diffusive PomZ fluxes into the cluster from either side, and the force exerted by a single PomZ dimer during its interaction with the cluster. This net force can account for the movement of the cluster to midnucleoid. In contrast, when the PomZ dimers diffuse slowly on the nucleoid, we observe oscillatory cluster movement.
We employ a stochastic lattice gas model, introduced in [5], to understand the dynamics of the PomXY cluster, i. e. the cluster consisting of PomX and PomY proteins, in M. xanthus bacterial cells [5]. In this model, both the nucleoid and the PomXY cluster are reduced to one-dimensional lattices of length L and Lc, respectively (Fig 1). Our model consists of two parts: first, the PomZ dynamics, and second the PomXY cluster movement due to its interactions with PomZ. We first describe the PomZ dynamics in the next paragraph. PomZ can occur in different configurations: it can be bound to ADP or ATP and in the latter case, form dimers. In our model we incorporate only the ATP-bound dimeric form of PomZ explicitly. We model the PomZ dimers effectively as springs with spring stiffness k to account for the elastic properties of the chromosome and the PomZ dimers. Each PomZ dimer spring has two binding sites, one which connects to the nucleoid, and one which connects to the PomXY cluster. ATP-bound PomZ dimers can bind with the first binding site to the nucleoid with rate kon (Fig 1A (1) ), except where the PomXY cluster is located, and diffuse on the nucleoid with diffusion coefficient Dnuc (Fig 1A (2) ). Because of thermal fluctuations, the relative position of the second binding site, which enables PomZ to bind to the PomXY cluster, is distributed according to a Boltzmann distribution with the energy of the spring. Therefore, PomZ dimers can attach to the PomXY cluster even if their nucleoid binding sites are not directly below the cluster (Fig 1A (3) and 1B). We include this factor in the model by multiplying the rate of attachment, k a 0, by the Boltzmann factor corresponding to the energy of the spring (as in [57]): k a = k a 0 exp [ - 1 2 k k B T (x i clu - x i nuc) 2 ]. (1) The positions of the cluster and nucleoid binding sites of the i-th PomZ dimer bound to the nucleoid and the PomXY cluster are given as x i clu and x i nuc, respectively (see Fig 1B). PomZ dimers bound to the PomXY cluster and the nucleoid are assumed to diffuse on both scaffolds (Fig 1A (4) ). This assumption is motivated by two experimental observations. First, fluorescently tagged PomZ brightly stains the entire cluster in fluorescence micrographs [5]. Second, in a mutant with PomZ dimers that cannot bind to DNA, PomZ is homogeneously distributed inside the cell, which suggests that PomZ dimers only bind to the PomXY cluster when they are nucleoid-bound [5]. Based on these two experimental findings it seems reasonable that PomZ dimers are also mobile on the PomXY cluster as otherwise the concentration of PomZ would be rather concentrated at the cluster edges. PomZ can diffuse on the PomXY cluster and nucleoid with different diffusivities: We assume that the hopping rates are ϵ hop, nuc 0 = D nuc / a 2 and ϵ hop, clu 0 = D clu / a 2, with the lattice spacing a, respectively, being weighted by a Boltzmann factor that accounts for the energy change of the spring due to the movement: ϵ hop = ϵ hop 0 exp [ - 1 4 k k B T ( (x i clu, to- x i nuc, to) 2 - (x i clu, from- x i nuc, from) 2) ]. (2) Here, x i clu, from, x i nuc, from and x i clu, to, x i nuc, to signify the position of the binding sites of the i-th PomZ dimer to the cluster and nucleoid before and after hopping, respectively. The additional factor of 1/2 in the exponent is chosen such that detailed balance holds for the hopping events and the rates for hopping to a neighboring site and hopping back are the inverse of each other (see [57]). A PomZ dimer is most likely to move in the direction that relaxes the spring (cf. exponential factor in Eq 2). We chose reflecting boundary conditions for diffusion of PomZ on both the nucleoid and the PomXY cluster. Next, we discuss the transition from the nucleoid-bound state of PomZ to the cytosolic state. In the experiments, PomX, PomY and DNA stimulate the ATPase activity of PomZ, which leads to a conformational change and finally to detachment of two ADP-bound PomZ monomers from the nucleoid [5]. In our model, we combine the processes of nucleotide hydrolysis and detachment into one rate by assuming that nucleoid- and cluster-bound PomZ dimers are released into the cytosol with hydrolysis rate kh (Fig 1A (5) ). The ADP-bound PomZ monomers must then exchange ADP for ATP and dimerize before they can rebind to the nucleoid. This leads to a delay between detachment from and reattachment to the nucleoid (Fig 1A (6) ). Because of this delay and rapid diffusion of PomZ in the cytosol [5] we assume that upon detachment of PomZ from the nucleoid, it can rebind to any lattice site of the nucleoid with the same probability. The total number of PomZ dimers is assumed to be constant and is denoted by Ntotal. So far we have described the stochastic dynamics