Alain Dautant
email: a.dautant@ibgc.cnrs.fr.

Philippe Meyer

Florian Georgescauld
email: georgescauld@ibpc.fr

Hydrogen/Deuterium Exchange Mass Spectrometry reveals Mechanistic Details of Nucleoside Diphosphate Kinases Activation by Oligomerization

Most oligomeric proteins become active only after assembly, but why oligomerization is required to support function is not well understood. Here, we address this question using the WT and a destabilized mutant (D93N) of the hexameric nucleoside diphosphate kinase from the pathogen Mycobacterium tuberculosis (Mt-NDPK). The conformational dynamics and oligomeric states of each were analyzed during unfolding/folding by Hydrogen/Deuterium exchange mass spectrometry (HDX-MS) at peptide resolution and by additional biochemical techniques. We found that WT and D93N native hexamers present a stable core and a flexible periphery, the latter being more flexible for the destabilized mutant. Stable but inactive species formed during unfolding of D93N and folding of WT were characterized. For the first time, we show that both of these species are native-like dimers, each of its monomers having a major subdomain folded, while a minor subdomain (Kpn/α 0 ) remains unfolded. The Kpn/α 0 subdomain, which belongs to the catalytic site, becomes structured only upon hexamerization, explaining why oligomerization is required for NDPK activity. Further HDX-MS studies are necessary to establish the general activation mechanism for other homo-oligomers.

INTRODUCTION

Depending on the organism, 40 to 60% of proteins assemble and become functional after forming oligomeric complexes. [START_REF] Nishi | Evolutionary, physicochemical, and functional mechanisms of protein homooligomerization[END_REF][START_REF] Marsh | Structure, dynamics, assembly, and evolution of protein complexes[END_REF] These proteins must, within a well determined tridimensional complex structure, subtly balance thermodynamic stability with conformational flexibility. [START_REF] Karshikoff | Rigidity versus flexibility: The dilemma of understanding protein thermal stability[END_REF][START_REF] Teilum | Protein stability, flexibility and function[END_REF] Thermodynamic stability is required to maintain a protein's activity throughout its lifetime while preventing proteolysis or aggregation. In parallel, flexibility is necessary for newly synthesized chains to acquire the native conformation, to assemble and afterwards for the protein to perform its function. How oligomerization induces a protein's "functionality" and how a protein's function, thermodynamic stability and conformational flexibility relate to each other are poorly understood, essentially due to lack of experimental information on the dynamic aspects. Since a single analytical method cannot provide information on a large panel of the physico-chemical properties of proteins, multi-technique approaches and development of new time and space resolved techniques appear essential in this understanding. [START_REF] Engen | Analysis of protein conformation and dynamics by hydrogen/deuterium exchange ms[END_REF][START_REF] Lossl | The diverse and expanding role of mass spectrometry in structural and molecular biology[END_REF][START_REF] Marcoux | Towards integrative structural mass spectrometry: Benefits from hybrid approaches[END_REF][START_REF] Mehmood | Mass spectrometry of protein complexes: From origins to applications[END_REF] Hydrogen/Deuterium exchange detected by mass spectrometry (HDX-MS) is a methodology applied to proteins in recent years for investigating their properties, at peptide resolution, such as dynamics, conformational changes, post-translational modifications, viral capsid assembly, spontaneous and assisted folding, and interactions with partners or ligands. [START_REF] Wales | Hydrogen exchange mass spectrometry for the analysis of protein dynamics[END_REF][START_REF] Engen | Analytical aspects of hydrogen exchange mass spectrometry[END_REF][START_REF] Pirrone | Applications of hydrogen/deuterium exchange ms from 2012 to 2014[END_REF][START_REF] Konermann | Mass spectrometry methods for studying structure and dynamics of biological macromolecules[END_REF][START_REF] Guttman | Isotope labeling of biomolecules: Structural analysis of viruses by hdx-ms[END_REF] The method is applied to proteins in solutions mimicking their physiological environment and relies on the natural property that proteins rapidly exchange their solvent accessible backbone amide hydrogens that are not involved in stable H-bonds. Exposure of the protein of interest into a D 2 O-containing solvent environment induces the replacement of labile accessible backbone hydrogen ( 1 H, H) atoms with deuterium ( 2 H, D). The exchange is measured by MS after protein digestion and interpreted in the view of the structure. The advantages of HDX-MS are that it requires only a small amount of material (in the pmol range), that it has great resolution and sensitivity, that it allows for an accessible mass range of up to 10 6 Da, that it allows high speed analysis and that it is a nonensemble technique; each single spectrum contains extractible information about all discrete coexisting states of the protein of interest. By massively increasing its interest in protein therapeutics within these last years, the biopharmaceutical industry became particularly interested in improving and developing spatially resolved analytical techniques like HDX-MS for its needs in both research and quality control. [START_REF] Marciano | Hdx-ms guided drug discovery: Small molecules and biopharmaceuticals[END_REF][START_REF] Leurs | Getting to the core of protein pharmaceuticals--comprehensive structure analysis by mass spectrometry[END_REF] Since so far HDX-MS was mostly implemented on monomeric proteins (16- 22) and only a few studies have been conducted on homo-oligomeric proteins [START_REF] Georgescauld | Groel/es chaperonin modulates the mechanism and accelerates the rate of tim-barrel domain folding[END_REF][START_REF] Singh | Structural effects of multiple pathogenic mutations suggest a model for the initiation of misfolding of the prion protein[END_REF][START_REF] Huang | Hydrogen/deuterium exchange and electron-transfer dissociation mass spectrometry determine the interface and dynamics of apolipoprotein e oligomerization[END_REF][START_REF] Wang | Continuous and pulsed hydrogen-deuterium exchange and mass spectrometry characterize csge oligomerization[END_REF][START_REF] Monroe | Hydrogen/deuterium exchange analysis of hiv-1 capsid assembly and maturation[END_REF] or protein complexes, [START_REF] Faleri | Two cross-reactive monoclonal antibodies recognize overlapping epitopes on neisseria meningitidis factor h binding protein but have different functional properties[END_REF][START_REF] Malito | Defining a protective epitope on factor h binding protein, a key meningococcal virulence factor and vaccine antigen[END_REF][START_REF] Shi | Rpn1 provides adjacent receptor sites for substrate binding and deubiquitination by the proteasome[END_REF][START_REF] Veesler | Architecture of a dsdna viral capsid in complex with its maturation protease[END_REF] efforts have to be continued for extracting the dynamic determinants of complexes.

