Kinetic principles of ParA2-ATP cycling guide dynamic subcellular localizations in Vibrio cholerae

Abstract Dynamic protein gradients are exploited for the spatial organization and segregation of replicated chromosomes. However, mechanisms of protein gradient formation and how that spatially organizes chromosomes remain poorly understood. Here, we have determined the kinetic principles of subcellular localizations of ParA2 ATPase, an essential spatial regulator of chromosome 2 segregation in the multichromosome bacterium, Vibrio cholerae. We found that ParA2 gradients self-organize in V. cholerae cells into dynamic pole-to-pole oscillations. We examined the ParA2 ATPase cycle and ParA2 interactions with ParB2 and DNA. In vitro, ParA2-ATP dimers undergo a rate-limiting conformational switch, catalysed by DNA to achieve DNA-binding competence. This active ParA2 state loads onto DNA cooperatively as higher order oligomers. Our results indicate that the midcell localization of ParB2-parS2 complexes stimulate ATP hydrolysis and ParA2 release from the nucleoid, generating an asymmetric ParA2 gradient with maximal concentration toward the poles. This rapid dissociation coupled with slow nucleotide exchange and conformational switch provides for a temporal lag that allows the redistribution of ParA2 to the opposite pole for nucleoid reattachment. Based on our data, we propose a ‘Tug-of-war’ model that uses dynamic oscillations of ParA2 to spatially regulate symmetric segregation and positioning of bacterial chromosomes.


INTRODUCTION
Chromosome segregation requires precise positioning of replicated chromosomes to opposite cell halves for inheritance of the genetic material. Eukaryotic cells use the spindle apparatus comprising polymerizing microtubules to pull sister chromatids apart during mitosis. Despite the lack of cytoskeletal structure in bacteria, chromosomes are spatially organized in defined patterns with specific segregation dynamics. Most bacteria harness a tripartite ParABS apparatus for chromosome segregation (1)(2)(3). First characterized in plasmid partitioning, the P arABS (P ar) system consists of centromere-analogous parS sequences that reside proximal to the replication origin. ParB proteins bind specifically onto parS sites and spread to form the partition comple x, which mar ks the chromosome (cargo) for segregation. P arA ATP ase binds to non-specific DNA and via ATP hydr olysis pr ovides for the moti v e force to position the cargo. Par proteins also contribute to other key cellular processes including DNA condensation, chromosome replica tion, transcription regula tion, cell division and motility (4)(5)(6)(7)(8). In bacteria with multipartite genomes such as Vibrio cholerae and Burkholderia cenocepacia , the Par system is essential for their survival and pathogenesis ( 9 , 10 ).
The ParA / MinD superfamily of P-loop ATPases (deviant Walker ATPases) are involved in gradient formation in many bacteria (11)(12)(13). ParA / MinD proteins selforganize to regulate the positioning of macromolecular complex es that ar e vital for DNA segr egation and cell division. These include chromosome and plasmid segregation dri v en by propagating ParA gradients (14)(15)(16)(17)(18)(19), positioning of cell division site by pole-to-pole oscillations of MinCDE in Esc heric hia coli ( 20 , 21 ), bipolar gradient formation of Caulobacter crescentus MipZ ( 22 , 23 ), fluxbased translocation of Myxococcus xanthus PomXYZ ( 24 ) and carboxysomes positioning by McdAB in cyanobacteria ( 25 ). Despite di v erse modes of ParA dynamic gradients, the mechanism of gradient-dri v en intracellular transport remains enigma tic. Investiga ting the P arA ATP ase cycle and its control by cofactors is the key to understand how ParA proteins hav e e volv ed to spatially organize various cellular complexes in different bacteria.
Initial models of plasmid or chromosome segregation were based on in vivo studies where ParA clouds appeared as helical filaments pushing or pulling partition complexes apart ( 14 , 15 , 26 , 27 ). This was supported by imaging of negati v ely-stained P arA fibr e bundles by electron microscopy (27)(28)(29)(30). Biochemical studies of ParA-DNA interactions and cell-free reconstitutions of the P1 and F plasmid partition systems have shifted the paradigm towards a dif fusion-ra tchet mechanism (31)(32)(33)(34). The dif fusionratchet model is based on ParA gradients providing a chemophor etic for ce, wher e collecti v e P arA-P arB tethers dri v e the processi v e motion of plasmids on the nucleoid ( 35 , 36 ). Variations of the diffusion-ratchet model have been reported in more recent studies on chromosome and plasmid segrega tion. C. cr escentus was proposed to utilise the intrinsic elastic properties of the chromosome to relay the partition complex across the cell ( 17 , 37 ). B. subtilis chromosomal loci and F plasmid hitch-hike on ParA bound to high-density regions within the nucleoid ( 18 ). Whilst plasmid TP228 has merged both concepts where ParF (ParA) polymers assemble into a 3D meshwork in which the plasmid is trapped and distributed ( 19 , 38 ). Howe v er, the underlying principles of ParA gradient formation and how the spa tial organiza tion of ParA gradients media te chromosome segregation, particularly in bacteria with multiple chromosomes, remain poorly understood.
V ibrio choler ae , the causa ti v e agent for cholera, has its genome split between a larger 3 Mb chromosome 1 (Chr1) and a plasmid-like 1 Mb chromosome 2 (Chr2). Each chromosome encodes its own ParABS system that functions in a chromosome-specific manner. The two Par systems display distinct and independent in vivo dynamics ( 6 , 13 , 14 ). Upon replication, the two Chr1 origins are distributed asymmetrically; one origin travels from the old to the new pole and the other stays at the old pole. When two-thirds of the Chr1 replication cycle is completed, then only Chr2 replica tion initia tes ( 39 , 40 ). The two Chr2 origins then relocate symmetrically from the mid-cell to quarter-cell po-sitions ( 14 , 41 , 42 ). Migration of the origins of Chr1 and Chr2 ar e dir ected by the partition complex, consisting of ParB-bound parS sites close to the origin ( 43 ). Chr1 ParA1 forms a retracting cloud, pulling one of the ParB1-parS loci towar d the ne w cell pole ( 14 , 44 ). This mimics a mitoticlike model where ParA1 filaments polymerize to the sister P arB1-parS complex es and r etract to pull them apart ( 14 ). The ParABS1 system is not essential for Chr1 segregation, which employs an unknown mechanism for segregation in absence of Par system ( 43 , 45 ). On the contrary, the ParABS2 system is essential for Chr2 segregation and V. cholerae viability. Deletion of parAB2 loci results in loss of Chr2 over time and activation of to xin-antito xin systems that kill cells, thereby causing loss of V. cholerae pathogenesis ( 9 ). ParA2 was initially shown by negati v e staining to form a left-handed helical filament on DNA ( 46 ). More recently the crystal structures of ParA2 apo and ADP-bound states, and the cryo-EM structure of ParA2 filament bound to DNA have been solved ( 47 ). ParA2 dimers undergo a dr amatic structur al rearr angement upon DNA binding that exposes an oligomerization interface, allowing it to form filaments. These studies provide a structural insight into how high-density ParA bound regions form on the nucleoid. There has been no reported ParB2 structures to date, although recent ParB structures from Bacillus subtilis and Myxococcus xanthus offer clues to the ar chitectur e of partition complexes (48)(49)(50). Despite the vital role of ParA2 in V. choler ae prolifera tion, very little is known about its subcellular dynamics and biochemical properties.
Here, we found that ParA2 forms asymmetric gradient that dynamically oscillates in V. cholerae cells. In vitro, ParA2 dimers upon binding ATP, undergo a rate-limiting conformational switch that is catalysed by DNA to achie v e DNA-binding competence. This activated ParA2 binds DNA cooperati v ely as higher-order oligomers to dynamically pattern the nucleoid. Nucleotide exchange is also slow, which might be controlling the rate of DNA rebinding and gradient formation. Nonetheless, the ATP cycling and DNA rebinding rates of ParA2 were significantly faster compared to those of plasmid ParAs. Based on these findings, we propose a 'Tug-of-war' model that uses ParA2 dynamic oscillations to regulate the symmetrical positioning and segregation of a large chromosomal cargo that is more responsi v e in coordinating segrega tion d ynamics with cell division.