of the PomZ dimers. The interactions of PomZ dimers with the PomXY cluster result in forces being exerted on the cluster, which cause it to move. The observable of interest is the cluster position, xc, over time. We approximate the cluster dynamics as overdamped, such that the equation of motion for xc is given by the following force balance equation γ c ∂ t x c = - k ∑ i = 1 N (x i clu - x i nuc), (3) with γc being the friction coefficient of the PomXY cluster in the cytosol and N the total number of cluster-bound PomZ dimers. Experiments show that the Pom cluster displays very little motion in M. xanthus cells that lack PomZ, whereas its mobility is increased if PomZ is present [5]. Based on this observation, we disregard movements of the cluster due to thermal noise and focus on the stochasticity in the interactions of PomZ dimers with the PomXY cluster, which in turn lead to stochastic forces acting on the cluster. Therefore, we do not include a Langevin noise term in Eq 3. Our simulations show that the model indeed yields a robust mechanism for stochastic midnucleoid positioning of the PomXY cluster. The underlying mechanism for midnucleoid localization is based on the flux of PomZ on the nucleoid, which can be described as follows. If the PomXY cluster is located to the left of midnucleoid, the average flux of PomZ dimers into the cluster from the right is larger than that from the left [36] (red arrows in Fig 1A). If particles that attach to the cluster typically exert a net force in the direction from which they reached the cluster, the flux imbalance leads to a net force towards the right. For a cluster that overshoots midnucleoid or is already positioned to the right of midnucleoid, the asymmetry in the fluxes is reversed and the cluster moves back towards midnucleoid. Overall, this leads to a self-regulating process that positions the PomXY cluster at midnucleoid. The stochastic simulations show midnucleoid positioning of the PomXY cluster (Fig 2, data shown in black) with physiologically relevant parameters (S1 Table, for the discussion of the parameters see S1 Text). To identify the parameter range that leads to midcell localization and investigate the role of each parameter on the cluster dynamics, we performed broad parameter sweeps. We varied the attachment rate of PomZ dimers to the nucleoid, kon, the binding rate of nucleoid-bound PomZ dimers to the PomXY cluster, k a 0, the ATP hydrolysis rate of PomZ dimers, kh, and the mobility of PomZ dimers on the nucleoid, Dnuc, and on the PomXY cluster, Dclu, over a broad range (Fig 2). We never restricted the PomZ dimer density on the nucleoid and bound to the cluster, as we can assume that these densities are low in the wild type situation with a total number of PomZ dimers of Ntotal ≈ 100 [5]. The parameter sweeps show that increasing the attachment rate to the nucleoid, kon, or the binding rate to the PomXY cluster, k a 0, decreases the time the cluster needs to reach midnucleoid (Fig 2A and 2B). In both cases, the trajectories become independent of the particular parameter when its value exceeds a certain threshold. We conclude that increasing the rate of attachment of PomZ to the nucleoid or the binding of PomZ to the PomXY cluster speeds up the positioning process until an optimum is reached. Next, we consider the effects of varying the rate of ATP hydrolysis by PomZ dimers associated with the PomXY cluster, which is important to maintain the cyclic flux of PomZ dimers between the cytosolic and nucleoid-bound state. This rate also sets the time scale for the interaction of PomZ dimers with the PomXY cluster. The simulations show that decreasing the hydrolysis rate (kh = 0. 01 s−1) reduces the velocity of the average cluster trajectory towards midnucleoid (Fig 2C). Qualitatively, large hydrolysis rates (kh = 10 s−1) have essentially the same effect (Fig 2C). Hence, there is a hydrolysis rate kh that minimizes the time the cluster needs to reach midnucleoid. Although the average cluster trajectory behaves similarly for large and small hydrolysis rates, we observe that the variance of the cluster distribution over time decreases with increasing hydrolysis rate (Fig 2C). Apart from the ATP hydrolysis rate, we expect the diffusivity of PomZ on the nucleoid to be a crucial parameter for cluster movement, because it determines the time needed for PomZ dimers to explore the nucleoid to the left or right of the cluster. Interestingly, when we reduce the diffusivity of PomZ on the nucleoid in the simulations, the clusters begin to oscillate around the midnucleoid position (Fig 2D, 2E and S1 Fig). Finally, we also decreased the diffusion constant of PomZ dimers on the PomXY cluster, while keeping the diffusion constant on the nucleoid fixed. In this case, the clusters take longer to reach midcell (Fig 2F). In addition to the parameter sweeps shown in Fig 2, we also considered the PomXY cluster trajectories when the spring stiffness k and the total PomZ dimer number Ntotal are varied. The cluster trajectories do not change significantly when the spring stiffness is altered over one