Nucleoside diphosphate kinase (NDPK) is a multifunctional protein which catalyzes the gammaphosphate transfer from nucleoside triphosphate to nucleoside diphosphate [START_REF] Lascu | The catalytic mechanism of nucleoside diphosphate kinases[END_REF] in order to maintain equilibrium between the pools of nucleoside triphosphates (NDP/NTP). Several isoforms are found in eukaryotes, [START_REF] Boissan | Learning about the functions of nme/nm23: Lessons from knockout mice to silencing strategies[END_REF] isoform 1 (called nm23A) being the first metastasis suppressor described in humans. [START_REF] Steeg | A critical evaluation of biochemical activities reported for the nucleoside diphosphate kinase/nm23/awd family proteins: Opportunities and missteps in understanding their biological functions[END_REF][START_REF] Takacs-Vellai | The metastasis suppressor nm23 as a modulator of ras/erk signaling[END_REF][START_REF] Marino | Insights into the biology and prevention of tumor metastasis provided by the nm23 metastasis suppressor gene[END_REF] Additionally, NDPK also acts as a protein histidine kinase, [START_REF] Attwood | Histidine kinases from bacteria to humans[END_REF][START_REF] Attwood | Nucleoside diphosphate kinase as protein histidine kinase. Naunyn-Schmiedeberg's[END_REF][START_REF] Srivastava | Histidine phosphorylation relieves copper inhibition in the mammalian potassium channel kca3.1[END_REF] and recent studies have shown that by interacting with and providing GTP to dynamins, NDPK is involved in membrane-remodeling events. [START_REF] Boissan | Membrane trafficking. Nucleoside diphosphate kinases fuel dynamin superfamily proteins with gtp for membrane remodeling[END_REF][START_REF] Schlattner | Mitochondrial nm23-h4/ndpk-d: A bifunctional nanoswitch for bioenergetics and lipid signaling[END_REF] Like many enzymes, NDPKs are non-allosteric oligomers, but in vitro and in vivo studies suggest that their assembly is required for kinase activity or metastasis suppression function. [START_REF] Lascu | Quaternary structure of nucleoside diphosphate kinases[END_REF][START_REF] Kim | Point mutations affecting the oligomeric structure of nm23-h1 abrogates its inhibitory activity on colonization and invasion of prostate cancer cells[END_REF][START_REF] Mocan | Protein phosphorylation corrects the folding defect of the neuroblastoma (s120g) mutant of human nucleoside diphosphate kinase a/nm23-h1[END_REF] Many aspects of the kinase activity mechanism are now understood from a structural point of view, [START_REF] Lascu | The catalytic mechanism of nucleoside diphosphate kinases[END_REF] namely, the role of amino acids involved in the catalysis, but the requirement for assembly to become functional is still unclear and remains to be elucidated. [START_REF] Lascu | Quaternary structure of nucleoside diphosphate kinases[END_REF][START_REF] Kim | Point mutations affecting the oligomeric structure of nm23-h1 abrogates its inhibitory activity on colonization and invasion of prostate cancer cells[END_REF] NDPKs contain ferredoxin-like folds [START_REF] Janin | Three-dimensional structure of nucleoside diphosphate kinase[END_REF] which assemble first as dimers and then as tetramers or hexamers. [START_REF] Moynie | The structure of the escherichia coli nucleoside diphosphate kinase reveals a new quaternary architecture for this enzyme family[END_REF][START_REF] Tokunaga | Residue 134 determines the dimer-tetramer assembly of nucleoside diphosphate kinase from moderately halophilic bacteria[END_REF][START_REF] Yonezawa | Dimeric structure of nucleoside diphosphate kinase from moderately halophilic bacterium: Contrast to the tetrameric pseudomonas counterpart[END_REF] Tetramers are present in gram-negative bacteria and archaea, while hexamers are present in gram-positive bacteria and eukaryotes. Hexamers represent the most abundant type of oligomerization, which specifically relies on the well conserved surface loop that carries the site of the natural "Killer of prune" mutation (Kpn-loop) (Figures 1,S1). In addition, the stabilization of hexamers by subunit interactions has been studied. [START_REF] Chen | X-ray structure of mycobacterium tuberculosis nucleoside diphosphate kinase[END_REF][START_REF] Karlsson | Nucleoside diphosphate kinase. Investigation of the intersubunit contacts by site-directed mutagenesis and crystallography[END_REF][START_REF] Mesnildrey | Coupling between catalysis and oligomeric structure in nucleoside diphosphate kinase[END_REF][START_REF] Giartosio | Thermal stability of hexameric and tetrameric nucleoside diphosphate kinases. Effect of subunit interaction[END_REF][START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] Crystallographic structures of several-NDPK mutants provide detailed understanding of the WT stabilization mechanism, [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF][START_REF] Georgescauld | Structure of mycobacterium tuberculosis nucleoside diphosphate kinase r80n mutant in complex with citrate[END_REF] but provide little understanding as to why oligomerization is necessary for function. So, we realized that by combining stability aspects with the study of their dynamic properties, we would improve this understanding.

FIGURE 1

Mycobacterium tuberculosis (Mt), the pathogenic bacteria responsible of tuberculosis, possesses only one NDPK isoform (Mt-NDPK), which is hexameric (Figures 1,S1). [START_REF] Chen | X-ray structure of mycobacterium tuberculosis nucleoside diphosphate kinase[END_REF] In addition to its kinase function, Mt-NDPK binds DNA [START_REF] Kumar | Nucleoside diphosphate kinase from mycobacterium tuberculosis cleaves single strand DNA within the human c-myc promoter in an enzyme-catalyzed reaction[END_REF] and also blocks phagosome maturation in murine macrophages leading to evasion of innate immunity. [START_REF] Sun | Mycobacterial nucleoside diphosphate kinase blocks phagosome maturation in murine raw 264.7 macrophages[END_REF] The escape mechanism is unknown, but efficient intracellular survival of Mt requires a strong thermodynamic stability of proteins. [START_REF] Ganaie | Thermostable hexameric form of eis (rv2416c) protein of m. Tuberculosis plays an important role for enhanced intracellular survival within macrophages[END_REF] We recently showed in vitro that isolated sub-hexameric assemblies of Mt-NDPK are inactive, complete assembly being essential for function. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] By combining crystallography and biochemistry, we also demonstrated that hexamer's high thermostability (76°C) is dependent on six inter-subunit R80-D93 salt bridges. [START_REF] Georgescauld | Structure of mycobacterium tuberculosis nucleoside diphosphate kinase r80n mutant in complex with citrate[END_REF] The point mutation D93N reduces the melting point by 28°C, without significant alteration of the hexameric 3D structure or enzymatic properties. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] The link between the protein's function, stability and oligomerization is still an open question.

Here, we show why the full oligomeric state is required for and affects protein function, as demonstrated by the study of WT and D93N Mt-NDPK. In addition to the classical biochemistry methods (fluorescence, enzymology, size exclusion chromatography), we use the HDX-MS technique to reveal the overall and local structure and dynamics of NDPK by monitoring amide hydrogen atoms in polypeptides. We show that both native hexamers are stable, without in/out subunit exchange, but that the D93N destabilizing mutation increases the flexibility at the hexamer surface. Inactive dimers which accumulate during folding of WT and D93N, as well as during D93N unfolding were analyzed. Strikingly, differences in conformational flexibility between hexamers and dimers explain why hexamerization is required for kinase activity.

MATERIALS AND METHODS

Reagents. Chemicals were of the highest purity grade from Sigma. Solutions of urea were freshly prepared for each experiment.

Proteins and enzymatic assays. WT and D93N Mt-NDPK were expressed without tag, using a pET24 vector (Novagen) in BL21-CodonPlus ® (DE3)-RIL (Stratagene) strain. Transformed cells were induced at 37°C with 1 mM IPTG for 6 h in the presence of 80 µg/mL kanamycin and the purification steps were carried out at 4°C as previously described. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] After sonication and centrifugation, the soluble fraction containing Mt-NDPK was DNase-treated, and the extract was applied to a Q-Sepharose column equilibrated with 100 mM Tris-HCl, pH 7.4. The enzyme was eluted with a linear gradient of 0 to 0.8 M NaCl in the same buffer. Active fractions were precipitated into 80% saturated ammonium sulfate (AS) and further purified by salting-out chromatography on a Sepharose 6B column equilibrated with 80% AS, 100 mM Tris-HCl, pH 7.4.

The protein was eluted by decreasing AS concentration from 80% to 20% in the same buffer. The active fractions were dialyzed against 100 mM Tris-HCl, pH 7.4, and further purified on a Source 15Q column, under the conditions described for the Q-Sepharose chromatography. The enzymes were precipitated by dialysis against a saturated solution of AS, recovered by centrifugation and further purified by size-exclusion chromatography on a Sephacryl S-200 column equilibrated with 0.2 M sodium phosphate buffer, pH 7.0. The final hexameric preparations were kept frozen in 50 mM Tris-HCl, pH 7.4 at -80°C.

The size-exclusion chromatography (SEC) was performed in 20 mM phosphate buffer, 150 mM NaCl using a Superdex 75 analytical column, in conditions described in the supplemental.

Molecular markers (cytochrome c, 12 400 Da; P105G Dd-NDPK, monomeric version from Dictyostelium discoideum, 16 800 Da; myoglobin, 17 000 Da; carbonic anhydrase, 29 000 Da; ovalbumin, 44 000 Da; BSA, 68 000 Da; aldolase, 158 000 Da) from GE Healthcare Life Sciences were used. The P105G Dd-NDPK, monomeric version from Dictyostelium discoideum, was a gift from Pr. Ioan Lascu and has an elution volume of 12.6 mL (data not shown).

NDPK enzymatic activity was measured using a coupled assay. [START_REF] Giartosio | Thermal stability of hexameric and tetrameric nucleoside diphosphate kinases. Effect of subunit interaction[END_REF][START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] The reactions of the test are shown below and the enzymes catalyzing each reaction are indicated in parentheses. In this assay, NDPK catalyzes phosphate transfer from ATP to 8-bromoinosine-5'-diphosphate (8-BrIDP).