Microscopy of ParA2 in V. cholerae
Cells were grown in 1 × M63 medium supplemented with 1 mM CaCl 2 , 1 mM MgSO 4 , 0.001% vitamin B1, 0.2% fructose, 0.1% casamino acids at 30 • C until an OD 600 nm of 0.3. When r equir ed, kanamycin was added to a final concentration of 12.5 g / ml. Expression of ParA2:GFP or ParA2 Nucleic Acids Research, 2023, Vol. 51, No. 11 5605 K124X:GFP (where X = Q, E or R) was induced by adding 0.0008% arabinose for 1 h at 30 • C with shaking. 10 l of the culture was plated on the centre of a glass P35 dish (Mat-Tek corporation, Ashland, MA), and overlaid with a 1% agarose disc pr epar ed with the same M63 medium described above but supplemented with 0.02% arabinose. Images were taken e v ery 30 secs on a Nikon Ti-Eclipse inv erted microscope with Nikon 100 ×/ 1.4 Oil Plan Apo Ph3 DM objecti v es, imageEM EMCCD camera (Hamamatsu, Japan) and Lumencor sola light engine (Beaverton, OR) set to 5% 475 nm laser output and 300 ms exposure.

SEC-MALS
Samples of 40 M ParA2 were incubated alone, or in the presence of 1 mM ATP or ADP, in 50 mM Tris-HCl pH 7.5+(210 mM NaCl, 5.0 mM MgCl 2 , 0.1 mM EDTA, 1.0 mM DTT, 1.0 mM NaN 3 ), for 20 min at 37 • C. SEC-MALS of ParA2 was performed with 15 l injections into a GE Super de x 200 10 / 300 GL SEC column at 0.75 ml / min equilibrated and run in 50 mM Tris-HCl pH 7.5 buffer+(100 mM NaCl, 5.0 mM MgCl 2 , 0.1 mM EDTA, 1.0 mM DTT, 1.0 mM NaN 3 ) using a Postnova AF2000 system with PN5300 autosampler. Protein elution was monitored with a Shimadzu Prominence SPD-20AV (PN3212) UV absorbance detector, PN3621b MALS detector and PN3150 Refracti v e Inde x Detector. Data analysis was conducted with No-vaFFF AF2000 2.1.0.1 (Postnova Analytics, UK Ltd) software and values plotted in Graphpad Prism 8.0.2. For protein concentration determination, a UV 280 nm molar extinction coefficient of 1.03 M −1 cm −1 was used and absolute molecular w eights w ere calcula ted using Zimm fits. Da ta was averaged from three repeat measurements.

ATPase activity
For pre-stead y sta te ATP ase activity measur ements, 1.5 M ParA2 or its mutant deri vati v es, 100 M ATP and 64 nM [ ␣-32 P]ATP were incubated in Buffer A. Where indicated, 1.5 M ParB2 and / or 0.1 mg / ml sonicated salmon sperm DNA were added. 10 l reactions were assembled on ice, incubated for the indicated time periods at 37 • C and quenched by the addition of 10 l 1% SDS and 20 mM EDTA. For stead y sta te activity assays, indicated concentrations of ParA2 were incubated in reactions set up as described above, at 37 • C for 30 min. 1 l from each sample was spotted onto a POLYGRAM CEL 300 PEI-TLC plate (Macherey-Nagel), and de v eloped with 0.5 M LiCl (Sigma) and 1 M formic acid (Alfa Aeser). Dried plates were exposed to a storage Phosphor screen and scanned with a phosphoimager (Typhoon FLA7000 IP) for quantification using ImageJ (NIH).

Circular dichroism spectroscopy (CD)
CD experiments were performed in filtered and degassed buffer containing 10 mM Tris pH 8.0+5 mM MgCl 2 . Reaction mixtures were prepared with 5 M ParA2 and 2 mM of either ATP , ADP , AMPPNP , ATP ␥ S, or no nucleotide. ParA2 with ATP in the absence of MgCl 2 was prepared in 10 mM Tris pH 8.0+2 mM EDTA. The samples were filtered by centrifugation using a 0.2 m Generon Proteus Clarification Mini Spin Column (GENMSF-500). The reactions were incubated at 23 • C for 15 min. Spectra were measured using a Jasco J-810 Spectropolarimeter in a 1 mm Hellma Analytics QS High Precision Cell. Measurements were collected from 300 to 200 ±2.5 nm, in 1 nm intervals with an 8 s integration time. Blank buffer solutions containing corresponding nucleotides were subtracted from the ParA2 spectra. Each experiment was repeated at least twice and each spectra is an average of 3 scans. ParA2 secondary conformation was monitored by CD at 220 ±2.5 nm with 8 s integration time, from 23 • C to 63 • C. The temperature was incr eased in 2 • C incr ements, and the sample was equilibrated to each temperature for 1 min before measurement of the signal.

Nucleotide binding, dissociation and e x change assays
Stopped flow measurements with MANT ( Nmethylanthranilo yl)-labeled n ucleotides (Jena) were performed at 23 • C using Applied Photophysics SX20. The excita tion monochroma tor w avelength w as 356 ±1.2 nm and emission filter was BLP01-405R-25 (Semrock). Nucleotide binding, dissociation, and e xchange e xperiments were performed in Buffer B with samples pr epar ed on ice. For nucleotide binding assays, 0.6, 1.25 or 2.5 M ParA2 was ra pidl y mixed with 25 M MANT-AXP (AXP = ATP or ADP) and fluorescence increase was monitored over time. For pseudo-first order reaction, 0.3125, 6.25, 1.25 or 2.5 M ParA2 was ra pidl y mixed with 3.125, 6.25, 12.5 or 25 M MANT-AXP in buffer B and their fluor escence incr ease monitor ed. The observed binding curves were fitted with single exponential increase to determine observed rate of binding, k obs . Plots of k obs versus substr ate concentr ation yielded k on and k off from the slopes and y-inter cepts, r especti v ely ( 51 ). For nucleotide dissociation assay, 2.5 M ParA2 and 5 M MANT-AXP wer e pr e-incuba ted a t 23 • C for 3 min, then ra pidl y mixed with 1 mM unlabelled AXP and their fluorescence decrease monitored. For nucleotide exchange assay, 0.625, 1.25 or 2.5 M P arA2 was pr e-incuba ted a t a 1:5 ra tio with 3.125, 6.25 or 12.5 M unlabelled AXP, respecti v el y, then ra pidl y mixed with 15.625, 31.25 or 62.5 M MANT-AXP at 5 × higher concentrations than AXP. All data were averages of at least two experiments. Values were reported as relati v e fluor escence incr ease or decr ease.

T ryptophan fluor escence experiment
For stead y sta te r eactions, 0.6 M P arA2 with 1 mM ATP, ADP or ATP ␥ S were incubated at 23 • C for 15 min in Buffer B. In the absence of MgCl 2 , a separate buffer was pr epar ed with 0.1 mM EDTA and without MgCl 2 . Tryptophan fluorescence signal was acquired using a SpectraACQ spectrafluorimeter set at 356 ±1.2 nm. FluorEssence V3.5 softw are w as used f or plotting data and GraphPad Prism f or data analysis. Stopped-flow measurements were performed at 23 • C using Applied Photophysics SX20 system. The excita tion monochroma tor w avelength w as 295 nm and emission filter was BLP01-325R-25 (Semrock). For kinetics experiment, 1.2 M ParA2 was ra pidl y mixed with 2 mM MANT-AXP in Buffer B and when present, 0.2 mg / ml DNA and 1.2 M ParB2. Final concentrations after mixing are half of initial concentrations. All results are averages of at least two experiments. Values were reported as relati v e fluor escence incr ease or decr ease.

