Heterogeneity in the M. tuberculosis β-Lactamase Inhibition by Sulbactam

Abstract For decades, researchers have been determined to elucidate essential enzymatic functions on the atomic lengths scale by tracing atomic positions in real time. Our work builds on new possibilities unleashed by mix-and-inject serial crystallography (MISC) 1–5 at X-ray free electron laser facilities. In this approach, enzymatic reactions are triggered by mixing substrate or ligand solutions with enzyme microcrystals 6 . Here, we report in atomic detail and with millisecond time-resolution how the Mycobacterium tuberculosis enzyme BlaC is inhibited by sulbactam (SUB). Our results reveal ligand binding heterogeneity, ligand gating 7–9 , cooperativity, induced fit 10,11 and conformational selection 11–13 all from the same set of MISC data, detailing how SUB approaches the catalytic clefts and binds to the enzyme non-covalently before reacting to a trans- enamine. This was made possible in part by the application of the singular value decomposition 14 to the MISC data using a newly developed program that remains functional even if unit cell parameters change during the reaction.

the chemical structure of the inhibitor lead to an inactivated enzyme (E-I * ). The structures of intermediates shown in Extended Data Fig. 1 are relevant to the observed results presented in this paper within the measured timescale. SUB has also been described as a substrate of class A β-lactamases and can be hydrolyzed albeit at much slower rate than β-lactam antibiotics 35,39 .
BlaC can be crystallized in a monoclinic space group with four subunits A -D in the asymmetric unit ( Fig. 2a) 1,40 . In this crystal form, the BlaC structure displays a large cavity of 30 Å diameter in the center.
Additional cavities as large as 90 Å are identi ed within adjacent asymmetric units that allow easy diffusion of ligand molecules through the crystalline lattice. The E-I * structure of the BlaC-SUB has been determined at cryogenic temperature after soaking BlaC crystals for several minutes with SUB 27,41−43 .
More recently, Pandey and colleagues captured an intermediate (presumably E:I) at a single time point (66 ms) after mixing BlaC crystals with SUB 34 . Although subunits B/D displayed already a covalently bound adduct, an intact, non-covalently bound SUB was observed in subunits A/C (Fig. 2a). Subunits A/C are rather inactive, since they did not participate at all in the reaction with CEF in earlier experiments 1,33 . Given the inactivity of subunits A/C, it was not clear whether the reaction with SUB takes more time to complete or proceed at all. Therefore, a time-series of MISC datasets is necessary, to further investigate both the binding of SUB inhibitor to and its subsequent reaction with the BlaC.
As in any time-resolved experiment, except in those performed on ultrafast time scales [44][45][46][47] , multiple states can mix into any single time point observed during the reaction 48 . As demonstrated for X-ray data 14 these mixtures can be characterized and potentially separated using the singular value decomposition (SVD). Within this context, SVD is an unsupervised machine learning algorithm 49 that can inform from time-resolved X-ray data the number of observable processes which is equivalent to the number of relaxation times and the number of structurally distinguishable time-independent reaction intermediates 14 . In addition, it can provide information regarding the kinetic mechanism and the energetics of the reaction [50][51][52][53][54] . SVD has never been applied to a time-series from a MISC experiment. Originally, the SVD method was developed to work with isomorphous difference maps assuming that the unit cells in the crystals do not change during a reaction. However, the unit cell parameters of the BlaC crystals vary after mixing (Extended Data Table 1). This makes a data analysis approach that relies on a stable volume of interest challenging. Therefore, a new suite of programs "pySVD4TX" was developed, that remains functional even when the unit cell parameters change.

Binding Of Sulbactam
The four subunits of BlaC found in the asymmetric unit arrange in shape reminiscent to a torus (Fig. 2a). The alternating subunits are display similar binding kinetics while signi cantly differing from the adjacent ones. Here, subunits A and C share similarities. As do subunits B and D. However, there are distinct differences between these pairs. Whereas the catalytic clefts of subunits B and D are wide open, those of subunits A and C are partially occluded by the neighboring subunits ( Fig. 2b; Supplementary Movies 1 and 2). Particularly the residues Gln109 B/D and Gln108 B/D (the superscript B/D denotes residues from the neighboring subunits B and D, respectively) prevent substrate diffusion from the center of the torus, and Gln112 B/D and Arg173 block access to the active site from the exterior.
