1. Chromosomes and Gene Expression
  2. Structural Biology and Molecular Biophysics
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ParB dynamics and the critical role of the CTD in DNA condensation unveiled by combined force-fluorescence measurements

  1. Julene Madariaga-Marcos
  2. Cesar L Pastrana
  3. Gemma LM Fisher
  4. Mark Simon Dillingham
  5. Fernando Moreno-Herrero  Is a corresponding author
  1. Centro Nacional de Biotecnología, Consejo Superior de Investigaciones Científicas, Spain
  2. University of Bristol, United Kingdom
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Cite this article as: eLife 2019;8:e43812 doi: 10.7554/eLife.43812

Abstract

Bacillus subtilis ParB forms multimeric networks involving non-specific DNA binding leading to DNA condensation. Previously, we found that an excess of the free C-terminal domain (CTD) of ParB impeded DNA condensation or promoted decondensation of pre-assembled networks (Fisher et al., 2017). However, interpretation of the molecular basis for this phenomenon was complicated by our inability to uncouple protein binding from DNA condensation. Here, we have combined lateral magnetic tweezers with TIRF microscopy to simultaneously control the restrictive force against condensation and to visualise ParB protein binding by fluorescence. At non-permissive forces for condensation, ParB binds non-specifically and highly dynamically to DNA. Our new approach concluded that the free CTD blocks the formation of ParB networks by heterodimerisation with full length DNA-bound ParB. This strongly supports a model in which the CTD acts as a key bridging interface between distal DNA binding loci within ParB networks.

https://doi.org/10.7554/eLife.43812.001

Introduction

Faithful segregation of bacterial chromosomes is mediated by partition systems that are classified depending on the type of NTPase involved (Gerdes et al., 2010). Type I partition systems, also known as ParABS include an ATPase motor protein, ParA, responsible for the movement of the replicated chromosomes to the distal pole of the cell, a DNA binding protein ParB, and a centromere-like DNA sequence, parS (Funnell, 2016). This type of partition system has been described in many bacteria, including Bacillus subtilis, where it is involved in chromosome segregation and sporulation (Mierzejewska and Jagura-Burdzy, 2012) (Yamaichi and Niki, 2000) (Sharpe and Errington, 1996) (Lin and Grossman, 1998) (Lee and Grossman, 2006). Importantly, the discovery that B. subtilis Structural Maintenance of Chromosome (SMC) proteins are loaded by ParB at parS sites in vivo and that parS sites act as condensation centers highlights the critical role played by ParB in chromosome organisation (Gruber and Errington, 2009) (Sullivan et al., 2009) (Umbarger et al., 2011) (Broedersz et al., 2014) (Taylor et al., 2015). Interestingly, apart from the specific binding to partition sites, ParB proteins can also bind non-specifically to DNAs spreading several kilobases around parS sites (Breier and Grossman, 2007) (Murray et al., 2006). This spreading is achieved by a limited number of ParB protomers, which has been rationalised by models of ParB-mediated bridging and condensation of DNA supported by in vitro analysis (Graham et al., 2014) (Taylor et al., 2015) (Fisher et al., 2017) (Broedersz et al., 2014) (Sanchez et al., 2015). The generation of these ‘ParB networks’ in vivo necessarily requires the presence of specific and non-specific DNA-binding domains in ParB as well as multimerisation interfaces between ParB proteins. Recent work has begun to unpick the structural basis for formation of these networks and has implicated both the N- and C-terminal domains in forming and maintaining bridging interactions (Leonard et al., 2004) (Fisher et al., 2017) (Song et al., 2017) (Chen et al., 2015).

In our previous work (Fisher et al., 2017), we studied the role of the C-terminal domain (CTD) of ParB in its multimerisation and DNA condensation. In particular, we confirmed its dimeric stoichiometry and identified a novel intermolecular non-specific DNA-binding site across the positively-charged β-sheet face of the CTD of ParB, which was critical for DNA condensation. We also showed that the presence of an excess of the CTD prevented DNA condensation by ParB and induced disruption of ParB networks in vitro and in vivo. This led us to propose three possible scenarios to explain CTD-mediated condensation inhibition (Figure 1). One possibility is that the CTD simply competes for the binding sites in the DNA molecule (Figure 1B, DNA-binding competition model). In this way, the CTD would displace ParB from the DNA and thus prevent DNA condensation, provided the CTD cannot condense DNA on its own. Another possible scenario is that the CTD is able to heterodimerise with free ParB in solution and these heterodimers are deficient for DNA binding, which directly impedes condensation (Figure 1C, DNA-binding inhibition model). Finally, it is possible that the CTD interacts with DNA-bound ParB, forming heterodimers that cannot condense but which may remain bound to DNA (Figure 1D, CTD capping model). This deficiency in condensation could be caused by the CTD ‘capping’ the ParB dimerisation interfaces.

Possible scenarios for CTD-induced decondensation of ParB-DNA networks.

(A) Model for ParB network formation and condensation via ParB-DNA and ParB-ParB interactions. ParB monomers comprise a central DNA-binding domain (CDBD) with specific and possibly non-specific DNA-binding activities and a carboxy-terminal domain (CTD) with non-specific DNA-binding capacity. The amino terminal domain (NTD) is not represented on the scheme for clarity. (B) In scenario 1, the CTD can bind dsDNA competing for the DNA-binding sites, displacing full length ParB and therefore de-condensing the DNA. (C) In scenario 2, the CTD can exchange with full length ParB forming heterodimers in free solution that are inactive so can no longer exchange with the ParB-DNA network. (D) In scenario 3, the CTD forms heterooligomers with DNA-bound ParB which retain DNA-binding activity but are not able to condense DNA because bridging interactions are ‘capped’ by the CTD. Possible scenarios for CTD-induced decondensation of ParB-DNA networks.

https://doi.org/10.7554/eLife.43812.002

Our previous experiments with CTD variants defective for DNA binding suggested that ParB-network disruption was unlikely to be mediated by direct competition for DNA binding sites (Fisher et al., 2017). However, because the ParB binding and DNA condensation signals were tightly coupled (Taylor et al., 2015) (Fisher et al., 2017), it was not possible to separate ParB binding from condensation or to correlate both measurements in parallel. This precluded a direct test of our favoured capping model, which predicts (uniquely) that the free CTD will inhibit DNA condensation but not DNA binding. For instance, in ensemble experiments using electrophoretic mobility shift assays and ParB-protein-induced fluorescence enhancement assays, the signal reported is concomitant to protein binding and condensation (Taylor et al., 2015). The same happens in single-molecule stretch-flow fluorescence experiments (Graham et al., 2014) requiring the use of condensation-deficient mutants to determine the kinetic parameters of DNA binding (Song and Loparo, 2015). Additionally, in stretch-flow experiments, condensation is restricted to the free end of the molecule due to a decreasing force gradient along the DNA. Magnetic Tweezers (MT) experiments overcome some of these difficulties in that the applied force is uniform and can be used to prevent DNA condensation. It is therefore possible to measure DNA condensation when the force is decreased to permissive levels (Taylor et al., 2015). Nonetheless, in MT it is not possible to visualise and determine the degree of protein binding to DNA molecules. Here, we have combined lateral MT with total internal reflection fluorescence (TIRF) microscopy (Madariaga-Marcos et al., 2018) to simultaneously control the force applied to DNA and to visualise ParB proteins by fluorescence. This allowed us to directly challenge the three possible scenarios to explain the condensation inhibition mechanism of the CTD of ParB. In addition, the combination of our MT-TIRF device with multilaminar flow technology facilitated the rapid exposure of DNA to different protein concentrations and provided a method to precisely capture the kinetics of binding and unbinding events. The experimental platform used here allowed us to confirm that the inhibition of ParB condensation by the free CTD occurs without displacement of full length ParB from DNA. Our data indicates that the free CTD of ParB blocks the formation of ParB networks via a dominant negative effect on a key protein-protein bridging interface.

Results and discussion

A new method for monitoring ParB DNA binding under conditions that are non-permissive for DNA condensation

We laterally stretched single ~24 kbps DNA molecules lacking the parS sequence (see Materials and methods) (Figure 2A) using a home-built magnet holder that can pull DNA molecules from one side (Madariaga-Marcos et al., 2018). This device can apply forces of ~1 pN on 1 μm beads, which is sufficient to prevent condensation of DNA upon introduction of ParB. We produced a fluorescent variant of ParB labelled with Alexa Fluor 488 (ParB2S68C-AlexaFluor488, hereafter ParB2AF), which retained wild-type DNA-binding properties in our established bulk assays (Figure 2—figure supplement 1). This allowed us to visualise protein binding while stretching the DNA at forces non-permissive for condensation (Figure 2B). Injection of the protein in the flow cell resulted in a gradual increase of fluorescence concomitant to the arrival and non-specific binding of the protein to the DNA (Video 1). Note, however, that sometimes a fraction of the DNA closer to the magnetic bead remained invisible in the fluorescence signal (Video 2). This is as expected and is due to the limited penetration of the excitation light produced by TIRF, the tilting angle of the DNA (≤5⁰), the attachment point of the DNA at the bead, and the fixed bead orientation under the magnetic field.

Figure 2 with 2 supplements see all
Combined lateral MT and TIRF microscopy to study ParB-DNA interactions.