order of magnitude, and an increase in the particle number increases the velocity of cluster movement towards midcell (S2 Fig). Furthermore, we verified that the results do not change qualitatively in an extension of the stochastic model that includes a small detachment rate of nucleoid-, but not cluster-bound PomZ dimers. For very large detachment rates that are above the expected values from experiments midcell localization of the cluster breaks down (see S1 Text and S3 Fig). To summarize, we observed that there exists an ATP hydrolysis rate that minimizes the time taken to reach midnucleoid. The diffusion constant of PomZ on the nucleoid determines whether the PomXY cluster moves towards or oscillates around midnucleoid. Moreover, the clusters move faster towards midcell if PomZ dimers diffuse faster on the PomXY cluster. In the following, we provide first an analytic approach that explains our observations regarding the cluster dynamics when the ATP hydrolysis rate and the diffusion constant of PomZ on the cluster is varied. We then consider the oscillatory cluster dynamics and give an estimate for the onset of oscillations. Our goal is to understand what generates the force behind midcell positioning in our model. We expect that the cyclic flow of PomZ dimers is at the root of this force: PomZ dimers attach to the nucleoid in their active state (as ATP-bound PomZ dimers), diffuse on the nucleoid and are released into the cytosol in their inactive state (ADP-bound PomZ monomers) after encountering the PomXY cluster. We describe how the cyclic flow can lead to a net force in the following. We assume that the PomZ dynamics is fast compared to the PomXY cluster dynamics (adiabatic assumption), and approximate the system by a stationary model, i. e. a system with a fixed cluster position. As we neglect exclusion effects on the nucleoid, PomZ dimers can only interact with each other via the PomXY cluster. However, when the cluster is stationary, no interaction between the cluster-bound PomZ dimers is possible, and thus the movements of different PomZ proteins are not correlated. Therefore, we can consider the interactions of PomZ dimers with the PomXY cluster as independent, which yields the following deterministic approximation for the total net force, F, acting on a cluster at position xc F (x c) = (N R (x c) - N L (x c) ) f, (4) with f being the time-averaged force exerted by a single PomZ dimer that attaches to the nucleoid on the right side of the cluster. For symmetry reasons, a PomZ dimer coming from the left then exerts a time-averaged force −f. NR and NL denote the numbers of PomZ dimers that are bound to the cluster and had originally attached to the nucleoid to the right and left of the cluster, respectively. These two numbers increase with the diffusive flux of nucleoid-bound PomZ dimers reaching the cluster region from the right and left side, jR/L, respectively, and decrease with the ATP hydrolysis rate, kh, as long as the attachment rate to the PomXY cluster is non-zero. Hence, we obtain N R / L (x c) = j R / L (x c) k h (5) in the steady-state. Inserting this into Eq 4, yields F (x c) = j R (x c) - j L (x c) k h f = f k h j diff (x c) ≡ C j diff (x c). (6) We conclude that the net force is proportional to the flux difference of PomZ dimers at the cluster, jdiff, and the proportionality constant is given by C = f/kh. Importantly, simulation results with a fixed cluster position confirm the observation that the total force exerted on the cluster is proportional to the PomZ flux difference (S4 Fig). Next, we investigate how the net force exerted on the PomXY cluster results in movement of the cluster. Notably, the PomZ dimers interacting with the cluster not only produce a net force on the cluster, they also reduce the mobility of the cluster by tethering it to the nucleoid. We assume that these two processes can be considered independently. We simulated the steady-state velocity with which a cluster moves when a fixed number of PomZ dimers are bound to it and an external force is applied to the cluster (see Materials and methods section for details). We found that this velocity varies linearly with the force (S5 Fig), which suggests that the force exerted on the cluster is balanced by a frictional force with effective friction coefficient γ (xc): F (xc) = γ (xc) v (xc). With Eq 6 we obtain the central equation of our analysis v (x c) = F (x c) γ (x c) = C j diff (x c) γ (x c), (7) which relates the average velocity of the cluster to the flux difference of PomZ dimers into the cluster, the proportionality constant C and the effective friction coefficient γ (xc) of the cluster. To obtain the average cluster trajectory, we need to integrate Eq 7 over time. In the following we derive analytical expressions for the flux difference into the cluster and the effective friction coefficient of the PomXY cluster. The constant C we determine from simulations. Since C does not change with the cluster position, xc, the dependence of the velocity on xc is given by an analytical expression, which can be integrated (numerically). We