Dynamic three-dimensional structures of a metal–organic framework captured with femtosecond serial crystallography

Crystalline systems consisting of small-molecule building blocks have emerged as promising materials with diverse applications. It is of great importance to characterize not only their static structures but also the conversion of their structures in response to external stimuli. Femtosecond time-resolved crystallography has the potential to probe the real-time dynamics of structural transitions, but, thus far, this has not been realized for chemical reactions in non-biological crystals. In this study, we applied time-resolved serial femtosecond crystallography (TR-SFX), a powerful technique for visualizing protein structural dynamics, to a metal–organic framework, consisting of Fe porphyrins and hexazirconium nodes, and elucidated its structural dynamics. The time-resolved electron density maps derived from the TR-SFX data unveil trifurcating structural pathways: coherent oscillatory movements of Zr and Fe atoms, a transient structure with the Fe porphyrins and Zr6 nodes undergoing doming and disordering movements, respectively, and a vibrationally hot structure with isotropic structural disorder. These findings demonstrate the feasibility of using TR-SFX to study chemical systems.


Kinetic analysis using kinetic modeling
An experimental difference electron density (DED) map at a certain time delay can be considered as a column vector, Δρ(p), where p is an index to denote the position in a three-dimensional lattice, as follows.The DED map at each time delay is conventionally expressed as a three-dimensional matrix, Δρ(px, py, pz).We reduced these three-dimensional matrices with a size of (nx, ny, nz) to a series of column vectors with a size of (nx × ny × nz, 1, 1) by reshaping the three-dimensional lattice (px, py, pz) into a single dimension of p. Consequently, the time-resolved DED maps, Δρ(px, py, pz, t), were also reduced into a two-dimensional matrix, Δρ(p, t), as follows: Δρ( , ) Δρ( , ) Δρ( , ) = [ , , , Δρ( , ) ] p t p t p t p t (2)   where each Δρ(p, ti) is a column vector for a DED map at a certain time delay, ti.Then, the measured DED map, Δρ(p, t), can be decomposed into two different components, one for the set of time-independent DED maps corresponding to each intermediate species or product and the other for their relative time-dependent contributions to the overall DED maps. Hence, where S is the number of reaction species (S = 3 for the specific system studied in this work), ci(tj) denotes the concentration of ith species at tj time delay, SADEDi(p) is the DED map corresponding to ith reaction species, i.e., species-associated DED map (SADED) of ith species, E(p) = [SADED1(p), SADED2(p), … , SADEDS(p)] and C(t) = [c1(t), c2(t), … , Here, we theoretically modeled the C(t) based on the kinetic model described above.
The corresponding SADED maps, E(p), are thus a solution of the following linear algebraic equation: T Δρ( , ) = E( )C( ) p t p t (4)  The transpose of this equation, C(t)E(p) T = Δρ(p, t) T , takes a well-known form of AX = B where A is C(t), X is E(p) T , and B is Δρ(p, t) T .Since the number of rows is larger than that of columns in C(t), the equation is over-constrained, whose solution is as follows: where C(t) + is the left pseudoinverse of C(t) defined as follows: In other words, we calculated the left pseudoinverse matrix of C(t) and multiplied its transpose to Δρ(p, t) to obtain E(p).The resulting three SADED maps are shown in Fig. 3A.
The calculated DED maps, obtained by calculating E(p)C(t) T of equation ( 4) are shown in Supplementary Fig. 5 (high contours) and Supplementary Fig. 6 (low contours), and the residual maps obtained by subtracting the calculated DED maps, E(p)C(t) T , from the experimental DED maps are shown in Supplementary Fig. 7.
Then, we analyzed the kinetics of the three species, Iosc, Itr, and Ihot.Specifically, we extracted the time profiles of the three species and applied a kinetic analysis on the extracted time profiles.To extract the time profiles, the DED map of each time delay was fitted as a linear combination of the three SADED maps.The time profiles for the Ihot, Itr, and Iosc were obtained from the coefficients of the linear combination.Finally, the time profiles were fitted as a convolution of an instrument response function (IRF) of ~200 fs full width at half maximum (FWHM) with a sum of exponential rise and decay functions and a damped cosine function.We note that, for the kinetic analysis of the time profiles for Itr and Iosc, the time constants obtained from the kinetic analysis of the 2nd RSVs of the Zr6 node and FeCO site were used without further optimization.In other words, for the fitting of the profiles of Itr and Iosc, the time constants for the rise (< 200 fs) and decay (47.1 ± 0.5 ps) of the 2nd RSVs and for the oscillatory motions in the RSVs (a period of 5.55 ± 0.01 ps and a damping constant of 2.68 ± 0.02 ps) were fixed.It can be confirmed that the time profiles of Itr and Iosc can be described well with these time constants obtained from the analysis of the 2nd RSVs.For the heating kinetics, the time profile of Ihot was fitted as a sum of two exponential rise functions convoluted with the IRF of ~200 fs FWHM.The two time constants for the exponential rise functions were optimized without a constraint.As a result, the time constants of 1.143 ± 0.005 ps and 11.32 ± 0.07 ps were obtained for the rise of the contribution of thermal motion.