The enzyme mix contained in a final volume of 0.8 mL: 20 mM Tris-HCl (pH 7.5), 5 mM MgCl 2 , 100 mM KCl, 1 mg/mL BSA, 1 mM de phosphoenol pyruvate, 0.1 mM NADH, 1 mM ATP, 0.2 mM 8-BrIDP, 2U/mL of pyruvate kinase and lactate dehydrogenase. We measured NADH disappearance at 340 nm and 25°C, using a Perkin-Elmer spectrophotometer. For such assay, the errors associated with the kinetic parameters are under 20%. [START_REF] Lascu | The catalytic mechanism of nucleoside diphosphate kinases[END_REF][START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] ATP + 8-BrIDP → ADP + 8-BrITP For full length experiments, protein samples were immediately injected into an Alltech analytical guard column, packed with POROS 20-R1 reversed-phase media (PerSeptive Biosystems). Each sample was washed within 20 s with 400 µL of 0.1% formic acid, ice cold. The guard column was then switched in line with the HPLC inject port with a flow rate of 200 µL/min of 15% acetonitrile/0.1% formic acid, pH 2.5 for 2 min. The protein was eluted at 40 µL/min using a 1.5 min gradient of 15%-75% acetonitrile directly into a Waters Synapt G1 mass spectrometer with a standard electrospray interface. The injector, column and all associated tubings were kept at 0°C using an ice bath to minimize back exchange. [START_REF] Wales | High-speed and highresolution uplc separation at zero degrees celsius[END_REF] All experiments were conducted under identical conditions to allow comparison of relative deuterium levels. [START_REF] Wales | Hydrogen exchange mass spectrometry for the analysis of protein dynamics[END_REF] Mass accuracy of less than 20 ppm was maintained by infusing horse heart myoglobin into the mass spectrometer at the end of each chromatographic gradient. Intact mass spectra were deconvoluted using MassLynx (Waters) and masses were corrected for back exchange (17%) and dilution effect (10%). As controls, unfolded and native Mt-NDPKs were subjected to D 2 O pulse labeling and LC-MS analysis. The backexchange of 17% was measured by diluting denatured Mt-NDPKs (6 M GuDCl, 1 mM DTT) into 20 mM Tris-HCl, 20 mM KCl buffer prepared with 99.9 D 2 O. After 10 s of deuteration, the reaction was quenched and immediately subjected to LC-MS at 0°C.

To obtain HDX data at peptide resolution, 200 µL of each acid-quenched sample were injected into a loop of 100 µL inside an HDX Waters nanoACQUITY Ultra Performance Liquid Chromatography (UPLC). [START_REF] Wales | High-speed and highresolution uplc separation at zero degrees celsius[END_REF] The sample passed through a Poroszyme-immobilized pepsin cartridge (Applied Biosystems) accommodated within the HDX manager at a flow rate of 100 µL min -1 and temperature of 20°C. Peptic peptides eluting from the pepsin column were trapped and desalted for 3 min at 100 µL min -1 , and then separated in 7 min with 8% to 40% acetonitrile:water gradient at 40 µL min -1 , all steps were carried out in 0.1% formic acid pH 2.5 before MS analysis.

The separation column was a 1.0×100.0 mm ACQUITY UPLC C18 BEH (Waters) containing 1.7 µm particles and the back pressure averaged 8000 psi. All chromatographic elements were held at 0°C for the entire time of the measurements. The average amount of back exchange was around 25% based on analysis of highly deuterated peptide standards. [START_REF] Wales | High-speed and highresolution uplc separation at zero degrees celsius[END_REF] Deuterium levels were corrected for 25% back-exchange and 10% dilution, all comparison experiments being done under identical experimental conditions. The UPLC steps were performed with protonated solvents, thereby allowing deuterium to be replaced with hydrogen from side chains and the amino/carboxyl terminus that exchange much faster than amide linkages. [START_REF] Engen | Analysis of protein conformation and dynamics by hydrogen/deuterium exchange ms[END_REF][START_REF] Wales | Hydrogen exchange mass spectrometry for the analysis of protein dynamics[END_REF] The error in the determination of the deuterium levels was ± 0.20 Da in this experimental setup consistent with previously obtained values. [START_REF] Iacob | Conformational disturbance in abl kinase upon mutation and deregulation[END_REF][START_REF] Georgescauld | Groel/es chaperonin modulates the mechanism and accelerates the rate of tim-barrel domain folding[END_REF][START_REF] Houde | The utility of hydrogen/deuterium exchange mass spectrometry in biopharmaceutical comparability studies[END_REF] Mass spectra were obtained with a Waters Synapt G1 using a standard ESI source (Waters Corp., Milford, MA, USA) over an m/z range of 50-1700. Mass accuracy was ensured by calibration with Glu-fibrinogen peptide, and was less than 10 ppm throughout all experiments. Identification of the peptic fragments was accomplished with at least 4 replicate MS E analyses [START_REF] Georgescauld | Groel/es chaperonin modulates the mechanism and accelerates the rate of tim-barrel domain folding[END_REF] using Identity Software (Waters Corp., Milford, MA, USA). MS E was performed by realizing series of low-high collision energies ramping from 5-30 V, therefore ensuring proper fragmentation of all the peptic peptides eluting from the LC system.

Pulse HDX. To characterize the unfolding pathway of D93N, the purified protein was first pre-incubated at 2 µM in 0-8 M urea, for 16 h at 10°C. After this step, the pulse labeling experiment was performed following the scheme (Figure S3C). 20 µL of each sample were deuterated for 12 s by dilution of the protein stock solution 10-fold into 20 mM Tris-HCl, 20 mM KCl, pD 7.5, 99.9% D 2 O, at 10°C. Deuteration was stopped by quenching the reaction with formic acid, final pH at 2.5, at 0°C. After the quenching step, samples were digested or not and analyzed as described in the previous paragraph concerning the continuous labeling.

The unfolding pathway of the WT was characterized in the same way as for the D93N one, except that we used a higher performance mass spectrometer (Synapt G2 Si, Waters) allowing a 9fold lower working protein concentration. Pre-incubation of WT was performed at about 220 nM final and the pulse labeling experiment followed the scheme (Figure S3C), all the other experimental aspects were identical to the ones explained for unfolding of D93N.

Spontaneous refolding of WT Mt-NDPK was initiated by 400-fold dilution from 7.2 M GuHCl into refolding buffer (20 mM Tris-HCl, pH 7.5, 100 mM KCl) at 10°C and final protein concentration of 220 nM. Residual GuHCl concentration was as low as 18 mM in order to not disturb the refolding process. The reaction scheme shown in Figure S4A was followed. Aliquots of refolded Mt-NDPK were deuterated for 12 seconds, quenched with formic acid at pH 2.5 and analyzed as described above for the continuous labeling experiments. A Synapt G2 Si mass spectrometer was used for these experiments. The unfolded references were arbitrarily given a refolding time of 3 s in these graphs.

Peptide Mass Analysis and Data Processing. Mass spectra were processed using DynamX 2.0 software (Waters) by centroiding the isotopic distribution corresponding to all charge states of each peptide. Automated selection of the isotope distribution was verified manually for all peptides and all charge states. The resulting relative deuterium levels (corrected for 25% back exchange and 10% dilution factor) were plotted versus the exchange-in time with respect to experimental described conditions. All comparison experiments were done under identical experimental conditions. NDPKs unfolded during 1 h in GuDCl 6 M were used as controls for 100% deuterated protein. The error of determining the deuterium levels was ± 0.20 Da for the experimental setup used. [START_REF] Iacob | Conformational disturbance in abl kinase upon mutation and deregulation[END_REF][START_REF] Georgescauld | Groel/es chaperonin modulates the mechanism and accelerates the rate of tim-barrel domain folding[END_REF][START_REF] Houde | The utility of hydrogen/deuterium exchange mass spectrometry in biopharmaceutical comparability studies[END_REF] For the pulse labeling experiments, when the isotope distributions were bimodal, all isotopes of the entire bimodal pattern were selected for processing. [START_REF] Georgescauld | Groel/es chaperonin modulates the mechanism and accelerates the rate of tim-barrel domain folding[END_REF] The relative deuterium incorporation was calculated by subtracting the centroid of the isotopic distribution for peptide ions of the native reference from the centroid of the isotopic distribution for peptide ions from each refolding sample (Figure S4B). All molecular structure figures were prepared using PyMol, [START_REF] Schrodinger | The pymol molecular graphics system[END_REF] and the alignment figure using ESPript (http://espript.ibcp.fr). [START_REF] Robert | Deciphering key features in protein structures with the new endscript server[END_REF] RESULTS AND DISCUSSION

Impact of destabilizing D93N mutation on Mt-NDPK hexamer dynamics

Here we perform continuous labeling HDX coupled to liquid chromatography (LC) and mass theoretically accessible H's, show that more than half of each chain is solvent accessible. At each incubation time, WT incorporates between 5 and 11 D's less than D93N indicating a higher overall rigidity for WT. The WT is thermodynamically more stable than the D93N mutant, clearly supporting the view that increased stability reduces flexibility. [START_REF] Karshikoff | Rigidity versus flexibility: The dilemma of understanding protein thermal stability[END_REF] To gain spatial information, we next performed continuous labeling HDX-MS at peptide level resolution by adding a pepsin digestion step before LC-MS (Figure S2A). 27 identical polypeptides covering 95.6% of the sequence were analyzed in both proteins (Figure S2B). 17 peptides had identical or almost identical behavior while 10 overlapping peptides (gray shaded panels) covering 27% of the total sequence, start and end with different exchange levels of at least 0.50 Da, representing more than 15% when normalized to the maximal incorporation value (Figure S2C).