observe a marked discrepancy between the simulated average cluster trajectory and our approximation when the diffusion constant of PomZ on the nucleoid is reduced and the cluster oscillates around midnucleoid (Fig 4D and 4E). Deviations from our theoretical predictions are to be expected in this situation, because we make an adiabatic assumption in our semi-analytical approach, i. e. we assume that the PomZ dimer dynamics on the nucleoid is fast compared to the cluster movement. This assumption no longer holds when PomZ dimers diffuse slowly on the nucleoid. In this case, the distribution of PomZ density along the nucleoid determined from simulations with a dynamic cluster deviates drastically from its steady-state distribution (S12 Fig). If the cluster initially lies to the left of midnucleoid and approaches midnucleoid from that side, our theory predicts a symmetric PomZ density, whereas the simulations show a higher density in front of the cluster. The flux difference also deviates from the stationary case: it increases as the cluster moves towards midnucleoid instead of vanishing at midnucleoid (S12 Fig). Both the asymmetric density and the non-zero flux difference at midnucleoid are in accordance with the observed oscillatory behavior. The switch between cluster localization at midnucleoid and oscillatory movement around midnucleoid is regulated by the relative time scales of PomZ dynamics and cluster dynamics: If the PomXY cluster is moving slowly or the PomZ dimers move fast, the latter have time to adjust to a change in the cluster position. On the other hand, if the cluster moves fast or the PomZ dimers move slowly, the PomZ dimer distribution deviates from the stationary case. The delay between the movement of the cluster and the build-up of the PomZ gradient, which in turn biases the movement of the cluster, leads to oscillations: the longer the delay, the larger the amplitudes of the oscillations. To investigate the oscillatory case further, we performed additional simulations in which the diffusion constant of the PomZ dimers and that of the PomXY cluster in the cytosol, which is inversely proportional to the friction coefficient, γc, according to the Stokes-Einstein relation, were varied. As expected, we find oscillatory behavior of the clusters for low diffusion constants of PomZ on the nucleoid (Fig 6). In the oscillatory regime we find both bimodal and monomodal cluster position distributions (Fig 6). As mentioned above, the onset of oscillations depends on the time scales of PomZ gradient formation and cluster movement. In order to understand how the parameters change the behavior of the cluster trajectory, i. e. lead to oscillatory movement or midcell positioning, we assume that the cluster is located at midnucleoid and search for a stability condition that distinguishes the two behaviors. The diffusion time for a PomZ dimer to explore a nucleoid of size L is given by t PomZ = L 2 D nuc. (16) In theory, the velocity of a cluster that starts from midnucleoid should be zero, because there should be no difference between the fluxes of PomZ dimers from both sides. However, due to stochastic effects, more particles may attach to the cluster from the right than from the left side, which will displace the cluster to the right. For our time scale argument, we consider an extreme case: we assume that PomZ dimers only arrive from one side, which we choose to be the right side without loss of generality. The time required for a cluster to move the length of the nucleoid is then given by t cluster = L v ≈ L γ (0. 5 L) C j right (0. 5 L), (17) with jright being the flux of PomZ dimers into the cluster from the right. Here, we approximate the velocity of the cluster by its effective description, Eq 7, using xc = 0. 5L, and replace the flux difference with the flux from the right only. According to Eq 17, the condition for stable positioning of the cluster at midnucleoid t PomZ ≪ t cluster (18) yields D nuc L ≫ C j right (0. 5 L) γ (0. 5 L) = C j right (0. 5 L) γ c + (k B T N (0. 5 L) ) / (D clu + D nuc). (19) For the parameter sweeps considered before (Fig 2 and S2 Fig), we find tcluster ≫ tPomZ for all cases except for small diffusion constants of PomZ on the nucleoid. With our time-scale argument, Eq 19, we can make further predictions as to which parameters should result in oscillations. First, we consider a change in the total particle number, Ntotal. Both jright as well as the number of cluster-bound proteins, N, are proportional to Ntotal, and C does not depend on Ntotal. Therefore, the right-hand side of Eq 19 is proportional to Ntotal for small values of Ntotal and converges to a constant for large values. From this we expect that oscillatory behavior may occur for large particle numbers. Simulations with 500 PomZ dimers and a smaller diffusion constant of PomZ on the nucleoid and the PomXY cluster compared to the parameters in S1 Table (Dnuc = Dclu = 0. 