Insights into Enzymatic Catalysis from Binding and Hydrolysis of Diadenosine Tetraphosphate by E. coli Adenylate Kinase

Adenylate kinases play a crucial role in cellular energy homeostasis through the interconversion of ATP, AMP, and ADP in all living organisms. Here, we explore how adenylate kinase (AdK) from Escherichia coli interacts with diadenosine tetraphosphate (AP4A), a putative alarmone associated with transcriptional regulation, stress, and DNA damage response. From a combination of EPR and NMR spectroscopy together with X-ray crystallography, we found that AdK interacts with AP4A with two distinct modes that occur on disparate time scales. First, AdK dynamically interconverts between open and closed states with equal weights in the presence of AP4A. On a much slower time scale, AdK hydrolyses AP4A, and we suggest that the dynamically accessed substrate-bound open AdK conformation enables this hydrolytic activity. The partitioning of the enzyme into open and closed states is discussed in relation to a recently proposed linkage between active site dynamics and collective conformational dynamics.


CHEMICALS
All adenosine nucleotides were purchased from Merck and used without any further purification. The non-canonical amino acid para-propargyloxy-L-phenylalanine (pPfF) was purchased from Iris Biotech GmbH. Azido-PROXYL was synthesized according to published procedures 1 .

Cloning
Wild-type E. coli adenylate kinase (AdK) was overexpressed from a self-inducing plasmid (pEAK91) using previously published procedures 2 , as well as the Val148Cys mutant used for PRE-NMR measurements 3 . For EPR measurements, a K50TAG was introduced as additional mutation for doublelabeling and subsequent EPR distance measurements. The amber-stop-codon (TAG) enabled the insertion of the non-canonical amino acid para-propargyloxy-L-phenylalanine (pPrF) 4 for subsequent labeling over Copper(I)-catalyzed azido-alkyne click reaction (CuAAC) 5 . The used primers are detailed in Table S1.

Protein expression and purification
Chemically competent BL21-gold (DE3) E. coli cells were transformed with the plasmid for E. coli AdK wild type according to published procedures 2,3 . The double mutant Lys50TAG/Val148Cys plasmid was co-transformed with an additional plasmid for the aminoacetyl-tRNA-syntethase/tRNA pair (pEVOL_pCNPhe provided by the Schultz lab 6 ) that can incorporate the non-canonical amino acid pPrF at the position of the amber stop codon 5 .
To obtain a uniformly 15 N-labeled enzyme for the 15 N-edited NMR experiments, expression cells were grown in minimal medium (M9) with 15 NH4Cl as the sole nitrogen source. Otherwise, LB-medium was used.
The expressed proteins were purified in two steps using first affinity chromatography over a Blue sepharose column (26 ml column volume) using 50 mM Tris-buffer with 1 M NaCl, pH 7.5 for elution, followed by gel filtration as second step. The protein concentration was determined via absorption measurement at 280 nm on a BioPhotometer (Eppendorf). Purified protein was stored in 30 mM MOPS, 50 mM NaCl buffer at pH 7.0, which was supplemented with 1 mM TCEP for the cysteine mutants to inhibit the formation of dimers through disulfide bond formation.

ISOTHERMAL TITRATION CALORIMETRY (ITC)
To determine the binding affinity of AdK to the AP4A ligand, ITC was carried out with wild-type AdK at a MicroCal ITC200 at 298 K. A 1.15 mM AP4A solution was titrated in 26 steps to a 168 µM solution of wild type AdK. As buffer, 30 mM MOPS, 50 mM NaCl, pH 7.0 was used. The dissociation constant Kd, binding enthalpy H 0 , binding entropy -TS 0 and the binding ratio (N) were determined using the analysis program provided by the ITC manufacturer. The results are shown in Figure S1.
CRYSTAL DATA COLLECTION AND REFINEMENT STATISTICS

