Molecular flexibility of DNA as a major determinant of RAD51 recruitment

The timely activation of homologous recombination is essential for the maintenance of genome stability, in which the RAD51 recombinase plays a central role. Biochemically, human RAD51 binds to both single-stranded DNA (ssDNA) and double-stranded DNA (dsDNA) with similar affinities, raising a key conceptual question: how does it discriminate between them? In this study, we tackled this problem by systematically assessing RAD51 binding kinetics on ssDNA and dsDNA differing in length and flexibility using surface plasmon resonance. By fitting detailed polymerisation models informed by our experimental datasets, we show that RAD51 is a mechanosensor that exhibits a larger polymerisation rate constant on flexible ssDNA compared to rigid ssDNA or dsDNA. This model presents a new general framework suggesting that the flexibility of DNA, which may increase locally as a result of DNA damage, plays an important role in recruiting repair factors that multimerise at sites of DNA damage.


Introduction
DNA double-strand breaks (DSBs) are cytotoxic lesions that can lead to chromosomal breaks, genomic instability and tumourigenesis in mammalian cells (Tubbs and Nussenzweig, 2017).
Homologous recombination (HR) can offer an error-free DNA repair mechanism to restore genetic information at DSB sites, and in this way, contribute to genome stability. During HR-mediated repair, single-stranded DNA (ssDNA) overhangs are generated and rapidly coated with the ssDNAbinding replication protein A (RPA) (Chen and Wold, 2014). ssDNA-bound RPA is then exchanged for RAD51, the central ATP-dependent recombinase that catalyses HR-mediated repair. RAD51 polymerises on ssDNA to form a ssDNA-nucleoprotein filament and guide homologous strand invasion and DSB repair (Baumann et al., 1996).
The central mechanism of HR is evolutionarily highly conserved, and the bacterial RAD51 ortholog RecA has clear preference to polymerise on ssDNA over dsDNA (Benson et al., 1994).
However, human RAD51 shows weaker binding affinity to ssDNA compared to RecA, and the mechanism by which RAD51 polymerises on ssDNA in preference to dsDNA remains enigmatic.
Earlier studies using electrophoretic mobility shift assay (EMSA) have suggested that RAD51 binds both ssDNA and dsDNA with similar affinities (Benson et al., 1994), albeit its preferential ssDNA binding was visible in the presence of ammonium sulphate (Shim et al., 2006). These endpoint assays, however, do not provide information as to whether binding kinetics may contribute to a potential RAD51-dependent ssDNA / dsDNA discrimination mechanism. Indeed, more recent kinetic studies have revealed that RAD51 polymerisation consists of two phases: a rate-limiting nucleation phase and a consecutive growth phase (Hilario et al., 2009;Miné et al., 2007;Van der Heijden et al., 2007). A minimal polymer nucleus, with a length of either two-to-three or four-to-five RAD51 protomers, is proposed to elicit the growth phase of RAD51 polymerisation. Nonetheless, it remains unknown whether there are any mechanistic differences between RAD51 polymerisation on ssDNA and dsDNA. Intriguingly, RAD51 is shown to display faster association kinetics on ssDNA (Candelli et al., 2014), whilst it also exhibits slower dissociation kinetics on dsDNA, indicating that the RAD51-dsDNA complex is stable once formed (Miné et al., 2007).
A key difference between ssDNA and dsDNA is their molecular flexibility: ssDNA is known to be more flexible compared to dsDNA. The flexibility of a DNA molecule can be characterised by its persistence length (L p ), a mechanical parameter quantifying polymer rigidity: the higher the persistence length, the more rigid the polymer. In the presence of monovalent or divalent salt, dsDNA displays an L p of ~ 30 -55 nm (Baumann et al., 1997;Brunet et al., 2015), while ssDNA is much more flexible, with an L p of 1.5 -3 nm (Chi et al., 2013;Kang et al., 2014;Murphy et al., 2004). These observations imply that ssDNA can explore a much larger configurational space compared to dsDNA. It follows that the formation of a structured (less flexible) RAD51 polymer on ssDNA will incur a large entropic energy penalty compared to the formation of a RAD51 polymer on dsDNA. Despite the clear thermodynamic implications of RAD51 polymerisation on DNA, the impact of DNA flexibility on RAD51 nucleoprotein filament formation has been largely overlooked.
In this study, we describe how RAD51 polymerises on DNA. Using a combination of surface plasmon resonance (SPR) and small-angle x-ray scattering (SAXS), we have assessed the RAD51 binding kinetics on DNA using ssDNA and dsDNA oligonucleotides differing in length and flexibility. Analyses of the SPR data using biochemical mathematical models revealed that RAD51 polymerisation required a minimal nucleus of four and two molecules on ssDNA and dsDNA, respectively. Interestingly, our analyses further uncovered that RAD51 is a mechanosensor that polymerises faster on flexible DNA. We hypothesised that the entropic penalty to polymerisation on flexible DNA is offset by a large enthalpic contribution and show this to be the case by perturbing the RAD51 protomer-protomer interaction. We propose the DUET model (Dna molecUlar flExibiliTy), which explains the ability of RAD51 to preferentially and stably polymerise on highly flexible ssDNA compared to dsDNA, enabled by the enthalpic energy of RAD51 polymerisation.