(A) Cartoon of the MT-TIRF setup used to visualise ParB-DNA interactions at the single-molecule level with controlled external force. A magnet pulls laterally on the distal end of a DNA molecule which is anchored to the coverslip. Fluorescently labelled ParBAF is excited in TIRF mode using 488 nm laser light and the emitted light is collected by an EM-CCD (Madariaga-Marcos et al., 2018). (B) TIRF image showing a laterally stretched DNA molecule under a force of 1 pN and in the presence of 500 nM ParBAF. Beads showed fluorescence due to additional binding of DNA molecules, which are further labelled by non-specifically bound ParBAF proteins. (C) Intensity of several DNA molecules (n = 15) in the presence of 500 nM ParBAF as a function of time. Even though the integrated intensity changes from molecule to molecule, intensity remains constant for more than 100 s, suggesting a dynamic and fast exchange between DNA-bound ParBAF and free ParBAF in the media. (D) Kymograph from the same experimental data shown in (B) highlighting changes in intensity along the DNA molecule through the entire experiment supporting the continuous exchange of the protein. Kymographs varied from molecule to molecule. All experiments were conducted in ParB reaction buffer supplemented with 4 mM Mg2+.

https://doi.org/10.7554/eLife.43812.003
Video 1
Movie showing that ParBAF binding increases the intensity along a DNA molecule.

Flow rate of 250 µl/min. Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.007
Video 2
Movie showing that intensity remains constant when imaging a ParBAF coated DNA molecule (250 nM).

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.008

The integrated intensity in the area of a DNA molecule remained constant for the longest time points tested (2 min) in the absence of an oxygen scavenger system, suggesting a continuous and fast exchange of ParB proteins with the surrounding media (Figure 2C). The integrated intensity differs between molecules due to the fraction of DNA imaged, which depends on the anchoring point of the DNA on the bead. This was further confirmed in kymographs taken along the DNA that showed variations in fluorescence intensity with time (Figure 2D, Video 1). As expected, we did observe a gradual decrease of fluorescence intensity when the protein was removed from free solution by flowing buffer alone (Video 3 and Figure 2—figure supplement 2A). Control experiments crosslinking ParB with formaldehyde indicated a photobleaching half-life time for Alexa Fluor of around 28 s, much shorter than the duration of our measurements (Figure 2—figure supplement 2B). The rapid exchange of the protein was further confirmed in fluorescence recovery after photobleaching (FRAP) experiments. They consisted of photo-bleaching all ParBAF molecules included in the visualisation area inside the TIRF field by using a high-laser power pulse, and then monitoring the fluorescence recovery of the protein-DNA complex in the presence of 250 nM ParBAF (Video 4 and Figure 2—figure supplement 2C). Altogether, these experiments demonstrate the dynamic interaction of ParB with DNA under forces that are non-permissive for condensation.

Video 3
Movie showing that the intensity of DNA coated with 250 nM ParBAF decays flowing buffer (250 µl/min).

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.009
Video 4
Movie showing that the intensity of DNA coated with 250 nM ParBAF is recovered in FRAP experiment.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.010

Association and dissociation kinetics of ParB measured at non-permissive forces for condensation

Prompted by the observation of fast ParBAF exchange, we attempted to elucidate the kinetics of the non-specific binding and unbinding of ParBAF. The simplest approach would be to repeatedly exchange protein-containing and protein-free buffer and to monitor changes in the fluorescence signal over time. However, the arrival of the protein at the site of interest in single-channel flow cells is gradual, typically requires tens of seconds to reach equilibrium, and it is difficult to assign arrival times (Gollnick et al., 2015; Liu et al., 2013). Although this effect can be minimised by using high flow velocities, it nevertheless would affect the measured kinetics of protein association. Moreover, flowing at high rates increases the force applied on the DNA as a result of the drag force on the bead and can lead to structural distortions of the DNA duplex, ultimately interfering with protein binding (Xiao et al., 2011).

In order to reduce reagent exchange times and the contribution of the flow on the total force on the DNA, we developed a fast buffer exchange system based on multilaminar flow cells (Liu et al., 2013) (Tan et al., 2007) (Brouwer et al., 2018). We used flow cells with two inlets, where two reagents are introduced (i.e. protein and buffer), and a single outlet (Figure 3A). The dimensions of our flow cells and the flow rates we used (≤500 µl/min) ensure a laminar flow regime under which the two reagents do not significantly mix (Materials and methods) (Brewer and Bianco, 2008). Fast alternation between buffer and protein was achieved by varying the flow rates of the syringes (190 µl/min and 10 µl/min) and maintaining a constant flow rate in the central channel (200 µl/min). This reduced reagent switching times, and allowed us to assign reagents arrival times while keeping a low total force (~3.5 pN) on the DNA (Materials and methods) which is non-permissive for condensation by ParB. Furthermore, protein concentrations could be accurately controlled as the DNA was exposed within a few seconds (see below) to the condition of interest, allowing one to measure association and dissociation constants.

Figure 3 with 2 supplements see all
ParB binding and unbinding kinetics.

(A) Scheme of the multilaminar flow system employed to fast-exchange of buffers. The fluid cell contains two inlets and a single outlet. Switching the velocities of both channels shifts the boundary of laminar flows resulting in the fast exchange of buffers. (B) Normalised integrated fluorescence intensity for a representative DNA molecule in a laminar-flow experiment with 250 nM ParBAF as a function of time (red) and for an equivalent background area (gray). Buffer injection (shadowed area) is correlated with decreasing intensity. Best fit curves to obtain koff (Equation 3) and kobs (Equation 2) are shown. Boundary exchange with fluorescein is also shown for comparative purposes (Figure 3—figure supplement 1D). (C) Unbinding rate koff measured at different initial ParB concentrations, obtained from fitting individual curves as shown in B, and then averaged at different concentrations. The values obtained are in the order of values published before (Song et al., 2016). (D) Observed binding rate kobs measured at different ParB concentrations, calculated as koff. Errors are SD. (n ~ 15-30 molecules). All experiments were conducted in ParB reaction buffer supplemented with 4 mM Mg2+.

https://doi.org/10.7554/eLife.43812.011

To characterise the switching between the two reagents, we first took advantage of magnetic tweezers to track a bead as it was driven by the flow alternation. We measured beads in both vertical (Figure 3—figure supplement 1A) and lateral (Figure 3—figure supplement 1B) configurations and demonstrated that changes in flow velocities correlate with the movement of the bead in the transverse direction. Importantly, the direction of the pulling force (x axis) was not disturbed when flow rates were switched (Figure 3—figure supplement 1B). Then, to directly monitor the boundary shifting, we measured fluorescence intensity changes in the background using a buffer containing 5 mM fluorescein at different flow rates (Figure 3—figure supplement 1C and D). Our results concluded that the boundary achieves a complete shifting in ~6.5 s.

We measured cycles of protein binding and unbinding by switching between channels containing ParBAF and buffer (Figure 3B, Video 5). The fluorescence intensity decreased when buffer was flowed (shadowed area) and it was recovered by the introduction of ParBAF. Note that despite the limitation of fluid exchange kinetics, the binding curve of ParBAF on the DNA was readily different from that of the background. We performed experiments at different protein concentrations to elucidate the kinetics of ParB (Figure 3—figure supplement 2). The binding of ParB to DNA can be formulated as (Goodrich and Kugel, 2007):

(1) DNA+ParB konkoffDNA:ParB

with kon and koff being the binding and unbinding rate constants, respectively. This is a simple model which cannot account for cooperativity in binding nor potential nucleation intermediates, and hence it may not capture the detailed (un)binding kinetics of ParB. However, it is still useful for semi-quantitative analysis of binding rates as well as for direct comparison of the values at different conditions. Moreover, the dynamics of the system can be easily solved assuming that the concentration of ParB remains constant in time (ParBt=ParB0). This is a reasonable assumption, since in our single-molecule experiments we expect an excess of protein compared with DNA binding sites, and our measurement is much longer than the boundary shifting time. Then, the final expression for protein binding reads:

(2) Ft=Fmax (1- e-kobst)

where F(t) states for the fluorescence signal and the observed binding rate kobs is defined as by kobs=kon ParB+ koff. Similarly, the unbinding kinetics can be obtained by considering [ParB] = 0, and giving:

(3) Ft=F0 e-kofft

where F0 is the initial fluorescence intensity (Goodrich and Kugel, 2007).

Video 5
Movie showing the laminar flow switching assay.

250 nM ParBAF binding increases intensity along a DNA molecule and unbinding decreases it (total flow rate 200 µl/min). Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.015

The measured binding constant koff (~0.1 s−1) was independent of the protein concentration and slightly lower than the previously reported value for the condensation-deficient ParB mutant ParBR82A, ~0.16 s−1 (Song et al., 2016). This might reflect localised condensation events in the wild type protein (Figure 3C). This is particularly relevant at the high concentrations used (≥500 nM), where ParB-ParB interactions are favored likely resulting in local condensates, which prevent unbinding of the protein. Regarding kobs we observed a saturation effect at high concentrations (Figure 3D). This cannot be explained by limitations in the temporal resolution of our assay (acquisition rate ~10 s−1) and we therefore favour the idea that it is limited by our fluid exchange kinetics. Nevertheless, the saturation observed can also be explained by rate limiting events after the formation of a collision complex (which are not considered in our simple model). We therefore considered kobs values obtained at low protein concentration to estimate kon (see Materials and methods). Our estimation (~4 x 105 M−1 s−1) was one order of magnitude below a published value for a mutant ParB, ~2.1 x 106 M−1 s−1 (Song et al., 2016) This discrepancy may reflect a different binding mechanism of the wild-type ParB protein employed here compared to the condensation-defective mutant, and/or simply the limitations of our laminar-flow-based technology.

The C-terminal domain of ParB does not compete for DNA binding with full length ParB in Mg2+ conditions

Motivated by previous work reporting a non-specific DNA binding of the free CTD of ParB and its important role in DNA condensation (Fisher et al., 2017), we investigated the effect of an excess of CTD in the ParBAF binding/unbinding reaction. Our previous work indicated distinct DNA-binding mechanisms for ParB in the presence of Mg2+ or EDTA (Taylor et al., 2015). In particular, the specific binding of the protein was only observed in Mg2+, while the non-specific binding was promoted by the presence of EDTA, especially at low ParB concentrations. In the work described below, we carried out assays in both Mg2+ and EDTA conditions and measured koff and kobs for full length ParB, and for ParB in the presence of free CTD.