EMSA
A standard reaction mixture (20 l) was prepared in Buffer A with 5 nM Cy3-labeled 69 bp DNA and 2 mM of ATP, ADP, ATP ␥ S or no nucleotide, with increasing concentrations of ParA2 as indicated. The reactions were assembled on ice, incubated for 30 min at 30 • C, and analysed by electrophoresis in 5% polyacrylamide gels in TBM (90 mM Tris, 150 mM Borate, 10 mM MgCl 2 ). Gel was pre-run at 120 V for 30 min at 4 • C, in a Mini-PROTEAN Tetra Cell, and then run at 120 V for 1 h at 4 • C. Gels were imaged using a Bio-Rad ChemiDoc ™ MP Imaging System using the Cy3 channel with 2 min exposure. Images were analysed with ImageJ (NIH).

TIRF microscopy
A home-built prism-based TIRF microscope was set up using a Ti-Eclipse (Nikon) microscope with a PlanApo 100x NA 1.45 oil-immersion objecti v e. Laser e xcitation light of 488 nm (Cobolt 06-MLD) at laser power 100 W was focused and aligned onto a prism placed on a quartz slide for TIRF illumina tion a t the centre of the objecti v e. The fluorescence emission light was filtered by a notch filter (Thorlabs NF488-15) and bandpass filter (Chroma ET535 / 70m). Fluorescence images were captured using a sCMOS camera (Prime 95B, Photometrics) with exposure time 100 ms, fr ame r ate 1 s at 16-bit depth. The camera bias of 100 arbitrary units was subtracted from measured intensity. Micromanager software was used for camera control and image acquisition. ImageJ (NIH) was used for image analysis. Syringe pumps (WPI) were controlled using RealTerm open software.

Flowcell assembly
Single-inlet and dual-inlet flowcells with Y-shaped channel were assembled and coated with biotinylated lipid bilay er, neutr avidin and DNA carpet as previously described ( 32 , 33 ).

ParA2-GFP binding and dissociation on DNA carpet
ParA2-GFP (10 M) was preincubated in Par Buffer (50 mM Tris pH 7.5, 100 mM NaCl, 5 mM MgCl 2 , 10% (v / v) glycerol, 1 mM DTT and 0.1 mg / ml ␣-casein) with 1 mM ATP, ADP or ATP ␥ S for 30 min at 25 • C. The sample was diluted to final protein concentrations as indicated with Par buffer. The sample was loaded into a 1 ml syringe (BD) and attached to inlet 1 of a dual-inlet flowcell. A separate syringe containing wash buffer (Par Buffer) was attached to inlet 2. The TIRF illumination field and microscope objecti v es were aligned to the Y-channel junction at the point of flow convergence. For ParA-GFP binding, sample and wash buffers were infused sim ultaneousl y into the flowcell at 20 l / min and 1 l / min respecti v ely. After 380 s, sample and wash buffer flow rates were switched to 1 l / min and 20 l / min respecti v ely, for protein dissocia tion. Da ta was anal ysed using Gra phPad Prism 7. Binding curves were fitted with 'one phase association' model whilst dissociation curves were fitted with a 'two phase decay' model.

Additional procedures
For plasmid construction and protein purification, see Supplementary Data.

ParA2 displays dynamic localization in V. cholerae
To monitor the spatiotemporal distribution of ParA2 in wild type V. cholerae cells, we expressed ParA2 with a Cterminally fused GFP (ParA2-GFP) from an inducible promoter, leaving the nati v e parA2 locus intact. The expression of the fusion protein affected the cell growth rate ( Figure  S5A), necessitating controlling the induction le v el and duration. At the le v el of e xpression used, cell-length distribution was not significantly altered, nor did chubby cells appear, indicating stability of Chr2 maintenance ( 9 ). We belie v e at the le v el of e xpr ession used, P arA2-GFP serves as a tracer for WT ParA2, since subcellular dynamics was seen in at least 50% of the cells by time-lapse microcopy, as we discuss below. We observed that almost all the cells that expr essed P arA2-GFP had an asymmetric distribution of the protein along the long-axis of the cell (98%, n = 61) (Figure 1 A). The distribution showed an asymmetric gradient with the highest intensity toward one of the cell poles that gradually decreased as it approached the cell center in predivisional cells, or to the septum in dividing cells. In timelapse imaging, the ParA2-GFP gradients showed variability in the spa tiotemporal d ynamics. About 50% of the cell population showed ParA2-GFP gradients either persisting at the same location or migrating to the opposite pole once in the same cell cycle (Figure 1 A). On the other hand, the remaining 50% of the cells showed more dynamic localizations of ParA2-GFP, a typical example of which is shown in Figure 1 B (Movie S1). In these cells, ParA2-GFP gradients started from one pole and transitioned to form the gradient at the opposite pole before switching back again. Se v eral periods of pole-to-pole oscillation occurred during each cell cycle. Kymo gra phs of the cells show their oscillation periods varied from 4 to 6 min and pole-to-pole transition times from 30-60 s with longer residence times at the cell poles (Figure 1 C, left and middle). In some dividing cells, ParA2-GFP gradients were observed at both the cell poles (Figure 1 C, right). In some of these cells, ParA2-GFP began to oscillate between the pole and the closing septum. A representati v e cell in Figure 1 C (right) shows an initial polar P arA2-GFP gradient r edistributing to bipolar loca tions a t 2 min, followed by oscillations of ParA2-GFP between the left cell pole and the dividing septa at 10 min and 18 min. W hile oscilla tory d ynamics are also exhibited by MinD and plasmid ParAs ( 15 , 19 , 20 , 26 , 52 ), periodic oscillations have not been consistently observed for chromosomal ParAs. In vivo dynamics of chromosomal P arAs ar e mor e varied; V. cholerae ParA1 of Chr1 forms a cloud that leads the ParB1-parS1 foci to the new pole at once per cell cycle, resembling that of C. crescentus ( 27 ). B. subtilis Soj localize at the septa and origin region in its replication-inhibitory state w hen stim ulated by Spo0J, and rebinds the nucleoid during replica tion initia tion ( 53 , 54 ). To understand how ParA2 dynamically localizes in V. cholerae and mediates Chr2 segregation, we characterized the biochemical and kinetic properties of interactions involving ParA2.

ParA2 dimerize without nucleotides
We purified wild type ParA2 and found it to be stable and soluble for in vitro assays (see Supplementary procedures and Figure S1A). To determine the oligomerization state of ParA2 with ATP-binding, we performed size exclusion chromato gra phy m ulti-angle light scattering (SEC-MALS) analysis. At 40 M, the majority of the popula tion alread y formed ParA2 dimers with an average MW of 91.1 kDa (theoretical MW 92.8 kDa) without nucleotides (Figure 2 A). The peak positions and widths of elution profiles remained relati v ely unchanged with ADP and ATP. The v ery low mean sample polydispersity index of 1.014 across all samples indica tes tha t a single species predominates. The peak MW remained slightly below that expected for a dimer, with a mean equivalent mass of 1.89 monomers. This lowerthan-expected mass is most likely the result of a small pop-ulation ( < 5%) of monomeric species present in equilibrium with dimers, regardless of the presence of ATP nucleotides. This is in contrast with other plasmid and chromosome ParAs, which exists as monomers with ADP (TP228 ParF) or without ATP (P1 ParA, C. crescentus ParA and Soj) ( 17 , 31 , 54-57 ). P1 ParA when analysed at similar concentrations, is in a monomer-dimer equilibrium without nucleotide and stabilizes as a dimer with ATP or ADP ( 31 ). This shows that ParA2 has a higher affinity than P1 ParA to dimerize without nucleotides. The X-ray structures of ParA2 dimer show that each ParA monomer binds 1 nucleotide to form a sandwich dimer ( 47 ). From this data, and published data on ParA homologs such as P1 / P7 ParA, F SopA and Hp Soj we infer that the binding stoichiometry of ParA2 monomer to nucleotide is 1:1.