Results
To evaluate how human RAD51 discriminates between ssDNA and dsDNA, we assessed the binding kinetics of untagged recombinant human RAD51 to a 50-mer mixed base ssDNA molecule (dN-50) and a 50-mer mixed base paired dsDNA molecule (dN-50p) by SPR. These analyses performed in the presence of ATP and Ca 2+ , which blocks RAD51 ATP hydrolysis (Bugreev and Mazin, 2004), showed 1) faster RAD51 association to ssDNA compared to dsDNA, and 2) similar RAD51 lifetimes on both ssDNA and dsDNA (Fig. S1A, B). These observations validated that RAD51 distinguishes ssDNA from dsDNA through faster polymerisation on ssDNA (Candelli et al., 2014).
For an advanced understanding of the mechanism by which RAD51 polymerises differentially on ssDNA and dsDNA, we set out a streamlined study comprises of the following processes; 1) perform SPR measurements to identify the kinetics of RAD51 binding to ssDNA and dsDNA, 2) develop a linear, ordinary differential equation (ODE) model, which infers RAD51 polymer formation on DNA, informed by respective SPR dataset, and 3) use Approximate Bayesian Computation coupled to Sequential Monte Carlo (ABC-SMC) (Toni et al., 2009) to fit the ODE model to all related SPR datasets. This strategy enables us to summarise our experimental SPR dataset with a mechanistic ODE model describing how RAD51 associates with naked DNA, forms a stable nucleus, and elongates to form polymers on DNA.

RAD51 nucleates poorly on ssDNA, but elongates efficiently to form long polymers.
We first aimed to define the initial steps of RAD51 association with ssDNA. To this end, we generated a series of mixed base short ssDNA oligos with different lengths, each of which can be bound by a restricted number of RAD51 molecules. As a single RAD51 molecule engages with three nucleotides of DNA (Short et al., 2016), a ssDNA consists of 5 nucleotides (dN-5), 8 nucleotides (dN-8), 11 nucleotides (dN-11), 14 nucleotides (dN-14) and 17 nucleotides (dN-17) can accommodate up to one, two, three, four and five RAD51 molecules, respectively (Fig. 1A). Our systematic SPR measurements of RAD51 binding kinetics revealed no RAD51 binding to the dN-5, dN-8 and dN-11, slow association and moderate dissociation with dN-14, and slightly faster association and slow dissociation with dN-17 (Fig. 1B). These observations suggested that three or less RAD51 molecules are unable to generate stable nucleus on ssDNA, four RAD51 molecules form a quasi-stable nucleus, and five or more RAD51 molecules are able to form a highly stable nucleus.
We then moved on to develop ODE models describing RAD51 polymerisation on ssDNA, informed by these SPR datasets. The distinct RAD51 nucleation states on ssDNA led us to hypothesise that RAD51 polymerisation is achieved by three sequential DNA-binding steps, namely adsorption, elongation and stabilisation (Fig. 1C). The adsorption step is characterised by the forward and reverse adsorption rate constants (k ad.f and k ad.r ) and describes the formation of an unstable RAD51 nucleus on ssDNA. The elongation step is characterised by the forward and reverse elongation rate constants (k el.f and k el.r ), and describes the formation of a quasi-stable nucleus. Finally, the stabilisation step is characterised by the same forward elongation rate constant (k el.f ) but a different stabilisation reverse rate constant (k s ). This step describes stable nucleus formation, and also nucleus elongation up to a maximum length of 16, which corresponds to the predicted maximum RAD51 molecules binding to a 50-mer ssDNA molecule. Additionally, we assumed that 1) RAD51 polymerisation mainly occurs via the addition of RAD51 dimers, and to a lesser extent via the addition of RAD51 monomers (Subramanyam et al., 2016), and that 2) RAD51 dissociation only occurs via the removal of monomers.
Having established the ODE model, we used ABC-SMC to simultaneously fit this model to all the RAD51 binding kinetics datasets and obtained values for the five model parameters (Fig.   1B, C, Fig. S2A, B). This model describes the experimental data accurately compared to two simpler models which respectively describe RAD51 polymerisation only via monomer addition (Fig.   S2C), or without taking account of the reverse elongation rate constant (i.e., k el.r = k s ) (Fig. S2D), as summarised in Fig. S2E. Importantly, our model identified a high forward elongation rate constant (k el.f = 53.7 ± 10.66 /µM/s) as the factor driving the rapid RAD51 polymerisation on ssDNA. Indeed, although our experimental data measured an average of six RAD51 molecules per dN-50 at the end of the SPR injection, our model predicts that most of the RAD51 forms long polymers of 16, 15 and 14 molecules on a fraction of DNA molecules during each injection (Fig.   S2F). Similarly, the model predicts that RAD51 forms mainly pentamers and tetramers on the dN-17 (Fig. S2G), and only tetramers on the dN-14 (Fig. S2H). These results support the notion that RAD51 polymer formation on ssDNA requires the nucleation of at least four RAD51 protomers, but is facilitated by a high elongation rate constant.

