Can a flux-based mechanism explain positioning of protein clusters in a three-dimensional cell geometry?
CCan a flux-based mechanism explain positioning of protein clusters in athree-dimensional cell geometry?
Matthias Kober, ∗ Silke Bergeler, ∗ and Erwin Frey † Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,Department of Physics,Ludwig-Maximilians-Universit¨at M¨unchen,Theresienstraße 37, D-80333 Munich, Germany
The plane of bacterial cell division must be precisely positioned. In the bacterium
Myxococcusxanthus , the proteins PomX and PomY form a large cluster, which is tethered to the nucleoid bythe ATPase PomZ and moves in a stochastic, but biased manner towards midcell, where it initiatescell division. Previously, a positioning mechanism based on the fluxes of PomZ on the nucleoid wasproposed. However, the cluster dynamics was analyzed in a reduced, one-dimensional geometry.Here we introduce a mathematical model that accounts for the three-dimensional shape of thenucleoid, such that nucleoid-bound PomZ dimers can diffuse past the cluster without interactingwith it. Using stochastic simulations, we find that the cluster still moves to and localizes at midcell.Redistribution of PomZ by diffusion in the cytosol is essential for this cluster dynamics. Ourmechanism also positions two clusters equidistantly on the nucleoid. We conclude that a flux-basedmechanism allows for cluster positioning in a biologically realistic three-dimensional cell geometry.
Keywords: midcell localization,
Myxococcus xanthus , cell division, stochastic simulation,flux-based mechanism
I. INTRODUCTION
In bacteria, intracellular positioning of proteins is im-portant for vital biological processes, including localiza-tion of the cell division machinery to midcell, as well aschromosome and plasmid segregation. The positioningsystems responsible often involve P-loop ATPases suchas ParA and MinD [1, 2]. These ATPases switch be-tween an ATP- and ADP-bound state, which alters theirsubcellular localization: the ATP-bound form typicallybinds as a dimer to the nucleoid or membrane, while theADP-bound form diffuses freely in the cytosol. Activat-ing proteins stimulate the ATPase activity of the ParA-like proteins, which results in detachment of the proteinfrom its respective scaffold in the ADP-bound form. In-tracellular patterns of these ParA-like ATPases dependon the binding properties of the ADP- and ATP-boundforms, the localization of the stimulating proteins, andthe cell geometry [3].Recently, a protein system that includes a ParA-likeATPase has been identified, which regulates the localiza-tion of the cell division site in the bacterium
Myxococcusxanthus [4–6]. The Pom system consists of three pro-teins: PomX, PomY and PomZ. The ATP-bound ParA-like ATPase PomZ binds as a dimer non-specifically toDNA. PomX and PomY form a macromolecular clus-ter, the PomXY cluster, which is tethered to the nu-cleoid via PomZ dimers. This tethering is transient, sincePomX, PomY and DNA synergistically stimulate the AT-Pase activity of PomZ, which results in two ADP-bound ∗ These two authors contributed equally † Correspondence: [email protected]
PomZ monomers being released into the cytosol. Fluo-rescence labeling shows that, shortly after cell division,the PomXY cluster moves from a position close to onenucleoid end towards mid-nucleoid in a biased randomwalk-like movement that depends on PomZ. Once at mid-nucleoid, the Pom cluster positively regulates FtsZ ringformation [4].Though the Pom system in
M. xanthus and the Minsystem in
Escherichia coli serve the same function (lo-calization of the cell division plane), they differ in impor-tant respects: Dimeric MinD-ATP binds to the cell mem-brane, and MinC inhibits FtsZ ring formation outsideof the midcell region, leading to negative regulation ofmid-plane localization. In this regard, the Pom system ismore similar to the Par system for low-copy-number plas-mid and chromosome segregation, and related systems,e.g. those used to position chemotaxis protein clusters [7]and carboxysomes [8]. These systems utilize ParA-likeATPases that bind in the ATP-bound form to the nu-cleoid and regulate the movement of a cargo (e.g. proteincluster, partition complex or plasmids) to the requiredintracellular positions [9].Previously, we introduced a one-dimensional mathe-matical model of the dynamics of the PomXY cluster in
M. xanthus [4, 10] that takes the elasticity of the nu-cleoid [11, 12] into account. Based on this model, weproposed a mechanism that relies on PomZ dimer fluxeson the nucleoid to generate the experimentally observedcluster dynamics, which is midcell localization. A flux-based mechanism was originally proposed by Ietswaartet al. [13] for Par-mediated plasmid positioning. Theyshowed that diffusive ParA fluxes can in principle ex-plain equidistant spacing of plasmids along the nucleoid,because only when this is the case will the fluxes fromeither side of each plasmid balance up. Hence, macro- a r X i v : . [ q - b i o . S C ] D ec molecular objects can be evenly distributed on the nu-cleoid if they move in the direction from which the largernumber of ParA proteins impinges upon them [13]. Inour model of the Pom system, an analogous mechanismcan localize the PomXY cluster to mid-nucleoid [4, 10].It remained unclear, however, whether such a flux-based mechanism can also localize midcell if the fullthree-dimensional geometry of the nucleoid is accountedfor (see Figure 1A). The reason is that, in contrast tothe one-dimensional case, PomZ dimers can now moreeasily pass the cluster by without interacting with it. Asa result, an asymmetry in the PomZ fluxes might be bal-anced. Here we show that the Pom cluster still local-izes at midcell by a flux-based mechanism even if PomZdimers can diffuse past the cluster. II. RESULTSA. Flux-based mechanism for midcell localization
The dynamics of the PomXY cluster on the nucleoidcrucially depends on the PomZ dynamics, as the clusteris tethered to the nucleoid via PomZ dimers [4]. Based onthe biochemical processes suggested by experiments [4],we model the dynamics of PomZ as follows (see Figure 1Band SI text for details). ATP-bound PomZ dimers bindto the nucleoid with rate k on (action 1 in Figure 1B).Once on the nucleoid, they diffuse with diffusion con-stant D nuc . The PomZ dimers are modeled effectively assprings to account for the elasticity of the chromosomeand the Pom proteins (as in [12, 14]). For simplicity wewill refer to the PomZ dimers as springs in the follow-ing, although the elasticity mainly originates from thenucleoid.A nucleoid-bound PomZ dimer can attach to the clus-ter with rate k a either ‘orthogonally’, or obliquely in anextended configuration (action 2 in Figure 1B). We as-sume that cluster-bound PomZ can diffuse on both thenucleoid and the PomXY cluster. However, the freedomof movement of nucleoid- and cluster-bound PomZ is re-stricted due to the energy cost involved in stretching thespring (for details see SI text). Cluster-bound PomZdimers remain attached to the cluster until they are re-leased into the cytosol upon ATP hydrolysis, which isstimulated by PomX, PomY and DNA. ATP hydrolysisleads to a conformational change in the PomZ dimer andtriggers the release of two ADP-bound PomZ monomersinto the cytosol [4]. In our model, we combine theseprocesses into one by using a single rate k h to describethe detachment of cluster-bound PomZ-ATP dimers intothe cytosol as monomers (action 3 in Figure 1B). Be-fore PomZ can rebind to the nucleoid, it must first bindATP and dimerize. This introduces a delay between de-tachment from and reattachment to the nucleoid, whichallows for spatial redistribution of the quickly diffusingcytosolic PomZ dimers in the cell (action 4 in Figure 1B).The cluster dynamics, which we approximate as over- damped, is determined by the forces exerted by the PomZdimers on the cluster and the friction coefficient of thecluster (see SI text). Previously we suggested a flux-based mechanism for its midcell positioning in a one-dimensional model geometry [4, 10], which can be sum-marized as follows. The PomZ dimers, which are mod-eled as springs, can exert net forces on the cluster intwo different ways. They can attach to the cluster in astretched configuration and thereby exert forces on thecluster (similar to the DNA-relay mechanism proposedfor Par systems, [12]). However, for PomZ dimers thatquickly diffuse on the nucleoid (as observed experimen-tally in M. xanthus cells) the initial deflection of thespring relaxes within a short time, such that only a smallnet force is exerted on the cluster. In this case, forces aremainly generated by cluster-bound PomZ dimers that en-counter the cluster’s edge (Figure 1A, bottom left) [10].Every time a nucleoid- and cluster-bound PomZ dimerreaches the cluster’s edge, the nucleoid binding site candiffuse beyond the cluster region, whereas the clusterbinding site is restricted in its movement, which on av-erage results in a net force on the cluster. These forcesare such that a protein that reaches the cluster from theright exerts a force, which ‘drags’ the cluster to the rightand vice versa.The fact that PomZ dimers can exert forces on thecluster is not enough to explain the movement of the clus-ter towards midcell. Here, the asymmetry in the PomZdimer density on the nucleoid, which prevails as long asthe cluster is located off center, i.e. not already at mid-nucleoid, is crucial. Since PomZ dimers can attach any-where on the nucleoid but can detach only when theymake contact with the cluster (as stimulation of theirATPase activity depends on PomX, PomY and DNA), anon-equilibrium flux of PomZ dimers in the system canbe maintained. In particular, this flux includes a dif-fusive flux of PomZ dimers along the nucleoid towardsthe cluster. For a cluster located off-center the diffusivePomZ fluxes into the cluster from either side will differ[13]. Since the average forces applied by PomZ dimersarriving from the right and left sides of the cluster actin opposite directions, the difference in the PomZ fluxesfrom the two sides determines the net force exerted onthe cluster [10]. Taken together, these factors combine todrive a self-regulated midcell localization process as longas the PomZ dynamics is fast compared to the clusterdynamics and, if this is not the case, lead to oscillatorycluster movements along the nucleoid [10].