Crystallization
For co-crystallization of AdK with AP4A, 23.3 mg/mL (980 µM) of AdK in 30 mM MOPS, 50 mM NaCl buffer at pH 7.0 was mixed with 4 µl 130 mM AP4A resulting in a protein to ligand ratio of 1:10.6 and a final concentration of AP4A of 0.9 mM. Crystallization was performed by sitting drop vapor diffusion method at 291 K. The AdK-AP4A mixture was mixed in a 1:1 ratio (0.5 µL + 0.5 µL) with the crystallization solution containing 20 % (w/v) PEG 3350 and 100 mM Bis-tris propane at pH 6.5. Grown crystals were flash-frozen in liquid nitrogen after a short soaking in a cryoprotection solution, consisting of the corresponding crystallization condition with the PEG 3350 concentration increased to 35 % (w/v). Crystals appeared after 3 -8 days and were mounted after 1 -3 weeks. It is noteworthy that this is in the same time-scale as the described AP4A hydrolysis.

Processing, phasing and refinement
The dataset was processed with the XDS program package 8 . For the subsequent steps, several programs from the CCP4 9 and Phenix 10 software suites were used. The structure was solved by molecular replacement with the program MOLREP 11 using a (unpublished) structure from a previously recorded dataset. The structure was once refined by a rigid body refinement as implemented in REFMAC5 12 and a simulated annealing refinement as implemented in phenix.refine 13 to reduce model bias. Further refinement was performed by several cycles of manual model correction in Coot 14 and automated refinement by either REFMAC5 12 or phenix.refine 13

CIRCULAR DICHROISM (CD)
To check whether spin-labeling affect the secondary structure of the AdK, CD spectra of wild-type protein, mutant and labeled mutant were recorded ( Figure S5). The protein samples were diluted to 25 µM in 30 mM MOPS, 50 mM NaCl, pH 7.0 and 80 µL were transferred to 0.5 mm demountable cuvettes. Spectra were recorded on a JASCO J-715 spectropolarimeter by continuously scanning from 280 to 180 nm with a scanning speed of 50 nm/min, response time of 4 s and a bandwidth of 1.0 nm at 293 K. The recorded spectra were averaged over 10 scans, baseline-and background-corrected with the same buffer. For the conversion of the spectrometer unit (CD-signal in mdeg) to the molar residue ellipticity (MRE), the following formula was used: With the molar mass M in g/mol, Naa as the number of amino acid residues in the protein, c in g/L, and d = 0.05 cm. The results are shown in Figure S5.

ENZYMATIC ASSAY
To assure that the AdK is still active when 2 spin labels are attached to the catalytically important ATPand AMPbd, an enzymatic assay was performed. The Michaelis constant KM and the catalytic turn-over rate kcat were determined with an enzymatic assay that couples the generation of ADP by the adenylate kinase to the (fast) reaction of the two enzymes pyruvate kinase and NADPH dehydrogenase 17,18 . In its course, NADH, H + is oxidized to NAD + which can be followed by UV/Vis spectroscopy because of the strong absorbance from NADH at 340 nm (340nm(NADH) = 6300 M -1 cm -1 in H2O, T = 298 K P NMR spectra at predetermined time intervals. Both spectra recording and incubation between the spectra were done at 295 K. The NMR spectra were recorded on a 600 MHz Bruker Advance III spectrometer ( 31 P frequency: 243 MHz) equipped with a 5 mm broadband cryo probe. The sweep width was 60 ppm (16384 Hz) and 256 scans were recorded for each spectrum. The concentration of each phosphorous species was determined by integration of each peak using Topspin 3.6.1 (Bruker Biospin), comparing it to the total integral under all peaks.

1 H 15 N HSQC spectra
1 H 15 N heteronucelar single quantum coherence (HSQC) spectra were recorded to monitor the overall protein structure and to detect possible local changes upon ligand addition. The spectra of 200 µM AdK in 30 mM MOPS, 50 mM NaCl buffer at pH 7.0 buffer were recorded on an 850 MHz Bruker Avance III spectrometer equipped with a 5 mm HCN cryoprobe with 1024*256 points over a 13*30 ppm sweep width with 8 scans used for each line. The spectra were processed using NmrPipe 19 and the peaks were picked using Ansig 20 or Sparky 21,22 . The assignments were based on previously published values for the apo form 23 . The kD was determined, based on induced chemical shifts in both the 1 H and the 15 N dimension, by fitting the one-site binding equation 24 to the determined chemical shift using in house developed MatLab (Mathworks) scripts.