01 μm2/s) indeed show oscillatory behavior, whereas simulations with the same parameters, but 100 PomZ dimers show midnucleoid localization (S13 Fig). However, for very large PomZ dimer numbers we expect exclusion effects, which are not considered here, to have an impact that will also affect the cluster dynamics. Second, we investigate the effects on the cluster dynamics when changing the nucleoid length L. Again, the constant C, which represents the force exerted by a single PomZ dimer on the PomXY cluster, does not depend on L. The number of cluster-bound proteins decreases with increasing L, because the relative size of the cluster Lc/L decreases and the total PomZ dimer number in the system is constant. Hence, also the flux of PomZ in the system is reduced, which leads to a decrease of the flux jright with increasing L. Bringing all terms in Eq 19 that depend on L to the right hand side yields a curve that first increases with L, then reaches a maximum and decreases again for large L. Hence, we expect no oscillations for small and large nucleoid lengths and oscillations might occur for intermediate lengths. Simulations with intermediate and large nucleoid lengths L indeed show this behavior (S13 Fig).
We analyzed how the cluster movement changes when the rates for the key biological processes are varied over a broad range. We found that there exists an optimal ATP hydrolysis rate of PomZ such that the time the cluster needs to move to midnucleoid is minimized. A parameter sweep of the diffusion constant of PomZ on the PomXY cluster shows that the mobility of PomZ dimers on the PomXY cluster is important for cluster movement towards midnucleoid. Qualitative changes in the cluster trajectories are observed when the diffusion constant of PomZ on the nucleoid is reduced: midnucleoid positioning of the cluster switches to oscillatory behavior of the cluster around midnucleoid. Hence, we conclude that positioning of the cluster in the flux-based model critically depends on the time scale for the cluster dynamics in comparison to the one for the PomZ dimer dynamics on the nucleoid. If the latter is slow compared to the cluster dynamics, the cluster will oscillate around midnucleoid. In contrast, fast PomZ dynamics on the nucleoid leads to midnucleoid localization of the cluster. In the latter case, we find that the average velocity of the PomXY cluster can be described by the PomZ flux difference into the cluster, which measures how far away the cluster is from midnucleoid, the force exerted by a single PomZ dimer on the cluster, and the effective friction coefficient of the cluster, which depends on the number of PomZ dimers bound to it (semi-analytical approach). This approach allows for further mechanistic insights into the cluster movement by PomZ dimer interactions. With it we can explain the dependence of the cluster dynamics on the model parameters as observed in our simulations. The mechanism we propose for midcell localization of the Pom cluster in M. xanthus is based on a flux-balance argument, which was previously proposed for positioning by the Par system [36] and also for self-organized positioning of protein clusters that dynamically form on the nucleoid [52]. In the model by Ietswaart et al. [36] and the model we present here, the cargo is a fixed structure, whereas Murray and Sourjik [52] consider a reaction-diffusion model for a protein that can form dynamic clusters on the nucleoid, which are positioned by the same protein due to a flux-balance argument. Necessary conditions for flux-based positioning are that the ATPase diffuses on the nucleoid (faster than the cargo) and cycles between a nucleoid-bound and cytosolic state [5,36,52]. Furthermore, the typical length an ATPase diffuses on the nucleoid before it detaches into the cytosol (without a preceded interaction with the cargo) has to be sufficiently large compared to the nucleoid length to ensure positioning of a cargo at midcell (see S3 Fig, S1 Text) [52,58]. How the forces are generated by the ParA-like ATPase to move the cargo (plasmid, partition complex or protein cluster) is still under debate. Lim et al. proposed that forces are generated due to the elasticity of the nucleoid [39], which we also assume here. Alternatively, a chemophoretic force has been suggested. Chemophoretic forces can explain the net movement of catalytic particles in the direction of an increasing concentration of a solute [58] and have also been applied to positioning of cargoes by the Par system [47,53–55,58]. To what extent a chemophoretic force and / or the elasticity of the nucleoid lead to the net force that moves the cargoes remains to be investigated. One important experimental observation that differs between the Pom system and several Par systems is that PomZ dimers accumulate at the cluster. In our model, we make two important model assumptions that affect the density profile of PomZ at the cluster: First, we assume that cluster-bound PomZ dimers can only detach from the cluster via ATP hydrolysis, such that the dimers are captured at the cluster until they are released into the cytosol. Second, we assume that cluster-bound PomZ dimers can diffuse on both the cluster and the nucleoid. These assumptions have important implications on how