B. A three-dimensional model for midcelllocalization
To understand how the geometry of the nucleoid andthe size of the PomXY cluster affect the cluster dynam-ics and thereby test whether a flux-based mechanism isfeasible in a biologically realistic three-dimensional geom-etry, we investigated the mathematical model illustrated
Figure 1.
Mathematical model for PomXY cluster positioning, which accounts for the three-dimensional ge-ometry of the cell. (A) Schematic representation of the geometry used in our model. Top: Sketch of a
M. xanthus cell.Cytosolic, ATP-bound PomZ can bind to the nucleoid (orange arrows towards the nucleoid), and diffuses on the nucleoid. Whenbound to the PomXY cluster, PomZ hydrolyzes ATP and is released into the cytosol (orange arrow away from the nucleoid).These dynamics lead to a net diffusive flux of PomZ on the nucleoid towards the cluster, which is larger from the side withthe larger cluster-to-nucleoid end distance (horizontal orange arrows) [13]. Bottom left: PomZ dimers, modeled as springs, canexert forces on the cluster by binding to the cluster in a deflected configuration and by encountering the cluster’s edge. Fora particular nucleoid binding-site position, x , the positions available to PomZ’s cluster-binding site are limited if the dimeris located close to the cluster’s edge (black cross). This asymmetry can result in a force exerted on the cluster (black arrow).Bottom right: The model geometry derived from the biologically realistic three-dimensional cell. The nucleoid is modeledas an open cylinder and the cluster as an object of fixed size of rectangular shape. The cytosolic PomZ-ATP distribution iseither assumed to be homogeneous or included effectively by modeling the cytosolic PomZ distribution along the long cell axis.(B) Schematic of the interactions of PomZ with the nucleoid and cluster considered in our model (see main text and SI textfor details). in Figure 1. Here, the nucleoid and the PomXY clusterare approximated as a cylindrical object and a rectangu-lar sheet, respectively. Since experiments in M. xanthus cells suggest that the cluster is large [4] we assume thatthe cluster, tethered to the nucleoid via PomZ dimers,moves over the nucleoid’s surface and does not penetratethe bulk of the nucleoid. Moreover, we assume that PomZdimers also bind to and diffuse on the nucleoid’s surfaceonly.The cylindrical geometry of the nucleoid is mathemat-ically implemented by a rectangular sheet with periodic boundary conditions for the PomZ movements along theshort cell axis ( y direction) and reflecting boundary con-ditions along the long cell axis ( x direction, Figure 1B).The cluster is modeled as a rectangle with reflectingboundaries at its edges for the PomZ dimer movement.We refer to the extension of the cluster along the longand short cell axis as the cluster’s length, l clu , and width, w clu , respectively (see Figure 1A).Besides the nucleoid and the cluster, the cytosol needsto be accounted for in our model, as PomZ dimers cy-cle between a nucleoid-bound and cytosolic state. Weexpect the cytosolic diffusion constant of PomZ to be ofthe same order as that of MinD proteins in E. coli cells, D cyt ≈ µ m s − [15]. For ParA ATPases involved inchromosome and plasmid segregation, a delay betweenthe release of ParA from the nucleoid into the cytosol andthe re-acquisition of its capacity for non-specific DNAbinding is observed experimentally [16]. Upon ATP hy-drolysis and release of two ADP-bound ParA monomersinto the cytosol, ParA must bind ATP, dimerize and thenregain the competence to bind non-specifically to DNAbefore it can reattach to the nucleoid. The last step, aconformational change in the ATP-bound ParA dimer, isthe slowest of these processes, occurring on a time scaleof the order of minutes, τ ≈ L diff = p D cyt τ ≈ µ m,which is significantly larger than the average cell lengthof a M. xanthus cell of 7 . µ m [4]. Hence, the assumptionof a well-mixed PomZ density in the cytosol is justified[3].However, it is not known how the cytosolic distribu-tion of PomZ affects the cluster dynamics in our pro-posed flux-based mechanism. In particular, does a re-alistic, non-uniform distribution increase or reduce thespeed of the cluster movement towards midcell? To ob-tain a qualitative answer to this question, we accountedfor the cytosolic PomZ distribution in a simplified way byfocusing on the variation in PomZ density along the longcell axis and approximating the density along the shortcell axis as uniform (Figure 1A, bottom right). In sec-tion II E we discuss how the cluster dynamics is affectedwhen PomZ’s diffusion constant in the cytosol is reduced(which leads to deviations from the uniform PomZ distri-bution), but for now we assume a homogeneous PomZ-ATP distribution in the cytosol. C. A flux-based mechanism can explain midcellpositioning in three dimensions
Our simulations using the three-dimensional model ge-ometry show that the net force exerted by the PomZdimers on a stationary cluster (i.e. a cluster with fixed po-sition) is still directed towards mid-nucleoid (Figure 2A).The asymmetry in the forces can be attributed to anasymmetry in the PomZ fluxes along the nucleoid intothe cluster. This finding indicates that the cluster move-ment is biased towards midcell in the three-dimensionalmodel geometry also. Hence, a flux-based mechanism isalso conceivable in three dimensions. Indeed, the sim-ulated PomXY cluster trajectories show movement to-wards and localization at mid-nucleoid (Figure 2B). If thecluster’s width does not cover the complete nucleoid cir-cumference, the movement towards midcell takes longerthan in the one-dimensional case (Figure 2B). The forces along the short cell axis direction balance (Figure 2A)and hence no bias in the cluster movement along theshort axis is expected for a slowly moving cluster andfast PomZ dynamics. The simulated cluster trajectoriesshow diffusive motion along the short cell axis directionfor the parameters given in Table S1 (see SI text).