Projection analysis
To quantify the structural changes induced by AP4A, a projection analysis of chemical shifts in the 1 H 15 N HSQC spectra of wild-type AdK was performed according to Selvaratnam and co-workers 25 . The projection analysis compares the vectors of chemical shift defined from the substrate free state to a ligand bound state. Two parameters are then extracted, the projection angle (θ) and the absolute value of the activation vector (|A|, main text, Fig. 4E, Figure S18A, B). The angle contains information of the similarity of the binding process for the two ligands that are compared, whereas the activation vector contains information on the statistical weight of the contributing states (here open and closed states). The spectrum of apo and AP5A saturated AdK was used as reference for the fully open and fully closed state, respectively. The projected apo-AP4A vector determines how much of the chemical shift induced by AP4A binding can be attributed to the same conformal change as AP5A binding. This procedure was performed for every assigned residue, but only residues where the angle θ between the apo-to-AP5A vector and apo-to-AP4A vector was smaller than 25.8 ° (= cos(0.9)) were taken into account 25 ( Figure  S18C, D). The average projected chemical shift induced by AP4A binding was 0.45 ± 0.13 of the chemical shifts induced by AP5A.

SITE-DIRECTED SPIN LABELING FOR EPR EXPERIMENTS
For EPR distance measurements, two paramagnetic labels are required to measure the distance between them. Here, one PROXYL label was introduced first at position Lys50 via Copper(I)-catalyzed azido-alkyne click reaction (CuAAC) and a second via cysteine-maleimide coupling at position Val148Cys (Scheme S1). Scheme S1. Spin labeling strategy and chemical structures of the used non-canonical amino acid and spin labels.

CuAAC
Coupling of azido-PROXYL to the non-canonical amino acid pPrF was performed as described previously by Kucher et al. 1 As first step, the buffer was exchanged to PBS with 1 mM TCEP, pH 7.5, as labeling in 30 mM MOPS, 50 mM NaCl pH 7.0 was found to be less efficient. Then, copper(II)-sulphate (CuSO4) and the ligand 2-(4-((bis((1-(tert-butyl)-1H-1,2,3-triazol-4-yl)methyl)amino)methyl)-1H-1,2,3-triazol-1yl)acetic acid (BTTAA) were mixed in deionized water in a 1:3 ratio. Then, labeling reagents were diluted with PBS buffer pH 7.5, the protein and azido-PROXYL in DMSO (100 mM stock solution) were added. Ascorbic acid was added in a 1:1 ratio to reduce Cu(II) ions to the catalytically active Cu(I) species. The final concentrations were: 1 mM copper sulfate, 3 mM BTTAA, 30 -50 μM protein, 1 mM spin label and 1 mM sodium ascorbate. The reaction was incubated for 40 min, 298 K and 400 rpm. After the reaction, a 5-fold excess of EDTA over copper was added to complex free Cu-ions. The free label and other reactants were removed by size exclusion chromatography using a Sephadex SEC 26/10 column (GE Healthcare).

C D B
A S8 2.7.2 Cysteine-maleimide coupling As second labeling step, maleimido-PROXYL was coupled to the introduced cysteine at position Val148. The protein concentration was adjusted to 20 -50 µM in PBS buffer, pH 7.5. A 5-fold excess was added in 2 steps: first, a 2.5-fold excess of maleimido-PROXYL was added for 30 min at 293 K, 300 rpm. Another 2.5-fold excess of maleimido-PROXYL was added for additional 30 min under the same conditions. The reaction was carried out in two steps to ensure low excess of label during the reaction to not promote labeling of the 2 nd naturally occurring cysteine at position 77. Free label was removed by size exclusion chromatography using a Sephadex SEC 26/10 column (GE Healthcare).
To prove that no significant labeling at Cys77 occurs, 4p-DEER measurements on AdK spin labeled only with maleimido-PROXYL were performed ( Figure S8). In case both Val148Cys and Cys77 positions were labeled with the above-described procedure, clear dipolar oscillations and modulation depth were expected. This was not the case ( Figure S8), proving efficient labeling only at position Val148Cys.
Labeling and spin-label concentrations were monitored by cw EPR (section 2.8.1). The influence of the labels on secondary structure was examined by CD spectroscopy (section 2.4) and the influence on ATP binding and activity via an enzymatic assay (Enzymatic assay & in-silico labelling Table S3).