P arA2 ATP ase activity is stimulated by P arB2 and DNA
ParA2 belongs to the family of P-loop ATPases that contains a deviant Walk er A motif ( K GGT G K S) that is involved in ATP-binding and hydrolysis. To determine the ATPase activity of ParA2, we incubated ParA2 with [ ␣ 32 P]-ATP and used thin layer chromato gra phy to separate the hydr olyzed pr oducts. We found that ParA2 is a weak AT-Pase with a catalytic constant, k cat that is higher than those of plasmid and chromosomal ParA homologs ( Figure S1B, C). Similar to plasmid ParAs, its pre-stead y sta te ATP hydrolysis rate was stimulated over 2-fold by nonspecific DNA and 3-fold by its cognate partner, ParB2 at equimolar concentra tions (Figures 2 B , S1B). The stimula tion increased to 8-fold in the presence of both DNA and ParB2. Steady state ATPase activity at increasing ParA2 concentrations showed these pre-steady state rates were close to maximum under these conditions (Figure 2 C). The amount of ATP hydrolyzed sa tura ted towards higher ParA2 concentrations due to depletion of [ ␣ 32 P]-ATP substrate (Figure 2 C). In summary, ParA2 ATPase activity is stimulated by both DNA and ParB2 to a k cat le v el higher than that of other plasmid and chromosomal P arAs (Figur e S1B , C , see Discussion).

ParA2 dimers exhibit slow nucleotide exchange rates
P1 ParA undergoes a slow multi-step conformational change upon ATP binding to switch to a DNA-binding state. This time delay in rebinding once released from DNA, was proposed to generate a ParA gradient surrounding the plasmid, driving plasmid mobility toward higher ParA concentrations ( 31 ). To test if V. cholerae uses a similar mechanism for ParA2 gradient f ormation, we in vestigated the rate-limiting-step in the ATPase cycle of ParA2. Us-ing fluorescently-labelled adenine nucleotides, MANT-ATP and MANT-ADP, we determined their binding affinities to ParA2. At stead y sta te, ParA2 bound MANT-ATP and MANT-ADP similarly with K D ∼11 M ( Figure S2). This affinity is 2-and 3-fold higher than that was determined in a previous study on ParA2: 22 M (ATP) and 34 M (ADP) ( 46 ).
We monitored the interactions of ParA2 with MANTlabelled nucleotides using stopped-flow kinetics. The relati v e fluorescence increased upon MANT-ATP binding and the extent of binding increased with increasing ParA2 concentrations (Figure 3 A). The binding curves took ∼30 s to r each an appar ent stead y sta te and fitted well to single exponential association with observed rates, k obs 0.09-0.13 s −1 ( Figure S3A). This timescale of binding ATP appears to be slow for a typical enzyme, suggesting that ParA2 may be undergoing dimerization or remodeling upon ATP-binding. Howe v er, at higher ParA2 concentrations, the MANT-ATP observed binding rate did not increase ( Figure S3A). This suggests that P arA2 alr eady exists as dimers at the lowest concentration tested (0.6 M), prior to binding ATP. This is also consistent with the SEC-MALS data (Figure  2 A). The pseudo-first-order rate constants, k on and k off were determined by titrations with increasing MANT-ATP concentrations (Figure 3 B). The observed binding rates of Plot of pseudo-first order rate constant k obs against MANT-ADP concentration. Samples were prepared as in (C), except MANT-ADP was 10x higher concentration than P arA2. ( E ) P arA2-MANT-AXP (AXP is ATP or ADP) dissocia tion kinetics. ParA2, a t indica ted concentra tions, was pr epar ed with MANT-AXP in a 1:2 ratio and pre-incubated at 23 • C for 3 min, then ra pidl y mixed with 1 mM AXP. Dissociation kinetics were measured as a decrease in relati v e MANT fluorescence. ( F ) Nucleotide exchange kinetics of ParA2. ParA2 and unlabelled AXP were pre-incubated to indicated concentrations at a 1:5 ratio, then ra pidl y mixed with MANT-AXP at a 5 × higher concentration than AXP. k obs plots versus substra te concentra tion yielded k on and k off of nucleotide exchange in table. MANT-ADP to P arA2 wer e slightly lower than MANT-ATP and did not increase with ParA2 concentrations (Figures 3 C and S3A). k on and k off of MANT-ADP to ParA2 were also similar to those of MANT-ATP (Figure 3 D), and their K D ( k off / k on ) were both ∼8 M, comparable to the stead y sta te measur ements (Figur e S2). We examined the nucleotide dissocia tion ra tes by pre-incuba ting ParA2 with MANT-AXP (ATP or ADP) before mixing with unlabelled nucleotides. The subsequent decrease in MANT-ATP fluorescence was multiphasic, with an initial fast phase followed by a slower phase of k obs 0.015-0.02 s −1 (Figures 3 E  and S3A), indicating that nucleotide release from ParA2 is slo wed do wn b y ATP hy drolysis. In comparison, MANT-ADP fluorescence dropped ra pidl y at k obs 0.07-0.08 s −1 , up to 5-fold faster than MANT-ATP dissociation ( Figure  S3A). In sum, both ATP and ADP bind ParA2 dimers with similar af finities, but ATP dissocia tion is slower due to ATP hydrolysis coupled with ADP release.
In growing cells, ParA2 dimers would undergo nucleotide exchange upon ATP hydrolysis. Depending on the rate of nucleotide exchange, this could limit the ATPase cycle. We inv estigated pseudo-first-or der rate constants of nucleotide ex change rates b y pr e-incubating P arA2 with unlabeled nucleotide and ra pidl y mixing with increasing MANT-AXP concentrations (Figure 3 F). k on of ADP → ATP exchange (6.5 ×10 −4 M −1 s −1 ) was similar to ADP → ADP exchange (6.1 ×10 −4 M −1 s −1 ), although much slower than A TP → A TP exchange (1.4 ×10 −3 M −1 s −1 ). In comparison, k on values for nucleotide exchange (Figur e 3 F) wer e much lower than for nucleotide binding (Figure 3 B, D). The slow rates for nucleotide exchange correspond to ADP release and subsequent MANT-AXP binding. The observed nucleotide exchange rates k obs increased with increasing ParA2 concentra tions, indica ting a higher order dependence on protein concentrations ( Figure S3B). Based on these data, ADP → ATP exchange appears to be a ratelimiting step in ParA2 ATPase cycle. This slow step in conversion of ParA2-ADP to ParA2-ATP dimers may contribute to the time delay in DNA rebinding, generating a ParA2 gradient on the nucleoid.