The highest flexibility is noticed in both hexamers for surface regions 39-62 (hairpin α A /α 2 ), 108-113 (helix α3'), and 132-136 (C-term end), which interestingly correspond to the most divergent sequences in NDPKs (Figures 2C-D, S2D). For the C-terminal end, the deuteration is already maximal at the first experimental time point after 10 s, as indicated by the corresponding flat plots.

Because of its biological relevance, original search for flexible regions in NDPKs started two decades ago using 1 H NMR. [START_REF] Xu | Nmr studies on the flexibility of nucleoside diphosphate kinase[END_REF] Here, our HDX-MS results show that as previously suggested, [START_REF] Karlsson | Nucleoside diphosphate kinase. Investigation of the intersubunit contacts by site-directed mutagenesis and crystallography[END_REF][START_REF] Xu | Nmr studies on the flexibility of nucleoside diphosphate kinase[END_REF] the hairpin α A /α 2 which is a part of the nucleotide binding site, is highly flexible and could undergo large movement in order to accommodate DNA, explaining the DNA binding function. [START_REF] Kumar | Nucleoside diphosphate kinase from mycobacterium tuberculosis cleaves single strand DNA within the human c-myc promoter in an enzyme-catalyzed reaction[END_REF] Both WT and D93N hexamers present with full deuteration after 10 min, with three quarters of their sequence being already deuterated at 10 s. The regions specifically affected by the D93N mutation are contiguously located at the hexamer surface (9-15: helix α 0 ) and at the trimer interface (85-113: 2C-D, left columns showing the surface). The region 108-113 is fully solvent accessible in the mutant, while in WT it becomes fully deuterated after 3 min exposure to solvent.

Kpn-loop) (Figures
The peptides covering region 114-130 present about 1 Da difference in deuterium incorporation, which corresponds to less than 15% difference indicating that mutation did not substantially affect this part of the structure. The strand β 2 (32-38) and the helix α 1 (18-27), which form the dimer interface, remain inaccessible to solvent indicating high stability for dimers (Figure 2D, right column showing the interface). In conclusion, differences in dynamics for D93N and WT Mt-NDPK hexamers correlate only with their difference in stability, while their 3D structures are very similar with an RMSD of 0.23 Å. Importantly, the decrease of thermostability of D93N is accompanied by the increase of the conformational flexibility at the hexamer surface, without affecting the stable core of the protein (Figure S2D).

D93N and WT unfolding pathways comparison

The next questions we addressed were whether the conformational flexibility of NDPK varies with the oligomeric state and whether this potential variation is the reason why oligomerization is required to support function. To find sub-hexameric species, unfolding and refolding pathways of WT and D93N were analyzed in the presence of urea. First, enzyme activity was measured after incubation of native and unfolded NDPK for 16 h in 0 to 7.2 M urea (Figure S3A). In both cases a hysteresis phenomenon was observed, since unfolding and refolding curves did not overlap, indicating that WT and D93N proteins were not at equilibrium. Second, the hydrodynamic radii of NDPK were measured by SEC in 0 to 7.2 M urea, after the same unfolding/refolding procedure (Figure S3B). Again, a hysteresis phenomenon was observed, since for the same urea concentration, the elution volume of NDPK differed between folding and unfolding. Taken together, these results

show an absence of equilibrium for WT and D93N during unfolding/folding experiments. They are in full agreement with our previous data on Mt-NDPK as well as with other similar studies on NDPK which show hysteresis after about 1 day of unfolding/refolding in presence of denaturing agents. [START_REF] Lascu | Quaternary structure of nucleoside diphosphate kinases[END_REF][START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] Having established that unfolding and refolding pathways were different for both WT and D93N

NDPKs, we next characterized the potential sub-hexameric species formed during unfolding. WT hexamers were incubated in 0 to 7.2 M urea during 16 hours and the oligomeric state was established by SEC for the different urea concentrations (Figure S3B). Two elution volumes of 9.65 and 10.2 mL were noticed, indicating respectively a direct passage from native hexamers (0 to 4.0 M urea) to unfolded monomers (>4 M urea). The WT unfolding pathway was next characterized by enzyme activity, which reflects integrality of quaternary structure and by the fluorescence of the unique tryptophan (W132), which reflects integrity of tertiary structure. Both curves overlapped, indicating again concomitant dissociation and unfolding (Figure 3B).

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Biochemistry

FIGURE 3

To obtain mechanistic details, pulse labeling HDX-MS at the peptide level was performed, following the scheme described in (Figure S3C). The level of deuterium which is detected in the peptides is a signature of the NDPK conformations during unfolding, providing a snapshot of the different coexisting populations. The 31 peptides analyzed, covering 98.5% of the sequence (Figure Confirmation of this unfolding pathway was obtained using two additional methods: enzyme activity is lost for urea concentrations higher that 0.5 M (Figure 3C, empty red squares), while from

Trp fluorescence, only one transition between native and unfolded species was observed around 2.7

M urea (Figure 3C, empty blue circles). Taken together, these results indicate that D93N hexamers disassemble into dimers before further concomitant dissociation and unfolding into monomers.

Pulse labeling HDX-MS at full-length protein level was performed to further characterization. In the absence of urea, hexameric full length D93N incorporates 36 D, while between 0.5 and 2.0 M urea, dimers incorporate additional 19 D (Figure 3C, green diamonds). From 3 to 8 M urea, progressive incorporation from 111 to 125 D over the 129 available amide protons reveals monomers with a slightly different level of unfolding, in agreement with SEC, fluorescence and activity data (Figure 3C).

To structurally determine why such dimers are enzymatically inactive, the D93N unfolding pathway was characterized by pulse labeling HDX-MS at peptide level, as described in the scheme of Figure S3C. 30 peptides covering 100% of the sequence (Figure S2C) were analyzed (Figures 3D, S3E-F). All spectra show bimodal isotope distribution, which corresponds to native or fully unfolded peptide, except peptides 108-113 and 132-136, which are permanently unfolded. Bimodal peptides had transitions between their folded and unfolded states at the same urea concentrations as the ones noticed for the loss of activity (0.5 M) or tertiary structure (2.7 M). The specific behavior of these two categories of peptide is illustrated by their mass spectra of peptides 9-21 and 25-35, respectively (Figure 4, right panels). Remarkably, at 0.5 M urea, the loss of enzyme activity is due to unfolding of the regions 85-107 (Kpn-loop) at the trimer interface and 9-16 (helix α 0 ). These regions are far in the sequence but close in the structure and constitute a separate subdomain of the rest of the monomer, further called Kpn/α 0 subdomain. The rest of the sequence including the dimer interface (region 17-40: helix α 1 , and strand β 2 ) is not altered up to urea concentrations higher than 2.7 M, when the dimers simultaneously dissociate and unfold. Taken together, these data show that the D93N inactive intermediate species correspond to native-like folded dimers, instead of monomers, as previously proposed. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] These dimers are well folded except for the Kpn/α 0 subdomain which is fully solvent accessible and by consequence lacks stable structure.

WT and D93N hexamers exhibit similar enzyme properties but different thermodynamic stabilities, which are responsible for their different unfolding pathways (Figure S3G). WT hexamers dissociate and unfold concomitantly indicating high cooperativity between all six subunits. D93N hexamers first dissociate into dimers before further concomitant dissociation and unfolding, indicating a partial loss of cooperativity between subunits, and so demonstrate the important role played by ionic bonds for cooperativity inside complexes. D93N dimers are inactive and much less stable compared to WT-hexamers (Figures 3A and3C, blue curves). Importantly, the lack of activity is only due to the lack of stable structure of the Kpn/α 0 subdomain, indicating that even a single-domain protein can have its dynamics uncoupled between its structural subdomains.