EPR 2.8.1 Continuous-wave (cw) EPR
To determine the spin labeling efficiencies of labeled AdK, cw EPR spectra were recorded. The spectra were recorded at r.t. (293 -297 K) on an EMX-Nano benchtop spectrometer (Bruker Biospin) at 9.4 GHz (X-band). Typically, 20 µL of the purified protein sample was filled into a glass capillary (HIRSCHMANN ringcaps®; inner diameter 1.02 mm). Spectra were recorded with a modulation amplitude of 0.8 G, 1 mW microwave power, a sweep width of 200 G and a sweep time of 80 s. To improve the signal-to-noise ratio, the spectra were averaged over 16 scans. Quantitative spin concentrations were obtained using the built-in EMXnano reference-free spin counting module (Xenon software, Bruker), from which the labeling efficiency, defined as the ratio of spin to protein concentration, was calculated.

Pulsed EPR
All pulsed EPR experiments were performed at 34 GHz (Q band) with a microwave attenuation of 0 dB. Echo signals were detected in integrator mode with a video bandwidth of 200 MHz. The shot repetition rate was set to 4080 µs.

Instrumentation
All pulsed experiments were performed on a Q-band Elexsys E580 spectrometer (Bruker Biospin). The device is equipped with a SpinJet-AWG unit (Bruker Biospin) and a 150 W pulsed TWT amplifier (Applied Systems Engineering), allowing the use of shaped microwave pulses. The temperature was maintained at 50 K with the EPR Flexline helium recirculation system (CE-FLEX-4K-0110, Bruker Biospin, ColdEdge Technologies), comprising a cold head (expander, SRDK-408D2) and a F-70H compressor (both SHI cryogenics), controlled by an Oxford Instruments Mercury ITC. For all pulsed EPR measurements, an overcoupled, commercial Q-band resonator (ER5106QT-2, Bruker Biospin) was used with a Q-value of approximately 200.

4p-DEER experiments
DEER samples were prepared by mixing the spin-labeled AdK with the respective ligands in deuterated 30 mM MOPS, 50 mM NaCl, pH = 7.0 and incubate it 15 min at room temperature. Then, 60 % of d8glycerol was added as cryoprotectant and each sample with a total volume of 60 µl and a final concentration of 20 µM AdK was transferred to a quartz tube with 3 mm outer diameter (fused quartz tubing, Technical Glass Products). The samples were flash-frozen in liquid nitrogen and stored at 193 K.
4p-DEER measurements were recorded with the standard pulse sequence: With interpulse delays of τ1 = 400 ns, τ2 = 6 µs (or more). The pump pulse position was incremented in t = 8 ns steps. Nuclear modulation artifacts were suppressed by averaging 8 traces with interpulse delay 1 varied by ∆ 1 = 16 ns. An 8-step phase cycling scheme {(x) [x] xp x} was performed to remove other echoes capable of interfering with the detected signal as suggested by Tait and Stoll 27 . Rectangular pulses with typical lengths of ca. t/2 = 15 ns and t = 30 ns were used as observer pulses. The pump pulse was a 90 MHz broad HS{1,1} pulse with t = 100 ns and /t = 8 that was set to match the nitroxide and resonator maximum at a frequency offset of +90 MHz to the observer pulses. It was calculated with MATLAB (version 2021b) using the function "pulse" of the easyspin software package 28 . Depending on the signal-to-noise of each sample, 2 -15 scans were recorded with 20 shots per point over 1 -10 h (Table S5).