The reaction of SUB with BlaC was followed by difference electron density (DED) maps obtained at MISC delays (Δt misc ) of 3 ms, 6 ms, 15 ms, 30 ms, 270 ms and 700 ms (Methods and Extended Data Table 1). At 3, 6 and 15 ms the DED features near the active site of subunit A are weak (Fig. 3a-c). Substantial displacements of the residues anking the active site are observed (Extended Data Table 2; Supplementary Movie 1). Particularly, the long side chain residues like Gln112 B and Arg173 (called here the guardian residues, Fig. 2b) move outward of the active site before relaxing back to original positions.
Next to these residues at 3ms, almost 10 Å away from Ser70, there are positive densities that are spatially more spread out than that of water. A SUB molecule can be placed in the electron density (Fig. 3a).
However, re nement of the SUB is di cult (see Extended Data Fig. 2a for an explanation). At 6 and 15ms, density features appear closer towards Ser70 (Fig. 3b-c). These could indicate an initial trace of SUB molecules migrating to the active site after being held up by the guardian residues. Up to 15 ms, these densities are too weak that a SUB molecule can be placed with con dence. At 30 ms, stronger DED features (max σ = 5.5) appear around 4.5 Å from the catalytically active Ser70 (Fig. 3d). An intact sulbactam can be modelled which reproduces and corroborates the ndings at 66 ms obtained from an earlier experiment 34 . The β-lactam ring is oriented away from the Ser70. Between 66 ms and 240 ms, the SUB must rotate so that the β-lactam ring is positioned towards the Ser70 at which point the nucleophilic attack occurs (Extended Data Fig. 1a). At 240 ms, the elongated DED feature that originates from the Ser70 directly supports the presence of a covalently bound trans-enamine (TEN) (Fig. 3e). The BlaC-TEN adduct structurally relaxes until 700 ms, the nal time point in the time series (Fig. 3f). These snapshots of the reaction in progress were assembled to a movie (Supplementary Movie 1) of an enzyme in action.
In subunit B, there is no evidence of an intact SUB that accumulates in the active site ( Fig. 3g-h). Even the features that appeared near Arg173 in subunits A/C are not present. However, at 15 ms, the presence of strong DED that extends from the Ser70 supports a covalently bound TEN (Fig. 3i, Extended Data Fig. 2b). The occupancy re ned to 85% in B and 86% in D. This suggests the reaction is close to completion. At later time points, no large changes in the structure of the BlaC-TEN complex are apparent. (Fig. 3j-l).
However, in subunits A/C, the reaction keeps progressing and structural changes of both the entire enzyme and the TEN are observed (Supplementary Movie 1).

Temporal Variation Of Difference Electron Density
The SVD analysis is required to identify the number of intermediates as well as the relaxation times from the time-series of DED maps 14 (see Methods). The right singular vectors (rSVs) obtained from the SVD analysis plotted as a function of MISC time delays represent the temporal variation of the reaction. It is important that the relaxation processes inherent to each rSV are accurately determined. The slow initial progress and sudden increase of the magnitude of DED values in the active sites require an appropriate function that could account for this behavior. An excellent t was obtained by Eq. 2 which consists of a (step-like) logistic function that accounts for the steep rst phase and an exponential saturation component. Detailed values of the parameters of Eq. 2 are shown in the Extended Data Table 3.
The rst two signi cant rSVs for each subunit are shown by blue and red squares respectively ( Fig. 4a-d).
They allow for the determination of relaxation times τ 1 and τ 2 by tting Eq. 2, shown by their respective colored solid line. The rest of the rSVs, shown by colored diamonds in Fig. 4a, are distributed closely around zero and do not contribute to the subsequent analysis. The process with relaxation time τ 1 results from the rst appearance of DED in the active sites. In subunits A/C, this process re ects the appearance of the non-covalently bound SUB which occurs at around 23.5 and 24.5 ms, respectively. In subunits B/D, a covalently bound TEN is observed at 9.6 ms and 10.2 ms respectively after mixing. In subunits A/C, the process τ 2 results from the transformation of intact SUB to covalently bound TEN which happens approximately 75 ms and 90 ms after mixing. In contrast, in subunits B/D TEN is present already during the rst phase ( Fig. 3g-l; Extended Data Fig. 2e-h) and no further chemical modi cation of the TEN is observed. Despite this, a second relaxation process is also observed which coincides with the SUB to TEN formation in subunits A/C.