ATP provides maximal stability to ParA2
As nucleotide exchange occurs at a slow rate, we wanted to investigate if nucleotide binding induces a conformational change in ParA2 dimers. Effects of various adenine nucleotides on ParA2 secondary structure and conformational stability were tested using circular dichroism (CD) spectroscop y. The pr esence of adenine nucleotides shifted the CD spectra considerably toward higher molar ellipticity ( ) at 208 nm and slightly at 220 nm, indicating a decrease in proportion of ␣-helices (Figure 4 A). Concurrently, a slight decrease at 218 nm showed an increase of ␤-sheets. The overall spectra resulted in a decrease of ParA2 helicity by 10% with ATP and by 20% with ATP ␥ S and ADP, indica ting conforma tional changes of ParA2 dimers upon nucleotide interactions. A similar spectral shift and loss of helicity was also observed for F SopA, suggesting analogous conformational changes upon nucleotide-binding ( 58 ). In contrast, P1 ParA showed a slight increase in helicity (4-5%) upon ADP binding ( 2 , 7 ). This was corroborated by re-cent crystal structures of apo and ParA2-ADP dimers showing slight shifts in the P-loop and the domain-swapping region ( 47 ). The effect of adenine nucleotides on ParA2 structural stability was examined by thermal melts and the molar ellipticity ( ) monitored at 220 nm between 23 • C and 63 • C (Figure 4 B). The dena tura tion of ParA2 occurred at T m = 44 • C and was irre v ersib le as P arA2 pr ecipita ted a t the end of the heating cycle. ATP stabilized ParA2 structure at most by 20%, conferring a maximal T m = 53 • C. Both ATP ␥ S and ADP raised T m to 50 • C and 51 • C, respecti v ely. AMPPNP or ATP in the absence of Mg 2+ had no effect on ParA2 stability. A similar effect of ATP and ADP on T m was also observed for P1 ParA and F SopA ( 58 , 59 ), indica ting tha t this may be a common feature among chromosome and plasmid P arAs, wher e nucleotide-binding stabilizes ParA conformation in general although the extent of stabilization may vary depending on the ParA in question.

ParA2 undergoes slow remodeling to ParA2*-ATP dimer
To further investigate ParA2 structural changes upon interactions with cofactors-adenine nucleotides, DNA or P arB2, we measur ed intrinsic tryptophan fluor escence of the protein. ParA2 monomer (407 aa), has six tryptophan residues at positions 111, 224, 267, 287, 308 and 401. Sequence alignment with P1 ParA shows that residue W224 of ParA2 corresponds to W216 of P1 ParA, located on the ␣-helix 11 and close to the P1 ParA-ADP dimer interface ( 31 , 60 ). This indica tes tha t intrinsic fluorescence could be used to monitor conformational changes of ParA2, as was the case with P1 ParA ( 31 ). Negati v e controls without ATP or Mg 2+ showed no change in tryptophan fluor escence (Figur e 4 C, bars 1, 2), while the presence of ATP and Mg 2+ increased fluorescence by 5% (Figure 4 C, bar 7). ADP and ATP ␥ S had little effect on the fluorescence change, indica ting tha t P arA2 r emodeling is ATP + Mg 2+ specific (Figure 4 C, bars 3, 5). Addition of DNA with ATP incr eased fluor escence by 13%, implying further r emodeling when ParA2-ATP dimers bind DNA (Figure 4 C, bar 8). Although ATP ␥ S did not initially cause a change in ParA2 tryptophan fluorescence, DNA was able to effect a slight increase, suggesting that DNA induces a ParA2 conformation that is different than with ATP (Figure 4 C, bar 6). As P arB2 interacts dir ectly with P arA2 to stimulate ATP hydrolysis (Figure 2 B, C), we wanted to investigate if interactions with ParB2 has an effect on ParA2 structure. ParB2 (323 aa) has two tryptophan residues at positions 246 and 268 but negati v e controls showed negligib le fluorescence change with ATP and DNA. This demonstra tes tha t ParB2 could be used as a cofactor in this assay without interfering with ParA2 tryptophan fluorescence (Figure 4 C, bars 11,12). Interestingly, we found that the fluorescence change of P arA2 with ATP r emained similar e v en in the presence of P arB2 (Figur e 4 C, bar 9). The fluor escence incr ease with DN A was slightl y dampened by ParB2, suggesting a separa te conforma tion of ParA2 DNA complex when interacting with ParB2 (Figure 4 C, bar 10).
To investigate the kinetics of ParA2 conformational change between different cofactors, we monitored ParA2 tryptophan fluorescence change using stopped flow. Consistent with the steady state measurements, the negati v e controls of ParA2 without ATP or Mg 2+ showed a prolonged decrease in fluorescence that is attributed to photobleaching (Figure 4 D). Similarly, the lack of fluorescence change by ADP and ATP ␥ S indicates that the nucleotides did not induce any noticeable conformational change despite binding to ParA2. On the other hand, ParA2 with ATP showed a surprisingly slow hyperbolic incr ease to r each an appar ent stead y sta te in ∼180 s (Figure  4 D). The curves fitted well to single-exponential with similar rates of k obs 0.015-0.018 s −1 and were not dependent on P arA2 concentrations. (Figur e S4A, S4E). The rates of conformational change were slower than MANT-ATP binding by at least 6-fold ( Figure S3A) and nucleotide ex change b y up to 3-fold a t higher ParA2 concentra tions ( Figure S3B). We hypothesize that this fluorescence change is due to a slow remodeling of ATP-bound ParA2 dimers to a distinct intermedia te sta te, ParA2* 2 -ATP 2 , analogous with P1 ParA. As ATP binding and nucleotide exchange hav e faster observ ed rates, we infer the slow remodeling to be the rate-limiting-step in the ATPase cycle. We predict that the presence of DN A will catal yticall y induce structur al rearr angement of P arA2 to the P arA2* 2 -ATP 2 intermedia te sta te tha t is acti v e for DNA binding.

DNA modulates kinetics of ParA2-ATP remodeling
We tested this hypothesis by ra pidl y mixing nonspecific DNA and ATP with P arA2. Tryptophan fluor escence of ParA2 showed an initial lag phase before rising sharply to reach stead y sta te a t ∼80 s (Figure 4 D). The peak fluorescence increased to more than 2-fold in intensity and reached at 2x faster rate than those for ParA2 with ATP alone. The initial lag phase was most likely due to nucleotidebinding step preceding the remodeling of ParA2-ATP dimers. Hence, the intensity curve was fitted with a single exponential excluding the initial lag phase. Strikingly, compar ed to P arA2 with ATP alone, the pr esence of DNA sped up k obs to 0.03-0.07 s −1 , increasing toward higher P arA2 concentrations (Figur e S4E). This dependence on ParA2 concentration suggest higher-order protein interactions with DNA that induce cooperati v e remodeling of ParA2-ATP dimers to the DNA-bound state (Figures 4 D,  S4C , S4E). W hen we mix ed P arA2 with ATP ␥ S, we found that ParA2 showed a slight fluorescence increase only with DNA, indicating that DNA induces P arA2 r emodeling that is not dependent on ATP hydrolysis. We belie v e that the r ecent structur e of DNA-bound P arA2-ATP ␥ S dimers r eflects that of the ATP-bound intermediate, where ParA2 dimers undergoes structural rearrangement in the crossdimer interaction to cooperati v ely oligomerize on DNA ( 47 ). Alto gether, these data impl y that DN A activates the ra te of conforma tional change of ParA2-ATP dimer, and lo wers the ener gy barrier to ward the ParA2*-ATP dimer intermedia te tha t is primed for DNA binding. ParA2 conformational change reaches steady state about 5-fold faster compared to P1 ParA, indica ting tha t ParA2 is able to switch more quickly from a non-binding state to an acti v e DNA-binding state. We suggest that this key feature distinguishes chromosomal from plasmid ParAs, enabling ParA2 to cooperati v el y rebind DN A more quickl y and to be more dynamic in exploiting and patterning the nucleoid as a scaff old f or Chr2 segregation.
When we investigated the effect of ParB2 on the kinetics of ParA2 tryptophan fluorescence, we found that surprisingly ParB2 did not change the rates of fluorescence increase, with or without DNA (Figures 4 D, S4B-E). From these data, we infer that DNA is the primary cofactor in modulating ParA2 conformational switch since ParA2-ParB2 interactions did not appear to change the conformational kinetics or extent of ParA2*-ATP dimer formation. We postulate that ParB2 interacts with ParA2 by inserting an arginine finger or ␣-helix at the ParA2 dimer interface to stim ulate ATP-hydrol ysis ( 56 , 61 , 62 ). Based on the overlay of TP228 P arA-P arB (5U1G) and pNOB8 ParA-DNA (5U1J) crystal structures, the location of ParB Nterminal helix at the ParA dimer interface clashes with the DNA-binding region ( 61 ). Thus, it was reported that P arB stabilizes P arA nucleotide sandwich dimer but not the DNA-binding state. Howe v er, when we overlay TP228 P arA-P arB (5U1G) ( 61 ) with Vc ParA2-ADP crystal structures (7NPE) and ParA2-ATP ␥ S-DNA cryo-EM structures (7NPF) ( 47 ), we found that the ParB helices do not clash with ParA DNA-binding regions ( Figure S4F). Instead, we infer that although ParB2 does not alter the kinetics or extent of P arA2 r emodeling, P arB2 helices can interact with P arA2-DNA complex es at the dimer interface and stimulate ATP hydrolysis without causing further conformational changes.