Distance analysis
Individual scans of the DEER measurements were phase-corrected and summed using MATLAB (version 2021b). Shown distance distributions were evaluated with the integrated deep neural network DEERNet (Generic). 29 Uncertainties in the 4p-DEER data analysis with DEERNet are obtained using an ensemble of independently trained neural networks and are shown as 95 % confidence intervals (shaded areas) in the distance distributions (P(r) in Figure 4, Figure S14 - Figure S10) 29,30 .
The distance constraints resulting from DEERNet analysis were compared with evaluation by Tikhonov regularization 31,32 with DeerAnalysis 33 (version 2019), which was the golden standard of model-free DEER distance evaluation before one-step approaches that fit the background and distance in a single step were introduced. For the analysis with Tikhonov, the zero time and the starting times for 3dimensional background fit were determined automatically. The  parameter was determined using the L-curve criterion 33 . All resulting distance distributions were similar to the results from analysis with DEERNet. Some examples of a comparison from DEERNet and Tikhonov L-curve analysis are shown in Figure S13. The resulting distance distributions in Figure S13 were validated using the validation tool included in the DeerAnalysis2019 software to assess the certainty of the distance evaluation. For this validation, new distance distributions were calculated on the same trace by adding up 1.5-fold noise (10 different values), varying the background start between 1 -3.5 µs (12 values) and the background dimension between 2.0 and 3.5 (4 values). Out of these 480 analyses, fits that exceeded 1.15 rmsd away of the best fit were excluded by using a prune level of 1.15. The remaining distance distributions gives an idea about the accuracy of the analysis method and are depicted as gray shaded areas in Figure  S13.
In addition, the newest and 3 rd method for DEER distance analysis was used for comparison in Figure  S13: DeerLab 34 . DeerLab provides in addition two methods for statistic validation of the resulting distance distributions: uncertainty estimation by covariance matrix and bootstrapping. 34 Figure S13 S10 shows the result from DEER lab in red, using 4-pulse DEER as experiment (ex_4pdeer), a 3D homogenous background function (bg_hom3d), a cutoff at 5 µs of the DEER trace and generalized cross-validation (gcv) criterion for determination 35  H 0 (kJ mol -1 ) -10.9 ± 1.8 -TS 0 (kcal mol -1 ) -17.1 ± 1.9 G 0 (kcal mol -1 ) -18.0 ± 3.7

CRYSTALLOGRAPHY DATA
Here, we provide a more detailed reasoning why we think that 2 ADP molecules resemble more closely to the electron density of the ligand when E. coli AdK is co-crystallized with AP4A ( Figure S2).
The electron densities for the ligands in the active centers of both monomers in the asymmetric unit are well-defined and thus allow a distinction between AP4A and ADP ( Figure S2A, C). In more detail: i) No electron density connecting the two β-phosphorus atoms (via an oxygen atom) are observedwhich would be expected to be noticeable in this well-defined density and in case of AP4A binding. Instead, two clearly separated electron densities are observed, which suggests two distinct molecules.
ii) The electron density for four (oxygen) atoms can be observed around the density of each βphosphorus atom, which is expected for the binding of two ADP molecules, whereas binding of AP4A would result in electron density for seven (oxygen) atoms around both β-phosphorus atoms. This is supported by modelling trials of ADP as well as AP4A. Placement of two ADP molecules ( Figure S2A) explains the observed electron densities considerably better than the placement of an AP4A molecule ( Figure S2C).
Furthermore, the modelling attempts show that the placement of AP4A would result in an unusual geometry around the two -phosphates, while building of two ADPs shows expected geometries in these molecules ( Figure S2B, D): iii) The O-P-O angle in the modelled AP4A ( Figure S2D) is for example larger than the comparable angles for ADP (130° compared to 107°). This bond angle seems not impossible when comparing to ab initio calculations on methyl di-and triphosphates that report O-P-O angles of 92° -136°. 37 However, comparing to similar angles in previously determined protein structures with AP4A, the angles are in the 102° -114° range (PDB ID: 2E20 and 2C9Y) 38 . In the crystal structure of E. coli AdK co-crystallized with AP5A, the OPO bond angles of the -or '-phosphates were between 105° and 116° (PDB ID: 1AKE) 39 .
iv) Another example in the modelled AP4A are the bond lengths of the β-, β'-phosphorus atoms and the oxygen atom bridging them, which are 1.72 and 1.73 Å. Here, ab initio calculations report bond lengths of 1.61 -1.72 Å 37 and in previously determined structures the P-O-P bonds are between 1.62 -1.65 Å long (PDB ID: 2E20 and 2C9Y) 38 . In other E. coli AdK structures co-crystallized with ADP, P-O-P bond lengths in the range of 1.63 -1.66 Å were reported 40 .
Consequently, building of two ADPs results in molecule geometries that are more in line with the expected values, than the strained geometries in the modelled AP4A molecule, which has both unusual bond angles and bond lengths. Therefore, we conclude that 2 ADP molecules rather than AP4A are bound to AdK in the crystal.    Compared to wild type AdK, the catalytic turn-over is reduced by factor 7 when the enzyme is doubly labeled with azido-PROXYL at position Lys50pPrF in the AMPbd and with maleimido-PROXYL at position Val148Cys (AdK_2x NO + ATP, section 2.7). The reduction in maximal catalytic rate is either because the label at K50 interferes with substrate binding or because the spin labels, even when sterically small, might slow down the motions required for efficient catalysis at the ATPlid and AMPbd.
From in-silico labelling 44 (i.e., attaching label rotamers to the crystal structures of open and closed AdK), we do not expect large steric hindrance ( Figure S6). This is supported by the KM value that increased by (only) a factor of 1.2 for the labeled mutant compared to the wild-type (increase in KM indicates a decreased binding strength). Thus, the binding affinity for the substrates ATP and AMP seems to be nearly unaffected by the double spin labeling.
Also, since only the protein conformation (and not the catalytic activity) was investigated using the doubly labeled AdK with excess of ligands, the found differences were not deemed important.    4p-DEER raw data normalized to the maximum (V(t)/V(0), left), after background correction (F(t)/F(0), middle) and resulting distance distribution normalized to the maximum (P(r)/P(max), right) for 20 µM of doubly PROXYL labeled AdK (apo 1, gray) and singly maleimido-PROXYL labeled AdK (apo (1x NO*, beige). Details are listed in Table S5. All DEER curves were analyzed with DEERNet 29 within the DeerAnalysis software 33 (version 2019). A low modulation depth of 0.7 % was found for the control labeled only with maleimido-PROXYL (1x NO*, beige), confirming a very low degree of unspecific labeling of E. coli AdK at C77 under given conditions.