For full length ParB, protein:DNA complexes were observed under both conditions (Figure 2B and Figure 4—figure supplement 1A), but we measured a lower koff in EDTA conditions compared to Mg2+ conditions (Figure 4A, Figure 4—figure supplement 2Table 1). The observed binding rate (kobs) did not show statistically significant differences between EDTA and Mg2+ (Figure 4A, Table 2). To result in similar observed binding rates and considering the definition of kobs, we infer that the association rate constant kon is higher in the presence of EDTA than in Mg2+. In summary, and in agreement with our published observations (Fisher et al., 2017), we conclude that the presence of EDTA strengthens interactions between ParB and non-specific DNA.

Figure 4 with 2 supplements see all
ParB binding kinetics in the presence and absence of Mg2+and the effect of the free CTD.

(A) Unbinding rate koff and observed binding rate kobs values for 250 nM ParBAF. We observed a reduction in koff (slower unbinding) in the case of EDTA but no significant difference was observed in kobs values. (B) Unbinding rate koff and observed binding rate kobs values for 250 nM ParBAF in the presence (red) and absence (gray) of 2.5 µM CTD, in Mg2+– or EDTA–containing buffer. We report faster unbinding in the presence of the CTD in EDTA. No effect of the CTD in the unbinding rate was observed in Mg2+ buffer, according to Student’s t-test. No significant difference was observed in kobs values in the presence or absence of the CTD in both Mg2+ and EDTA conditions. All errors are SD. (n ~ 12-28 molecules). Data were accounted for statistical differences following a Student's t-test (see Table 1 for koff  p-values and Table 2 for kobs p-values).

https://doi.org/10.7554/eLife.43812.016
Table 1
Student’s t-test p-values for koff values.
https://doi.org/10.7554/eLife.43812.019
WT - Mg2+WT + CTD Mg2+WT - EDTAWT + CTD - EDTA
WT – Mg2+10.12355E-40.0019
WT + CTD Mg2+100
WT - EDTA16E-5
WT + CTD - EDTA1
Table 2
Student’s t-test p-values for kobs values.
https://doi.org/10.7554/eLife.43812.020
WT - Mg2+WT + CTD Mg2+WT - EDTAWT + CTD - EDTA
WT - Mg2+10.0790.95420.1139
WT + CTD Mg2+10.02430.0103
WT - EDTA10.1129
WT + CTD - EDTA1

We next produced a fluorescently labeled CTD protein (CTDAF) and studied its binding to DNA in the presence of EDTA or Mg2+ (Figure 2—figure supplement 1). We observed protein:DNA complexes at high CTDAF concentrations (10 μM) in the presence of EDTA but no complexes were visible in the presence of a magnesium-containing buffer at the same concentration (Figure 4—figure supplement 1A). The fluorescence intensity of CTDAF complexes remained constant over time, suggesting continuous and rapid exchange between bound and free CTD proteins (Figure 4—figure supplement 1B and Video 6). Further experiments, where CTDAF complexes were photobleached, confirmed the rapid exchange of the protein in a similar manner to ParBAF (see Figure 4—figure supplement 1C and D for data and discussion). Altogether, these results show that the CTD of ParB can bind and unbind rapidly from DNA, but only under EDTA conditions. This implies that the free CTD will not inhibit the interactions of full length ParB with DNA by simple competition under Mg2+ conditions. We next tested this more directly.

Video 6
Movie showing that intensity remains constant when imaging 10 µM CTDAF, similar to ParBAF.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.021

In the presence of excess free CTD and under Mg2+ conditions, the koff and kobs values for full length ParB were indeed unchanged when compared to full length ParB alone (Figure 4B). An excess of the CTD in EDTA-containing buffer facilitated ParBAF unbinding, yielded a higher koff  (Figure 4B, Figure 4—figure supplement 2), and resulted in lower intensity CTDAF-DNA complexes. Therefore, as expected based on the properties of the free CTD alone, it can compete for DNA but only under EDTA conditions, which are not considered to have any particular biological relevance.

These results strongly suggest that scenarios 1 and 2 (Figure 1) can be discounted, because the CTD does not reduce DNA binding by full length ParB under Mg2+ conditions. However, these experiments were performed at restrictive forces to uncouple binding and condensation. At permissive forces, the Mg2+ conditions would support condensation. We next investigated the dominant negative effect of the free CTD under conditions that were permissive for condensation.

The free CTD prevents ParB-dependent condensation at permissive forces without affecting its binding to DNA

Previous experiments showed that an excess of the CTD of ParB prevents the condensation of DNA tethers and also promotes decondensation of preformed ParB-DNA networks (Fisher et al., 2017). However, as mentioned, we could not unequivocally distinguish between the possible mechanisms for condensation prevention, because binding and condensation could not be uncoupled. Here, using our MT-TIRF setup we could apply changes in force in a controlled way while at the same time visualising ParB binding and condensation.

We first monitored the condensation activity of ParBAF in a buffer supplemented with Mg2+. Condensation events were initiated by ParBAF by reducing the force from 1 to 0.2 pN and were directly visualised by fluorescence microscopy in laterally stretched DNA molecules (Figure 5A, Video 7). This process could be stopped by increasing the force (see Video 8) but could not be reversed, possibly due to a strong interaction of the protein with the surface and the smaller range of available forces applied by the lateral magnet in comparison to previous conventional MT experiments (Taylor et al., 2015). In general, the condensation process did not show any brighter features along the DNA that could be associated with the formation of large loops or the presence of nucleation regions (Figure 5A, Video 7). Nevertheless, we should keep in mind that our setup has a limited resolution of about 1 kbp as determined from measurements on single fluorophores (data not shown). Therefore, local condensation at scales lower than this could not be resolved.

Competition of the CTD in ParB binding and unbinding kinetics, and effects in condensation.

(A) Visualisation of condensation of a single DNA molecule induced by 250 nM ParBAF binding. The volume of the bead causes the DNA to be slightly tilted with respect to the surface such that emission of fluorescence is limited by the penetration depth of the excitation field. The DNA does not condense while held at non-permissive force of 1 pN. However, as in conventional magnetic tweezers experiments, at a permissive force of 0.2 pN a progressive reduction of the DNA length is observed. (B) Condensation experiment as in Figure 5A but in the presence of 2.5 μM non-fluorescent CTD showing that at a constant force of 0.2 pN, ParB is not able to condense to the inhibition of the CTD. Co-incubation with the CTD does not inhibit ParB binding to DNA, as indicated by clearly visible fluorescence filaments. (C) Condensation experiment as in Figure 5B. The CTD inhibits ParB-dependent DNA condensation, even at forces as low as 0.07 pN. Note that some residual condensation was observed at very low forces, visible at the anchoring point (arrow) as shown by the bright dot. (D) Percentage of DNA molecules that were condensed (more than 90% extension reduction), partially condensed or not condensed by ParBAF (less than 10% extension reduction), in the presence (n = 47) or absence of an excess of the CTD (n = 36). (E) Integrated fluorescence intensity in a laminar flow experiment. Intensity remains constant throughout the cycles of ParBAF and ParBAF + CTD (shadowed area), indicating that the excess of CTD is not competing for the binding sites on the DNA, but rather ‘capping’ the DNA-bound ParBAF. Intensity changes when alternating ParBAF with a combination of ParBAF and unlabelled ParB (shadowed areas). (F) Integrated intensity of ParBAF plotted versus the integrated intensity of ParBAF + CTD or ParBAF + ParB. ParBAF + CTD shows a slope of 1.00 ± 0.02 corresponding to unchanged intensity, while ParBAF + ParB shows a slope of 1.23 ± 0.02. (n ~ 10–16 molecules). All experiments were conducted in ParB reaction buffer supplemented with 4 mM Mg2+.

https://doi.org/10.7554/eLife.43812.022
Video 7
Movie showing that 250 nM ParBAF is able to condense DNA under permissive forces.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.024
Video 8
Movie showing that 250 nM ParBAF does not decondense upon force increase, contrary to ParB in previously published magnetic tweezers assays.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.025

In vivo studies have shown that a very small number of ParB molecules are actually responsible for chromosome segregation in the bacterial cell (Graham et al., 2014). Inside the cell, the entire chromosome and all ParB dimers are of course compartmentalised leading to a situation in which [ParB] and [DNA-binding sites] are both well above the Kd values we observe for DNA binding, but are present in a stoichiometry such that there are many more DNA-binding sites than there are ParB dimers. Thus, ParB presumably ends up in foci around parS because of the high-affinity interaction between ParB and its specific site mediated by the HTH domain. This situation cannot possibly be achieved in vitro and in order to observe ParB:DNA binding we had to raise [ParB] to concentration over 100 nM. The minimum concentration tested that resulted in condensation was 250 nM ParBAF.

We repeated condensation experiments using 250 nM ParBAF in the presence of 2.5 μM CTD in Mg2+ conditions (Figure 5B, Video 9). Only a very small fraction of DNA molecules showed some degree of condensation. Indeed, the time-lapse in Figure 5C (Video 10) shows that, even though the DNA molecule remains fully coated by ParBAF, it did not condense, even at very low (permissive) forces. For the totality of molecules analyzed in this study, (47 in the case of ParBAF and 36 in the case of ParBAF + CTD), we found that 83% of the molecules were condensed by ParBAF, whereas only 19% were condensed when the CTD was also present (Figure 5D). Moreover, when condensation was observed in the presence of the CTD, it was never complete. Typically, the extension of the molecule was not recovered upon re-application of high force, and a bright dot appeared at the anchoring point. The fact that normal condensation data (i.e. those collected with full length wild-type ParB only) do not show any such condensation foci (Figure 5A) and that the brighter dot in the CTD experiment appears at the anchoring point, suggests that this represents non-specific binding of the DNA to the surface that cannot be prevented at the low applied forces. The same bright dot was sometimes observed in the case of ParBAF at forces non-permissive for condensation, confirming our hypothesis (Video 11). We conclude that the presence of free CTD has converted the full length ParB into a form that can still bind, but can no longer condense DNA, which can only be explained by the capping model (Figure 1D).