forces are generated at the cluster in our model. We find that the PomZ dimer springs not only exert forces when they attach to the cluster in a stretched configuration (as in the DNA-relay model, [39]), but instead forces can be generated every time a cluster-bound PomZ dimer encounters the cluster’s edge. Our simulations show that the latter contribution to the overall force of a single PomZ dimer is much more important than binding in a stretched configuration, for the parameters we consider. This is in stark contrast to the situation in the DNA-relay model, where only the initial deflection of the ParA dimer from its equilibrium position when binding to the cargo accounts for the generated force. Another important difference between our model and previously proposed models for the Par system that include the elasticity of the nucleoid [39,48,51,59] is how mobile the ATPase is compared to the cargo. In contrast to the aforementioned Par models, the ATPase (PomZ) diffuses rapidly on the nucleoid and the cargo (Pom cluster) only moves due to its interactions with PomZ dimers, in our model. Fast diffusion of PomZ dimers on the nucleoid and the relatively large spring stiffness of PomZ dimer springs explain the small force exerted on the cluster due to the initial deflection of the spring: this deflection is only small and quickly reduced by diffusion of PomZ. We conclude that force generation based on the elasticity of the nucleoid can be sufficient for cargo translocation even if the mobility of the transporting proteins is higher than the mobility of the cargo. Our observation of an oscillatory cluster movement when the dynamics of the PomZ dimers is slow compared to the dynamics of the PomXY cluster is in agreement with findings for the Par system [48,53], despite differences between their models and ours. Similar to our finding that an intermediate ATP hydrolysis rate of PomZ minimizes the time the cluster needs to reach midcell, Hu et al. observed that an intermediate detachment rate of the ATPase from the cargo leads to the most persistent movement of the cargo [59]. However, their model differs from our model as they consider the movement of a fast diffusing cargo on a two-dimensional DNA-carpet to mimic an in vitro Par system [47]. In contrast, our model for the in vivo Pom system accounts for the nucleoid as an object of finite size. Since the Pom cluster diffuses slowly compared to the PomZ dimers, the diffusive fluxes of PomZ into the cluster need to be accounted for when determining the dependence of the cluster dynamics on the ATP hydrolysis rate (Fig 5). The model we present here yields a mechanistic understanding of midcell localization of the Pom cluster. So far, not all model parameters are determined experimentally in M. xanthus cells. Hence, it would be important to measure the remaining biological rates, such as the nucleoid attachment rate, the diffusion constants and the cluster binding rate in vivo. Another limitation of our current model is that it is one-dimensional. How the cluster dynamics changes in a three-dimensional geometry is an interesting question for further research. Furthermore, in the current model we do not account for the PomXY cluster formation, but consider the cluster as a fixed structure. This is motivated by the experimental finding that PomX forms filaments in vitro and a high fraction of fluorescently labelled PomX was observed in the cluster in vivo [5]. However, it remains unclear how the cluster is formed in vivo and how the size of the cluster is maintained from one cell generation to the next. Our model for the Pom cluster positioning makes three important predictions, which would be interesting to test experimentally: First, the cluster starts to oscillate if PomZ dimers diffuse slowly on the nucleoid. We hypothesize that this might be tested experimentally by increasing the binding affinity of PomZ dimers to the DNA and in this way decreasing the mobility of PomZ on the nucleoid. Second, we predict that there is an optimal ATP hydrolysis rate to minimize the time the cluster takes to reach midnucleoid. Decreasing the rate of ATP hydrolysis by PomZ dimers associated with the PomXY cluster in experiments reduced the velocity of cluster movement towards midcell [5]. It would be interesting to test whether the velocity of the cluster is also reduced for an enhanced ATP hydrolysis rate in in vivo experiments. Finally, we predict that the mobility of the PomZ dimers on the Pom cluster can increase the velocity of the cluster movement. To test this model prediction, experiments to uncover the dynamics of PomZ dimers bound to the cluster are needed. The research presented here gives insights into the dynamics of the Pom cluster in M. xanthus, which is determined by its interactions with the nucleoid-bound PomZ dimers. With our semi-analytical approach we gain a better mechanistic understanding of the net force generation in our model. This approach might also prove to be useful for the related ParABS systems or other stochastic, out of equilibrium systems to position intracellular cargoes.