Quantification of the lid-to-lid EPR distance measurements
Single Gaussians were fitted to the distance distributions representing the open and closed AdK conformation. We used the AP5A bound AdK as reference for closed enzyme. For the open conformation, we observed a shoulder at smaller distances (3 nm) for the enzyme without ligand that was pronounced to different extend between the samples (compare apo 1 and apo 3, Figure S9). We therefore assumed that part of the enzyme is closed even in absence of any ligand, as reported earlier 45 . Therefore, we used a Gaussian that takes only the long distances into account (dashed red line, Figure S9).  Table S5. All DEER curves were analyzed with DEERNet 29 within the DeerAnalysis software 33 (version 2019). Interestingly, the shoulder at approximately 3 nm in the distance distribution P(r) of apo 1 is less pronounced in the sample apo 3. A single Gaussian was fitted to the main peak (red dotted lines) for quantification of the open conformation.
We used a Gaussian analysis of a two-state model to describe the distance distribution for the AP4Abound AdK as superposition of the closed and open state. To this end, the mean value and standard deviation of the 2 single Gaussians were kept fixed (Table S4) and only the amplitudes were varied. The resulting populations of open and closed states are listed in Figure S10 A.

Gauss 2 (open AdK) rmean (nm)
3. coli AdK conformation that results from Gaussian fitting of distance constraints obtained from 4p-DEER EPR distance measurements. The mean ± the difference between two independent measurements (see Figure S14, Table S5