ParA2*-ATP dimers bind DNA cooperatively
In general, ParAs bind DN A nonspecificall y, coating the nucleoid to mediate plasmid or chromosome segregation. ParA2 forms various structured helical filaments on DNA that is nucleotide-dependent ( 46 , 47 ). To evaluate ParA2 binding properties on DNA, we performed EMSAs using Cy3-labeled 69 bp non-specific DNA and adenine nucleotides. A sharp retarded band of ParA2-DNA complexes was seen with increasing intensities at higher ParA2 con-centrations (Figure 5 A). The binding curves showed that ParA2 had similar affinities with ATP (46 nM) and ATP ␥ S (34 nM), demonstra ting tha t ATP hydrolysis is not r equir ed for DNA binding (Figure 5 B) or conformational change (Figure 4 D). The binding curves were sigmoidal, showing that ParA2 binds DNA cooperati v ely in the presence of ATP with a Hill coefficient, n = 4. This implies that two dimers of ParA2*-ATP bind to 69 bp DNA ( ∼30 bp / dimer) and suggests that ParA2 has a propensity to form higheror der comple xes on DNA via dimer-dimer interactions. With ADP, ParA2 could not achie v e full binding e v en at 1.2 M and had 8-fold lower affinity for DNA (378 nM). Without nucleotide, the bands wer e smear ed throughout the lanes, showing that ParA2 by itself binds DNA too weakly (1 M) to be stable during electrophoresis ( Figure  5 A, B). To test for ParA2 dissociation from DNA, unlabeled sonicated salmon sperm DN A (sssDN A) was added to preformed ATP-bound ParA2-DNA complexes ( Figure  S6A, C). sssDNA facilitated disassembly of the complex in a concentration dependent manner. This was also observed with ATP ␥ S, indicating that ParA2 is able to dissociate from DNA without ATP hydrol ysis. To gether, the results show that upon ATP binding, ParA2 dimer undergoes slow remodeling to ParA2*-ATP dimer, a DNA-binding competent sta te. DNA ca talyzes this slow step and licenses ParA2 to cooperati v ely bind onto DNA to form higher order [ParA2* 2 -ATP 2 ] n oligomers to d ynamically coa t the nucleoid.

In vivo dynamics and in vitro interactions of K124 variants of ParA2 with ATP and DNA
We further examined the roles of ATP binding and hydrolysis in ParA2 function and their effects on in vivo dynamics. Thr ee P arA2 mutants wer e constructed by substituting a conserved lysine residue at 124 in the Walker A box with glutamine (K124Q, uncharged side chain), glutamic acid (K124E, negati v ely charged side chain), or arginine (K124R, positi v ely charged side chain). These are analogous to the P-loop P1 ParA mutants that showed defecti v e ATPase activity ( 63 ). All purified ParA2 mutant proteins showed defects in ATP hydrolysis (Figure 6 A). K124Q was the onl y m utant that maintained some ATPase activity and reduced stimulation by DNA compared to WT ParA2. K124R and K124E displayed no ATPase activity and no stimulation by DNA. End-point fluorescence measurements showed that K124R bound MANT-ATP to a lower extent than WT ParA2 (Figure 6 B). K124Q showed fluorescence quenching, suggesting interaction with MANT-ATP. K124E had negligible MANT-ATP fluorescence change, indicating lack of ATP binding. DNA-binding activities of the mutants were determined by EMSA using a Cy3-labeled 69 bp nsDNA fragment and ATP (Figure 6 C). K124R had a similar affinity for DNA as WT ParA2 ( K D = 47.1 nM), despite being deficient in ATP hydrolysis. K124Q had a 3fold decrease in DNA affinity ( K D = 137.5 nM) that is attributed to aberrant ATP-binding. K124E had the lowest DNA affinity ( K D = 452 nM) due to deficient ATP binding. EMSAs of K124R and K124Q nucleoprotein complexes showed more resistance to dissociation, while K124E complexes dissociated completely at lower competitor DNA concentrations ( Figure S6B). This corroborates the cryo-EM data where K124R and K124Q but not K124E, formed filaments on DNA with ATP ( 47 ). These findings indicate that binding to and dissociation from DNA are not dependent on ParA2 ATPase activity. ParA2-ATP is thus able to exchange on DNA without hydrolysing ATP.
To investigate the effects of K124 substitutions on ParA2 in vivo , we expressed C-terminally GFP-fused K124 variants from plasmids in the presence of WT ParA2 in V. cholerae cells and performed time-lapse imaging as before (Figure 6 D). Most of the cells appeared to grow and divide, albeit slower than cells expr essing P arA2-GFP, indica ting tha t transient expression of the mutants are not grossly toxic to the cells. Intriguingly, K124R-GFP formed punctate foci at the mid or quarter cell positions that remained immobile over time (Figure 6 D, top). We infer that K124R-GFP binds to the ParB2-parS2 complexes but fails to turnover and dissociate from the partition complexes, sequestering K124R-GFP from patterning the surrounding chromosomal DNA and pre v enting ori2 segregation. This is analogous to the P1 par PD (propagation defecti v e) mutant ParA[K122R] that irre v ersib ly anchors reconstituted partition complexes to the DNA carpet and blocks complex dynamics ( 32 ). K124Q-GFP was localized throughout the cells (Figure 6 D, middle). We did not observe any dynamic oscillations as with ParA2-GFP. Howe v er, in some cells, K124Q-GFP appeared as patches and seemed to 'flicker', suggesting some binding and dissociation from the nucleoid. This demonstra tes tha t the r esidual ATP ase activity of K124Q is insufficient to produce consistent dynamic oscillations in cells. K124E-GFP appeared diffused and more e v enly distributed throughout the cells (Figure 6 D, bottom). Again, no protein oscillations was observed. Together with the biochemistry data these results highlight the importance of the ATPase activity of ParA2 for its dynamic oscillations.