Inhibitor Diffusion
The kinetics of SUB binding to subunits B/D is evaluated rst, since it allows for an estimate of the ligand concentration in the unit cell that is required to analyze the observations for subunits A/C. The total concentration of BlaC monomers in the crystals (E free ) is ~ 16 mM. If only subunit B is considered, the effective [E free ] is 4 mM. The apparent diffusion time of SUB into the microcrystals is ~ 7 ms (Extended Data Table 4). It needs to be pointed out that MISC does not directly measure diffusion inside crystals.
Instead, free ligand concentrations, and related to them the diffusion time, are estimated only indirectly through the rate equations (Eqs. 3 and 4) which ultimately must reproduce the observed (re ned) ligand occupancies 34 . The unknown parameters in these equations must be varied until the concentration of species [E:I] and [E-I * ] best match their respective occupancy values (Fig. 5a). Additionally, relaxation times derived from the concentration pro les of intermediates must also agree with those obtained from SVD analysis of the DED maps (compare Extended Data Tables 3 and 4). The concentrations of the free SUB within the microcrystals rise slightly slower than the values estimated in the central ow of the injector (Extended Data Table 4, compare Fig. 5a blue solid line and blue squares). The use of a logistic function (Eq. 3) that describes the ligand increase in the unit cell is justi ed in particular by (i) the steep rst phase observed in the rSVs (Fig. 4), but also by (ii) the very rapid increase of the ligand in the inner ow of the injector constriction where saturation occurs already after 15 ms. The covalently bound TEN accumulates rapidly with a characteristic time of 11.5 ms (Extended Data Table 4) which is in line with the result from the SVD analysis (~ 10 ms). No additional intermediate is observed.
In subunits A/C, the apparent diffusion time of SUB necessary to reproduce the occupancies of the noncovalently bound intermediate is 20 ms (Fig. 5b and Extended Data Table 4). This is much longer than that observed in subunits B/D (7 ms). This lag can only be explained by a restricted access to the active site.
The guardian residues open the active site after about 6 ms (Supplementary Movie 2) which corresponds to about 35 mM outside ligand concentration (Fig. 5a). Entry to active site is controlled by an additional rate coe cient, k entry . k entry was modelled (Eq. 5) by an exponential function that depends on the (time-dependent) concentration difference between the outside and the inside of the active site, respectively, and a characteristic concentration difference ΔI c set to 25 mM (Extended Data Table 4). I out is the SUB concentration in the unit cell, which is known from the substrate  Fig. 4a, c). By 240 ms, more than 9 % of the crystal is occupied by TEN (Fig. 5b).

Discussion
While it is established that all the subunits take part in the reaction despite the heterogeneity, they do so at different rates. In subunits A/C, BlaC is inhibited by SUB via a two-step mechanism. The non-covalent intermediate can accumulate since the SUB is not properly oriented (Fig. 3d) and can react to TEN only after an additional time delay. SUB displacements will be restricted by interactions with the surrounding residues (Extended Data Table 2). To react with the active Ser70, the SUB must re-orient to expose the βlactam ring towards the serine. The catalytic opening of the β-lactam ring and the unfurling of the thiazolidine ring all happen in succession inside the narrow reaction center cavity. This severely lowers the rate of TEN formation.