RAD51 nucleates efficiently on dsDNA, but elongates poorly to form long polymers.
We next evaluated the initial steps of RAD51 association with dsDNA. We generated an analogous series of mixed base dsDNA consists of 5 base pairs (dN-5p), 8 base pairs (dN-8p) and 11 base pairs (dN-11p), which can accommodate up to one, two or three RAD51 molecules, respectively ( Fig. 2A). Our SPR measurement detected no RAD51 binding to the dsDNA dN-5p, as was the case for the ssDNA dN-5. To our surprise, however, we detected slow association and moderate dissociation of RAD51 with the dN-8p, and even faster association and slower dissociation with the dN-11p (Fig. 2B). These observations suggested that one RAD51 molecule is unable to associates stably with ssDNA, but two RAD51 molecules form a quasi-stable nucleus, and three or more RAD51 molecules are able to form a highly stable nucleus.
An ODE model describing RAD51 polymerisation on dsDNA was then developed using the same assumptions applied to the RAD51-ssDNA ODE model, with one exception: the model is simplified to include two, instead of three, sequential DNA-binding steps, namely adsorption and stabilisation (i.e. with k el.f = k s ) (Fig. 2C). Our ABC-SMC simulation was able to provide defined values for the four parameters of the model (Fig. 2C, Fig. S3A, B), validating that this model explains the SPR data well. Conversely, the assumption of RAD51 polymerisation only via monomer addition (Fig. S3C) or the addition of a reverse elongation step (k el.r ≠ k s ) (Fig. S3D), do not improve the model (Fig. S3E). Importantly, our model highlighted a considerably low forward elongation rate constant (k el.f = 0.953 ± 0.48 /µM/s) compared to ssDNA, describing the unproductive RAD51 polymerisation on dsDNA. This is reflected by our model prediction that most RAD51 does not form long polymers on the dN-50p (Fig. S3F). Moreover, our model predicted that RAD51 forms dimers with a small amount of trimers on the dN-11p (Fig. S3G), and both monomers and dimers on the dN-8p (Fig. S3H). Collectively, these analyses revealed that a RAD51 dimer is sufficient to form a stable nucleus on dsDNA, but its polymer formation is restricted by a lower elongation rate constant.

RAD51 polymerises faster on flexible DNA
Overall, our analyses suggest that RAD51 nucleates more efficiently on dsDNA but elongates significantly faster on ssDNA (56-fold higher ssDNA k el.f compared to dsDNA k el.f ). We speculate that the difference in RAD51 elongation speed is due to the higher flexibility of ssDNA compared to dsDNA. Explicitly, we propose a Bend-To-Capture (BTC) mechanism to explain how DNA flexibility can impact polymerisation kinetics (Fig. 3A). A free RAD51 molecule in solution would need to generate two sequential, non-covalent interactions to be incorporated into the growing polymer: the interaction with the exposed interface of existing RAD51 polymer and with the exposed scaffold DNA. To avoid steric clashes, we reasoned that each of these interactions requires naked DNA immediately next to the existing RAD51 polymer in a configuration that can bend away from the preferred direction of the polymer. In this way, flexible DNA is expected to explore more conformations compatible with the further addition of RAD51 per unit time.
To test this notion, we designed an experiment to measure the kinetics of RAD51 binding to DNA oligos of varying flexibility. It has been shown that poly-dT ssDNA, which is widely used for RAD51 binding assays, is highly flexible, while poly-dA ssDNA is highly rigid due to base stacking interactions (Sim et al., 2012). Consistently, our SAXS-derived persistence length measurements showed ssDNA dT-50 is the most flexible oligo, followed by dN-50, dA-50 and dsDNA dN-50p ( Fig.   S4 and Table 1). By measuring RAD51 binding kinetics to these DNA oligos (Fig. 3B), we found that RAD51 indeed displayed faster association to the dT-50 compared to the dN-50, and the model fit suggests this is due to a higher elongation rate (k el.f ) (Fig. 3C, D). Furthermore, RAD51 displayed very slow association to the dA-50, comparable to that of dsDNA dN-50p, which is explained by a lower elongation rate (Fig. 3C, D). Similarly, individually fitting the adsorption forward rate (k ad.f ) or adsorption reverse rate (k ad.r ) can explain the ssDNA dT-50 and dA-50 data ( Fig. S5A, B). Conversely, individually fitting the elongation reverse rate constant (k el.r ) or the stabilisation rate constant (k s ) was unable to explain the dT-50 and dA-50 data (Fig. S5C, D).
Meantime, we observed that the lifetime of the RAD51 polymers was independent of the flexibility of the underlying DNA (Fig. 3E). Taken together, these observations suggest that although RAD51 is able to form a stable nucleus with fewer molecules on dsDNA compared to ssDNA (two versus four), it elongates and/or absorbs more efficiently on ssDNA because of its higher flexibility, consistent with our proposed BTC mechanism.