AP4A binding mode
Because AP4A is a symmetric molecule, binding to both binding sites (i.e., to AMPbd or ATPlid separately or simultaneously) might be possible.
To test whether AP4A only binds to one of the two binding sites, we compared the EPR distance distribution obtained for the AP4A-bound enzyme with (i) AdK bound only to AMP and (ii) AdK bound only to ATP ( Figure S11 and Figure S12).
Hereby, we assumed that the AMP and ATP bind to their respective binding domains (i.e., the AMPbd and the ATPlid) and not to both binding domains. This was shown before by NMR, and AMP or ATP     Table S5. All DEER curves were analyzed with DEERNet 29 within the DeerAnalysis software 33 (version 2019). The distance distributions for ATP-bound AdK ressembles that of AP4Abound AdK.
From the Figure S11 and Figure S12 above, discrimination between the conformations of apo AdK, AMP-, ATP-and AP4A-bound AdK is possible.
(i) It becomes clear that closing of the AMPbd alone does not lead to a conformation that is comparable with the AP4A-bound state ( Figure S11), as the distance distribution is narrower and shifted towards longer distances for AMP-bound AdK compared to AP4A-bound AdK. Thus, AP4A binding to only the AMPbd is not what we observe. Therefore, we conclude that binding of AP4A occurs at the ATP binding site.
(ii) Figure S12 shows that AP4A-binding lead to a similar distance distribution than ATP-binding but shifted slightly (< 0.1 nm, Table S4) towards smaller distances, indicating more closed enzymes in the presence of AP4A.
From the EPR data, we cannot tell whether this is due to a higher population of enzymes with closed ATPlid or because both substrate binding domains closed simultaneously in some enzymes, while other remained fully open. However, the NMR data shown in Figure 4F in the main text indicate chemical shift changes in both the ATP and AMP binding domains; thus, closing of only the ATPlid can be excluded.
Summarized, we think that both the AMPbd and ATPlid close when binding AP4A, which than leads to hydrolysis.  Table S5. Analysis was performed with different methods: DEERNet 29 (in respective colors as shown in figures before), with the conventional Tikhonov approach using the automated L-curve criterion 33 (L-c, black) and the newest approach using DeerLab 34 , here with the GCV criterion (GCV, red). Details and parameters for the different analysis methods are given in the text above. Similar background fits and resulting distance distributions ( Figure  S13) imply that the resulting distance distributions are independent on the analysis approach. For the following DEER data evaluations, we used the DEERNet method.  Table S5. All DEER curves were analyzed with DEERNet 29 within the DeerAnalysis software 33 (version 2019). The distance distributions P(r) were reproducible for different labeling batches with different double labeling degrees that consequently led to different modulation depths for the form factor F(t)/F(0). Note that different number of scans were accumulated, see Table S5 for details.  Table S5. All DEER curves were analyzed with DEERNet 29 within the DeerAnalysis software 33 (version 2019). The similar distance distributions imply that higher excess of AP4A does not change the enzyme conformation any further and that saturation is reached already with 2 mM AP4A.   Overlay of 1 H 15 N HSQC spectra acquired for wild-type 15 N AdK without ligand (apo, black), with 1 mM of AP4A (blue) and 1.7 mM of the inhibitor AP5A (red). Some amino acid residue peaks with large chemical shift differences are circled as guide to the eye, showing that the chemical shifts differences from apo are highest in the AP5A bound form. Exemplary residues Val111 and Lys200 for Figure S18 are highlighted by boxes. The comparison of these three spectra hints that AdK adopts a conformation that is neither open, apo-like, nor fully closed as it is the case when AP5A is bound. (Same spectra as in the main text, figure 4D). . The angle (θ) between the black reference vector between the apo signal (blue) and the AP5A signal (red), and the yellow activation vector between the apo and the AP4A signals determines how much of the chemical shift induced by AP4A binding can be attributed to the same conformal change as AP5A binding. A cosine of this angle above 0.9 denotes that the angle is small enough to be reliable included in the analysis. In this case for lysine 200, the cos(θ) is 0.99 and for Val111, the cos(θ) is 0.81. The dashed black/yellow part of the reference vector is the projected AP4A vector and the ratio between the length of this and the length of the full vector. (C) All cosine of the projection angles θ between AP4A and AP5A projection vectors. The black dots represents projections that were included in the analysis, thus fulfilling the requirements of a cos (θ) higher than 0.9 and a minimum shift difference of 0.05 ppm. Gray dots represent residues excluded from the analysis, with squares marking residues with a cos (θ) lower than 0.9 and triangles marking residues with an absolute value of the chemical shift lower than 0.05 ppm. (D) Size of the projected AP4A chemical shift vector on the AP5A chemical shift vector of all residues included in the analysis. The average projected chemical shift induced by AP4A binding is 0.44 ± 0.13 of the chemical shifts induced by AP5A.