Video 9
Movie showing that 250 nM ParBAF in the presence of an excess of CTD (2.5 µM) is not able to condense DNA.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.026
Video 10
Movie showing that 250 nM ParBAF in the presence of an excess of CTD (2.5 µM) is not able to condense DNA even in the absence of magnetic force.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.027
Video 11
Movie showing that a bright dot sometimes appears at the anchoring point of a DNA molecule in the presence of 250 nM ParBAF.

Movie is 5X accelerated.

https://doi.org/10.7554/eLife.43812.028

We performed additional experiments using our laminar-flow system combined with direct visualisation of DNA-protein complexes. We switched between a buffer containing 250 nM ParBAF and another containing 250 nM ParBAF and 2.5 μM CTD in Mg2+ conditions, whilst keeping the DNA stretched with lateral MT. The period of buffer exchange (T=60 s) is long enough to account for protein exchange since we measured a koff for ParBAF of 0.08 s−1, corresponding to an occupation half-time of around 9 s (Figure 3C). Importantly, the intensity did not decrease when the conditions were switched from ParB alone to ParB co-incubated with CTD (Figure 5E, upper panel), again confirming a lack of competition for the DNA binding sites. Additionally, a control experiment where we substituted the excess of CTD for an equivalent excess of unlabelled ParB displayed a noticeable decrease in intensity attributed to the expected competition for DNA-binding sites between non-fluorescent ParB and ParBAF (Figure 5E, lower panel). To quantify this effect, we represented the average integrated intensity of ParBAF against the average integrated intensity of ParBAF + CTD/ParB for several molecules (Figure 5F). A deviation from a slope ~1 indicates changes in fluorescence intensity as a result of the competition for DNA-binding sites, validating the capability of our assay to qualitatively resolve differences in protein binding. Together, these results allowed us to conclude that condensation is impeded (or decondensation promoted) by the CTD as a result of capping a ParB-ParB dimerisation interface (Figure 1D).

Conclusions

Here, we have built upon our previous work to incorporate MT and TIRF microscopy to visualise the formation of ParB-DNA complex structures at controlled forces that can be either permissive or non-permissive for condensation. Our novel approach allowed us to visualise the dynamic nature of the ParB binding with evidence for fast and continuous exchange of proteins with the surrounding media. This dynamic behavior is consistent with results described not only for BsParB (Graham et al., 2014) (Taylor et al., 2015) (Song et al., 2017) (Fisher et al., 2017), but also for plasmid ParB (Debaugny et al., 2018; Sanchez et al., 2015) (Le Gall et al., 2016), and for other proteins, at the single molecule level (Gibb et al., 2014). Implementation of a multilaminar flow exchange system to the combined MT-TIRF setup provided a measurement of binding and unbinding reactions in the presence of the free CTD leading to the conclusion that it only affects binding and unbinding of ParB in a buffer supplemented with EDTA. Fluorescence experiments with the CTD in a buffer supplemented with Mg2+ strongly supported a mechanism for its dominant negative effect involving decondensation by direct disruption of ParB-ParB interfaces. Interestingly, the fact that the CTD prevents condensation by capping a dimerisation interface strongly suggests that the CTD in the full length ParB in solution is somehow inaccessible, unlike the DNA-bound ParB. This could explain why multimeric forms of ParB are not found in solution. The mechanism by which the CTD becomes accessible upon DNA binding is unclear and will be the subject of future research.

Materials and methods

Combined lateral MT-TIRF apparatus

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Experiments were performed in two MT-TIRF setups similar to one described before (Madariaga-Marcos et al., 2018). In brief, 488 nm laser source (Vortran Stradus) was focused on the back focal plane of a high numerical aperture objective (Olympus UAPON TIRF 100×). We used two separate detectors to visualise the emission of the fluorophores in the sample and the magnetic beads; an EM-CCD temperature-controlled camera (Andor Ixon Ultra 897/Andor Ixon Ultra 888) and a CCD or CMOS camera (Pulnix 6710 CL/Mikrotron MC1362) for bright-field video microscopy. Lateral Magnetic Tweezers consisted of a pair of permanent magnets (Q-05-05-02-G, Supermagnete) connected to a linear motor (Piezomotor). The MT setup was controlled by a custom-written code in Umbarger et al. (2011). Fluorescence camera was controlled by Andor Solis software.

Flow cells (simple and double channel)

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In this work, two kinds of flow cells were used (with one or two inlets). In both cases, glass coverslips (Menzel-Gläser, #1) were cleaned by 30 min of sonication in acetone followed by 30 min in isopropanol, and dried using compressed air. The clean surface was coated with 1% polystyrene dissolved in toluene. The top cover glass contained two or three holes drilled with a laser engraver, as well as two-inlet paraffin wax (Parafilm M, Bernis USA) gaskets (VLS2.30, Universal Laser Systems). One-inlet gaskets were manually cut. The two cover glass slides and a gasket were sandwiched and heated up for a few seconds at 120°C to assemble the flow cell. The cells were then incubated with an Antidigoxigenin (25 ng/μl) solution (Roche) overnight at 4°C and were passivated for at least 2 hr using BSA (NEB). The cells were stored in a humid and sealed container at 4°C until further use.

DNA substrate (λ/2)

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λ/2 molecules were fabricated as described elsewhere (Madariaga-Marcos et al., 2018) (Camunas-Soler et al., 2013). Briefly, CosR-tail and CosL-tail oligonucleotides were biotin tailed and the XbaI-A oligonucleotide was digoxigenin tailed using Terminal Transferase (NEB) and either BIO-dUTP or DIG-dUTP (Roche). The modified oligonucleotides were purified using a Qiaquick nucleotide removal kit (Qiagen). N6-Mehtyladenine-free λ DNA (NEB) was cleaved with XbaI, giving two ~ 24 kbp fragments. These fragments and the tailed oligonucleotides in addition to the XbaI-B oligonucleotide were subsequently annealed and ligated overnight using T4 DNA ligase (NEB).

Protein purification and labelling

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B. subtilis ParB and the truncated CTD variant were recombinantly overexpressed in E. coli and subsequently purified using no extrinsic tags as described previously (Taylor et al., 2015) (Fisher et al., 2017).

ParB contains no native cysteine residues and so design of point substituted variants for covalent coupling to maleimide-conjugated fluorophores was pursued. Mutation S68C was selected because it is located in a surface-accessible putative loop region of poorly conserved sequence and length. ParBS68C was purified as described for the wild type protein with an additional labelling step. Following elution from a Hiload 16/600 Superdex S75 gel filtration column (equilibrated in the absence of DTT), pooled concentrate was added to a 1 mg aliquot of Alexa Fluor 488 C5 Maleimide (Invitrogen) at a molar ratio of 1:5. This reaction was rotated end-over-end at 4°C for 16 hr before quenching by the addition of DTT to 5 mM. The conjugate solution was then loaded onto a Hiload 16/600 Superdex S75 gel filtration column (in the presence of 1 mM DTT), to separate labeled ParB (ParBAF) and free fluorophore. ParBAF was pooled and concentrated before storage as described for wild type ParB. Apparent labelling efficiency was assessed by spectrophotometry as described by the vendor, by comparing the absorbance from the protein and the Alexa Fluor 488 dye, including a correction factor for absorbance of the dye at 280 nm (0.11). Native and collision-induced dissociation (CID) or electron-transfer dissociation (ETD) MS were used to assess labelling efficiency and stoichiometry, and to confirm covalent attachment of the fluorophore (not shown). ParBAF was fluorescently labelled at an approximate efficiency of 2 dyes per ParB monomer.

Similarly, the structure of the CTD dimer (PDB ID: 5NOC) was used to design a suitable site for cysteine substitution and fluorophore attachment. The residue S278 was selected as it is found in a relatively unconserved region located at the end of the final α-helix. Labelling reactions were performed as described before with the following modifications. Prior to labelling, the purified protein was treated with freshly prepared TCEP (at a final concentration of 1 mM), at room temperature for 1 hr with end-over-end rotation. A near stoichiometric molar ratio (~1:1.5) of the maleimide-conjugate dye was used over a 16 hr reaction course. Purification, quantification and storage of the protein-fluorophore conjugate were completed as described before (Fisher et al., 2017). CTDAF was labelled with a final efficiency of 0.82 dyes per CTD monomer.

ParB/CTD binding and visualisation experiments

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Tethers of λ/2 DNA molecules were obtained by mixing DNA with 1 μm size magnetic beads (Dynabeads, MyOne streptavidin, Invitrogen) in a buffer containing 100 mM NaCl, 50 mM Tris (pH 7.5), 0.2 mg/ml BSA, and 0.1% Tween 20, supplemented with 4 mM MgCl2 (ParB reaction buffer) or supplemented with 1 mM EDTA (ParB-EDTA reaction buffer). DNA molecules were laterally stretched at non-permissive forces for condensation (over 1 pN). Then, 250 nM ParBAF or 10 µM CTDAF was injected into the cell at a flow rate of 250 µl/min and the DNA molecules were imaged using Andor Solis software. Images were acquired at a frequency of 9.52 Hz, using an EM level of 100 and cooling the sensor to −60 or −80°C. For condensation experiments, the lateral magnet was moved away from the flow cell to apply a force of 0.2 pN, while recording the fluorescence image. Fluorescence data analysis was performed in Andor Solis and Origin and movies were generated using ImageJ. For representation, images were exported from Andor Solis as 8-bit gray tiff files. First, a background was subtracted using a 50 pixel radius (sliding paraboloid, smoothing disabled) and then brightness and contrast were adjusted by visual inspection to enhance the signal. Finally, a custom look up table in green scale was applied. For the production of videos, fluorescence movies were exported as RGB single tiff files and assembled as videos using a custom ImageJ script (Source code 1 provided). Both biological (new sample preparations from a fresh stock aliquot) and technical (single-molecule measurements) repeats were undertaken.