D. The cluster’s linear dimensions determine thetime taken to reach midcell
In this section we ask how the arrival time of the clus-ter at mid-nucleoid depends on the linear dimensions ofthe cluster. Our simulations show that the larger thecluster’s length or width, the faster the cluster moves to-wards midcell (Figure 3A). In the following we discusshow these two observations can be explained heuristi-cally.The net force applied to the cluster by PomZ dimersdepends on the average force exerted by a single PomZdimer and on the flux difference in PomZ dimers arrivingat the cluster [10]. The latter can be regarded as thefrequency of PomZ interactions with the cluster that leadto a net force contribution.The flux difference into the cluster increases for largeror broader clusters for two reasons. First, increasingthe length or width of the cluster results in an increasednumber of cluster-bound PomZ dimers, as the cluster be-comes more accessible to nucleoid-bound PomZ dimers.This increase in cluster-bound PomZ dimers leads toa larger turnover of PomZ dimers cycling between thenucleoid-bound and cytosolic state, which also increasesthe flux difference. Second, the larger and wider the clus-ter, the less likely it is that PomZ dimers will diffuse pastit without attaching, which would otherwise reduce theflux difference into the cluster along the long cell axis.The forces exerted by PomZ dimers also depend on thelinear dimensions of the cluster. Increasing the clustersize increases the average force exerted by a single PomZdimer, until a maximal force is reached (Figure 3B). Thisdependence can be explained by diffusion of PomZ dimersbound to the cluster: The smaller the cluster size, themore likely it is that a PomZ dimer attaching close toone edge of the cluster will reach the other edge beforedetaching into the cytosol. Since the PomZ dimer thenexerts forces in both directions along the long cell axis,the time-averaged force it applies to the cluster is re-duced.Based on these observations, we can rationalize thedependence of the arrival time at mid-nucleoid on thelength of the cluster as follows. An increase in clusterlength (while keeping its width constant) increases thenumbers of PomZ dimers interacting with the cluster,and hence the flux difference along the long cell axis.Indeed, our simulation results show that the longer thecluster, the larger the difference in the PomZ dimer fluxesfrom either side along the direction of the long cell axis(Figure 3C). To test whether the increased flux of PomZ time [min] x - po s i t i on [ % ] B y - po s i t i on [ % ] A |F y | [pN] x-position [%] | F x | [ p N ] Figure 2.
A flux-based mechanism can explain midcell localization in a three-dimensional cell geometry. (A) Fora fixed cluster position (here at 20% nucleoid length), the average forces exerted by PomZ dimers on the cluster are plottedper nucleoid lattice site. The color code shows the magnitude of the average force vector, which is highest at the cluster’sedges (darker red indicates higher values). At these edges, the average force vectors are plotted, ~F = ( F x , F y ). The average x - and y -component of the force, F x and F y , (summed over all y - and x -positions, respectively), are shown in the lower andright panels, respectively. (B) Comparison of the average cluster trajectories along the cell’s long axis, obtained from thethree-dimensional model and its one-dimensional counterpart studied previously [10] (see also Figure S3). We averaged overan ensemble of 100 simulations. The shaded regions depict one standard deviation around the mean density value. In all cases,the cluster is initially positioned at the left edge of the nucleoid such that it overlaps entirely with the nucleoid (at 7% of thenucleoid length). Mid-nucleoid is indicated by the horizontal black line. The parameter values used in the simulations are givenin Table S1. dimers in the system is the main determinant for theobserved increase in the flux difference, we scaled thefluxes by the number of PomZ dimers in the cytosol, N cyt ,which is proportional to the flux of PomZ dimers onto thenucleoid. Upon rescaling, the flux differences decay ap-proximately linearly with the cluster position (Figure 3C,right). However, longer clusters still show the largest fluxdifferences. This phenomenon can be attributed to thefact that for shorter clusters PomZ dimers are more likelyto diffuse past the cluster.For a longer cluster not only the diffusive flux of PomZdimers into the cluster, but also the force exerted by a sin-gle PomZ dimer on the cluster is increased (Figure 3B).Hence, frequency and magnitude of forces exerted on thecluster are increased, implying a larger net force, whichexplains the shorter arrival times of the cluster at mid-nucleoid (Figure 3A).Next, we discuss the dependence of the arrival time onthe cluster width. As in the case of an increase in length,the overall turnover of PomZ dimers increases with clus-ter width. The width of the cluster determines how manyPomZ dimers approach and attach to the cluster fromthe directions of the long and the short cell axes, respec-tively, and how many pass the cluster without interactingwith it. The broader the cluster, the smaller the flux ofPomZ dimers that can diffuse past the cluster without at-taching. Our simulation results show an increased PomZflux difference when the cluster width is increased from16% to 100% of the nucleoid’s circumference (Figure 3D).When the cluster covers the entire circumference of thenucleoid, we call it a ring. Rescaling of the flux differ-ences with the number of PomZ dimers in the cytosol, N cyt , leads to values that are still larger for wider clus-ters, due to the reduced flux of PomZ dimers past the cluster (Figure 3D, right). The forces exerted by singlePomZ dimers in the direction of the long cell axis direc-tion are not affected by a change in the cluster width.Hence, the decrease in arrival time for wider clusters canbe explained by the increased PomZ flux difference intothe cluster along the long cell axis alone. E. Cytosolic diffusion ensures fast midcellpositioning of the cluster
So far, we have assumed that the cytosolic PomZ distri-bution is spatially uniform. Now we investigate how thecluster dynamics change when spatial heterogeneity inthe cytosolic PomZ distribution is included in the model.To this end, we explicitly incorporate the cytosol by ap-proximating the cytosolic volume as a one-dimensionallayer of the same length as the nucleoid (see Figure 1A)and formulate reaction-diffusion equations for the densityof cytosolic PomZ-ADP ( c D ) and PomZ-ATP ( c T ) alongthe long cell axis. For simplicity, we consider only anactive (ATP-bound) and inactive (ADP-bound) confor-mation of PomZ, and disregard any explicit monomericand dimeric states of PomZ in the cytosol (for detailssee SI text). The stationary solution for c T ( x ; x c ) devi-ates most from a uniform distribution, the smaller thecytosolic diffusion constant (Figure 4A). In the limit ofinfinitely large cytosolic PomZ diffusion constants, thecytosolic PomZ distribution becomes spatially uniform.To investigate the effect of the cytosolic PomZ distribu-tion on the cluster’s trajectory, we replaced the spatiallyuniform cytosolic PomZ-ATP distribution by c T ( x ; x c ),in our three-dimensional model. Since M. xanthus cellsare rod-shaped with the length being much larger than * *
Figure 3.