In subunits B/D the covalent binding of SUB can apparently be explained by a one-step mechanism. However, the two-step mechanism described in Fig. 1 is also consistent with the observations when the non-covalent binding of substrate to the enzyme is much slower than the formation of TEN. By applying the two-step mechanism (Eq. 4), a k ncov value of ~ 1.5 mM − 1 s − 1 and a large k cov value of ~ 8000 s − 1 are estimated (Extended Data Table 4) so that the non-covalently bound E:I complex does not accumulate. The open binding pocket accommodates for large chemical and structural changes which result in the large k cov . The rate determining step is controlled by k ncov although the non-covalent BlaC-SUB complex never accumulates. In a one-step scenario (the free SUB reacts directly to TEN), the pseudo one-step rate coe cient would have to be the same as the k ncov , which was con rmed with a separate calculation (not shown). The two-step mechanism explains the enzymology in the active sites of all subunits in a consistent way.
A second relaxation phase is observed in the SVD analysis of DED maps from subunits B/D ( Fig. 4b and d), although the reaction to TEN has already taken place. It appears as if this is a result of the continuing reaction in subunits A/C and, related to this, an ongoing relaxation of the entire protein structure

Rapid Reaction With Sulbactam In BlaC Microcrystals
By taking into account the molecular volume that is enclosed by the van der Waal's surface, SUB (181.7 Å 3 ) is 2.5 times smaller than CEF (444.9 Å 3 ). For CEF, 50% occupancy of the enzyme substrate complex has been observed at Δt MISC = 5 ms in subunits B and D 34 . Due to the smaller size, SUB should diffuse faster into the crystals than CEF and the signature of an enzyme inhibitor complex is expected to appear already at the earliest time points (3 ms and 6 ms). However, the rst event that could be identi ed in the DED maps is the appearance of the TEN at 15 ms in subunits B/D (Fig. 3i). The concentrations of SUB at the active site of subunits B/D, which allow direct, unrestricted access, can be taken as an estimate of the SUB concentration inside the BlaC crystals. The slight lag between the estimated SUB concentration in the inner ow of the injector constriction and the concentrations in the crystals at early time points (Fig. 5a, compare blue squares with the blue line) is expected, since diffusion in BlaC crystals is slowed down in crystals 34 compared to water. The apparent diffusion time (7ms, Extended Data Table 4) of the SUB is almost the same as that for the antibiotic substrate CEF (~ 5 ms) 34 . Still, contrary to expectations based on the size of the SUB, no electron density has been present at the earliest time points. This can be explained by the smaller second order binding coe cient, k ncov (1.5 mM − 1 s − 1 for SUB compared to 3.2 mM − 1 s − 1 for CEF), that prevents the accumulation of electron density at earlier times.
The characteristic times observed in both classes of subunits for the formation of the covalently bond TEN species (around 10 ms and 90 ms, respectively, Fig. 5) are quite fast in comparison to earlier suggestions that the reaction might take minutes to complete 27,41 . The fast reaction provides an advantage when β-lactam substrates and the SUB inhibitor are competing for the same active site. For example, the non-covalent enzyme-substrate complex with CEF persists for up to 500 ms 33 . During this time, the CEF can leave the enzyme and be replaced rst competitively and then irreversibly by a quickly reacting SUB molecule. Since the inhibitor competes with co-administered antibiotics for the active site of BlaC, one can imagine that the covalent bond formation with an inhibitor must occur as fast as possible to effectively eliminate β-lactamase activity in the presence of substrate. This is in addition corroborated by Jones and coworkers who reported that SUB has a ten times higher a nity and binding constant for plasmid mediated class A β-lactamases compared to cefoperazone 55 , which is a third-generation cephalosporin-based antibiotic similar to CEF.

Ligand Gating, Induced Fit And Conformational Selection
Our results show that ligand binding to enzymes may be more complicated than initially thought 27,37,38 .
Only after a delay the ligand penetrates into the active sites of subunits A/C controlled by the guarding residues (Fig. 5b). The narrow entrance to active site ( Fig. 2b; Supplementary Movie 2), and the displacements of the guardian residues (Extended Data Table 2; Supplementary Movies 1 and 2) are reminiscent of a substrate tunneling-and-gating 7-9 like mechanism which has not yet been discovered in published structures of BlaC. Supplementary Movie 2 shows, that the movement of the guardian residues open the entrance, thereby controlling the access of ligand. More work is needed to determine the mechanism that drives the displacement of these residues. As these displacements were not observed when reacting with CEF 33,34 , an allosteric mechanism that links the position of the guardian residues to the covalent binding of SUB in adjacent subunits is unlikely. However, electrostatic interactions 56 of the negatively charged sulbactam with the positively charged Arg173 or even polar interaction with the Gln112 B,D , respectively, may plausibly induce these structural changes.