RAD51 protomer-protomer interaction compensates for the entropic penalty of flexible DNA
The BTC mechanism explains how higher DNA flexibility can induce faster RAD51 polymerisation.
However, the formation of a rigid RAD51 polymer on highly flexible DNA is expected to be associated with a large entropic penalty, because RAD51 polymerisation can dramatically reduce the number of spatial configurations that a flexible DNA polymer can explore. Hence, RAD51 polymerisation on flexible ssDNA is expected to incur a higher entropic penalty compared to its polymerisation on rigid dsDNA. To explain this paradox, we reasoned that the favourable enthalpic energy of RAD51 polymerisation is sufficient to compensate for any entropic penalty, referred to as the entropy-enthalpy compensation (EEC) mechanism (Fig. 4A).
To directly test this hypothesis, we assessed the kinetics of a RAD51 mutant, which confers a reduced enthalpic contribution compared to wild-type (WT) RAD51. We predicted that, with such a mutation, the stability of the mutant RAD51 polymers on DNA would become more dependent on the flexibility of the underlying DNA (Fig. 4A, B). We took advantage of a phenylalanine to glutamate substitution at RAD51 residue 86 (F86E), which reduces the RAD51 protomer-protomer interaction affinity (Esashi et al., 2007;Yu et al., 2003). Indeed, using size exclusion chromatography with multi-angle light scattering (SEC-MALS), we confirmed that F86E RAD51 is primarily monomeric in solution ( Fig. S6A-C). We then used SPR to measure the binding kinetics of F86E RAD51 to ssDNA dT-50, dN-50, dA-50, and dsDNA dN-50p, and fit our models to these data under the assumption that F86E RAD51 polymerisation takes place only via monomer addition (Fig. 4C, D). It was immediately evident that F86E RAD51 shows significantly reduced affinity for ssDNA and dsDNA compared to WT RAD51, with no detectable binding to DNA at the concentration of 300 nM or 3 µM (Fig. S6D, E). Nonetheless, at 30 µM, we observed that F86E RAD51 displays faster elongation on more flexible DNA, as we observed for WT RAD51 (Fig. 4D).
Strikingly, F86E RAD51 displayed reduced lifetimes (1/k s ) that inversely correlated with the flexibility of DNA (Fig. 4E). This observation is in sharp contrast with WT RAD51, which formed stable, long-lived polymers independently of DNA flexibility (Fig. 3E). Taken together, these observations demonstrate that the enthalpic contribution of WT RAD51 polymer formation is sufficient to offset the large entropic penalty associated with polymerisation on flexible DNA. As a result, RAD51 is able to polymerise more rapidly on flexible DNA despite incurring a larger entropic penalty.