Characterization of refolding and assembly pathway of WT Mt-NDPK

Stable but inactive species accumulate for WT Mt-NDPK only during refolding at low protein concentration (Figure 5A, blue circles), [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] and their characterization by size exclusion chromatography indicates they are dimers (Figure S3B). diluted 400-fold into refolding buffer to a final concentration of 220 nM for 0-3 h. Aliquots were deuterated, quenched and digested, and the peptide masses were immediately measured (Figure S4A). It is important to note that the residual 18 mM GuHCl in the refolding buffer is insufficient to unfold any peptide and did not affect the folding/assembly pathway. Strikingly, after 10-30 seconds refolding, all peptides became native, except for the previously described Kpn/α 0 subdomain (84-107 and 9-21), which stays fully unfolded during at least 3 h (Figure S4B). The weak hydrophobicity of the Kpn/α 0 subdomain, when compared to the rest of the NDPK sequence, is probably not sufficient to allow its burying/folding by hydrophobic collapse. We conclude that, in agreement with stopped-flow data, [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] WT monomers are rapidly folded and assembled into inactive dimers (Figures 5B,S4C) similar to the ones described for D93N unfolding. WT and D93N dimers are native-like except for the Kpn/α 0 subdomain, which gets buried and structured only during assembly into hexamers, inducing enzyme activation.

During two decades and based on spectroscopic methods, it has been proposed that structurally single-domain proteins -like NDPKs -present only two conformations (folded or unfolded), the interconversion between them being performed within one single step, without formation of stable intermediates. [START_REF] Englander | Protein folding-how and why: By hydrogen exchange, fragment separation, and mass spectrometry[END_REF] Here we show that such methods are insufficient to fully characterize properties of folding/unfolding intermediates, mostly when such intermediates present many but not all physical properties the native protein possess. We propose that as for a few proteins, [START_REF] Hu | Cytochrome c folds through foldon-dependent native-like intermediates in an ordered pathway[END_REF] Mt-NDPK folding/unfolding appears by blocks or subdomains, which are called foldons in the literature. [START_REF] Englander | Protein folding-how and why: By hydrogen exchange, fragment separation, and mass spectrometry[END_REF] Clearly in our study the Kpn/α 0 subdomain is indeed such a foldon. We could demonstrate that the Kpn/α 0 foldon is unstructured in monomers/dimers and acquires stable conformation only upon full hexameric assembly. Since the Kpn/α 0 foldon is part of the catalytic site, this explains why full hexamerisation is required for enzyme function acquisition. Finally, our study demonstrates that during the phenomena of folding/unfolding of NDPK, the Kpn/α 0 foldon is stabilized/destabilized in a cooperative sequential order.

CONCLUSIONS

In this study we have applied a combination of biochemical and biophysical methods to Mt-NDPK including the emerging technique of structural mass spectrometry to investigate the link between a protein's function and several of its physical parameters: conformational dynamics, stability and oligomerization. Even though NDPK research today comprises more than 2500 publications, the HDX-MS methodology has only marginally been applied to study this important class of proteins. [START_REF] Kim | Structure of nm23-h1 under oxidative conditions[END_REF] Here, for the first time, its application provides an explanation of why oligomerization is required for the kinase function. We show that in order to be active, hexameric NDPK requires a structured Kpn/α 0 foldon, while to be structured the foldon requires the assembly of the NDPK. NDPKs are oligomeric enzymes devoid of allosteric regulation. Many structures of NDPKs have been solved in the presence of different ligands (ADP, non hydrolysable ATP analogues, small molecules, …), but the apo and in complex structures are identical and do not show conformational changes. Here, we show that the Kpn/α 0 foldon is an activation region and a critical control switch, through its shifting between the structured or unstructured conformation.

Since not all NDPKs are hexameric, the question arises as to how bacterial tetrameric and dimeric ones are active. The analysis of crystal structures shows that in the type 2 tetramer, [START_REF] Moynie | The structure of the escherichia coli nucleoside diphosphate kinase reveals a new quaternary architecture for this enzyme family[END_REF] the Kpn/α 0 foldon is located at the dimer-dimer interface and has the same control switch role. On the contrary, in type 1 tetramer [START_REF] Boissier | An intersubunit disulfide bridge stabilizes the tetrameric nucleoside diphosphate kinase of aquifex aeolicus[END_REF] and dimer, [START_REF] Tokunaga | Residue 134 determines the dimer-tetramer assembly of nucleoside diphosphate kinase from moderately halophilic bacteria[END_REF][START_REF] Arai | A structural mechanism for dimeric to tetrameric oligomer conversion in halomonas sp. Nucleoside diphosphate kinase[END_REF][START_REF] Yonezawa | Nucleoside diphosphate kinase from psychrophilic pseudoalteromonas sp. As-131 isolated from antarctic ocean[END_REF] the Kpn/α 0 foldon is fully exposed and could not have such a regulatory role. High eukaryotes have several isoforms of NDPK with additional functions: metastasis suppression in humans [START_REF] Takacs-Vellai | The metastasis suppressor nm23 as a modulator of ras/erk signaling[END_REF][START_REF] Marino | Insights into the biology and prevention of tumor metastasis provided by the nm23 metastasis suppressor gene[END_REF][START_REF] Rosengard | Reduced nm23/awd protein in tumour metastasis and aberrant drosophila development[END_REF][START_REF] Giraud | Crystal structures of s120g mutant and wild type of human nucleoside diphosphate kinase a in complex with adp[END_REF] and involvement in animal development. (71- 73) Interestingly, natural mutants for these two hexameric NDPKs present folding or assembly defects. [START_REF] Lascu | A pro/ser substitution in nucleoside diphosphate kinase of drosophila melanogaster (mutation killer of prune) affects stability but not catalytic efficiency of the enzyme[END_REF] Since our present study on Mt-NDPK shows that flexibility is increased by the hexamer's destabilization, we suggest these mutants are inactive because of aberrant conformational dynamics. Potentially, their stabilization could be enhanced by rational design of their most dynamic parts, a promising but actually unexplored strategy in cancer or developmental research. In conclusion, our results illustrate the power of MS-based technologies to uncover new concepts in studying structure-activity relationships, in complement to biochemical techniques, X-ray crystallography and NMR.

ASSOCIATED INFORMATION

Sequence of Mt-NDPK (136 aa) with the secondary structure elements, the Kpn-loop, the active site residues, the inter-subunit salt bridge, the solvent accessibility of each residue (Figure S1A), scheme of continuous labeling HDX-MS experiment at peptide level (Figure S2A), peptide maps (Fig S2B ), kinetics of deuterium incorporation for peptic peptides of WT and D93N (Figure S2C), local dynamics of WT and D93N (Fig S2D ), unfolding and refolding of WT and D93N followed by activity (Figure S3A) and size-exclusion chromatography (Figure S3B), scheme of the pulse labeling HDX-MS experiment for the D93N and WT after incubation in urea (Figure S3C), deuterium incorporation for peptic peptides of WT (Figure S3D) and D93N (Figure S3E), pulse labeling HDX data for D93N in 0 and 1 M urea mapped onto the WT structure (Figure S3F), pulsed labeling HDX-MS data for WT and D93N in 0-6 M urea were mapped onto the monomer WT structure (Figure S3G), scheme of the refolding/assembly experiment followed by pulsed labeling HDX-MS at peptide level (Figure S4A), kinetics of deuterium incorporation for peptic peptides of WT (Figure S4B), pulsed labeling HDX-MS data for refolding and assembly pathway of WT mapped onto the monomer WT structure. Unfolding was monitored by intrinsic fluorescence (blue circles), residual enzymatic activity (red squares) [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] (A, C). The values are normalized to the experiment in the absence of urea. D93N unfolding was also monitored by a 12 sec pulse labeling deuterium incorporation at full-length level (green diamond) (C). The relative percentage of deuterium incorporation measured at peptide level with a 12 sec D 2 O pulse for WT in 0, 3, 4, and 5 M urea (B) and D93N in 0, 1, 2 and 3 M urea (D) was mapped onto the crystal structure of Mt-NDPK using a blue (0%) to red (100%) scale (Figure S3). A top view (left) and a split dimer interface (right) were shown. When the hexamer was dissociated or unfolded, it was split into three dimers or six randomly oriented monomers, respectively. The white contour on columns two and four outlines the fingerprint of the dimer interface. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] The lines do not represent theoretical models but were drawn to help the reader. (B) Refolding and assembly pathway of WT Mt-NDPK.