Origin of the absorption of PCN-224(Fe) at 400 nm
The UV-visible absorption spectrum of PCN-224(Fe) exhibits a resemblance to that of Fe-porphyrin, particularly around ~400 nm, indicating that the absorption at 400 nm in PCN-224(Fe) is primarily attributed to the Fe-porphyrin group 2,3 .Furthermore, the similarity in UV-visible absorption spectra between PCN-224(Fe) and Fe-porphyrin implies that the effect of the ordered structure of the chromophores inside the MOF on UV-Vis absorption, specifically the electronic interaction between the chromophores, is not significant.The estimated distance between the Fe-porphyrin groups falls within the range of approximately 10 to 20 Å, which is relatively long.Additionally, as the presence of the Zr6 node between the Fe-porphyrin groups and the perpendicular geometry of adjacent Fe-porphyrin groups impede π conjugation among the Fe-porphyrin group, interactions among the Fe-porphyrin groups through conjugation are expected to be small.Consequently, these negligible interactions between the Fe-porphyrin groups would have minimal effects on the UV-visible absorption.
It is widely known that the absorption of porphyrin at around 400 nm corresponds to the π-π* transition, localized within the porphyrin orbitals.Thus, it is reasonable to infer that the ordered structure of the MOF and CO binding do not substantially affect the absorption.
Taking these findings into account, we conclude that the transient motions observed at the Zr6 node in the MOF are likely induced by the photon-absorbing Fe-porphyrin sites.

Data reproducibility and errors in difference electron density maps
We conducted repeated measurements, referred to as "runs," under the same conditions to enhance the signal-to-noise ratio of the data.Here, a "run" indicates a collection of multiple diffraction patterns at the covered time delays, spanning from negative to positive time delays.To assess data reproducibility, we examined the consistency between the measured runs.A total of seven runs were collected as summarized in Supplementary Table 8.
The data presented in the main text represents the average of all seven runs.Ideally, analyzing each individual run independently to ensure consistent results would be preferable.
In reality, however, the number of indexed images in each individual run is insufficient for standalone analysis.Therefore, we categorized the measured data into a total of four subsets and averaged the data corresponding to each subset.The subsets were composed as follows: Subset A: runs 2, 3, 4, 6, and 7; Subset B: runs 1, 2, 4, 5, and 7; Subset C: diffraction images from all runs whose indices are odd-numbered; Subset D: diffraction images from all runs whose indices are even-numbered.To examine the consistency between these four subsets, we generated the DED maps for each subset and performed singular value decomposition (SVD) analysis to compare the resulting right singular vectors (RSVs).The resulting RSVs are presented in Supplementary Fig. 8, where we compared the 1st and 2nd RSVs for all subsets.As discussed in the main text, the 1st RSV primarily describes the kinetics of Ihot and the 2nd RSV characterizes the kinetics of Iosc and Itr.Remarkably, the comparison reveals a high level of consistency in both the 1st and 2nd RSVs among the subsets.This consistency confirms the reproducibility of all subsets, which supports the overall reproducibility of our data.
In addition to the confirmation of the data reproducibility, the error of a DED map was estimated using the subset DED maps.For that purpose, here we assumed that the standard error of the mean (SEM) of four subset DED maps can represent the error of the DED map shown in our manuscript.We calculated the SEM of four subset DED maps for each time delay to examine the error of our DED maps.To represent the magnitude of resulting errors, isosurface plots were generated for comparison between the DED map and its corresponding error, defined as the SEM calculated from the subset DED maps, at a representative (1 ns) time delay as an example (Supplementary Fig. 9).We note that similar magnitudes of errors were observed for the other time delays as well.The comparison demonstrates that the error values are significantly smaller compared to the DED values, indicating a high degree of reproducibility of the measured data.