Discussion
In this study, we have analysed the kinetics of RAD51 binding to ssDNA and dsDNA oligos of varying length and flexibility via a combination of SPR, SAXS and mathematical modelling to understand how RAD51 discriminates between ssDNA and dsDNA. Based on our observations, we propose the DUET (Dna molecUlar flExibiliTy) model, which describes how RAD51 polymerises on ssDNA in preference to dsDNA: 1) RAD51 has a faster polymer elongation rate on flexible DNA because flexible DNA explores more conformations compatible with RAD51 binding per unit time (the BTC mechanism) (Fig. 3A), and 2) the enthalpic contribution of RAD51 polymerisation enables RAD51 to overcomes the entropic penalty to binding to flexible DNA (the EEC mechanism) (Fig. 4A).
The diploid human genome consists of 6.4 billion base pairs and ~ 50 endogenous DSBs are estimated to occur in every cell cycle (Vilenchik and Knudson, 2003). In human cells, the resection step of HR-mediated DSB repair can generate ssDNA overhangs up to 3.5 k nucleotides in length (Zhou et al., 2014), which then serve as a platform for RAD51 polymerisation. Given that ssDNA overhangs would constitute only ~ 0.005 % of the total genomic DNA (3.5 k nucleotides x two overhangs x 50 DSBs / 6.4 billion base pairs), the polymerisation of RAD51 on resected ssDNA needs to be greatly directed. The average nuclear concentration of RAD51 is estimated at ~ 100 nM (Reuter et al., 2014), and several RAD51 mediators, such as BRCA2, PALB2 and RAD52, contribute to increase the RAD51 local concentration at DSBs and/or RAD51 binding to ssDNA, while preventing its association with dsDNA (Buisson et al., 2010;Carreira and Kowalczykowski, 2011;Jensen et al., 2010;Ma et al., 2017a;Miyazaki et al., 2004;Zhao et al., 2015). Nonetheless, to date, in vivo evidence that these RAD51 mediators are sufficient to promote RAD51 polymerisation on DSB-derived ssDNA in preference to the bulk of undamaged dsDNA is limited.
It is important to note that resected ssDNA is first bound by RPA prior to RAD51 polymerisation. RPA is a heterotrimer complex with six-OB folds, four of which can associate tightly with ssDNA in a stepwise manner (Fan and Pavletich, 2012;Zou et al., 2006). RPA binding to ssDNA is believed to eliminate ssDNA secondary structures to facilitate RAD51 polymerisation.
Significantly, previous kinetic studies have demonstrated a rapid exchange of ssDNA-bound RPA with free RPA available in solution, a phenomenon known as 'facilitated exchange' (Ma et al., 2017b). Given that one RPA heterotrimer has a footprint of 30 nucleotides and one RAD51 protomer binds to 3 nucleotide of ssDNA (Short et al., 2016), the dissociation of a single RPA heterotrimer would expose enough ssDNA to accommodate up to ten RAD51 molecules. Hence, it is reasonable to speculate that RPA dissociation provides highly flexible ssDNA, which in turn can promote the adsorption and elongation steps of RAD51 polymerisation.
We initially aimed to model RAD51 polymerisation by stepwise addition of RAD51 monomers, but with this approach we were unable to explain the ssDNA data or the dsDNA data ( Fig. S2C and S3C, respectively). The revised assumption of RAD51 polymerisation via dimers addition, presented in this study, is in line with the model proposed by Subrayaman et al. (Subramanyam et al., 2016). Although the multimeric state of RAD51 in solution is not fully characterised, it is conceivable that RAD51 mediators and/or the increased local concentration RAD51 may support the availability of soluble RAD51 dimers at DSB sites.
Another major unanswered question regarding RAD51 polymerisation is how RAD51 can distinguish ssDNA generated by DSB resection from ssDNA formed during normal cellular processes, such as DNA replication and transcription. Spontaneous RAD51 polymerisation on all ssDNA would be problematic, as it may trigger undesired, toxic recombination events or disruption of DNA replication and transcription, causing genomic instability. In this context, the DUET model is particularly appealing: ssDNA generated during transcription and replication does not have free ends and could therefore be less flexible compared to resected ssDNA with a free 3' overhang.
This in turn may limit RAD51 polymerisation on transcription-and replication-derived ssDNA, while promoting it on flexible ssDNA at DSBs.
This study also suggests a more general mechanism for the recruitment of DSB repair factors. Indeed, numerous DNA repair machineries are comprised of multiple subunits with several binding interfaces to DNA and other co-factors. Broken DNA ends may enhance recruitment of such repair complexes simply due to the increased DNA flexibility. Further studies in combination with RAD51 mediators, such as RAD51 paralogs and BRCA2, which are shown to stimulate RAD51 filament formation (Shahid et al., 2014;Taylor et al., 2016), will be important to further advance our full understanding of the RAD51 polymerisation mechanism. This study provides evidence for a previously unappreciated role of DNA flexibility in the process of DNA repair.
The suspension was sonicated and spun at 20k rpm with a 45 Ti rotor (Beckman). The supernatant was slowly mixed with 0.1 % polyethylenimine at 4°C for 1 hour and spun at 20K rpm using a 45 Ti rotor to remove DNA. The supernatant was then slowly mixed with an equal volume of 4 M ammonium sulphate (2 M final concentration) at 4°C for 1 hour and spun at 10 K rpm using a JA-17 rotor (Beckman). Pellets containing RAD51 were suspended in 25 ml of resuspension buffer (0.5 M KCl, 50 mM Tris-HCl pH 7.5, 1 mM EDTA, 2 mM DTT, 10 % glycerol) and spun again at 20k rpm to remove residual DNA. The supernatant was then dialysed overnight at 4°C in TEG buffer (50 mM Tris pH 7.5, 1 mM EDTA, 2 mM DTT, 10 % glycerol) containing 200 mM KCl (TEG200). For F86E RAD51, the cell lysate was prepared as for WT RAD51, and dialysed overnight at 4°C in TEG buffer containing 50 mM KCl (TEG75).
RAD51 purification was carried out by chromatography at 4°C using the AKTA Pure Protein Purification System (GE Healthcare). For WT RAD51, the dialysed WT RAD51 containing sample was loaded onto a 5 ml HiTrap Heparin column (GE Healthcare) and eluted via a linear gradient of 200 mM -600 mM KCl. The peak fractions were pooled and dialysed in TEG200, and concentrated on a 1 ml HiTrap Q column (GE Healthcare) followed by the isocratic elusion with TEG buffer containing 600 mM KCl (TEG600). Peak fractions containing WT RAD51 were dialysed in SPR buffer (150 mM KCl, 20 mM Hepes pH 7.5, 2 mM DTT, 10 % glycerol), aliquoted, snap frozen and stored at -80°C. Similarly, the dialysed F86E RAD51 containing cell lysate was fractionated through a 5 ml HiTrap Heparin column, but with a linear gradient of 75 mM -600 mM KCl. The flow-through and F86E RAD51 peak fractions were pooled and dialysed in TEG buffer containing 50 mM KCl (TEG50) and reloaded onto a 5 ml HiTrap Heparin column. Following a 50 mM -600 mM KCl linear gradient elution, F86E RAD51 peak fractions were pooled and dialysed in TEG buffer containing 100 mM KCl (TEG100). The sample was concentrated on a 1 ml HiTrap Q column followed by TEG600 isocratic elution. Peak fractions were applied on a 24 ml Superdex200 10/300 GL size-exclusion column (GE Healthcare), and fractions containing monometic F86E RAD51 were pooled and dialysed in TEG100. F86E RAD51 was then re-concentrated using a 1 ml HiTrap Q column and TEG600 isocratic elution. The peak fractions were dialysed in SPR buffer, aliquoted, snap frozen and stored at -80°C.