LEGENDS OF FIGURES

The percentage of deuterium incorporation after 12 seconds D 2 O pulse at different refolding time is plotted onto the structure (Figure S4B). Six monomers in random orientation are drawn when unfolded. For each time point, a top view (left) and a split dimer interface of three split dimers (right) are shown. Residues with no data are colored in black. Note for refolding experiments that at this low protein concentration, only dimers are formed. Hexamers are formed at ten time higher concentrations. [START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF]

  Unfolding and refolding. The unfolding/refolding experiments were performed as previously described.[START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] 10 µg/mL final concentration of native or unfolded Mt-NDPK were incubated for 16 h in 0-8 M urea or 0-5 M GuHCl, and 20 mM phosphate buffer, pH 7.0 at 25°C. Fluorescence intensities of the single tryptophan residue Trp132 were measured at 335 nm with an excitation at 295 nm. Continuous HDX. Experiments were performed at full length and peptide resolution, by following the schemes (Figures 2A and S2A). Stock solutions of 2.0 µM native Mt-NDPKs (WT and D93N) were prepared in 20 mM Tris-HCl pH 7.5, 100 mM KCl, H 2 O. Deuterium exchange was initiated by dilution of the Mt-NDPK stock solution 10-fold into 20 mM Tris-HCl, 20 mM KCl, pD 7.5, 99.9% D 2 O, 10°C. The KCl salt concentration was reduced from 100 mM to 20 mM in order to obtain better MS spectra quality. At different times of incubation, a 200 µL aliquot from the exchange reaction was removed (from 10 s to 90 min) and labeling was quenched by adjusting the pH to 2.5 with formic acid, at 0°C.

FIGURE 2 Page

 2 FIGURE 2

  spectrometry (MS) to characterize the overall conformational dynamics of purified hexameric WT and D93N Mt-NDPKs (Figure2A). Samples kept at 10°C are incubated for 10 s to 90 min into 90% D 2 O final concentration and acid quenched on ice to stop the H/D exchange. On line injection of full length protein to a LC-MS system allows immediate mass measurement. Deuterium incorporation in WT and D93N hexamers follows apparent first-order exponential kinetics from 10 s to 40 and 60 min respectively, when two maximum values are reached (Figure2B). These values of 67 and 75 D's (after correction for back exchange (17%) and dilution effect (10%)), on 129

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 29

Page 13 of 29 ACSFigure 4 (

 294 Figure 4 (left panels). This result clearly shows concomitant dissociation and unfolding of WT hexamers (Figure 3B), in full agreement with SEC, tryptophan fluorescence and activity data.

FIGURE 4 Page 14 of 29 ACS

 429 FIGURE 4When the same strategy of characterization by SEC, enzyme activity and tryptophan fluorescence, followed by HDX-MS analysis was applied to the D93N mutant, a different unfolding

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 29 FIGURE 5

Figure 2 .

 2 Figure 2. Overall and local dynamics of WT and D93N Mt-NDPK. (A) Scheme of full length continuous labeling HDX-MS experiment. (B) Plot of deuterium incorporation over time, after correction for back-exchange and dilution. On top, arrows refer to analysis at peptide level on panels C and D. (C, D) Relative percent of deuterium incorporation into each peptide (Figure S2C-D) mapped onto the protein surface using a blue (0%) to red (100%) scale. For WT (C) and D93N (D), a top view along the 3-fold axis of the 6-mer (first column) and the dimer interface after splitting the two trimers (second column) are shown. The white contour outlines the fingerprint of the dimer interface. Residues with no data are coloured in black.

Figure 3 .

 3 Figure 3. Unfolding pathway at full length and peptide level of D93N (A-B) and WT (C-D) Mt-NDPK. Each sample was pre-incubated during 16 hours at different concentrations of urea.Unfolding was monitored by intrinsic fluorescence (blue circles), residual enzymatic activity (red squares)[START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] (A, C). The values are normalized to the experiment in the absence of urea. D93N unfolding was also monitored by a 12 sec pulse labeling deuterium incorporation at full-length level (green diamond) (C). The relative percentage of deuterium incorporation measured at peptide level with a 12 sec D 2 O pulse for WT in 0, 3, 4, and 5 M urea (B) and D93N in 0, 1, 2 and 3 M urea (D) was mapped onto the crystal structure of Mt-NDPK using a blue (0%) to red (100%) scale (FigureS3). A top view (left) and a split dimer interface (right) were shown. When the hexamer was dissociated or unfolded, it was split into three dimers or six randomly oriented monomers, respectively. The white contour on columns two and four outlines the fingerprint of the dimer interface.

Figure 4 .

 4 Figure 4. Representative HD-XMS spectra, obtained during peptic peptide analysis for WT and D93N Mt-NDPK after a 12 second pulse at different urea concentrations. The detailed experimental scheme is shown in the figure S3B. Mass spectra of the +2 charge state of peptides 9-21 and 25-35

Figure 5 .

 5 Figure 5. Unfolding and refolding pathways of WT Mt-NDPK. (A) Refolding (blue lines) versus unfolding (red lines) pathways measured by intrinsic fluorescence (full circles) and enzyme activity (empty squares) after 16 hours pre-incubation in GuHCl.[START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF] The lines do not represent theoretical models but were drawn to help the reader. (B) Refolding and assembly pathway of WT Mt-NDPK. The percentage of deuterium incorporation after 12 seconds D 2 O pulse at different refolding time is plotted onto the structure (FigureS4B). Six monomers in random orientation are drawn when unfolded. For each time point, a top view (left) and a split dimer interface of three split dimers (right) are shown. Residues with no data are colored in black. Note for refolding experiments that at this low protein concentration, only dimers are formed. Hexamers are formed at ten time higher concentrations.[START_REF] Georgescauld | Intersubunit ionic interactions stabilize the nucleoside diphosphate kinase of mycobacterium tuberculosis[END_REF]

.

  Structure of Mt-NDPK (A) View of a Mt-NDPK monomer with labelled secondary structural elements. (B) Side view of a dimer showing the dimer interface (residues 17,19-21, 23- 24, 27, 33-38, and 72) and the active site pocket (K10, Y50, R104, N114, H117, S119, E128). (C) Top view of a trimer. At the trimer interface (residues 16, 25, 28-31, 79-80, 83-84, 87-88, 93-96, 98-102, and 105-110) the Kpn/α 0 foldon (the Kpn-loop and the α 0 helix are colored in magenta and pink, respectively), and the R80-D93 salt bridge (sticks) are involved in hexamer assembly. The trimers stack 'head-to-head' and not 'head-to-tail' such that the Kpn/α 0 is exposed on either side of the hexamer. (D) Side view of the surface of the 6-color hexamer Mt-NDPK. The active site is colored in yellow. In (B) and (C), the chains are colored as in (D).

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ACKNOWLEDGEMENTS

The authors wish to thank Pr Anna Giartosio, Pr Ioan Lascu, Dr Erin Henninger and the European Synchrotron Radiation Facility (ESRF, Grenoble, France). This work was supported by the Fondation ARC pour la Recherche sur le Cancer (FR) to F.G and P.M (Grant SFI20101201793) and by Agence Nationale de la Recherche to A.D. (Grant ANR-12-BSV8-024) and P.M. (Grant ANR-11-BSV8-015-03 and ANR-16-CE11-0032). P.M. was supported by the LABEX DYNAMO (Grant ANR-11-LABX-0011) and EQUIPEX CACSICE (ANR-11-EQPX-0008CACSICE). Keywords: structure-activity relationship · protein folding · hydrogen deuterium exchangemass spectrometry · nucleoside diphosphate kinase · oligomerization

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Biochemistry

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Chun Yeung Lee
email: chun.lee@education.ox.ac.uk

Factors influencing teachers' decision making on reasoning-and-proving in Hong Kong

Keywords: decision making, Hong Kong, reasoning-and-proving, teachers' perceptions

A project is ongoing to develop an intervention that promotes the teaching and learning of reasoning-and-proving in teacher education in Hong Kong. Since the area of reasoning-and-proving in Hong Kong has not been widely studied, this paper reports the first research cycle of the project: to investigate the potential factors that limited reasoning-and-proving activities in Hong Kong classrooms. In this cycle, seven secondary mathematics teachers were interviewed. The participating teachers' perceptions of reasoning-and-proving, students and learning cultures in Hong Kong were examined, and the results indicate factors that may contribute to the marginal situation of reasoning-and-proving in Hong Kong school mathematics.