Dependence of the time needed to reach midcell on the size of the cluster. (A) Average first passage timeof the PomXY cluster to reach mid-nucleoid for different cluster sizes. In all simulations the cluster starts at 13% of nucleoidlength, which corresponds to the leftmost position possible such that for all cluster sizes considered a full overlap with thenucleoid is ensured. The error bars show the standard error of the mean. (B) Ensemble average of the time-averaged forceexerted by a single PomZ dimer on the cluster for a one-dimensional model geometry and different cluster lengths. (C)-(D) ThePomZ flux difference into the cluster, j diff , along the long cell axis for a cluster at a fixed position, which is varied from 10%to 50% of nucleoid length, is shown (see also Figure S2). In (C), the cluster’s length is varied and in (D) the cluster’s width.For a ring-shaped cluster, w clu = w nuc , the simulation results agree with those from the one-dimensional model (black crosses).To understand how the cluster’s length and width affect the flux difference, the values are scaled with the number of PomZdimers in the cytosol, N cyt , for each cluster position separately (Figures on the right). In (D) an analytical estimate for the fluxdifference is plotted (dashed line), which agrees with the simulation results for a ring-shaped cluster (see SI text for details). Ifnot given explicitly, the parameter values used are those listed in Table S1 (data marked with a star). the width, we approximate the cytosolic PomZ-ATP dis-tribution along the short cell axis as uniform. With thisspatially heterogeneous attachment rate to the nucleoid,our simulations show that for a larger deviation of thecytosolic PomZ distribution from a spatially uniform one (decreasing D cyt ), the movement of the clusters is lessbiased towards mid-nucleoid (Figure 4B). We concludethat the cytosolic PomZ distribution has an impact onthe cluster trajectories and the velocity of the cluster to-wards mid-nucleoid is maximal for a uniform cytosolic
10 20 30 40 50 cluster x-position [%] a sy mm e t r y C A cyt A flux (w clu = 0.7 m)A flux (w clu = 2.2 m) x-position [%] cy t. P o m Z - A T P c l u s t e r N left N right A D cyt largeD cyt = 0.5 m /sD cyt = 0.1 m /s time [min] x - po s i t i on [ % ] B Figure 4.
Fast cytosolic diffusion accelerates midcell lo-calization. (A) Cytosolic PomZ-ATP distribution along thelong axis ( x axis) for different cytosolic PomZ diffusion con-stants D cyt (see also Figure S1). Integrating the distributionsalong the long cell axis left and right of the cluster yields thetotal number of PomZ-ATP proteins left and right of the clus-ter, N left and N right . (B) Average cluster trajectories alongthe x direction for the different cytosolic PomZ diffusion con-stants in (A). The shading denotes the regions of one standarddeviation around the average trajectories. Mid-nucleoid is in-dicated by the solid black line. In the simulations, the clustersare positioned initially such that the left edge of the clustercoincides with the left edge of the nucleoid. (C) Asymme-try measure of the number of cytosolic PomZ-ATP left andright of the cluster, A cyt (Equation 1) (solid lines), comparedto the PomZ flux asymmetry into the cluster, A flux (Equa-tion 4), for different cytosolic PomZ diffusion constants. Thedots indicate the flux asymmetry into a cluster that formsa ring. Crosses indicate the asymmetry for a cluster with awidth of 32% of the nucleoid’s circumference (same value as inTable S1). We averaged over 100 runs of the simulation. Theparameter values given in Table S1 are used if not explicitlystated otherwise. PomZ-ATP distribution.How does the cytosolic PomZ distribution affect thecluster’s movement? The flux of PomZ dimers onto thenucleoid region to the left of the cluster and the diffu-sive flux of PomZ dimers into the cluster from the leftare equal for a stationary cluster in the one-dimensionalmodel geometry, and vice versa for the right side (see SItext). Since the total flux of PomZ dimers onto the nu-cleoid to the left and right of the cluster depends on thecytosolic PomZ-ATP distribution, we expect the diffusivePomZ fluxes into the cluster to depend on this distribu-tion as well. To investigate this further, we define thefollowing asymmetry quantity A cyt = N right − N left N right + N left , (1)which compares the total numbers of cytosolic PomZ-ATP left and right of the cluster (see Figure 4A): N left = Z x c − l clu / c T ( x ; x c )d x , (2) N right = Z l nuc x c + l clu / c T ( x ; x c )d x . (3)The corresponding asymmetry in the PomZ fluxes intothe cluster from the left, j left , and right, j right , sides isgiven by A flux = j right − j left j right + j left . (4)We measured this flux asymmetry in our simulationsfor two scenarios. First, for a PomXY cluster that formsa ring around the nucleoid and second, for a cluster thatdoes not entirely encompass the nucleoid’s circumfer-ence. We find that the asymmetry in the flux, A flux ,obtained from simulations for a cluster that forms a ring,and the corresponding asymmetry in the cytosolic PomZ-ATP density, A cyt , agree nicely (Figure 4C). Hence, anasymmetry in the cytosolic distribution of PomZ-ATP isdirectly reflected in the diffusive PomZ fluxes into thecluster. For an infinitely large cytosolic PomZ diffusionconstant, both asymmetry measures decay linearly whenthe cluster position is varied from far off-center towardsmid-nucleoid. Decreasing the cytosolic PomZ diffusionconstant results in asymmetry curves that decay fasterthan linearly towards zero (Figure 4C). For a cluster thatdoes not cover the whole nucleoid’s circumference, theasymmetry in the PomZ fluxes into the cluster is smallerthan the asymmetry in the cytosolic PomZ-ATP concen-tration (Figure 4C). This can be attributed to the reduc-tion in the diffusive fluxes of PomZ dimers into the clusteralong the long cell axis, as discussed before. The reducedasymmetry in the diffusive PomZ fluxes explains the lessbiased movement of the cluster towards mid-nucleoid forsmaller cytosolic PomZ diffusion constants (Figure 4B). F. Two clusters localize at one- and three-quarterpositions
Motivated by equidistant positioning of multiple car-goes, such as plasmids [13, 17–20], we also consideredthe dynamics of two PomXY clusters in the three-dimensional model geometry. Our simulation resultsshow equidistant positioning of the two clusters (see SItext, Figure S4).
III. DISCUSSION
In this work we investigated a mathematical modelfor midcell localization in
M. xanthus using a biolog-ically realistic three-dimensional geometry for the nu-cleoid. Whether or not a flux-based mechanism can po-sition macromolecular objects when the ATPases (herePomZ) can diffuse past a cargo (here PomXY cluster)has been questioned, because the fluxes into the cargomight equalize [13]. We showed that if PomZ dimerscan diffuse past the PomXY cluster, there is still a fluxdifference into the cluster (for a cluster positioned off-center), which leads to a bias in the cluster movementtowards mid-nucleoid. Hence, we conclude that a flux-based mechanism can explain midcell positioning of oneor equidistant localization of several Pom clusters even ifPomZ can diffuse past the cluster.To investigate the effect of the flux of PomZ dimerspast the cluster, we studied the dependence of clusterdynamics on its width and length. We find that increas-ing the cluster length or width shortens the time taken forthe cluster to reach mid-nucleoid. This can be attributedto an overall increase in the flux difference, reduced fluxpast the cluster and the larger forces single PomZ dimersexert, on average, on larger and wider clusters.Our simulation data further demonstrates that fast cy-tosolic diffusion of PomZ proteins reduces the time takento find midcell. This finding is in agreement with previ-ous results for dynamic protein clusters in bacterial cells[21]. The asymmetry in the cytosolic PomZ density leftand right of the cluster along the long cell axis is re-flected in the diffusive flux difference of PomZ dimersinto the cluster, and thus influences the cluster dynam-ics. If PomZ diffuses quickly in the cytosol, the cytosolicPomZ-ATP distribution becomes spatially uniform. In this case, the flux of PomZ dimers onto the nucleoid scaleswith the length of the nucleoid regions left and right ofthe cluster. This results in the largest PomZ flux dif-ferences (for an off-center cluster) compared to spatiallynon-uniform cytosolic PomZ-ATP distributions. Inter-estingly, spatial redistribution of proteins in the cytosolis also found to be important for other pattern-formingsystems, including Min protein pattern formation [22, 23]and ParA-mediated cargo movement [16].Le Gall et al. [24] have shown that partition complexesas well as plasmids move within the nucleoid volume. Incontrast, based on the large size of the PomXY cluster,we assumed that the movement of the cluster, tethered tothe nucleoid via PomZ dimers, is restricted to the surfaceof the nucleoid. To verify our assumption, the positionof the cluster relative to the nucleoid needs to be mea-sured in vivo using e.g. super-resolution microscopy. Inaddition, since PomZ dimers are much smaller than thePomXY cluster, they might be able to diffuse within thenucleoid volume even though the cluster does not. Itwould be interesting to investigate this aspect further.In summary, we have shown that a flux-based mecha-nism can explain midcell localization of one, and equidis-tant positioning of two clusters in a model geometrythat allows the ATPase PomZ to diffuse past the clusterson the nucleoid. This observation is also important forother positioning systems, such as the Par system, whichequidistantly spaces low-copy-number plasmids along thenucleoid. Understanding the differences and similaritiesbetween these positioning systems will help us to under-stand the generic mechanisms underlying the localizationpatterns of cargoes inside the cell.