Multi-Angle Light Scattering
To confirm the monomeric status of F86E RAD51, the peak size-exclusion chromatography F86E RAD51 elution fraction was serially diluted (5, 1:1 serial dilutions) and loaded onto a Superdex200 10/300 GL size exclusion column (GE Healthcare) equilibrated with MALS Buffer (150 mM KCl, 50 mM Tris pH 7.5, 1 mM EDTA, 2 mM DTT). Each elution was analysed using a Wyatt Heleos8+ 8angle light scatterer linked to a Shimadzu HPLC system comprising LC-20AD pump, SIL-20A Autosampler and SPD20A UV/Vis detector. Data collection was carried out at the Department of Biochemistry, University of Oxford. Data analysis was carried out using the ASTRA software (Wyatt).

DNA Oligo and Duplex Synthesis
The dN-5p, dN-8p and dN-11p dsDNA sequences were designed as two complimentary ssDNA sequences connected via two units of hexaethylene glycol (flexible linker). The dN-5, dN-8, dN-11 ssDNA molecules were designed using one of the two corresponding dsDNA annealing sequences. The dN-14 and dN-17 ssDNA molecules were designed by extending one of the two dN-11p dsDNA annealing sequences. Standard DNA phosphoramidites, solid supports, 3'-Biotin-TEG CPG and additional reagents were purchased from Link Technologies Ltd and Applied Biosystems Ltd. All oligonucleotides were synthesized on an Applied Biosystems 394 automated DNA / RNA synthesizer using a standard 1.0 µmole phosphoramidite cycle of acid-catalyzed detritylation, coupling, capping, and iodine oxidation. Stepwise coupling efficiencies and overall yields were determined by the automated trityl cation conductivity monitoring facility and in all cases were >98.0%. All β-cyanoethyl phosphoramidite monomers were dissolved in anhydrous acetonitrile to a concentration of 0.1 M immediately prior to use. The coupling time for normal A, G, C, and T monomers was 60 s, and the coupling time for the hexaethylene glycol phosphoramidite monomer (from link) was extended to 600 s. Cleavage of the oligonucleotides from the solid support and deprotection was achieved by exposure to concentrated aqueous ammonia solution for 60 min at room temperature followed by heating in a sealed tube for 5 h at 55 °C. Purification of oligonucleotides was carried out by reversed-phase HPLC on a Gilson system using a Brownlee Aquapore column (C8, 8 mm x 250 mm, 300 Å pore) with a gradient of acetonitrile in triethylammonium bicarbonate (TEAB) increasing from 0% to 50% buffer B over 20 min with a flow rate of 4 ml/min (buffer A: 0.1 M triethylammonium bicarbonate, pH 7.0, buffer B: 0.1 M triethylammonium bicarbonate, pH 7.0 with 50% acetonitrile). Elution of oligonucleotides was monitored by ultraviolet absorption at 295 or 300 nm. After HPLC purification, oligonucleotides were freeze dried then dissolved in water without the need for desalting. All oligonucleotides were characterised by negative-mode HPLC-mass spectrometry using either a Bruker micrOTOFTM II focus ESI-TOF mass spectrometer with an Acquity UPLC system, equipped with a BEH C18 column (Waters) or a Waters Xevo G2-XS QT mass spectrometer with an Acquity UPLC system, equipped with an Acquity UPLC oligonucleotide BEH C18 column (particle size: 1.7 µm; pore size: 130 Å; column dimensions: 2.1 x 50 mm). Data were analysed using Waters MassLynx software or Waters UNIFI Scientific Information System software.