Introduction

Proof can verify the truth of a mathematical statement and may also promote students' sense making and conceptual understanding in mathematics by providing epistemological justifications [START_REF] Harel | Types of epistemological justifications, with particular reference to complex numbers[END_REF][START_REF] Knuth | Secondary school mathematics teachers' conceptions of proof[END_REF]. Consequently, many mathematics educators suggest that teaching and learning proof and proof-related reasoning activities should spread through school mathematics at different levels and in different content areas. Noting that teachers are the key decision makers of what the students will experience in the classrooms, many studies have been conducted to examine teachers' knowledge and perceptions of proofs (e.g. [START_REF] Knuth | Secondary school mathematics teachers' conceptions of proof[END_REF]). Yet, reasoning-and-proving (RP, see [START_REF] Stylianides | An analytic framework of reasoning and proving[END_REF] tends to have a marginal place in school mathematics. Whilst Western mathematics educators attribute this situation to teachers' counterproductive mathematical knowledge and beliefs about RP, and established mathematical cultures of the schools (e.g. [START_REF] Stylianides | Prospective teachers' challenges in teaching reasoning-and-proving[END_REF], studies on factors that influence teachers' decisions around RP activities are scarce in East-Asian countries (e.g. Hong Kong, Singapore, South Korea). The author has begun a project aimed at developing an intervention that promotes teaching and learning RP in teacher education in Hong Kong. Since the area of RP in Hong Kong has not been widely studied, the first research cycle focused on the context, Hong Kong. This paper intends to report the findings of this cycle. This cycle aimed at investigating the factors that limited the teachers' decisions on teaching and learning RP in the situation of Hong Kong classrooms.

The context: Hong Kong

According to the official documents of the Education Bureau, the mathematics education curriculum in Hong Kong aims to develop students' "(a) ability to think critically and creatively, to conceptualise, inquire and reason mathematically, […]; (b) ability to communicate with others and express their views clearly and logically in mathematical language; […]" (CDC & HKEAA, 2015, p. 2). Despite the lack of an explicit statement, readers may find some implicit indications of teaching RP among these curricular aims.

Like other East-Asian countries, teachers in Hong Kong are often the primary decision makers of what students would experience in mathematics class through lesson preparation and in-the-moment teaching approaches. Teachers' decisions can be influenced by their perceptions and interpretation of the situation, for example, mathematics and RP, students, and the learning culture [START_REF] Stylianides | Prospective teachers' challenges in teaching reasoning-and-proving[END_REF]. Whilst learning culture in Hong Kong is said to be examination-oriented, which may have influenced decisions on school curriculum and assessments [START_REF] Brown | Assessment for student improvement: understanding Hong Kong teachers' conceptions and practices of assessment[END_REF][START_REF] Choi | The process and factors shaping the New Senior Secondary Mathematics Curriculum decision making in Hong Kong[END_REF], studies have not been widely conducted in Hong Kong to investigate how a teacher's lesson preparation and in-the-moment teaching approaches are influenced by this culture. This study aims at examining the factors limiting or promoting teachers' decisions around teaching and learning RP, with respect to their perceptions of teaching RP, students and the learning culture. As a research cycle of a design-research project, the findings in this study will inform the development of an intervention.

Theoretical backgrounds

To investigate teachers' decision making around teaching, researchers often examined the relationship between teacher's beliefs, knowledge and practice. Conner (2017) examined the differences in argumentation between two U.S. classes, and suggested these differences might be attributed to the teachers' intentions regarding the components of rationality. [START_REF] Kempen | Pre-service teachers' perceptions of generic proofs in elementary number theory[END_REF] investigated 12 pre-service teachers' perceptions of the generic proof, and reported that participants' preference for formal proof over generic proof in a task might be attributed to the situations that they adopted the construction of a formal proof as a norm for solving the task during their time in school, or they (mis)interpreted that construction of a formal proof was asked in the task. These results indicated that teachers' belief constructs could be complex and influenced teachers' teaching practices. However, the process of decision making and transition from teachers' belief constructs to practices were not thoroughly discussed. Blömeke, Gustafsson, and Shavelson (2012) and [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF] attempted to model teachers' decision making. In the model of Blömeke et al. (2012), when teachers' perceptions are affected by their belief and knowledge constructs, their interpretation of the context may mediate between their perceptions and decision making (and practices). [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF] further elaborated the process of transition and decision making using the conceptualisation of goals, orientations and resources; he defines resources (with a focus on knowledge) as "the set of intellectual, material, and contextual [information] available to the teacher" (p. 10) and orientations as "an inclusive term encompassing a group of related terms such as dispositions, beliefs, values, tastes, and preferences" (p. 29, italics added). I adopted the models of Blömeke et al. (2012) and [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF]: if teaching RP is a routine practice of a teacher, the teacher will continue this routine practice; otherwise, if teaching RP is an unfamiliar practice of the teacher, the teacher will perform a cost-benefit analysis on different options using the variables (perceptions and interpretation), and the decision will be based upon the outcomes of the calculation of expected values on the options. Figure 1 depicts a model of a teacher's decision adopted from Blömeke et al. (2012) and [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF].

Methods

Seven secondary mathematics teachers participated in this study. These teachers were selected because: (a) they obtained related undergraduate degrees and gained their teacher qualification through postgraduate teacher training programmes; (b) they have been serving local schools differed on locations, school cultures and students' academic achievements: in Hong Kong, secondary public/government-aided schools and students are divided into three school bands regarding their academic achievement. One participant was serving high-achieving schools, two in mid-achieving schools, two in low-achieving schools, and two were serving private schools that do not belong to any school band; (c) they had worked at schools for at least three years; and (d) they had served at least two schools. Their diverse backgrounds resulted in responses that covered a range of concerns of teachers when considering teaching and learning RP in the situation of Hong Kong classrooms.

The teachers participated in task-based interviews. The tasks and questionnaire items were adopted from the studies of [START_REF] Almeida | A survey of mathematics undergraduates' interaction with proof: Some implications for mathematics education[END_REF], Keceli-Bozdag, Ugurel and Bukova-Guzel (2014), [START_REF] Kotelawala | Exploring teachers' attitudes and beliefs about proving in the mathematics classroom[END_REF], and [START_REF] Yoo | Effects of traditional and problem-based instruction on conceptions of proof and pedagogy in undergraduates and prospective mathematics teachers[END_REF] to capture teacher's knowledge, attitudes and beliefs about RP and their instructional practices. Table 1 lists some sample items. Each interview lasted about an hour, and was conducted and audio-recorded in Cantonese, the teachers' mother tongue to enable them to express their opinions easily and effectively. The teachers were asked to complete the questionnaire and tasks using think-aloud strategy and then respond to follow-up queries.

Origins of the items Sample items [START_REF] Almeida | A survey of mathematics undergraduates' interaction with proof: Some implications for mathematics education[END_REF] Decide which 'proofs' you find (a) convincing, (b) not convincing, or (c) incorrect. Proposition:

(1 + 1/2 + 1/2 2 + 1/2 3 + … + 1/2 n ) → 2 as n → ∞.
Assumptions/assertions that may be used in the proofs: The audio-recorded data were transcribed into written Chinese. The transcripts were then analysed using the model of the teacher's decision (Figure 1) regarding two themes: (a) teachers' perceptions of RP and teaching and (b) their interpretation of the contexts. Based on the teachers' responses, their interpretations of the contexts were further divided into (i) interpretation of students as learners and (ii) interpretation of learning cultures. To report the findings, selected excerpts were further translated into English using word-to-word strategy (that is, English grammatical rules are not strictly followed) assisted with appropriate interpretation.

A1. If |a| < 1, then an → 0 as n → ∞. 'Proof' 1 Let S n = 1 + 1/2 + 1/2 2 + 1/2 3 + … + 1/2 n .

Results

Perceptions of RP and teaching

All participants related proofs to logical reasoning and its promotion. For example, when asked which items of learning activities are related to proof, Participant 2 stated, "if the activities enable to develop [students'] logical reasoning, I think the options are [formulating conjecture], [explaining why answers or steps make sense] and [working on one's own methods for solving unfamiliar problem]." He also added that making a conjecture may be a precursor of a proof.