At Δt MISC of 30 ms and 66 ms 34  BlaC. This is a different form of conformational selection 12,13 , in a sense that there is not a particular "preferred" protein conformation that reacts with a substrate, but here, a particular (active) ligand orientation is required and "selected" by the enzyme for further reaction. It is the rate of the reorientation (rotation of the SUB) that seems to control the rate of this reaction. Once a favorable orientation is reached, further reaction to TEN is instantaneous on the timescale of the observation (90 ms) (Extended Data Table 4). This informs the design of improved (faster) inhibitors that consist of symmetric active moieties 57,58 or are engineered to enter the active sites in the correct orientation.
In subunits B/D, the inhibitor is brought in rapidly by diffusion and reacts instantaneously (< 1.5 ms) on the timescale of observation (> 15 ms). Neither an induced t nor conformational selection can be observed or distinguished which previously led to intense discussions for other enzymes 11 . The active site structures relax in unison with the extent of covalently bound inhibitor in all subunits as explained above.
The Fate of the Trans-Enamine It has been proposed that on longer timescales (> 30 min) a second nucleophilic attack by a nearby serine can occur in other, structurally closely related Ambler Class A β-lactamases 35 . This serine (Ser128 in BlaC) may react with the C5 position of the TEN (Extended Data Fig. 1). This is followed by the loss of the opened thiazolidine ring fragment (Extended Data Fig. 1d). A covalent bond may be formed between C5 and Ser128 of BlaC (Extended Data Fig. 2c) leading to the prolonged inhibition of the enzyme 35,41 . It has also been suggested that only the transient inhibition by TEN is responsible for the medical relevance of SUB as any reaction that lasts longer than one hour is irrelevant due to the bacterial lifecycle of ~ 30 minutes 41 . However, the life cycle of Mtb is around ~ 20 hours 59 . The permanent inhibition achieved only after the second nucleophilic attack might be the ultimate factor for SUB's clinical usefulness in ghting antibiotic resistance in slow growing bacteria like Mtb. Inspection of the soaked structure may give an answer. Covalently bound TENs were observed in all four subunits when BlaC was soaked with SUB for 3 hours (Extended Data Fig. 2e-h). The B-factors of the fragment beyond N4 that would be cleaved off (displayed in pale colors in Extended Data Fig. 2c) are consistently higher by 20 Å 2 than that of the part which would form the cross-linked species. However, it is more plausible that higher B-factors are caused by the dynamic disorder of the long TEN tail and not by the presence of mixture of intact and fragmented TEN. There is no clear evidence of TEN fragmentation, and TEN remains the physiologically important species for BlaC inhibition for hours.
A phosphate group binds to a speci c site immediately adjacent to the active serins 70 and 128 (Extended Data Fig. 2d) with multiple hydrogen bonds to surrounding residues. This location appears to be conserved among all published BlaC structures where others have also reported sulfate and acetate molecules in the same position (Extended Data Fig. 2d) 27,60−62 . Naturally occurring compounds with phosphate like group, for instance adenosine phosphates, might also interact with BlaC. β-lactamase production increases in some bacteria grown in a phosphate enriched medium 63 while phosphate can also promote the hydrolysis of the clavulanate inhibitor by BlaC 62 . More structures are required after soaking with high SUB concentrations for longer periods of time, perhaps days, to observe potentially fragmented TEN. Since the phosphate is replaceable 1,33,34 , the TEN might indeed react further. Larger inhibitors like tazobactam and clavulanic acid might also be able to displace the phosphate molecule directly. Recently, trans-enamine intermediates of tazobactam were identi ed in the serine β-lactamase TEM-171 at a position similar to that of TEN in BlaC and at another that is occupied by the phosphate in BlaC 64 . The diverse chemistry that is already observed very early on in BlaC may extend to much longer time scales.