Multichannel laminar-flow experiments for kinetics measurements

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Tethers of λ/2 DNA were obtained by mixing DNA with 1 µm sized superparamagnetic beads in ParB or ParB-EDTA buffer supplemented with 1 mM Trolox, 20 mM glucose, 8 µg/ml glucose oxidase and 20 µg/ml catalase. The mixture was incubated and then flowed into the flow cell. After tethers were formed, unbound beads were extensively washed away. Tethers were laterally stretched at a force of 1 pN using lateral magnets. Fluorescence images were acquired using Andor Solis software. Images were acquired at a frequency of 9.52 Hz, using the EM level of 100 and cooling the sensor to −60 °C. 4000 frames were recorded. Syringes were controlled with the neMESYS UserInterface software. Briefly, a square-wave pattern was set for syringes to alternate flow-rates between 10 and 190 µl/min, keeping the flow-rate in the central channel to 200 µl/min. Employed ParBAF concentrations were 125, 250 and 500 nM and CTDAF was 10 µM. Data analysis was performed using Andor Solis and Origin. For each molecule of interest, a main region of interest (ROI) was selected around the DNA molecule in Andor Solis. Four additional ROIs around the main ROI were also selected for background correction by subtracting the mean intensity of the four ROIs to the DNA ROI (in counts). Intensities were then analyzed in Origin. kon was estimated by fitting the first two points (corresponding to 125 nM and 250 nM) on Figure 3D to the equation kobs=kon ParB+ koff, which yielded a value of ~3.95 x 105 M−1 s−1.

Maximum Reynolds number in our microfluidic device

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The Reynolds number is calculated as:

(4) Re=2rρ vmean η

where  vmean is the average linear velocity at the applied flow (vmean ≈ 10−3 m s−1 for the maximum flow rate of 200 µl/min), η the dynamic viscosity of the fluid (10−3 Pa s) and ρ the density of the fluid (1 g cm−3). We can determine the equivalent hydrodynamic radius of a square duct as  r=d w/d+w, where d and w are the height and the width of the cross section of the liquid cell, respectively. In this case, d ≈ 200 μm and w ≈ 3 mm and we obtain a radius r ≈ 188 μm. This gives a Reynolds number of Re ≈ 0.001 << 2000. Consequently, our system is always under laminar flow conditions.

Calculation of the total force exerted on the DNA molecule while flowing under lateral magnetic pulling

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The total force applied to the DNA is the sum of two components, the magnetic force exerted by the magnets and the drag force applied by the flow (Madariaga-Marcos et al., 2018):

(5) Ftotal= Fmagnetic2+ Fdrag2

The experiments reported here were done at a 1 pN pulling force, Fmagnetic1 pN. The drag force was deduced from the flow rate using a previously measured curve of force versus flow rate, in flow stretch assays Fdrag3.36 pN. This led to a total force Ftotal3.5 pN.

Fluorescence recovery after photobleaching (FRAP) experiments

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We laterally stretched DNA tethers at a force of 1 pN. We then flowed 250 nM (or 1 µM for comparison with CTDAF) ParBAF. The imaging protocol consisted of imaging a DNA molecule at a low laser power for 200 frames before increasing the laser power for 100 frames to photobleach all ParBAF molecules. Fluorescence recovery was then allowed for 1200 frames at the initial laser power. Images were acquired at a frequency of 9.52 Hz, with an EM gain of 100 and with the sensor cooled to −60 or −80°C. The area (ROI) around each DNA molecule was analyzed using Andor Solis. Integrated fluorescence intensities were analysed using Origin.

References

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    Binding and Kinetics for Molecular Biologists
    1. JA Goodrich
    2. JF Kugel
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    Cold Spring Harbor Laboratory Press.
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Decision letter

  1. Sebastian Deindl
    Reviewing Editor; Uppsala University, Sweden
  2. John Kuriyan
    Senior Editor; University of California, Berkeley, United States

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article "ParB dynamics and the critical role of the CTD in DNA condensation unveiled by combined force-fluorescence measurements" for consideration by eLife. Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by John Kuriyan as the Senior Editor. The reviewers have opted to remain anonymous.

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

This manuscript, submitted here as a Research Advance, follows up on a previous study published in eLife last year demonstrating that the Bacillus subtilis ParB C-terminal domain (CTD) binds DNA non-specifically in addition to dimerization, and that this domain was required for DNA condensation. One of the key experiments showed a loss of ParB-induced condensation upon addition of free CTD domain. However, the exact mechanism underlying this process was not clear, as DNA binding could not be uncoupled from ParB-induced DNA condensation. In this current manuscript, this obstacle is overcome by simultaneously and independently probing ParB DNA binding and condensation using a combination of lateral magnetic tweezers and TIRF microscopy. These experiments clearly demonstrate that the CTD of ParB specifically inhibits its condensation activity without substantially affecting DNA binding when added in trans to full-length ParB. This leads the authors to the conclusion that the CTD interferes with ParB condensation through 'capping' of CTDs within ParB complexes as both the NTD and CTD dimerisation interfaces are needed for network formation.

Therefore, the manuscript represents a valuable addition and conclusion to the previous eLife paper, "The structural basis for dynamic DNA binding and bridging interactions which condense the bacterial centromere" by Fisher et al., (2017). The experiments are well-designed and overall, the results are presented clearly and support the authors' main conclusions. Nevertheless, a number of important points need to be addressed by the authors before the manuscript could be published.

Essential revisions:

1) The measurements of binding and unbinding kinetics are not consistent/convincing for a number of reasons. First, since kobs = koff + kon, it seems strange that in a number of cases (e.g., in Figure 3C and D, as well as 4A with Mg), koff is either indistinguishable or higher than kobs. Based on the data from Fisher et al., 2017, Kd for ParB non-specific DNA binding is on the order of few hundred nM, and therefore in the tested concentration range kon and koff are expected to be comparable. Also, the fact that the equilibrium fluorescence level of ParB (Figure 3—figure supplement 2) changes with ParB concentration means that (assuming that koff is independent of ParB concentration) kon has to change at least 3-fold, whereas it doesn't seem to significantly change based on Figure 3C. Combined with the fact that the estimated kon is an order of magnitude lower than a previously published result for a ParB mutant (Song et al., 2016), this raises the question of how accurate the kinetic measurements presented in this manuscript might be. One possible explanation would be that the measurement is dominated by liquid exchange kinetics. The authors should provide data on the concentration of ParB in solution during buffer exchange (from the background intensity). An independent measurement of ParB-DNA binding kinetics using a different method would be needed to support the TIRF kobs and koff measurements. Generally, the statistics for these experiments are insufficient and the authors need to improve them and address above issues. That being said, the kinetic measurements are not critical for the main conclusion of the paper.

2) Previously, the authors have demonstrated that CTD forms a highly stable dimer in solution with a characteristic exchange time of hours to days (Fisher et al., 2017), but the proposed capping mechanism of ParB condensation inhibition involves a CTD monomer binding to full-length ParB. The authors should demonstrate the CTD monomer exchange that seems to be necessary for the proposed model. One way to do this would be to use a CTD labeled with a quencher or a FRET acceptor for A488 (e.g., Cy3). In that case, the formation of heterodimers with A488-CTD and their exchange in the presence of unlabeled partners could be monitored using bulk fluorescence (or even TIRF microscopy). Otherwise, the authors need to provide a convincing explanation for this apparent contradiction.

3) The author's conclusion that the ParB CTD might be inaccessible in solution is appealing, as it echoed with the observation that Bacillus subtilis and Thermus thermophilus ParB both exist in solution in a monomer-dimer equilibrium while the CTD alone is a dimer (Leonard et al., 2006, Taylor et al., 2015). Additional experiments could be carried out by the authors to strengthen and develop their hypothesis, using hydrogen-deuterium exchange mass spectrometry (HDX) for example. Since the current manuscript is meant to be a single advance, this is best kept for a future, more in-depth study.

4) The authors have mentioned that the fluorescent labeling of proteins was controlled using mass spectrometry. These data need to be included in the paper, in particular in light of the puzzling reported labeling efficiency for ParB (2 dye molecules per monomer containing a single cysteine as stated in the captions for Figure 2A). The authors should discuss the possible reasons for a labeling efficiency that is substantially higher than 1.

5) An intriguing result presented by the authors is the lack of DNA-binding activity from the CTD in an Mg2+-supplemented buffer (Figure 4—figure supplement 1A), contrary to ParB in the same conditions. Although not addressed by the authors in their manuscript, this is highly reminiscent of their hypothesis (Fisher et al., 2017) of a second non-specific DNA binding site (possibly on the CDBD). In that case, this second site would still contribute to non-specific DNA binding in Mg2+ conditions while the CTD does not bind anymore, resulting in different binding kinetics which could be those measured in Figure 4A in Mg2+-supplemented buffer. This idea could be tested with labelled NTD-CDBD and/or labelled CDBD. One could also try to characterise the binding kinetics of the CTD in EDTA to compare it to ParB's.