Small Angle X-Ray Scattering
The flexibilities of the ssDNA dT-50, ssDNA dN-50, ssDNA dA-50 and the dsDNA dN-50p molecules were assessed using small-angle x-ray scattering (SAXS) at the Diamond Light Source (Harwell, UK). SAXS data were collected using a size exclusion KW 402.5 (2.4 ml) column (Shodex). 50 µl of DNA sample was injected and elution was carried out at 37°C at 75 µl/min using SPR running buffer in the absence of glycerol and BSA. The flexible cylinder model was fit to the four scattering data sets within the 0.0037 -0.27 [1/Å] range to derive the persistence lengths.
Data plotting and fitting was carried out using the SasView software for SAXS data analysis.

Surface Plasmon Resonance
The binding kinetics of WT and F86E RAD51 on ssDNA and dsDNA were assessed via surface plasmon resonance (SPR) using a Biacore T200 (GE Healthcare). CM5 SPR chip flow cells were activated by injecting 100 µl of a 1:1 N-Hydroxysuccinimide (NHS), ethyl(dimethylaminopropyl) carbodiimide (EDC) mix at 10 µl/min. 100 µl of purified streptavidin (1 mg/ml) was then injected over the flow cells at 10 µl/min. Unbound amine groups were then deactivated by injecting 50 µl of 1 M ethanolamine-HCl. 50 µl of 10 mM glycine pH 2.5 was injected to remove uncoupled, sterically bound streptavidin. For all experiments, ~ 8 -15 RU of biotinylated DNA substrates were immobilised onto either flow cells 2, 3 or 4. After DNA immobilisation, 10 µl of purified biotin was injected over all four flow cells at 10 µl/min to block the remaining free streptavidin binding sites.
Chip activation and ligand immobilisation steps were carried out at 25°C. Using high performance injections, the CM5 chip surfaces were then primed via 10 injections of 10 µl SPR running buffer (150 mM KCl, 20 mM Hepes pH 7.5, 2 mM DTT, 5 mg/ml BSA, 2.5 mM ATP pH 7.5, 10 mM CaCl 2 , 10 % glycerol) at 30 µl/min. WT or F86E RAD51 diluted in SPR running buffer to the specified concentration was injected at 30 µl/min (association), followed by the injection of SPR running buffer for 20 minutes at 30 µl/min (dissociation) at 37°C.

SPR Data Processing
SPR traces were processed using the BIAEvaluation software (GE Healthcare) as following. First, the negative control flow cell trace (Fc1 or Fc3) was subtracted from the experimental flow cell trace. The trace was then vertically and horizontally aligned, such that the start of the protein injection occurs at time = 0 s, Response = 0 RU. Subsequently, the drift trace between the negative control flow cell and the experimental flow cell during the final priming injection was subtracted from the experimental flow cell trace (double-referencing) (Myszka, 1999). All experimental curves were normalised using the following equation:

Mathematical Model Development and Analysis
During SPR injections, the concentration of free analyte (i.e. RAD51) in solution in the flow cell remains constant because only a very small proportion of analyte binds to the surface. For this reason, the ordinary differential equation (ODE) models were formulated using first order mass action kinetics via the rule-based modelling language BioNetGen (Harris et al., 2016) , and analysed using MATLAB 2016b lsqcurvefit unless otherwise specified. The ABC-SMC simulations were carried out in MATLAB 2016b to determine a unique set of parameters that can explain the SPR data of ssDNA (Fig. S2A, B) and of dsDNA (Fig. S3A, B). The ABC-SMC simulation for the ssDNA was unable to determine a unique value for k ad.r and k ad.f , but explained the data when either higher or lower values are concurrently applied to both k ad.r and k ad.f . This result is due to the fact that only a lower bound can be determined for the adsorption dissociation constant (K ad = k ad.r / k ad.f ), as RAD51 binding to the ssDNA dN-5, dN-8 or dN-11 was undetectable. Nonetheless, the accurate fit for the ssDNA dN-50 (R 2 = 0.95) suggests that the model captures the essential features of RAD51 polymerisation on ssDNA. The ABC-SMC simulation for the dsDNA model suggests all four model parameters are well defined, as the k ad.f and k ad.r can be fitted directly based on the dsDNA dN-8p data. All scripts are freely available at: https://www.dropbox.com/s/xh1z2h7eo9rmybw/RAD51%20Polymerisation%20Models.zip?dl=0

Statistical Analysis
Unpaired, two-tailed t-tests were carried out to test for significant differences between the different fitted RAD51 elongation rates and lifetimes (Fig. 3D, E, 4D and E). All t-tests were carried out using GraphPad Prism 7.      Fig. 1C and 2C, and k el.f was fitted using lsqcurvefit (MATLAB). In panel E, all parameters except k s and k el.f were fixed to the mean values as identified in Fig. 1C and 2C, and k s and k el.f were fitted using lsqcurvefit (MATLAB). Mean ± 1 SD of 3 ODE model fits (n=3). Asterisks indicate unpaired, twotailed t-tests: * p < 0.05; ** p < 0.01. ns = non-significant. . Asterisks indicate unpaired, two-tailed t-tests: * p < 0.05; ** p < 0.01, *** p < 0.001. ns = non-significant. Table 1. Persistence lengths (L p ) ± 95% confidence intervals are estimated from fitted Kuhn lengths for the ssDNA dN-50, dA-50, dT-50 and the dsDNA dN-50p SAXS plots. All data manipulation and model fitting were done using SasView. It is likely that the relatively low dsDNA dN-50p L p of 7.34 nm compared to the average dsDNAp L p of 30 -55 nm (Baumann et al., 1997;Brunet et al., 2015) is due to the low GC content of the dN-50p (36%).