There are other classes of β-lactamases that are more concerning than BlaC such as the metallo βlactamases (MBL). They are capable of hydrolyzing almost all clinically available β-lactam antibiotics and inhibitors 65,66 . Similar work to the one presented here and earlier 33

Declarations Data Availability
Data has been deposited in the Coherent X-ray Imaging Database (CXIDB) with accession code 209. The structure factors and the re ned coordinates of XFEL structure of BlaC mixed with sulbactam for 15ms, 30 ms, and 240ms, and the cryo-soaked structure have been deposited in the PDB as entries 8EBI, 8EBR, 8EC4 and 8ECF, respectively.

Code Availability
Code for each step of SVD calculation, and the characterization of concentrations can be obtained from the authors upon request. These custom codes will also be available on GitHub upon publication. The columns of the m × n matrix U, are called the left singular vectors (lSVs). They represent the basis (eigen) vectors of the original data in data matrix A. S is an n × n diagonal matrix, whose diagonal elements are called the singular values (SVs) of A. These non-negative values indicate how important or signi cant the columns of U are. The columns of the n × n matrix V, called the right singular vectors (rSVs), contain the associated temporal variation of the singular vectors in U. S contains n singular values in descending order of magnitude.
The number of signi cant singular values and vectors can inform how many kinetic processes can be resolved 14,72 . The SVD results can then be interpreted by globally tting suitable functions to the rSVs, that can consist, in the simplest form, of sums of exponentials. The rSVs contain information on the population dynamics of the species involved in the mechanism 14,72 .
The earlier program SVD4TX 14 and a newer version 73 could not be applied to X-ray data when large unit cell changes occur during the reaction since these implementations relied on a region of interest that is spatially xed. This is not given when the unit cell changes. In order to accommodate changing unit cells, a new approach was coded by a combination of custom bash scripts and python programs described below.

Adapting SVD for MISC Datasets with Changing Unit Cell Parameters
The DED map are calculated (Supplementary Methods) in the CCP4 le format 74 and cover the entire unit cell of the crystal. The maps are represented by a three-dimensional (3D) array with m x , m y and m z grid points for each unit cell axis, respectively. Each 3D grid point (voxel) contains the magnitude of the difference electron density at that given position (Extended Data Fig. 3 a). In such a DED map, positive features indicate regions where atoms have shifted away from their position in the reference model. Negative features are then found on top of the atoms in the reference model. Most of the map contains only spurious noise except in ROIs such as the active sites where larger structural changes are expected due to the binding or dissociation of a ligand (Extended Data Fig. 3 b). The noise within the majority of the difference map would interfere with the SVD analysis. To avoid this, a ROI was isolated individually for each subunit and an SVD performed only on the DED within. When multiple active sites are present, each active site can be investigated separately. Extended Data Fig. 4 shows a ow chart of the steps required to prepare the data matrix A. The steps are described in detail below.
Step 1: The coordinates of the atoms of the amino acid residues and the substrate of interest are speci ed in a particular subunit. This de nes the ROI. For the present work, four different ROIs were de ned, one for each subunit A to D, respectively, and investigated separately.
Step 2: A mask is calculated that covers the selected atoms plus a margin of choice (Extended Data Fig.3  b). The density values outside of the mask are set to 0 while the ones inside are left unchanged. This results in a masked map with the dimensions of original map with density values present only around the emerging DED in the active site (Extended Data Fig.3 c). This mask was evolved later (after step 4) by allowing only grid points that contain DED features greater or smaller than a certain sigma value (for example, plus or minus 3 σ) found at least in one time point 14 .
When the unit cell parameters do not change during the reaction, the difference maps at all time points will have the same number of voxels and the voxel size is also constant. However, once the unit cell dimensions change, either the voxel size will change, if the number of voxels is kept constant, or the number of voxels will change, if the voxel size is kept constant. If the voxel size changes, the DED value assigned to each voxel position will also change which will skew the SVD analysis. If the voxel numbers change, the SVD algorithm will fail as it requires that all the maps are represented by arrays of identical sizes. Accordingly, both conditions, (i) a constant voxel size and (ii) a constant number of grid points in the masked volume must be ful lled when the unit cell changes.
