**Singular value decomposition**

A kinetic analysis was performed by the application of the singular value decomposition (SVD) to the X-ray data 14 using the DEDomit maps (for data collection and difference map calculation, see the Supplementary Information, SI). A region of interest (ROI) was determined that corresponds roughly to the volume of an individual active site selected from the four subunits in the asymmetric unit. The ROI covers the volume occupied by the SUB and TEN ligands and the entire side chain of Ser70. It touches the terminal atoms of residues Lys 73, Glu168, Thr239, Asn172, Arg173, Ser128, Ser102 and Gln112 (from B/D) in the case of subunits A/C. Difference electron density values found in grid points (voxels) of the ROI were assigned to a m-dimensional vector. N = 6 of these vectors were obtained for measured Δtmisc and assembled to a m x n dimensional matrix **A**, called the data matrix. SVD is the factorization of matrix **A** into three matrices: **U**, **S** and **V****T** according to

**A = U·S·V** **T** (1)

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 significant 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 significant singular values and vectors can inform how many kinetic processes can be resolved 14,72. The SVD results can then be interpreted by globally fitting 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 fixed. 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 file format 74 and cover the entire unit cell of the crystal. The maps are represented by a three-dimensional (3D) array with mx, my and mz 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 flow 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 specified in a particular subunit. This defines the ROI. For the present work, four different ROIs were defined, 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 fulfilled when the unit cell changes.

**Step 3:** In order to fulfill (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 fulfilled, 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 defines the final 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 fit to the significant rSVs to determine relaxation times and the minimum number of intermediates involved in the reaction (see e.g., Ihee et al., 200553).

**Global fit of the Significant rSVs**

For a simple chemical kinetic mechanism with only first-order reactions, relaxations are characterized by simple exponentials 48. For higher order reactions, the rSVs have to be fitted 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 significant rSVs were fitted 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 fit was weighted by the square of the corresponding singular values Si.

$$S_{i}^{2}\,rS{V_i}(t)={A_{0,i}}+\frac{{{A_{1,i}}}}{{1+{e^{ - \lambda (t - {\tau _i})}}}}+{A_{2,i}}\left( {1 - {e^{ - \frac{t}{{{\tau _2}}}}}} \right)$$

2

While the amplitudes A’s are varied independently for each significant rSVi (i = 1…n), the parameter λ and the relaxation times, τ1 and τ2 are shared globally. The number of relaxation times (here 2) is equal to the number of significant rSVs and to the number of distinguishable processes.

**Species Concentrations**

The diffusion of SUB molecules into the BlaC crystals and subsequently into the active sites triggers the reaction. Concentrations of SUB in the central flow were estimated according to Calvey et al., 2019 76. At the longest Δtmisc= 700 ms, the SUB concentration was 100 mM, which was used as the maximum ligand concentration for all calculations. The resulting evolution of concentration of SUB at the active sites was modeled by Eq. 3.

$$I(t)=\frac{{{I_{\hbox{max} }}}}{{1+{e^{ - \mu \left( {t - {t_0}} \right)}}}}$$

3

I(t) is the concentration of SUB at the active sites averaged over all unit cells in the crystal as a function of time and Imax is the maximum (100 mM) SUB concentration (Extended Data Table 5). Eq. 3 is a logistic function where µ is the growth rate and t0 is the midpoint value of the growth.

Once the SUB molecule reaches the active site of BlaC, the first 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 coefficient for non-covalent complex formation (kncov). This step is usually reversible defined by both the forward rate coefficient (k1) and the backward rate coefficient (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 k1 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 modification, and a product is formed where the enzyme is covalently bound to the irreversibly modified inhibitor (E- I*). kcov is the apparent rate coefficient 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.

$$\begin{gathered} d\,[E:I]=[{E_{free}}]({t_i}) \cdot [I]({t_i}) \cdot {k_{n\operatorname{cov} }} \cdot dt \hfill \\ [{E_{free}}]({t_{i+1}})=[{E_{free}}]({t_i}) - d[E:I] \hfill \\ d[E - I*]=[E:I]({t_i}) \cdot {k_{\operatorname{cov} }} \cdot dt \hfill \\ [E - I*]({t_{i+1}})=[E - I*]({t_i})+d[E - I*] \hfill \\ [E:I]({t_{i+1}})=[E:I]({t_i})+d\,[E:I] - d[E - I*] \hfill \\ {t_{i+1}}={t_i}+dt \hfill \\ \end{gathered}$$

4

d[E:I] is the change in concentration of the non-covalent BlaC-SUB complex at any given time (ti) and depends on the free enzyme concentration, [Efree], the second order rate coefficient of non-covalent binding (kncov), and the inhibitor concentration, [I]. [I] is calculated from Eq. 3. As the concentration of [E:I] increases, [Efree] decreases. d[E-I*] is the increase of the covalently bound TEN. It depends on the available concentration of the non-covalent BlaC-SUB complex [E:I], and the rate coefficient kcov. [E:I] decreases by the same rate [E- I*] increases.

The increase of the SUB concentration in the active site (Iin) is delayed relative to that of the SUB concentration in the unit cell (Iout). To account for this delay, the rate coefficient that determines the entry into the active site (kentry, Fig. 2 c) is assumed to be dependent on the concentration difference \(\Delta I(t)={I_{out}}(t) - {I_{in}}(t)\) between outside and inside the active site and a characteristic difference \(\Delta {I_c}\). It is modeled by an exponential function

$${k_{entry}}={k_{{\text{max,entry}}}}\left( {1 - {{\exp }^{ - \frac{{\Delta I(t)}}{{\Delta {I_c}}}}}} \right)$$

5

The relevant SUB concentrations within the active site (Iin) are generated by solving the following rate equation:

$$\begin{gathered} d{I_{in}}={k_{entry}} \cdot {I_{out}} \cdot dt \hfill \\ {I_{in}}({t_{i+1}})={I_{in}}({t_i})+d{I_{in}}\,. \hfill \\ \end{gathered}$$

6

Eqn. 6 is used in lieu of Eq. 3 to calculate the relevant inhibitor concentration: Iin(t) is fed as [I](t) to Eq. 4 to calculate the concentrations of the non-covalently and covalently bound species shown in Fig. 5 b. At early MISC delays kentry is small. The channel opens, and kentry is large only when sufficient SUB has accumulated in the unit cell (Fig. 2 d). All relevant parameters are listed in Extended Data Table 4.