Participant 2: Like Pythagoras' theorem, he (the prover) makes a conjecture. If he did not have a conjecture, he would not have tried to prove [it]. Therefore, [we] must first have conjectures.

Participant 4 perceived that, through teaching and learning proofs, students may be provided with epistemological justification of a mathematical statement for promoting conceptual understanding in mathematical facts and ideas. When discussing 'proofs' of "square root of 6 is an irrational number", Participant 4 expressed his intention and motivation of teaching the proof: to foster students' conceptual understanding of irrationality of a number:

Participant 4: … because learning a proof like this means to understand what "irrational" [an irrational number] is… because something is true… like this then [we] can dig out the meaning of "irrational". Therefore, I will teach [the proof].

The realisation of educational values of proofs may initiate teachers' motivation, like Participant 4, of teaching RP: the teachers initiated the process of benefit-cost analysis on teaching RP, but their interpretation of students and learning cultures may mediate between their perceptions, and decisions.

Interpretation of students as learners

Most participants perceived that their students were not capable of nor interested in learning any proof of a mathematical proposition in general.

Participant [I] sometimes, perhaps, find some smarties, and then ask them to do this conjecturing. Therefore, the proportion [of making conjectures] would be far less… in the real life (my actual teaching practices).

To facilitate some RP activities in classroom, the teachers tended to select proofs that they found easy to understand, but there were no systematic selection criteria; instead, the teachers selected a proof by their feelings of its length and complexity.

Participant 5: [I prefer proofs that are] easy to "see". [The students] all can "see"; that is, easy to explain and understand.

Participant Whilst teachers' realisation of the educational values of teaching RP (perceptions) may promote their use of RP when teaching (decisions), the participants demonstrated how teachers' counterproductive interpretation of students' learning capabilities and attitudes may mediate between their perceptions and practices of teaching RP. Participant 1 demonstrated that his counterproductive interpretation interacted with his perception of what students needed to learn/master and teacher-centred disposition. The choice of a "simple and easy" proof (decision) may refer to his interpretation that (a) the length and complexity of the selected proof were aligned with students' capabilities, and/or (b) the main purpose of the selected proof was to convince students of the truth of a proposition with less effort. Participant 7 demonstrated that he developed a positive perception of teaching RP and an ideal practice, but his counterproductive interpretation contributed to investing less in involving students in RP activities (e.g. making a conjecture). As a result, teaching RP was limited in their classes.

Interpretation of learning cultures

Apart from their interpretation of students, teachers' interpretation of the learning cultures where they are situated may also mediate between their perceptions of RP and teaching, and decision making. Like other East-Asian countries, examination-oriented culture is well established in Hong Kong [START_REF] Brown | Assessment for student improvement: understanding Hong Kong teachers' conceptions and practices of assessment[END_REF][START_REF] Choi | The process and factors shaping the New Senior Secondary Mathematics Curriculum decision making in Hong Kong[END_REF]. The participating teachers' responses revealed that this examination-oriented culture greatly influenced their decisions on teaching RP and mathematics. (attitudes) and that of the examination-oriented learning culture. Participant 7 added that the examination-oriented culture has also contributed to the discrepancy between his ideal RP lesson (perceptions of RP and teaching) and his actual implementation.

Participant 7: … But [the actual lessons focused] more on whether they could "do the maths" (solve the problem tasks), which is more about examination. Of course, I am not completely saying [that my lessons are] examination-oriented; my lessons are really not so examination-oriented. However, [my lessons] in fact tend more to be the side of calculation. Yes, therefore, when having the term "ideally" here, my… my [ideal] practice will be a bit different [from the actual practice].

The participating teachers perceived that preparing students for public examinations is a main role of being a teacher in Hong Kong (a goal); they also held an interpretation that the public examination in Hong Kong was calculation-oriented. Any decision that may strengthen students' accurate calculation and problem-solving skills was prioritised over the teachers' ideal practices if the decision and the ideal practices were different. For example, Participant 4 perceived that the examination-oriented culture was deeply rooted in his teaching environments, and, as a result, students expected mathematics class to be oriented towards examination preparation. He prioritised the goal of preparing his students for examination highly over the goal of fostering students' conceptual understanding of mathematics. Since teaching RP was not his routine practice, according to [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF] model, a "cost-benefit" calculation was performed by Participant 4. His cost of practising RP in his mathematics class outweighed the benefits that he perceived when teaching RP. Therefore, he found it difficult to practise any RP activity in his mathematics class. Therefore, they invested more in involving students in calculation and problem-solving activities, rather than RP.

Discussion

This study examined the three interrelated factors that may influence Hong Kong secondary mathematics teachers' decisions around RP: teacher's perceptions of teaching RP, interpretation of their students as learners, and interpretation of the learning cultures. Whilst the teachers' productive perceptions of teaching RP, for example, recognising that teaching proofs may foster students' conceptual understanding of mathematical ideas and logical reasoning, may promote and develop their ideal practices, their counterproductive interpretation of students' learning needs: in particular, they did not think proofs are needed or suitable for all students (cf. [START_REF] Frasier | Proofs in Mathematics Education: Expectations and Reality[END_REF][START_REF] Varghese | Secondary-level student teachers' conceptions of mathematical proof[END_REF], and the learning cultures (examination-oriented culture) may interact with each other and mediate between their perceptions (of students' learning needs) and actual practices.

The teachers perceived examination-oriented culture to be a main factor that limited their decisions on teaching RP; this culture influenced the teachers' interpretation of students' attitudes and their learning needs. Examination-oriented culture is deeply embedded in Hong Kong school mathematics and teachers' interpretation, and the public examination in Hong Kong is interpreted as calculation-oriented. As a result, the teachers deviated from researchers' expectation of making RP spread through school mathematics, and prioritised calculation-oriented teaching materials over RP.

If teachers avoid RP in their teaching because they perceive that proofs are less important and/or not suitable for all students in understanding mathematics and preparing for examination, professional development should address teachers' interpretation of students' learning needs and the social contexts (school and learning cultures) and be focused on fostering teachers' reflection on their teaching philosophy and broadening their horizons for learning other trends and visions. Educational reforms including frameworks of public examination and teacher education may also be considered when RP activities should be central of mathematics education in Hong Kong, and worldwide.

Figure 1 :

 1 Figure 1: Model of teacher's decision adopted from Blömeke et al (2012) and Schoenfeld (2011)

Table 1 : Sample items of the tasks

 1

		'Proof' 2
		Let S = 1 + 1/2 + 1/2 2 + 1/2 3 + … + 1/2 n + …	(1)
		Multiplying both sides of the equation above by 1/2 gives
		S/2 = 1/2 + 1/2 2 + 1/2 3 + … + 1/2 n + …		(2)
		Subtracting equation (2) from (1) gives: S/2 = 1 → S = 2.
		This proves the proposition.
	Keceli-Bozdag et	I think that theorems and proofs are the foundations of mathematics.
	al., 2014	Strongly agree Agree		Neutral		Disagree		Strongly
									disagree
	Kotelawala, 2007	Please divide 100 units of time among the following activities, consider a typical junior
		secondary mathematics class (12-15 years old) that you teach.
		Items							Units of time
		1. Receiving instructions from teacher on how to solve a problem
		2. Practising calculations for recall, accuracy, and/or speed
		3. Student solving problems using steps from the day's lesson or
		the weeks lessons
		4. Students formulating conjectures
		5. Student explaining to group why their answers or steps make
		sense
		6. Students working on their own methods for solving unfamiliar
		problems
	Yoo, 2008	Ideally in a proof-based mathematics lesson, students should (a) make conjectures,
		investigate them, and construct their own proofs, or (b) practise proving statements or
		conjectures that they know have been proven to be true before by an expert.
		(a)	towards (a)	both	towards (b)	(b)	neither
		1	2	3	4	5	6	7	8

Table 1 (con't): Sample items of the tasks

 1

  Of course, I have to know the characters of my class, then I will evaluate if they can understand what I say… evaluate if they will "accept" (be convinced by and/or understand) this proof. If, after I complete the evaluation, I think [it is] worth saying (teaching)[it], then I will try to say (teach)[it]. If, after evaluation, I think [it is] not worth saying (teaching) [it], then I won't.

1: [I choose to teach proofs which are] simple and easy to follow. [Interviewer: What do you mean by simple and easy to follow?] This depends on [my] feelings… This is sometimes my feelings… If, if I am a teacher and have a judgement, if I see this [proof] and I think this can be understood by the majority, from my gut feeling…