Step 3: In order to ful ll (i) the total number of grid points in the DED map is changed proportionally to the unit cell change. When the volume of the ROI is not changed, condition (ii) is automatically ful lled, and a suitable data matrix A can be constructed. However, when the unit cell parameters change, the ROI is also changing position. This must be addressed in addition.
Step 4: A box is chosen that will cover the density that was just masked out (Extended Data Fig.3 d). The box will include the ROI which is saved as a new map. The size of the box must be large enough such that the ROIs can be covered at all time points. The box must be calculated with reference to a stable structure (usually the protein main chain). As the protein chain displaces as a result of the change of the unit cell, the box will also move accordingly to cover the correct ROI (Extended Data Fig.3 e). As mentioned, the DED within the moving box can be used to evolve the mask that de nes the nal ROI as indicated in step 2.
Step 5: All m voxels in the evolved mask are converted to a one-dimensional (1D) column array, a vector in high (m) dimensional space. How the conversion is achieved does not matter as long as the same convention is applied to all the n maps. N of the m-dimensional vectors are arranged in ascending order of time to construct the data matrix A.
Step 6: SVD is performed on matrix A according to Eqn. 1.
Step 7: Trial functions are globally t to the signi cant rSVs to determine relaxation times and the minimum number of intermediates involved in the reaction (see e.g., Ihee et al., 2005 53 ).

Global t of the Signi cant rSVs
For a simple chemical kinetic mechanism with only rst-order reactions, relaxations are characterized by simple exponentials 48 . For higher order reactions, the rSVs have to be tted by suitable functions which must explain the changes of the electron density values in a chemically sensible way 14,29,75 . In our case, the signi cant rSVs were tted by Eq. 2 which, apart from a constant term, consists of a logistic function that accounts for abrupt changes of the electron densities observed in the active sites, and an additional saturation term. Further, the t was weighted by the square of the corresponding singular values S i .  Table 5). Eq. 3 is a logistic function where µ is the growth rate and t 0 is the midpoint value of the growth.
Once the SUB molecule reaches the active site of BlaC, the rst step is the formation of a non-covalent enzyme inhibitor complex (E:I) (Fig. 1). The process depends on the free BlaC concentration inside the crystal, and the rate coe cient for non-covalent complex formation (k ncov ). This step is usually reversible de ned by both the forward rate coe cient (k 1 ) and the backward rate coe cient (k − 1 ). However, the mounting concentrations of inhibitor inside the crystals forces more molecules towards the active site. At least initially, the binding rate depends on k 1 alone. The non-covalent E:I complex is the reactant for the next phase of the reaction where the β-lactam ring opens. The resulting covalently bound acyl-enzyme complex (E-I) (Fig. 1) is so short lived that it never accumulates in the timescale of the measurement. The SUB undergoes rapid modi cation, and a product is formed where the enzyme is covalently bound to the irreversibly modi ed inhibitor (E-I * ). k cov is the apparent rate coe cient which describes the velocity of E-I * formation directly from the E:I complex (Fig. 1). Ligand concentrations were determined by numerically integrating the following rate equations that describe the mechanism in Fig. 1 to calculate the concentrations of the non-covalently and covalently bound species shown in Fig. 5 b. At early MISC delays k entry is small. The channel opens, and k entry is large only when su cient SUB has accumulated in the unit cell (Fig. 2 d). All relevant parameters are listed in Extended Data Table 4.  Figure 1 A simpli ed two-step mechanism of BlaC inhibition by sulbactam. The rst step is the formation of the non-covalent enzyme inhibitor complex whose rate of formation depends on the concentration of the   site. An intact SUB molecule is placed there. (e) At 240 ms, the SUB has reacted with Ser70 to form TEN giving rise to an elongated density. (f) At 700 ms, the elongated density of the TEN is fully developed. Additional hydrogen bonds between the TEN and other side chains are shown. Subunit B, bottom row: (gh) At 3 and 6ms, no interpretable density was present in the catalytic center. (i) At 15ms, the SUB has already reacted with Ser70 to from TEN. (j-l) TEN densities as observed at Δ misc from 30 ms to 700 ms.
Gln109 A and Arg173 are marked in j and k, respectively.