6) The authors studied DNA condensation with 250 nM ParBAF. In vivo studies (Graham et al., 2014) have shown that a very small number of ParB molecules was actually responsible for chromosome segregation in the bacterial cell. In their setup, as the authors state, "an excess of protein [is expected] compared to DNA binding sites". Did the authors test what was the lowest concentration of ParB they could get to condense DNA? It would also be interesting to compare condensation between the λ/2 DNAs used in this study and DNAs harbouring one or several parS sites, but this is beyond the scope of this study.

7) It is not very clear if the experimental conditions for condensation inhibition are the same here than in the previous authors' published work. Indeed, previous experiments seem to have been performed in buffer without Mg2+ and without EDTA if I understand correctly ("100 mM NaCl, 50 mM Tris-HCl or HEPES-KOH pH 7.5, 100 mg/ml BSA and 0.1% Tween 20 (v/v)" according to Materials and methods section from Fisher et al., 2017). Please clarify.

https://doi.org/10.7554/eLife.43812.040

Author response

Essential revisions:

1) The measurements of binding and unbinding kinetics are not consistent/convincing for a number of reasons. First, since kobs = koff + kon, it seems strange that in a number of cases (e.g., in Figure 3C and D, as well as 4A with Mg), koff is either indistinguishable or higher than kobs. Based on the data from Fisher et al., 2017, Kd for ParB non-specific DNA binding is on the order of few hundred nM, and therefore in the tested concentration range kon and koff are expected to be comparable. Also, the fact that the equilibrium fluorescence level of ParB (Figure 3—figure supplement 2) changes with ParB concentration means that (assuming that koff is independent of ParB concentration) kon has to change at least 3-fold, whereas it doesn't seem to significantly change based on Figure 3C. Combined with the fact that the estimated kon is an order of magnitude lower than a previously published result for a ParB mutant (Song et al., 2016), this raises the question of how accurate the kinetic measurements presented in this manuscript might be. One possible explanation would be that the measurement is dominated by liquid exchange kinetics. The authors should provide data on the concentration of ParB in solution during buffer exchange (from the background intensity). An independent measurement of ParB-DNA binding kinetics using a different method would be needed to support the TIRF kobs and koff measurements. Generally, the statistics for these experiments are insufficient and the authors need to improve them and address above issues. That being said, the kinetic measurements are not critical for the main conclusion of the paper.

We thank the reviewers for raising this point regarding the accuracy of our kinetic measurements. Our kinetic measurements used a simple bimolecular association model as the framework for the analysis. If this model holds then koff is expected to be independent of [ParB], whereas kobs (the pseudo first order binding rate) should increase linearly with [ParB] above the value for koff. This is not the case and both rates appear to be limited to ~0.1 s-1.

This could be interpreted in a number of ways. On the one hand, this could indicate that the model we are using to describe the system is too simplistic (which is almost certainly the case given that DNA binding is apparently coupled to protein association). However, the reviewers are quite correct to highlight that another possible explanation for this behavior would be an artefactual rate-limiting event for both processes associated with fluid exchange. Indeed, we were concerned ourselves about this, and have performed additional experiments to determine the time of fluid exchange in our setup.

Although our Magnetic Tweezers measurements showed an apparent instantaneous boundary exchange (on the order of 500 ms, see Figure 3—figure supplement 1), further characterization of our experimental system shows that the effective fluid exchange rate is slower. We recorded the fluorescence at the surface under TIRF illumination to characterize the boundary switching between 5 mM fluorescein and buffer (Figure 3—figure supplement 1C-D) and found that it is achieved within 3-6.5 s (90% of saturating intensity), at 600-200 μl/min flow rates. We fit the Taylor-Aris model (Taylor, 1953) (Aris, 1956) for a rectangular channel as described in (Niman et al., 2013). The fit was very good as expected for a boundary fluid exchange, in contrast with the exponential rise expected for protein association. Additional experiments using 1 mM fluorescein and 100 nM SYTOX Green showed very similar results (not shown). These measurements were also found to be dependent on the quality of the surface, which varies between flow cells. Additionally, our measurements include effects from fluorophore adsorption and photobleaching which are difficult to account for. All of these factors contribute to the measured signal and highlight the difficulty in measuring the liquid exchange kinetics. However, we found that the kinetics of protein binding to DNA were readily different from those observed in the background, and in the fluorescein experiment. In the example shown in new Figure 3B, the koff value for the DNA was ~0.09 s-1 whereas for the background it was ~0.12 s-1. Similarly, kobs for the DNA was ~0.06 s-1 and ~0.09 s-1 for the background. These figures have been added to Figure 3—figure supplement 1 and the new information to the kinetics section of the text (subsection “Association and dissociation kinetics of ParB measured at non-permissive forces for condensation”).

Nevertheless, we concede that the limitation in the non-instantaneous reagent exchange may limit an absolute determination of the kinetic rates in our measurements as may also be the case for similar approaches based on DNA flow stretching and visualization by fluorescent dyes and/or proteins.

Considering that the binding and unbinding rate of ParB to and from DNA can be clearly differentiated from that of the background rate, our semi-quantitative measurements are still useful for comparison purposes. Moreover, the results of kobs/koff in the presence of Mg2+/EDTA and in the presence or absence of the CTD (Figure 4) are compatible with the results obtained in bulk (Figure 2—figure supplement 1) and in equilibrium single-molecule experiments (Figure 2 and Figure 4—figure supplement 1). We have replaced Figure 3B by an updated version Figure that includes the background signal and modified the main text accordingly (subsection “Association and dissociation kinetics of ParB measured at non-permissive forces for condensation”).

Additionally, we have calculated the average intensity at equilibrium in kinetic binding experiments and have represented them as a function of protein concentration (new Figure 3—figure supplement 2B). On top of these data, we generated a theoretical Hill plot considering the Kd and Hill values obtained by bulk binding measurements in Taylor et al., 2015 and the two sets of data are clearly in reasonable agreement (see new Figure 3—figure supplement 2). Bulk stopped-flow measurements of binding kinetics were also reported in the Supplementary Information of Taylor et al. These were complex, but like the single molecule measurements very slow, with an observed halftime of ~10 s at 1 µM ParB. Note however, that these measurements are not comparable as the PIFE measurement likely measured the kinetics of both binding and condensation as opposed to our single molecule measurements which measure binding only under conditions non-permissive for condensation.

Finally, we appreciate the reviewers’ comment concerning the statistics. However, as pointed out also by the reviewers, the values of the rates determined from our semiquantitative kinetic measurements are not essential for the general message of the paper. Nevertheless, they are important qualitatively to support for instance that CTD is only capable of binding to DNA in the presence of EDTA (Figure 4B).

2) Previously, the authors have demonstrated that CTD forms a highly stable dimer in solution with a characteristic exchange time of hours to days (Fisher et al., 2017), but the proposed capping mechanism of ParB condensation inhibition involves a CTD monomer binding to full-length ParB. The authors should demonstrate the CTD monomer exchange that seems to be necessary for the proposed model. One way to do this would be to use a CTD labeled with a quencher or a FRET acceptor for A488 (e.g., Cy3). In that case, the formation of heterodimers with A488-CTD and their exchange in the presence of unlabeled partners could be monitored using bulk fluorescence (or even TIRF microscopy). Otherwise, the authors need to provide a convincing explanation for this apparent contradiction.

This question was already a topic of discussion for our last paper and the short answer is (somewhat surprisingly), yes, it is possible, and there is no apparent contradiction between the model and the data. Our NMR experiments showed previously that CTD monomers can exchange. However, the reviewers’ intuition says that, since the timescale of CTD exchange is on the orders of hours/days, this surely cannot be consistent with the relatively rapid exchange (on the order of seconds/minutes) proposed to occur in the MT experiments.Using kinetic simulations, we rigorously demonstrate below that it is possible to have rapid monomer exchange even though the dimer is extremely tight. This requires two special conditions to be satisfied: (A) one partner in the exchange must have an artificially elevated koff and (B) the other partner must be at a relatively high concentration and large stoichiometric excess. Both of these conditions are satisfied in our experiment.

A) The ParB on the DNA experiences a restrictive force against condensation which, by definition, will increase the off rate for whatever molecular interaction bridges between ParB molecules to promote condensation (we propose that these include interactions between ParB CTDs). koff increases exponentially with applied force (Schwesinger et al., 2000), such that relatively small force increases can result in very large effects on molecular interactions. Note that, the force in our experiment can easily reach a level (just a few piconewtons) at which the molecular interactions holding the condensate together are instantaneously broken (i.e. koff is much faster than the experimental timescale).

B) The free CTD in solution is present at 5 µM, and in a huge stoichiometric excess compared to the ParB bound to DNA on the coverslip.

The kinetic simulations below use the same model to compare monomer:monomer exchange between tight dimers under four different scenarios.

1) The exchanging partners (dimers) are present at equimolar concentration in the μM range and there is no opposing force on either. The exchange is very slow as would be expected, as is observed in NMR experiments, and as the referees’ intuition suggests.

2) As 1, but one of the exchanging partners is present at a dramatically lower concentration. The exchange is still very slow.

3) As 1, but one of the exchanging partners experiences a restrictive force against the dimer interface thereby elevating its koff by 1000-fold. Nevertheless, the exchange remains very slow because the system is rate limited by the slowly dissociating partner.

4) As 1, but one of the exchanging partners is present at a dramatically lower concentration AND one of the exchanging partners experiences a restrictive force against the dimer interface thereby elevating its koff by 1000-fold. The exchange is now very fast.

This phenomenon can be rationalised by considering the fact that only a tiny fraction of the free CTD dimer must spontaneously dissociate in order to cap the full length ParB proteins bound to DNA, whose CTDs are frequently exposed due to the applied force. Thus, although the rate constant for CTD monomerization is very slow indeed (in units s-1), the absolute rate of monomer formation (nM s-1) is sufficiently high (given the high relatively concentration of the CTD dimer) to facilitate fast exchange and efficient capping.