ATP stimulates ParA2*-GFP binding and dissociation on DNA carpet
To visualize ParA2 interactions on DNA, we constructed and purified ParA2 with C-terminally fused GFP. ParA2-GFP was onl y mildl y toxic in vivo and its functionality in vitro was equivalent to biochemical properties as WT ParA2 ( Figure S5). We coated a two-inlet flowcell surface with a DNA carpet acting as a biomimetic chromosome and imaged ParA2-GFP with a TIRF microscope (Figure 7 A). The two-inlet flowcell allows us to instantly switch between flowing sample solution and wash buffer to monitor realtime binding and dissociation kinetics of ParA2-GFP on the DNA carpet (see Materials and Methods). ParA2-GFP was shown by EMSA to have similar DNA binding activity compared to that of WT ParA2 ( Figure S6). We preincubated 10 M ParA2-GFP with 2 mM of various adenine nucleotides and diluted the mixture 10-fold (to 1 M P arA2-GFP) befor e infusing into the flowcell. This helps to sa tura te nucleotide-binding to ParA2 dimers so tha t the observed kinetic changes would be attributed primarily to DNA-binding of dimers. Upon infusing with ATP, ParA2-GFP bound to the DNA carpet instantly and exponentially a t ra te of 0.05 s −1 and reached steady state within 1 min (Figure 7 B, Table S1). When we switched to wash buffer at 6 min, ParA2-GFP intensity dropped immediately, dissociating at twice the rate of binding (0.1 s −1 ). ParA2-GFP coated the DNA carpet thoroughly and uniformly, and dissociated e v enly with wash buf fer, indica ti v e of highly efficient re v ersib le binding (Movie S2). In the presence of ATP ␥ S, although ParA2-GFP bound the DNA carpet to a similar extent as with ATP, the binding and dissociation rates were both slower ( ∼0.02 s −1 , Table S1). This supports our EMSA da ta tha t ATP hydrolysis is not r equir ed for DNA interactions but does stim ulate ParA2-DN A binding (3-fold) and dissocia tion (7-fold). W hen infused with ADP or without nucleotides, ParA2-GFP showed negligible binding to DNA carpet, contrary to a previous study using negati v e staining ( 46 ). We belie v e that the apparent low affinity binding in the flowcell is due to high concentration ratio of nonspecific DNA to protein, a condition likely to mimic intracellular environment more closely.
Incr easing P arA2-GFP concentrations doubled the DNA binding from 250 to 500 nM but showed higher order concentration dependence at 1 M, suggesting cooperati v e binding (Figure 7 C). All three concentrations had a similar binding rate of 0.05 s −1 and dissociation rates of 0.11-0.19 s −1 (Table S1). These dissociation rates are 5-8-fold faster than those obtained with P1 ParA (0.8-1.8 min −1 ) and F Sop A (1.9 min −1 ) ( 31 , 32 ). The dissocia tion ra tes slightly increased in the presence of ParB2 and DNA in the wash b uffer, b ut whether this slight differences are significant remain to be established. If the differences prove to be real, they could be due to stimulated ATP hydrolysis and DNA dissociation by the cofactors (Figure S7).
From our li v e-cell imaging of V. cholerae , ParA2-GFP took 30-60 s to dissociate from the nucleoid and transit across the cell (Figure 1 ). These fast dynamic oscillations of ParA2-GFP observed in the cell corroborate well with the rapid dissociation rates measured on DNA carpet that is ATP-stim ulated. In contrast, w hen we mix ed P arA2-GFP and ATP just before infusing into the flowcell in an 'ATPstart' experiment, there was an initial lag time of 1 min before intensity increased ( Figure S8). We attribute this lag to ATP-binding. Notably, the extent of binding dropped Representati v e FRAP curv e (b lack line) of P arA2-GFP:P arB2 mix ed at 1:2 ratio was fitted to a double exponential (red line). The fitted time constants correspond to a fast species (8.2 s) and slow species (308 s). All FRAP data are listed in Table S2. 5-10-fold and the binding rates were slower by 3-fold compar ed to pr e-incubation experiments. These r esults support our tryptophan fluorescence data, showing a time delay switch upon ATP-binding for ParA2-GFP to transition to an acti v e DNA-binding state.

Rapid e x change of ParA2*-GFP on DN A carpet is stabilized by ParB2
To characterize the mobility of ParA2-GFP on DNA carpet, we performed fluorescence recovery after photobleaching (FRAP). If ParA2 forms extended stable filaments on DNA, we would expect a large fraction of unr ecover ed to slowly r ecover ed species on the DNA carpet depending on protein turnover. If instead there is fast recovery of ParA2-GFP on the carpet, this indicates protein exchange on DNA, where ParA2 dimers unbind and rebind DNA from solution phase. ParA2-GFP was infused into the flowcell at two different densities (28% and 100% at steady state), which are in excess of estimated physiological density of ∼1% ( 32 ). A spot was photobleached for < 1 s on the ParA2-GFP-coated carpet and the fluorescence recovery monitor ed (Figur e 7 D, Movie S3). At 28% density, we found only a small fraction of immobile species (13%). Majority of ParA2-GFP bleached spots belonged to a fast species and r ecover ed ra pidl y at 0.43 s −1 (64%, Figure 7 E and Table S2). As most of the proteins were DNA-bound and recovered by exchanging with ParA2-GFP from solution phase, the photob leaching recov ery rate is e xpected to be dependent on the rate of protein unbinding. At 100% density or steady state binding, P arA2-GFP r ecovery rate decr eased to 0.12 s −1 and a fraction of fast species shifted towards slower and immobile species (Movie S4 and Table S2). This bimodality in dissociation suggests that with increasing protein to DNA ratio, there is an increasing population of ParA2-GFP that is forming higher-order complexes with conformational changes on DNA that is exchanging at a slower rate. The exchange rate is also consistent with the dissocia tion ra tes determined from the wash e xperiments (Tab le S1). When compared to plasmid ParAs, the rate of recovery of ParA2 was 15-and 6-fold faster than that of P1 ParA (1.8 min −1 ) and F Sop A (4.7 min −1 ), respecti v ely ( 32 , 33 ). These data show that the exchange rate of ParA2 on DNA is clearly faster compared to plasmid P arAs. Her e, we did not observe any indication of stable ParA2-ATP filaments on DNA. Howe v er, it is likely that higher protein densities lead to cooperati v e binding of ParA2 to form higher order oligomers or filaments on DNA ( 46 , 47 ).
As ParB2 stimulates ParA2 ATPase activity, we wanted to test if ParB2 also affects ParA2 exchange on DNA. We found that ParB2 lowers ParA2-GFP density on DNA carpet and inhibits protein exchange, increasing recovery time by up to 4-fold at higher ParB2 to ParA2 ratio (Figure 7 F, Table S2). Furthermore, the major fraction of faster species was reduced in the presence of ParB2, converting it to the immobile fraction. This indicates that ParB2 stabilizes ParA2 binding on DNA and slows down its unbinding. A similar effect was also reported for P1 ParA and F SopA ( 32 , 33 ). These results imply the presence of two separate populations of ParA: one that is DNA-bound and another that is interacting with ParB on the partition complex. We infer that ParB2, when bound to the partition complex at high concentrations, slows down ParA2 exchange on the partition complex and its vicinity, allowing for longer-li v ed depletion of ParA2 from DNA. Collecti v ely, our data -fast ParA2-DNA exchange and the absence of stable filaments on DNA carpet --contradict the filament model where ParA2 forms extended polymers on DNA that is stable enough to exert a pulling force on the chromosome. Instead, our data support a dif fusion-ra tchet based mechanism where ParA2 cooperati v ely binds Chr2 as dynamic oligomers to pattern the nucleoid, mediating its movement and bidirectional segregation.

Kinetics of ParA2 ATPase cycle and DNA rebinding
Her e, we r eport the rate constants of ATP ase cycle and DNA binding of a chromosomal ParA that mediates segregation of V. cholerae Chr2 (Figure 8 ). ParA2 forms spontaneous dimers at < 0.6 M. Upon binding ATP ( k 1 , k -1 ), ParA2 2 -ATP 2 under goes slo w r emodeling to P arA2 2 *-ATP 2 sta te ( k 2 ) tha t licenses the dimers to bind DNA. This transition to the DNA binding state is accelerated in the presence of DNA by 2-to 5-fold. Once competent for DNAbinding, ParA2 2 *-ATP 2 loads onto DNA and induces cooperati v e binding of mor e P arA2 dimers via dimer-dimer interactions to form higher-order oligomers on DNA, [ParA2 2 *-ATP 2 ] n ( k 3 , k -3 ). Although, ParB2 did not appear to influence ParA2 conformational change, ParB2 inhibited ParA2 exchange by stabilizing ParA2-DNA interactions. High local concentrations of ParB2 on the partition complex triggers ParA2 dissociation by stimulating ATP hydr olysis. Upon hydr olysis and release of two inorganic phospha tes, ParA2 2 -ADP 2 dissocia tes from DNA and diffuse away ( k 4 ). Once ADP dissociates from ParA2 dimers ( k 5 , k -5 ), the dimers are ready to rebind available ATP for the next round of DNA binding and the cycle restarts.
ParA2 dimers undergo nucleotide exchange ( k 6 , k -6 ) at much slower rates than ATP binding ( k 1 , k -1 ). Although the rate of ParA2 rebinding DNA is primarily determined by the rate-limiting conformational switch to the acti v e state, there is also a secondary dependence on the nucleotide exchange r ate. Particular l y, once DN A activates P arA2 r emodeling, slow nucleotide exchange regulates DNA rebinding and dynamic localization. This allows more time for ParA2 to diffuse and redistribute in the cell before rebinding the nucleoid. This delay in ATP recovery inhibits ParA2 from rebinding to the same location from where it dissociated and promotes P arA2 r edistribution to the opposite pole. We predict that lower nucleotide exchange rate, lead to slow er DNA re binding, generating faster ParA2 oscillations. In MinCDE system, it has been shown that nucleotide exchange rates regulate the spatial distribution of MinD rebinding and oscillations ( 64 ). Thus, it is likely that ParA2 d ynamic pa tterns on the nucleoid are spatiotemporally regulated by ParA2 remodeling as well as nucleotide exchange.