Figure S1. Experimental setup.
A. Depiction of the experimental setting. A biotinylated DNA oligo is immobilised onto a surface plasmon resonance (SPR) CM5 chip via biotin-streptavidin interaction. RAD51 protein is injected over the DNA-coated SPR matrix to measure polymerisation kinetics. Throughout this period, association and dissociation of RAD51 take place simultaneously. Following protein injection (stop), running buffer is injected to measure dissociation kinetics. B. WT RAD51 SPR curves for ssDNA dN-50 and dsDNA dN-50p. RAD51 was injected at 150 nM in the presence of 2.5 mM ATP (pH 7.5) and 10 mM CaCl 2 . Curves were normalised to the R max , whereby the normalised response value of X (vertical axis) indicates the average number of RAD51 molecule bound to a single DNA oligo.

Figure S2. Evaluation of models describing RAD51 binding to ssDNA and dsDNA
A. ABC-SMC simulation for the ssDNA model. Heat maps indicate particle frequency and describe pair-wise correlations between model parameters (k ad.f , k ad.r , k el.f , k el.r , and k s ). Each vertical and horizontal axis of heat maps is labelled with the colour of the corresponding parameter (k ad.fpurple, k ad.r -brown, k el.f -orange, k el.r -black, k s -green). A high particle frequency corresponds to a good model fit to the data for the specified parameter pair. k ad.r and k ad.f are correlated. The probability densities for individual parameters are presented along the top left diagonal and were estimated using a Gaussian kernel. Particles were initialised via uniform priors with the following lower and upper bounds: 0.0001 < k ad.f < 0.01, 0.001 < k ad.r < 20, 1 < k el.f < 1000, 0.0001 < k el.r < 0.01, 0.00001 < k s < 0.005. B. Adaptive threshold for the ABC-SMC simulation. The plateau at 10 iterations means that the model has reached a best fit to the data. The simulation was carried out using in-house MATLAB ABC-SMC scripts with 10 iterations and 2000 particles. C. ssDNA_dN-X data fit as in Fig. 1B, except assuming that RAD51 polymerisation occurs only via the addition of RAD51 monomers. D. ssDNA_dN-X data fit as in Fig. 1B, except in the absence of the elongation step (k el.r = k s ). In panels C and D, parameter values are derived using the best fit ABC-SMC particles (described in Fig. 1). E. Sum-of-squares residuals (SSRs) for the three ssDNA models, described in Fig. 1C, S2C and S2D. The chosen model (shown in Fig. 1C), circled in red, best describes the experimental data. F. Experimental data of ssDNA dN-50 from Fig. 1 (dotted lines) and predicted polymers with defined length on each single DNA oligo using the model from Fig. 1 (solid lines). Predicted polymer curves were calculated by multiplying the concentration of a specific polymer by its length (i.e. [n-mer](t) * n). G and H. As in F, except ssDNA dN-17 (G) and ssDNA dN-14 (H) experimental data and predicted polymers are shown.

Figure S3. Evaluation of models describing RAD51 binding to dsDNA
A. ABC-SMC simulation for the dsDNA model. Heat maps indicate particle frequency and describe pair-wise correlations between model parameters (k ad.f , k ad.r , k el.f , and k s ). Each vertical and horizontal axis of heat maps is labelled with the colour of the corresponding parameter (k ad.fpurple, k ad.r -brown, k el.f -orange, k s -green). A high particle frequency corresponds to a good model fit to the data for the specified parameter pair. k ad.f , k ad.r , k el.f , and k s are well determined. The probability densities for individual parameters are presented along the top left diagonal and were estimated using a Gaussian kernel. Particles were initialised via uniform priors with the following lower and upper bounds: 0.0001 < k ad.f < 0.01, 0.001 < k ad.r < 20, 0.1 < k el.f < 10, 0.00001 < k s < 0.001. B. Adaptive threshold for the ABC-SMC simulation. The plateau at 10 iterations means the model has reached a best fit to the data. C. dsDNA_dN-Xp data fit as in Fig. 2, except assuming polymerisation only via RAD51 monomer addition. D. dsDNA_dN-Xp data fit as in Fig. 2, except in the presence of the elongation step (i.e. k el.r ≠ k s ). In panel F and G, parameter values are derived using the best fit ABC-SMC particles (described in Fig. 2). E. Sum-of-squares residuals (SSRs) for the three dsDNA models, described in Fig