For the particular case of scenario 4 (similar to our experimental conditions) we obtain a halftime of about 10 s in contrast with scenarios 1-3 that give values of 7000 s. Details of the kinetic model and simulations for all scenarios are included at the end of our responses under the heading Kinetic Model.

For some time, we also attempted to directly demonstrate experimentally the binding of the CTD monomers to ParB. We initially tried the experiment of injecting CTDAF into a ParB-coated DNA molecule. However, the large concentrations of the CTD required caused a very high fluorescence background which prevented us from being able to observe any fluorescence that might have occurred on the ParB-coated DNA. The alternative idea suggested by the reviewer of using a quencher is appealing but these materials would have to be fabricated and properly tested before use. Instead, we have followed an alternative approach similar to the procedure described in the manuscript to determine the photobleaching rate of ParBAF (Figure 2—figure supplement 2B). In brief, we co-incubated ParB (non-labeled) with a large excess of CTDAF in Mg2+ conditions (which are restrictive for effective binding of CTD alone to DNA) together with 1% formaldehyde. Formaldehyde will crosslink DNA and proteins. After 15 minutes we washed the non-bound complexes by injecting a protein-free buffer. As expected based on our favored model, we observed fluorescent proteins (CTDAF) crosslinked to the DNA by the formaldehyde, which progressively photobleached (Author response image 1). Note that this method gives a very high signal-to-background ratio because it allows one to work with a minimal fluorescence background. This experiment shows that the CTD binds preferentially to DNA-bound ParB, under Mg2+ conditions (Author response image 1A right panel). This experiment was however imperfect, in that we observed some binding of CTDAF alone to the DNA with formaldehyde (Author response image 1A left panel), albeit much lower than when a mixture of ParB and CTDAF was used. These experiments support the CTD capping model where CTD dimers dissociate and bind to ParB DNA-bound proteins inducing decondensation (Scenario 3, Figure 1D).

Author response image 1
CTDAF binds to full-length ParB bound to DNA.

(A) Fluorescence signal of CTDAF crosslinked with formaldehyde in Mg2+ conditions in the absence of ParB (left) and in the presence of ParB (right). (B) Quantification of fluorescence signal in both experiments. This experiment supports the model of CTDAF dimers dissociate to bind to ParB proteins bound to DNA (Scenario 3, Figure 1D).

https://doi.org/10.7554/eLife.43812.032

3) The author's conclusion that the ParB CTD might be inaccessible in solution is appealing, as it echoed with the observation that Bacillus subtilis and Thermus thermophilus ParB both exist in solution in a monomer-dimer equilibrium while the CTD alone is a dimer (Leonard et al., 2006, Taylor et al., 2015). Additional experiments could be carried out by the authors to strengthen and develop their hypothesis, using hydrogen-deuterium exchange mass spectrometry (HDX) for example. Since the current manuscript is meant to be a single advance, this is best kept for a future, more in-depth study.

We agree with the reviewers’ point here. The idea that the CTD is somehow shielded from intermolecular interaction in the absence of DNA, and becomes available for interactions that lead to multimerization in the presence of DNA is indeed attractive; it can explain why a protein with two (perhaps three) protein:protein interfaces can exist as a dimer when alone in free solution but assemble into large networks in the presence of DNA. It is essentially the model presented by Leonard et al. in their structural paper (albeit in a slightly different context because we now have different thoughts about the nature of ParB networks). Dillingham’s PhD student is currently performing FPOP- and HDX-coupled MS analysis of ParB in the absence and presence of DNA. This is a collaboration with the Sobott group (Leeds) and the data will not be available and/or processed on a useful timescale for this re-submission.

4) The authors have mentioned that the fluorescent labeling of proteins was controlled using mass spectrometry. These data need to be included in the paper, in particular in light of the puzzling reported labeling efficiency for ParB (2 dye molecules per monomer containing a single cysteine as stated in the captions for Figure 2A). The authors should discuss the possible reasons for a labeling efficiency that is substantially higher than 1.

ParBS68C was purified in the same manner as wild type ParB and labelled using a standard (vendor-supplied) protocol using a maleimide reagent. This yielded fully-functional (i.e. DNA binding proficient) ParB with an average of 2.4 dyes/monomer based on absorbance measurements, which was unexpected given that the protein contains a single engineered Cys (Author response image 2). Native mass spectrometry shows that the monomer contains between 1-4 labels (with 2 labels being by far the most abundant population) and that the dimer contains between 2 and 7 labels (with 4 labels being the most abundant) (Annika Butterer and Frank Sobott, personal communication). This is independent of whether the protein is folded or not and so these labels are covalently attached to the protein. Note that maleimides also react with amines, especially if they are activated by their local environment, and this is the most likely explanation for the observed labelling. This over-labelling is nevertheless unusual (e.g. our laboratory has labelled a very large number of different proteins using this chemistry and we have never come across this extent of non-specific labelling before). The over-labelling is reproducible and we also know that the wild type protein (with no cysteines) is efficiently labelled with ~ 1 dye/monomer, presumably reflecting the same phenomenon. However, regardless of this non-specific labelling activity, it is critical to note that all of our labelled full length ParB proteins retain wild type DNA binding activity.

Since our proteins display wild type activity, the over-labelling detected for ParB was not considered to be relevant.

Author response image 2
Preparation of functional fluorescently labelled ParB.

(A) SDS-PAGE of finally purified ParBS68C and wild type ParB, selectively and non-selectively, respectively, conjugated to Alexa 488. (B-D) Representative TBM- and TBE-EMSAs to assess the activity of the fluorophore-labelled proteins. Wild-type-like specific and non-specific DNA-binding activity is retained. In-gel detection of the fluorescent protein corresponds to the pattern of nucleoprotein complexes.

https://doi.org/10.7554/eLife.43812.033

5) An intriguing result presented by the authors is the lack of DNA-binding activity from the CTD in an Mg2+-supplemented buffer (Figure 4—figure supplement 1A), contrary to ParB in the same conditions. Although not addressed by the authors in their manuscript, this is highly reminiscent of their hypothesis (Fisher et al., 2017) of a second non-specific DNA binding site (possibly on the CDBD). In that case, this second site would still contribute to non-specific DNA binding in Mg2+ conditions while the CTD does not bind anymore, resulting in different binding kinetics which could be those measured in Figure 4A in Mg2+-supplemented buffer. This idea could be tested with labelled NTD-CDBD and/or labelled CDBD. One could also try to characterise the binding kinetics of the CTD in EDTA to compare it to ParB's.

Originally, a CTD deletion construct was designed based on secondary structure predictions, in which a stop codon was introduced following residue E222 (ParBΔCTD E222). Somewhat surprisingly, TBM- and TBE-EMSA analysis showed that this construct was defective in parS DNA binding, even when using concentrations 10-fold higher than required by wild type ParB. However, our structural model building later suggested that E222 resides within a structured α-helix that is part of the CDBD (helix 11). Modelling instead suggested that E227 may be a better truncation point for a clean removal of the CTD, leaving the NTD/CDBD domains intact (ParBΔCTD E227). This position appears to be in the linker between the final α-helix of the CDBD and first β-sheet of the CTD. Expression trials confirmed that this construct was soluble, and for ease of purification, an N-terminal His-tag with a HRV 3C cleavage site, was used (Author response image 3A). EMSA analysis showed that ParBΔCTD E227, similar to ParBΔCTD E222, had impaired parS and nonspecific DNA-binding activity (Figure Author response image 3B-C). Given that the HTH motif seems to be the major structural determinant for parS binding, this result suggests it does not function and/or is not folded correctly, in the absence of the CTD. Clearly, the functional interplay of domain organisation of B. subtilis ParB is complex. We have never attempted to make a CDBD only construct, given the complication, with making a functional NTD-CDBD only construct.

Finally, we tried, as the referee suggests, to measure the binding kinetics of the CTD in EDTA conditions. However, it was not possible to obtain reliable (un)binding kinetic curves of CTD in EDTA due to the low signal-to-background ratio observed for these particular conditions that include high CTD concentrations (Figure 4—figure supplement 1A).

Author response image 3
The CTD of ParB is required for the correct folding of the N-terminal and/or central DNA-binding domains.

(A) SDS-PAGE of wild-type ParB and ParBΔCTD E227. 2-6 µg of each was loaded as indicated. (B-C) Representative TBM- and TBE- EMSAs for ParBΔCTD E227 monitoring binding of parS-containing or non-specific 147 bp DNA.

https://doi.org/10.7554/eLife.43812.034

6) The authors studied DNA condensation with 250 nM ParBAF. In vivo studies (Graham et al., 2014) have shown that a very small number of ParB molecules was actually responsible for chromosome segregation in the bacterial cell. In their setup, as the authors state, "an excess of protein [is expected] compared to DNA binding sites". Did the authors test what was the lowest concentration of ParB they could get to condense DNA? It would also be interesting to compare condensation between the λ/2 DNAs used in this study and DNAs harbouring one or several parS sites, but this is beyond the scope of this study.

We have previously published an in-depth study of the relationship between ParB concentration, DNA binding, and DNA condensation (Taylor et al., 2015). We think the reviewers are implying that the condensation observed in vitro may not occur under more physiologically relevant conditions. This touches upon an important technical issue around the study of ParB in vitro which we acknowledge. Inside the cell, the entire chromosome and all ParB dimers are of course compartmentalized, leading to a situation in which [ParB] and [DNA binding sites] are both well above the Kd values we observe for DNA binding, but are present in a stoichiometry such that there are many more DNA binding sites than there are ParB dimers. Thus, the ParB in the cell must “make a choice” about where in the chromosome it will bind, and presumably ends up in foci around parS because of the high affinity interaction between ParB and its specific site mediated by the HTH domain. This situation cannot possibly be achieved in vitro. In order to observe ParB:DNA binding we must raise [ParB] to ~1 µM concentration, at which point it will always be in a large stoichiometric excess compared to the binding sites in our radiolabelled or immobilised DNA substrates. It is critical that in vitro experiments are interpreted in the light of this issue, which also highlights the importance of recent superresolution imaging approaches for ParB in vivo in order to study ParB networks. We have added a section to the Discussion section to acknowledge this point.