Comparison of Vc ParA2 with ParA homologs
Our studies re v ealed many similarities but also significant differences in the activities of ParA2 with other plasmid and Nucleic Acids Research, 2023, Vol. 51, No. 11 5617 Figure 8. Kinetic pathway and rates for ATP control of P arA2-DNA interactions. P arA2 dimers (magenta and pink) bind ATP to form a closed sandwich dimer ( k 1 , k -1 ). ParA2 2 -ATP 2 under goes a slo w remodeling to acti v e ParA2 2 *-ATP 2 state ( k 2 ), an open dimer that licenses ParA2 dimers to bind DNA. This slow transition is accelerated in the presence of DNA. Once competent for DNA-binding, ParA2 2 *-ATP 2 loads onto DNA (orange) and induces cooperati v e binding of more ParA2* dimers via dimer-dimer interactions to form oligomers on DN A ( k 3 , k -3 ). Ortho gonal ParA2 dimers are coloured green and white for clarity. ParB2 inhibits ParA2* exchange on DNA as it stabilizes P arA2*-DNA interactions. P arA2 2 -ADP 2 dissociates from DNA ( k 4 ) upon ATP hydrolysis that is stimulated by DNA and ParB2. Once ADP dissociates from ParA2 dimers ( k 5 , k -5 ), the cycle restarts and ParA2 dimers diffuse away and are ready to rebind ATP for the next round of DNA-binding. ParA2 dimers undergo nucleotide exchange ( k 6 , k -6 ) at much slower rates than ATP-binding. ParA2 recruitment on DNA and its d ynamic oscilla tions are primarily controlled by the ra te-limiting conforma tional switch to its acti v e state, and the secondary control by the nucleotide exchange rate.  ( 66 ). Significantly, Vc ParA2 shows higher k cat than other plasmid and chromosomal ParAs, with greater ATP turnover rate (Figure S1B, S1C) ( 17 , 54-56 , 65 , 67-69 ). The stimulatory effects by ParB2 and DNA are also more pronounced (except Bs Soj). Despite ParA2 and P1 ParA sharing the same ratelimiting step, ParA2 remodeling is faster, leading to faster DNA-rebinding. We infer that in the cell, the fast exchange rates of ParA2 on the nucleoid will lead to fast on and off rates with its cognate partner ParB2. These faster transient interactions between ParA2 and the nucleoid, as well as ParA2 and ParB2-parS2 , allow for the chromosome partition complex to be more dynamic in interacting with the nucleoid, relati v e to plasmids. Ov erall, the faster reaction rates compared to other ParA homolo gs impl y that Vc ParA2 is a more efficient and robust enzyme, and selection of its d ynamic fea tures may have been necessita ted to ef ficiently segregate a larger chromosomal cargo in coordination with Chr1 segregation and the cell cycle.

Tug-of-war model of Chr2 segregation driven by oscillating P arA2 w aves
Based on our biochemical characterization, we propose a tug-of-war model for V. cholerae ParA2 dynamic gradients and how this could be driving Chr2 movement and positioning ( Figure 9 ). We found that DNA rebinding could be r egulated by thr ee factors: the r emodeling to acti v e ParA2*-ATP dimers that bind DNA, the nucleotide exchange rate and the suppression of DNA ex change b y ParB2. Overall, the rate of DNA rebinding is dir ectly corr elated to the concentration of ParA2 or the concentration ratios of P arA2:P arB2 in the cell. In a young cell when ori2 is at midcell, ParB2 binds specifically with high local concentrations onto parS2 sites to form the partition complex (Figure 9 A). Upon stimulation of ATP hydrolysis by ParB2-parS2 locus, P arA2-ADP dimers ar e r eleased from the partition complex and surrounding DNA. Prior to Chr2 replica tion, the persistent localiza tion of ParB2-parS2 a t midcell depletes and suppresses ParA2 from rebinding DNA at the midcell region. The delayed ATP recovery from slow remodeling to ParA2* and slow nucleotide exchange inhibits DNA rebinding and promotes ParA2 distribution to the polar r egion. Once P arA2*-ATP dimers r ebind the nucleoid, they cooperati v el y accum ulate as oligomers, setting up a protein gradient that is highest towards the cell poles. As P arB2-parS2 locus r emains at mid-cell r elati v e to the dynamic oscillating ParA2 wave, we speculate that in young cells, ParA2 wav e cooperati v ely binds onto the nucleoid as dynamic oligomers and constantly sweeps from one pole to the opposite pole to 'tug' at ParB2-parS2 locus via ParA2-ParB2 interactions. This 'tug-of-war' motion fine-tunes and maintains the locus at midcell position with each periodic oscilla tion. Once Chr2 replica tion initia tion is licensed a t two-thirds of Chr1 replication cycle ( 40 ), the doubling of ParB2-parS2 loci activates further depletion of ParA2 at midcell, r edistributing the P arA2 w ave tow ards elongating cell poles (Figure 9 B). In older predivisional cells, the oscillating ParA2 waves 'tug' at the sister loci to bidirectionally segregate to quarter-cell positions, followed by the progressi v e separation of the bulk of Chr2. The chromosome terminus is held in place at midcell by MatP-matS until unlinking of sister chromosomes and segregation leads to septal ring formation (Figur e 9 C) ( 70 ). P arA2 oscillation r e-establishes between the two cell halves and repositions the chromosome loci back to midcell positions in each of the daughter cells (Figure 9 D).
Ther e ar e se v eral factors that compound the spatiotemporal dynamics of ParA2 that is r equir ed for chromosome segr egation compar ed to plasmid partition. First, Chr2 cargo is 10x larger than plasmids, making the large chromosomal cargo less diffusi v e, allowing for more directed mobility than smaller plasmids ( 36 ). Second, Chr2 replication and segregation cycle has to spatiotemporally coordinate with Chr1 replication and cell division cycles ( 39 , 40 ). Chr2 has then to progressi v ely segregate towar ds quartercell positions before licensing of septation by SlmA ( 71 ).
Nucleic Acids Research, 2023, Vol. 51, No. 11 5619 Third, as the partition complex drives the bulk segregation of Chr2, the underlying structure of the nucleoid scaffold is also getting remodeled as a result of concomitant replication and segregation of Chr1 and Chr2. We belie v e that the fast kinetics of ParA2 ATPase form the basis of the oscillating ParA2 waves that dri v e the symmetrical positioning and segregation of Chr2, as well as increase the dynamic response to coordinate with Chr1 segregation and cell division. It will be important to test how variations of the kinetic principles could determine spatiotemporal dynamics tha t regula te chromosome and plasmid segrega tion d ynamics across different bacteria.

DA T A A V AILABILITY
The authors confirm that the data supporting the findings of this study are available within the article [and / or] its supplementary materials.