ACKNOWLEDGEMENTS
The authors thank Isabella Graf, Emanuel Reith-mann, Christoph Brand, Dominik Schumacher and LotteSøgaard-Andersen for helpful discussions. This researchwas supported by a DFG fellowship through the Grad-uate School of Quantitative Biosciences Munich, QBM(SB, EF), the Deutsche Forschungsgemeinschaft (DFG)via project P03 within the Transregio Collaborative Re-search Center (TRR 174) “Spatiotemporal Dynamics ofBacterial Cells” (SB, EF), and the German ExcellenceInitiative via the program “Nanosystems Initiative Mu-nich” (EF). [1] K. Gerdes, M. Howard, and F. Szardenings, “Push-ing and pulling in prokaryotic DNA segregation,”
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Supplemental Material
I. DETAILS ON THE MATHEMATICAL MODEL1. Attachment rate of PomZ to the nucleoid
In our simulations we either assume that the PomZ density in the cytosol is homogeneous such that the flux ofPomZ dimers onto the nucleoid is constant along the nucleoid or we account for the cytosolic PomZ distribution ina simplified manner. In the former case, a cytosolic PomZ dimer attaches to each lattice site of the nucleoid, alsowhere the cluster is located, with the same rate. This rate is given by the attachment rate to the entire surface ofthe nucleoid, k on , divided by the number of lattice sites l nuc · w nuc /a . By a we denote the lattice spacing, whichhas the same value in x - and y -direction. In the latter case, when we account for the cytosolic PomZ distribution,we replace the homogeneous with the steady-state cytosolic PomZ-ATP distribution in our model (see section I 5).Here, we use a one-dimensional solution for the cytosolic PomZ-ATP density, which describes the variation of theprotein density along the long cell axis. In our model, which has a three-dimensional geometry, we assume that thecytosolic PomZ-ATP density is uniform along the short cell axis and changes along the long cell axis according to theone-dimensional distribution.
2. Attachment of PomZ to and detachment from the PomXY cluster
A nucleoid-bound PomZ dimer can bind to a lattice site on the PomXY cluster. Since PomZ dimers are modelledas springs to account for the elasticity of the nucleoid, they can attach to the cluster not only ‘orthogonally’, but alsoin a stretched configuration. We approximate the rate for a PomZ dimer, bound to the nucleoid at position ~x nuc , toattach to the cluster at position ~x clu , as follows: k a ( ~x clu ) = k a · exp (cid:20) − βk ( ~x clu − ~x nuc ) (cid:21) . (S1)The rate for attachment to a lattice site at position ~x clu on the cluster is then given by k a ( ~x clu ) · a . In the formulaabove, we multiply the constant rate k a with a Boltzmann factor corresponding to the distribution of the elongationof a spring in a thermal heat bath with temperature T . Here, k denotes the effective spring stiffness of the PomZdimers and β is the inverse of the thermal energy, β = 1 /k B T . The positions of the cluster and nucleoid binding sitesof the PomZ dimer are denoted as ~x clu and ~x nuc , respectively.We expect that the position of a cluster, which is tethered to the nucleoid, does not change remarkably in thedirection perpendicular to the nucleoid’s surface because the cluster is unlikely to penetrate into the nucleoid’s volumedue to its large size and, on the other hand, cannot move far away from the nucleoid due to the tethering. Hencewe neglect the forces that a PomZ dimer exerts on the cluster in the direction perpendicular to the nucleoid’s surfaceby approximating the positions of the cluster and nucleoid binding site by their projections on the rectangular sheetrepresenting the nucleoid, i.e. ~x clu = ( x clu , y clu ), and ~x nuc = ( x nuc , y nuc ).The rate for a PomZ dimer located at ~x nuc to attach, with its second binding site, to any site of the cluster (“totalattachment rate”) is then given by integration of k a ( ~x clu ) over all possible cluster binding sites: k tot a = Z A clu k a · exp (cid:20) − βk ( ~x clu − ~x nuc ) (cid:21) d ~x clu ≈ ( k a · πβk , if ~x nuc ∈ A clu , , otherwise . (S2)Here, A clu denotes the area on the nucleoid that is covered by the cluster. Since the Boltzmann factor decays quickly(1 / √ βk = 0 . µ m for the spring stiffness used, see Table S1), we can neglect the boundaries of the region A clu for ~x nuc ∈ A clu and approximate the attachment rate of a PomZ dimer bound to the nucleoid outside of the cluster regionby zero.Due to the fast decay of the exponential factor with an increase in | ~x clu − ~x nuc | , we can also save computationtime by introducing a cut-off distance above which we set the attachment rate to zero. The cut-off is defined as thesmallest distance ∆ x = | ~x clu − ~x nuc | for which the attachment rate per lattice site, k a · a , is smaller than 10 − s − .This value is chosen such that the average number of times this event occurs during the time the cluster takes toreach midcell ( ≈ . a r X i v : . [ q - b i o . S C ] D ec into the cytosol) into one effective detachment process and denote the corresponding rate as the ATP hydrolysis rate k h , which we assume to be independent of the degree of stretching of the dimer.
3. Hopping of PomZ dimers on the nucleoid and the PomXY cluster
In our model we assume that nucleoid-bound PomZ dimers can diffuse on the nucleoid and, when the dimer isattached to the cluster, also on the PomXY cluster with diffusion constants, D nuc and D clu , respectively. Thesetwo diffusive processes are implemented as stochastic hopping events that occur with rate k = D nuc /a and k = D clu /a . More concretely, a nucleoid-bound PomZ dimer at site ( i, j ), with i denoting the lattice sitein x -direction and j in y -direction, can move to site ( i ± , j ) or ( i, j ±
1) with hopping rate k . As we areusing periodic boundary conditions in y -direction, a particle may also hop from site ( i, N nuc ,y ) to the site ( i,
1) andvice versa; here N nuc ,y = w nuc /a denotes the number of sites along the y -axis. In x -direction we assume reflectingboundary conditions for the PomZ dimer movements.If a PomZ dimer is bound to both cluster and nucleoid, a hopping event leads to gain or loss in elastic energy. Inthis case we multiply the constant hopping rates by exponential factors, which are chosen such that detailed balanceholds (see Lansky et al. [25]): k hop,clu = k · exp (cid:20) − βk (cid:2) ( ~x clu, new − ~x nuc ) − ( ~x clu, old − ~x nuc ) (cid:3)(cid:21) , (S3) k hop,nuc = k · exp (cid:20) − βk (cid:2) ( ~x clu − ~x nuc, new ) − ( ~x clu − ~x nuc, old ) (cid:3)(cid:21) . (S4)The labels “old” and “new” refer to the positions of the binding sites before and after the hopping event.