In the simulations to determine the cluster dynamics, all PomZ dimers are initially in the cytosol. The PomXY cluster position is kept fixed for tmin = 10 min with all possible actions of the PomZ dimers allowed. As a result, the initial condition resembles the stationary distribution of PomZ dimers. The initial position of the cluster is such that the left edge of the cluster and the nucleoid coincide. To derive PomZ flux and density profiles at specific cluster positions, the simulated fluxes and densities are recorded only if the cluster is within a certain distance of a predefined position of interest. For example, to get the PomZ flux / density for clusters at 20% nucleoid length, recording begins when the PomXY cluster is in the region 20 ± 0. 2% and stops if it leaves the region 20 ± 1%. Averaging is performed over all times at which the cluster resides within the specific region, weighting each density or flux profile with the corresponding time spent by the cluster in that specific region. To estimate the difference in PomZ flux into the PomXY cluster from either side, the maximal and minimal flux values in the average flux profile of PomZ dimers bound to the nucleoid, but not the PomXY cluster, are determined. These values are typically found a short distance from the edge of the PomXY cluster region, because PomZ dimers can attach to the cluster in a stretched configuration. The two extreme flux values of opposite sign are added together to get the average flux difference of PomZ dimers into the cluster. Simulations with a fixed position of the PomXY cluster are performed to measure the force exerted by a single PomZ dimer on the cluster (“one-particle simulations”) or to measure the PomZ dimer flux into the cluster and the forces exerted on the cluster for an arbitrary number of PomZ dimers in the system. In these simulations, the PomZ dimer (s) are initially in the cytosol. When the adiabatic assumption holds true, the results from the stationary cluster simulations can be used as approximations for the PomZ dynamics in the dynamic cluster simulations.
In order for the rod-shaped bacterium M. xanthus to reproduce, its genetic content must be duplicated, distributed equally to the two cell halves and then the cell must divide precisely at midcell. Three proteins, called PomX, PomY and PomZ, are important for the localization of the cell division site at midcell. PomX and PomY form a cluster and PomZ tethers this cluster to the bacterial DNA or nucleoid (region containing the chromosomal DNA) and is important for the movement of the cluster from the nucleoid pole towards midcell. We are interested in the question how the cluster trajectories change when the PomZ dynamics is varied. To address this question we investigate a previously developed mathematical model that incorporates the nucleoid, the cluster and PomZ. We simulated the cluster trajectories for different model parameters, such as different diffusion constants of PomZ on the nucleoid. Interestingly, when PomZ diffuses slowly on the nucleoid, we observed oscillatory cluster movements around midcell. Our results provide general insights into intracellular positioning of proteins.
dimers (chemical physics)
classical mechanics
enzymes
cell cycle and cell division
cell processes
enzymology
phosphatases
simulation and modeling
research and analysis methods
proteins
chemistry
adenosine triphosphatase
physics
biochemistry
hydrolysis
cell biology
biology and life sciences
chemical reactions
physical sciences
chemical physics
cytosol
atp hydrolysis
Regulation of Pom cluster dynamics in Myxococcus xanthus