In our MT experiments there is an interplay between force and concentration. The minimum concentration tested in condensation experiments involving ParBAF was 250 nM.

With respect to the suggested experiments using long DNAs including parS sites, we do have some preliminary experiments indicating an accumulation of fluorescence around the region of parS sites However, we are still working on this and should regard our current data as too preliminary to publish.

7) It is not very clear if the experimental conditions for condensation inhibition are the same here than in the previous authors' published work. Indeed, previous experiments seem to have been performed in buffer without Mg2+ and without EDTA if I understand correctly ("100 mM NaCl, 50 mM Tris-HCl or HEPES-KOH pH 7.5, 100 mg/ml BSA and 0.1% Tween 20 (v/v)" according to Materials and methods section from Fisher et al., 2017). Please clarify.

In the previous work (Fisher et al., 2017) all Magnetic Tweezers experiments were performed in a buffer containing 4 mM MgCl2, 100 mM NaCl, 50 mM Tris-HCl or HEPES-KOH pH 7.5, 100 mg/ml BSA and 0.1% Tween 20 (v/v).

All experiments in the present study were performed either including 4 mM MgCl2 or 1 mM EDTA. We have explicitly included this information in the manuscript in the captions of Figure 2, Figure 3, and Figure 5.

References

Aris, R. (1956). On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. London Ser. A 235, 67-77.

Fisher, G.L., Pastrana, C.L., Higman, V.A., Koh, A., Taylor, J.A., Butterer, A., Craggs, T., Sobott, F., Murray, H., Crump, M.P., Moreno-Herrero, F., and Dillingham, M.S. (2017). The structural basis for dynamic DNA binding and bridging interactions which condense the bacterial centromere. eLife 6.

Niman, C.S., Beech, J.P., Tegenfeldt, J.O., Curmi, P.M., Woolfson, D.N., Forde, N.R., and Linke, H. (2013). Controlled microfluidic switching in arbitrary time-sequences with low drag. Lab on a chip 13, 2389-2396.

Schwesinger, F., Ros, R., Strunz, T., Anselmetti, D., Guntherodt, H.J., Honegger, A., Jermutus, L., Tiefenauer, L., and Pluckthun, A. (2000). Unbinding forces of single antibody-antigen complexes correlate with their thermal dissociation rates. Proc Natl Acad Sci U S A 97, 9972-9977.

Taylor, G. (1953). Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. London Ser. A 219, 186-203.

Kinetic Model:

Let AA dimers represent the free CTD in solution

Let BB dimers represent the hypothesised CTD dimers that bridge FLParBs in the condensate, and against which force is applied in the magnetic tweezers.

Let AB dimers represent the hypothesised CTD-FLParB heterodimers proposed to cap the ParBDNA complexes.

We are interested in the rate of formation of AB dimers upon mixing of AA and BB.

Kinetic scheme for a simple dimer exchange:

AAk1k2A+A (Eq. R1)

BBk3k4B+B (Eq. R2)

ABk5k6A+B(Eq. R3)

Rate equations

dAAdt=k1×AA+0.5×K2×A×A(Eq. R4)

dAdt=2K1AAk2AAk6AB+k5AB(Eq. R5)

dBBdt=k3BB+0.5k4BB(Eq. R6)

dAdt=2K3BBk4BBk6AB+k5AB(Eq. R7)

dABdt=k5AB+k6AB(Eq. R8)

We will use units of nM and seconds throughout.

We will assume that kon (i.e. k2, k4 and k6) is diffusion limited in all cases (kon = 1 nM-1 s -1).

We will assume that the free CTD is a very tight dimer (koff = 0.0001 s-1 equivalent to time constant of about 3 hours, and an equilibrium dissociation constant of 0.1 pM)

We will assume in all cases that at time zero there is no free monomer whatsoever. Thus AB can only be produced by dissociation (of AA and BB) followed by re-association (to form AB).

Scenario 1 (an example Berkeley Madonna syntax in shown below)

[A] = 0 nM

[B] = 0 nM

[AA] = 5000 nM

[BB] = 5000 nM

[AB] = 0 nM

k2 = k4 = k6 = 1 nM-1 s-1

k1 = k2 = k3 = 0.0001 s-1

Upon mixing, A and B will exchange to make ~5000 nM AB. The halftime is ~7000 seconds.

Scenario 2

[A] = 0 nM

[B] = 0 nM

[AA] = 5000 nM

[BB] = 1 nM

[AB] = 0 nM

k2 = k4 = k6 = 1 nM-1 s -1

k1 = k2 = k3 = 0.0001 s-1

Upon mixing, A and B will exchange to make ~2 nM AB. The halftime is ~7000 seconds.

Scenario 3

[A] = 0 nM

[B] = 0 nM

[AA] = 5000 nM

[BB] = 5000 nM

[AB] = 0 nM

k2 = k4 = k6 = 1 nM-1 s -1

k1 = k2 = 0.0001 s-1

k3 = 0.1 s-1 (k3 increased by 1000-fold to mimic applied force, note that BB is still 90% dimer at equilibrium with B)

Upon mixing, A and B will exchange to make ~10000 nM AB. The halftime is ~7000 seconds.

Scenario 4

[A] = 0 nM

[B] = 0 nM

[AA] = 5000 nM

[BB] = 1 nM

[AB] = 0 nM

k2 = k4 = k6 = 1 nM-1 s -1

k1 = k2 = 0.0001 s-1

k3 = 0.1 s-1 (k3 increased by 1000-fold to mimic applied force, note that BB is still 90% dimer at equilibrium with B)

Upon mixing, A and B will exchange to make ~2 nM AB. The halftime is ~10 seconds.

Syntax for Berkeley Madonna Simulation (Scenario 1 shown)

METHOD RK4

STARTTIME = 0

STOPTIME = 40000

DT = 1

init AA = 5000

init A = 0

init BB = 5000

init B = 0

init AB = 0

d/dt(AA) = -k1*AA + 0.5*k2*A*A

d/dt(A) = 2*k1*AA - k2*A*A - k6*A*B + k5*AB

d/dt(BB) = -k3*BB + 0.5*k4*B*B

d/dt(B) = 2*k3*BB - k4*B*B - k6*A*B + k5*AB

d/dt(AB) = -k5*AB + k6*A*B

k1 = 0.0001

k2 = 1

k3 = 0.0001

k4 = 1

k5 = 0.0001

k6 = 1

https://doi.org/10.7554/eLife.43812.041

Article and author information

Author details

  1. Julene Madariaga-Marcos

    Department of Macromolecular Structures, Centro Nacional de Biotecnología, Consejo Superior de Investigaciones Científicas, Madrid, Spain
    Contribution
    Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing—original draft, Writing—review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-1458-3369
  2. Cesar L Pastrana

    Department of Macromolecular Structures, Centro Nacional de Biotecnología, Consejo Superior de Investigaciones Científicas, Madrid, Spain
    Contribution
    Conceptualization, Data curation, Software, Formal analysis, Investigation, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
  3. Gemma LM Fisher

    DNA:Protein Interactions Unit, School of Biochemistry, Faculty of Life Sciences, University of Bristol, Bristol, United Kingdom
    Contribution
    Data curation, Formal analysis, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
  4. Mark Simon Dillingham

    DNA:Protein Interactions Unit, School of Biochemistry, Faculty of Life Sciences, University of Bristol, Bristol, United Kingdom
    Contribution
    Conceptualization, Supervision, Funding acquisition, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-4612-7141
  5. Fernando Moreno-Herrero

    Department of Macromolecular Structures, Centro Nacional de Biotecnología, Consejo Superior de Investigaciones Científicas, Madrid, Spain
    Contribution
    Conceptualization, Supervision, Funding acquisition, Methodology, Writing—original draft, Project administration, Writing—review and editing
    For correspondence
    fernando.moreno@cnb.csic.es
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-4083-1709

Funding

Basque Government (PRE_2013_11_1174)

  • Julene Madariaga-Marcos

Biotechnology and Biological Sciences Research Council (1363883)

  • Gemma LM Fisher

Wellcome (100401)

  • Mark Simon Dillingham

Wellcome (077368)

  • Mark Simon Dillingham

Agencia Estatal de Investigación (BFU2017-83794-P)

  • Fernando Moreno-Herrero

H2020 European Research Council (681299)

  • Fernando Moreno-Herrero

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We are thankful for financial support from AEI/FEDER, UE (BFU2017-83794-P) and from European Research Council (ERC) under the European Union’s Horizon2020 Research and Innovation programme (grant agreement no. 681299). JM-M acknowledges a predoctoral PhD fellowship from the Basque Country Government Department of Education, Language Policy and Culture (ref. PRE_2013_11_1174). GLMF was supported by the Biotechnology and Biological Sciences Research Council (1363883). MSD was supported by the Wellcome Trust (100401 and 077368).

Senior Editor

  1. John Kuriyan, University of California, Berkeley, United States

Reviewing Editor

  1. Sebastian Deindl, Uppsala University, Sweden

Publication history

  1. Received: November 23, 2018
  2. Accepted: March 9, 2019
  3. Version of Record published: March 25, 2019 (version 1)

Copyright

© 2019, Madariaga-Marcos et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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