4. Movement of the PomXY cluster
Cluster-bound PomZ dimers can exert forces, which lead to a net movement of the cluster. Let us denote theposition of the nucleoid binding site of the i -th PomZ dimer as ~x i, nuc , and the position of the cluster binding siteas ~x i, clu = ~x c + ∆ ~x i, clu . With ~x c = ( x c , y c ) ∈ R we denote the position of the midpoint of the cluster. Here, wedecomposed the position of a cluster binding site into two parts: the cluster position, ~x c , and an additional vector∆ ~x i, clu . The reason for this decomposition is that we are interested in the equation of motion for the cluster position, ~x c , and, as long as the PomZ dimer does not diffuse on the nucleoid or the cluster, the two vectors ~x i, nuc and ∆ ~x i, clu are constant.A single PomZ dimer, bound to the nucleoid and cluster at fixed positions relative to both scaffolds, exerts a force ~F i ( t ) on the cluster: ~F i ( t ) = − k ( ~x i, clu ( t ) − ~x i, nuc ) = − k ( ~x c ( t ) + ∆ ~x i, clu − ~x i, nuc ) . (S5)In a friction dominated regime, the sum over all forces exerted by N b cluster-bound PomZ dimers has to balance withthe friction force acting on the cluster (friction coefficient γ ): γ ˙ ~x c ( t ) = N b X i =1 ~F i ( t ) = − k N b X i =1 ( ~x c ( t ) + ∆ ~x i, clu − ~x i, nuc ) . (S6)This equation is solved by separation of variables, yielding ~x c ( t ) = ( ~x c ( t ) − ~x f ) exp (cid:18) − N b kγ ( t − t ) (cid:19) + ~x f , (S7)with ~x f = 1 N b N b X i =1 ~x i, nuc − ∆ ~x i, clu ! . (S8)The initial time is denoted as t . We find that the cluster approaches the position ~x f , at which no net force is actingon the cluster, exponentially fast with characteristic time t clu = γ/ ( N b k ).
5. Cytosolic PomZ distribution
To include the cytosolic PomZ dynamics in our model, we reduce the ATPase cycle to three processes: attachmentof PomZ-ATP to the nucleoid, detachment of PomZ-ADP at the cluster, and nucleotide exchange. We model thedynamics of PomZ-ATP and PomZ-ADP in the cytosol as one-dimensional reaction-diffusion equations as described inthe following. At the position of the cluster, x c ( t ), PomZ-ADP is released into the cytosol. Since the PomZ dynamicsis a lot faster than the cluster dynamics, we can assume that the cluster is stationary, x c ( t ) = x c on the time scale ofthe PomZ dynamics. The local increase in cytosolic PomZ-ADP at the cluster position due to detachment facilitatedby ATP hydrolysis is approximated as a point source: s δ ( x − x c ). The constant s depends on the hydrolysis rate k h and the amount of PomZ dimers bound to the cluster. However, our final result, the normalized steady-state PomZ-ATP distribution will not depend on this constant. In the cytosol, PomZ-ADP exchanges ADP for ATP nucleotideswith an effective rate k ne . We assume that cytosolic PomZ in both nucleotide states diffuses with the same diffusionconstant, D cyt . However, only the ATP-bound form of PomZ can attach to the nucleoid with a rate k on . In total,we obtain the following coupled partial differential equations for the cytosolic PomZ-ATP ( c T ) and PomZ-ADP ( c D )density: ∂ t c D ( x, t ) = D cyt ∂ x c D ( x, t ) − k ne c D ( x, t ) + s δ ( x − x c )Θ( t ) , (S9a) ∂ t c T ( x, t ) = D cyt ∂ x c T ( x, t ) + k ne c D ( x, t ) − k on c T ( x, t ) . (S9b)We solved these two differential equations with no-flux boundary conditions for the stationary case. The solutionfor PomZ-ATP, given the cluster is at position x c , reads: c T ( x ; x c ) = ˜ c (cid:20) − λ T cosh (cid:18) L λ T (cid:19) cosh (cid:18) L + xλ T (cid:19) sinh (cid:18) Lλ D (cid:19) ++ λ D cosh (cid:18) L λ D (cid:19) cosh (cid:18) L + xλ D (cid:19) sinh (cid:18) Lλ T (cid:19)(cid:21) , for − x c ≤ x ≤ , (S10) c T ( x ; x c ) = ˜ c (cid:20) − λ T cosh (cid:18) L λ T (cid:19) cosh (cid:18) L − xλ T (cid:19) sinh (cid:18) Lλ D (cid:19) ++ λ D cosh (cid:18) L λ D (cid:19) cosh (cid:18) L − xλ D (cid:19) sinh (cid:18) Lλ T (cid:19)(cid:21) , for 0 ≤ x ≤ l nuc − x c , (S11)with ˜ c = 4 s λ T e L (1 /λ D +1 /λ T ) D cyt ( λ D − λ T ) (cid:0) e L/λ D − (cid:1) (cid:0) e L/λ T − (cid:1) . (S12)We chose the coordinate system such that the cluster position, x c ∈ [0 , l nuc ], is shifted to the origin. The lengths of thecluster-to-nucleoid end distances left and right of the cluster are given by x c and l nuc − x c , respectively. Furthermore,we defined the diffusive length scales for PomZ-ADP until it exchanges its ADP for ATP, and PomZ-ATP until itattaches to the nucleoid as λ D and λ T , respectively: λ D = r D cyt k ne and λ T = r D cyt k on . (S13)The above solution for the cytosolic PomZ-ATP density holds true for λ T = λ D . If the two length scales are equal,˜ c becomes singular and hence this case needs to be considered separately. For λ D = λ T ≡ λ the solution is given by: c T ( x ; x c ) = ˜ c (cid:20) (2 L − x ) cosh (cid:18) L + xλ (cid:19) − x cosh (cid:18) L + xλ (cid:19) ++ (2 L + x ) cosh (cid:16) xλ (cid:17) + (2 L + x ) cosh (cid:18) L − xλ (cid:19) ++ 4 λ cosh (cid:18) L λ (cid:19) cosh (cid:18) L + xλ (cid:19) sinh (cid:18) Lλ (cid:19)(cid:21) , for − x c ≤ x ≤ , (S14) c T ( x ; x c ) = ˜ c (cid:20) (2 L + x ) cosh (cid:18) L − xλ (cid:19) + x cosh (cid:18) L − xλ (cid:19) ++ (2 L − x ) cosh (cid:16) xλ (cid:17) + (2 L − x ) cosh (cid:18) L + xλ (cid:19) ++ 4 λ cosh (cid:18) L λ (cid:19) cosh (cid:18) L − xλ (cid:19) sinh (cid:18) Lλ (cid:19)(cid:21) , for 0 ≤ x ≤ l nuc − x c , (S15)with ˜ c = s D cyt sinh ( L/λ ) . (S16)With the analytical solution for the PomZ-ATP density in the cytosol, we can now define the attachment rate of aPomZ dimer to the nucleoid, accounting for the cytosolic PomZ distribution. We normalize the steady-state solutionfor c T ( x ; x c ) to obtain a probability density: p T ( x ; x c ) = k on s c T ( x ; x c ) . (S17)The rate for a PomZ dimer to attach to position ( x, y ) on the nucleoid is then defined by: k on ( x, y ; x c ) ≡ k on p T ( x ; x c ) p T ( y ) = k on w nuc p T ( x ; x c ) , (S18)with a uniform distribution, p T ( y ), along the short cell axis direction. Approximately, the attachment rate per latticesite is then given by this value multiplied with the lattice spacing a squared. x-position [%] p T Point clusterExtended cluster
FIG. S1.
Cytosolic PomZ-ATP distribution.
Comparison of the steady-state solutions for the cytosolic PomZ-ATP densityif release of PomZ-ADP into the cytosol at the PomXY cluster is modelled as a point source (red line) or a source with thesame extension as the cluster (blue line). The cluster is at position x c = 30% of nucleoid length. We used the parametersshown in Table S1. In the presented derivation of the cytosolic PomZ-ATP density we reduced the cytosol to a one-dimensional lineand the PomXY cluster to a point source. To investigate how the PomZ-ATP density changes when the cluster’sextension is accounted for, we solved equations S9a and S9b with the Dirac delta distribution replaced by a Heavisidestep function Θ( x − x c + l clu / x c + l clu / − x ) /l clu , numerically. We find that the steady-state solution for thePomZ-ATP density, when the cluster is included as a point source (Equations S10 and S11), is a good approximationto the solution considering a one-dimensional cytosolic lane and an extended cluster (Figure S1). The PomZ densityprofiles only deviate significantly in close proximity to the cluster, which can be attributed to the different shapes ofthe cluster used. II. DISCUSSION OF PARAMETERS USED IN THE SIMULATIONSParameter Variable 1D model 3D model
Nucleoid length l nuc . µ m 5 . µ mNucleoid width (circumference) w nuc - 2 . µ mPomXY cluster length l clu . µ m 0 . µ mPomXY cluster width w clu - 0 . µ mEffective spring stiffness of a PomZ dimer k k B T µ m − k B T µ m − Attachment rate of cytosolic PomZ to nucleoid k on . − . − Attachment rate of nucleoid-bound PomZ to cluster (unstretched) k a
500 s − µ m − . × s − µ m − Diffusion constant of PomZ on nucleoid D nuc . µ m s − . µ m s − Diffusion constant of PomZ on cluster D clu . µ m s − . µ m s − ATP hydrolysis rate of PomZ at the cluster k h − − Total number of PomZ dimers in the cell N
100 100Diffusion constant of the PomXY cluster in the cytosol D cluster × − µ m / s 4 × − µ m / sDiffusion constant of PomZ in the cytosol D cyt - 0 . µ m s − , 0 . µ m s − Nucleotide exchange rate of PomZ k ne - 6 s − Lattice spacing a . µ m 0 . µ mTABLE S1. Parameters used in the simulations.
The parameters used in the one-dimensional model are the same as in[10]. In the three-dimensional model we used the same parameter values when possible.
The total PomZ dimer number, the length and width of the cluster and the length of the nucleoid are chosen inaccordance with experimental observations in
M. xanthus [4]. For the ATP hydrolysis rate we use a value of 1 s − assuggested by FRAP experiments where PomZ in the cluster is bleached [4]. The attachment rate of cytosolic PomZ tothe nucleoid is approximated by literature values for the related Par system for chromosome and plasmid segregation(50 s − [13] and 0 .
03 s − [12]). To get a double attachment rate that is comparable to the one-dimensional case, wechose k a such that the total attachment rate k tot a is equal in the one-dimensional model studied previously [10] andthe three-dimensional model studied here. This implies that the rate used in the one-dimensional model has to bemultiplied by a factor of p βk/ π to account for the additional dimension.The diffusion constant of PomZ on the nucleoid and on the PomXY cluster is approximated by the effectivediffusion constant of ParA dimers on the nucleoid used in models for the Par system. However, these values vary alot: 0 . µ m / s – 1 µ m / s [13, 26]. The friction coefficient of the cluster, γ , is related to its diffusion constant, D cluster ,via Stokes-Einstein: γ = k B T /D cluster . We approximate the diffusion constant D cluster by the corresponding literaturevalues for plasmids. However, since the size of the PomXY cluster is larger than the typical size of a single plasmid,these values are only an upper bound for the diffusion constant of the Pom cluster: D cluster ≤ − µ m s − [13, 14].The effective spring stiffness of the PomZ dimers, k , we approximate by the stiffness of a bond between a plasmidand the nucleoid via ParA dimers [26].We approximate the nucleotide exchange rate of PomZ-ADP to PomZ-ATP by the corresponding rate for MinDproteins, 6 s − [15]. Based on the large diffusion constant of Min proteins in the cytosol (on the order of 10 µ m s − ,[15]), and the fast dynamics of PomZ in the cytosol as observed in FRAP experiments [4], we expect the cytosolicdiffusion constant of PomZ to be large. When we tested the effect of a non-homogeneous PomZ-ATP distribution inthe cytosol on the cluster dynamics, we chose diffusion constants that are two orders of magnitudes smaller (0 . µ m / sor 0 . µ m / s). III. DETAILS ON THE STOCHASTIC SIMULATION1. Initial PomZ distribution
Initially, all PomZ proteins are in the cytosol. Then we let the simulations run for a time t min of at least 10 minuteswith a fixed cluster position such that the PomZ proteins can approach their steady-state distribution. The time t min is chosen such that it is larger than the typical time scales for the PomZ dynamics, i.e. the time scale for attachment tothe nucleoid (1 /k on = 10 s), the time scale for PomZ to explore the whole nucleoid by diffusion ( l / D nuc = 125 s),and the time scale for cluster-bound PomZ to detach (1 /k h = 1 s). After the initial time, t min , recording starts. Thecluster can now start to move or is kept at a fixed position (“stationary simulation”) during the entire simulation.
2. Gillespie algorithm
We implemented our model using the Gillespie algorithm [10, 27], a stochastic simulation algorithm. Since thecluster position, ~x c ∈ R , changes over time according to the forces cluster-bound PomZ dimers exert on the cluster(Equation S7), all rates that depend on the position of the cluster binding site of a PomZ dimer, depend on time.These include the attachment rate of a nucleoid-bound PomZ dimer to the cluster and the hopping rates of a PomZdimer bound to the nucleoid and the cluster. However, if the cluster only moves slightly in one time step of theGillespie algorithm, we can approximate the time-dependent rates as constant.To quantify the effect of the time dependence of the rates, let us consider a PomXY cluster with N b PomZ dimersbound to it such that a non-zero net force acts on the cluster. According to Equation S7 the time scale for the clusterto relax to the force-free position, ~x f , is given by t clu = γ/ ( N b k ). The number of cluster-bound PomZ dimers, N b ,changes with the position of the cluster along the nucleoid. For a cluster positioned at 10% of nucleoid length andone at mid-nucleoid the number of cluster bound PomZ dimers, as obtained from simulations, leads to t clu ≈ .
18 sand t clu ≈ .
09 s, respectively (for the parameters as in Table S1).Next, we consider the time step, ∆ t , until the next event happens in the Gillespie algorithm. The most frequentevent is hopping of PomZ dimers on the nucleoid for the parameters we consider (Table S1). Hence, the time step∆ t can be approximated by the time until a PomZ dimer bound to the nucleoid, hops on the nucleoid. The ratefor the event that any of the nucleoid-bound PomZ dimers, N nuc , hops in any of the four possible directions on thenucleoid (ignoring the boundaries) is given by 4 k , nuc N nuc . The typical time until the next event happens can thenbe approximated by the inverse of this rate. Again, the number of nucleoid-bound PomZ dimers, N nuc , varies with theposition of the cluster. For a cluster at 10% of nucleoid length and one at mid-nucleoid, we get ∆ t ≈ × − s and∆ t ≈ × − s, respectively. Since the typical time until the next event happens, ∆ t , is much smaller than the timescale for the movement of the cluster, t clu , we can approximate all rates in the Gillespie algorithm as time-independent,which significantly improves the computational speed of the algorithm. IV. PROCESSING OF SIMULATED DATA1. PomZ flux on the nucleoid
The PomZ flux along the nucleoid for a specific cluster position is determined by recording the PomZ flux at anytime the cluster is in a small region ( ± .
5% of nucleoid length) around the x -position of the cluster of interest. Toobtain the flux of PomZ into the cluster along the long cell axis direction, the fluxes are averaged over the values iny-direction, but only considering the region of the nucleoid that corresponds to the extension of the cluster regionalong the long cell axis (yellow region in Figure S2A). Additionally, the data is averaged over an ensemble of about100 simulations. An example for such an averaged flux profile is shown in Figure S2B. To obtain the difference inthe PomZ fluxes into the cluster from each side along the long cell axis, the maximal / minimal values of the averageflux profile left / right of the PomXY cluster are determined (red lines in Figure S2B). Finally, the two flux values ofdifferent signs are added together to obtain the flux difference. V. FLUX DIFFERENCE INTO THE CLUSTER
PomZ dimers detach from the nucleoid into the cytosol upon ATP hydrolysis, which breaks detailed balance andleads to a net flux of PomZ in the system. In the following we consider the fluxes of PomZ for the one- and three- y x
NucleoidClusterAveraging region Averaging along this direction A x-position [%] c u rr en t [ / ( s m ) ] BDC cytosol nucleoidcytosol nucleoidclusterI IIIIIIVIV w clu w nuc l clu l nuc j j j j j j j j j j R right R left FIG. S2.
Flux of PomZ in the three-dimensional model geometry. (A) Sketch of the region used for averaging theflux. Only hopping events in the yellow region are taken into account when determining the PomZ flux profile along the longcell axis. The fluxes are averaged over the values in y -direction. (B) Simulated flux profile of nucleoid-bound PomZ alongthe long cell axis with averaging performed as illustrated in (A). The flux into the cluster is given by the maximal value left,and the minimal value right of the cluster (red horizontal lines). We used the same parameters as in Table S1. (C) In theone-dimensional model, nucleoid and cluster are incorporated as one-dimensional lattices. The cluster region is shown as a redrectangle. PomZ dimers attach to and diffuse on the nucleoid with reflecting boundary conditions at the nucleoid ends. Thearrows show the different PomZ fluxes from one region (cytosol, cluster and nucleoid left and right of the cluster) to another.(D) Similar to (C), but the fluxes in the three-dimensional model geometry are shown. The grey region shows the nucleoid ofsize l nuc × w nuc and the red region the cluster of size l clu × w clu . The area of the nucleoid regions left and right of the clusterare denoted by R left and R right , respectively. dimensional model geometry (see Figure S2C,D). Since the cluster is a lot less mobile than the PomZ dimers, we canapproximate the cluster position as stationary. In this case, the fluxes in and out of each region (cytosol, cluster andnucleoid region left and right of the cluster) have to balance in the steady state. In the one-dimensional geometry,the fluxes of PomZ dimers to the nucleoid right and left of the cluster, j and j , balance the fluxes into the clusterregion, j and j , respectively (Figure S2C). If PomZ is homogeneously distributed in the cytosol, the fluxes onto thenucleoid scale with the lengths of the respective nucleoid regions: j = k on l nuc − x c − l clu / l nuc N cyt , (S19) j = k on x c − l clu / l nuc N cyt , (S20)with x c the position of the cluster, and N cyt the number of PomZ dimers in the cytosol. This results in the followingformula for the flux difference of PomZ into the cluster j diff = j − j = j − j = k on N cyt (cid:18) − x c l nuc (cid:19) . (S21)The flux difference is proportional to the attachment rate of PomZ to the nucleoid, k on , and the number of PomZdimers in the cytosol, N cyt . It is important to note, that N cyt also depends, among other parameters, on the positionof the cluster x c .In the three-dimensional model geometry there are additional fluxes compared to the one-dimensional geometry ifthe cluster does not encompass the entire nucleoid circumference, i.e. if w clu < w nuc holds (see Figure S2D). Nucleoid-bound PomZ dimers in region I or II can leave these regions either by entering the cluster region and then attachingto the cluster, or by diffusing into the region in the extension of the cluster along the short cell axis (region IV inFigure S2D). In the latter case, the PomZ dimers can enter the cluster region along the short axis, diffuse back intothe region they came from or diffuse past the cluster. In the steady state, the fluxes in and out of each region haveto balance. For region I and II this implies: j = j ,a + j ,b , (S22) j = j ,a + j ,b . (S23)If we assume that the fluxes into the cluster region (region III) and into region IV scale with the extensions of therespective regions along the short cell axis, i.e. j ,a /j ,b = w clu / ( w nuc − w clu ) and similarly for j ,a and j ,b , the fluxdifference into the cluster reads: j diff = j ,a − j ,a = k on N cyt w clu w nuc (cid:18) − x c l nuc (cid:19) , (S24)which agrees with the formula for the one-dimensional system if the cluster is ring-shaped, i.e. w clu = w nuc . Thisanalytical expression fits well with our simulation results for a ring-shaped cluster (see Figure 3D). However, if thecluster does not cover the full nucleoid’s width, it deviates from the simulation results. This deviation can be attributedto the fact that PomZ dimers that diffuse into region IV are not absorbed here, but can diffuse back into regions I orII. Hence, the fluxes into the cluster are larger than the values obtained from the simple estimate we used before. Forthe fluxes into the cluster from the right, we have j ,a j ,b > w clu w nuc − w clu . (S25)Furthermore, PomZ dimers can pass the cluster by diffusion from region I to II or vice versa. However, this flux onlymatters if the cluster is small both in length and width. VI. CLUSTER MOVEMENT ALONG SHORT CELL AXIS DIRECTION
In addition to the cluster dynamics along the long cell axis, as discussed in the main text, we also consideredthe dynamics along the short cell axis. Because of the rotational symmetry the clusters do not have a preferreddirection along this axis, on average. However, both diffusive and persistent, unidirectional motion is conceivable. Weexpect persistent movement if PomZ diffusion on the nucleoid is slow compared to the cluster dynamics, such thatthere is a delay between the cluster movement and the PomZ gradient reaching its steady state [10, 14, 28 - 30]. Inthis case, the initial direction of the cluster’s movement along the nucleoid’s circumference is chosen stochasticallyby the interactions of the PomZ dimers with the cluster. Once the cluster started to move in one direction, it ismore likely to continue in this direction, because the asymmetry in the PomZ density gradient is maintained. Incontrast, for fast PomZ diffusion on the nucleoid, we expect an approximately symmetric PomZ distribution aroundthe cluster, resulting in equal likelihood for the cluster to move in each direction, suggesting diffusive motion. Forthe parameter set we considered (Table S1) the clusters show diffusive behavior in y -direction as indicated by amean-square displacement that grows linearly in time (Figure S3, inset). time [min] y - po s i t i on [ % ] time [min] M S D t FIG. S3.
Cluster movement along short cell axis direction.
Average cluster trajectory in y -direction using an ensembleof 100 simulations (solid blue line). The shaded region indicates one standard deviation above and below the average trajectory.The inset shows the mean-square displacement, which increases linearly in time, indicating diffusive motion. VII. DYNAMICS OF TWO POM CLUSTERS
Motivated by equidistant positioning of plasmids by ParAB S systems we investigated the dynamics of two Pomclusters in the realistic three-dimensional cell geometry. We find that two clusters localize at the one- and three-quarterpositions along the nucleoid, i.e. at equidistant positions (Figure S4A). time [min] x - po s i t i on [ % ] FIG. S4.
Two clusters are localized at the one- and three-quarter positions.
Averaged positions of the two clustersalong the x -direction (solid lines, ensemble of 100 runs). The shaded regions indicate the average cluster positions plus andminus one standard deviation. Initially, the two clusters are positioned side by side along the long cell axis such that the leftedge of one and the right edge of the other cluster are positioned at mid-nucleoid. We used the parameter set given in TableS1.-direction (solid lines, ensemble of 100 runs). The shaded regions indicate the average cluster positions plus andminus one standard deviation. Initially, the two clusters are positioned side by side along the long cell axis such that the leftedge of one and the right edge of the other cluster are positioned at mid-nucleoid. We used the parameter set given in TableS1.