Comparison of the structural dynamics of PCN-224(Fe) and myoglobin
One of the representative systems containing Fe-porphyrin is myoglobin (Mb), and its structural dynamics have been investigated with TR-SFX.Although PCN-224(Fe) and Mb share similar Fe-porphyrin groups, they exhibit significant differences in their molecular We term the estimated temperature as Tlight, as this temperature is derived only from the ΔB of light atoms.
In contrast, our second method utilized time-resolved changes of isotropic B-factors (ΔB) of heavy atoms (Fe and Zr).Specifically, the isotropic displacement parameter was calculated from ΔB, with an average value of 0.118 Å 2 observed for the heavy atoms (Fe and Zr) over time delays ranging from -4 to -0.3 ps.Subsequently, the temperature scale parameter (kB) value was derived using a previously described method 7 .The temperature changes were then calculated based on two key assumptions: (1) kB is directly proportional to temperature, and (2) photo-induced movements, excluding those contributing to the temperature rise, account for approximately 20% of the kB value.The resulting temperature, denoted as Theavy, deduced from the ΔB of heavy atoms is depicted in Supplementary Fig. 10, where the temperature is estimated to increase by approximately 60 K at 1 ns.This estimate significantly exceeds the rise estimated from the isotropic B-factor of carbon atoms.
The significant difference in the estimated temperature increments derived from the two methods can be attributed to the differing atomic focus.The first method's estimation relies on the behavior of carbon atoms, whereas the second method is focused on heavy atoms.The difference underscores the pivotal role that the choice of atoms plays in these estimations, emphasizing the complexities inherent in precisely determining temperature changes in photoexcited materials.

Supplementary Table 1. Crystallographic parameters and complete experimental and refinement statistics for TR-SFX data from PCN-224(Fe)-CO for laser off and time
delays from -3.9 ps to 0.1 ps.

Response to alert
To prevent the intensities of the Bragg spots in the low-resolution region from exceeding the detector's saturation limit, the X-ray flux had to be limited.Consequently, the intensity of the Bragg spots in the high-resolution region was insufficient for detection, resulting in the limited resolution.

Response to alert
In serial femtosecond crystallography, diffraction images are acquired from a large number of different crystals with different sizes and crystallinity.Consequently, errors are introduced while merging data obtained from different crystals.Such errors result in a high wR2 value for this crystal structure.Moreover, the crystallographic analysis of PCN-224 reveals the presence of MOF-525 structural motifs, which manifest as defects within the crystal matrix.These defects are presumed to be a contributing factor to the elevated wR2 values noted in the analysis.

029_ALERT_3_B _diffrn_measured_fraction_theta_full value Low
CCDC reference number for the results related to this warning 2308209

Response to alert
Serial femtosecond crystallography employs multiple crystals for data collection.In an ideal scenario with ample experiment time and random crystal orientations, all the crystal orientations can be covered.However, a fraction of crystal orientation, as well as a fraction of the corresponding Bragg spots, may remain unmeasured due to either the limited experiment time or the presence of a preferred orientation of the crystals.Another factor contributing to the unmeasured fraction of Bragg spots is the weak intensities of the Bragg spots in the high-resolution region.
Supplementary Table 8.Summary of statistics for each of the seven measured runs.We repeated measurements seven times under the same experimental conditions and for the same or highly similar time delay points, and the data presented in the main text is the average of these measurements (runs) This table provides information for each run, including the time delay range, the number of time delays, the number of measured diffraction images per time delay, and the mean number of indexed images per time delay.

4 .
photoexcitation, CO dissociation and heme doming occur within their time resolution (~250 fs)4 .In our study, the structural species Itr, where the CO dissociation and structural changes of Fe-porphyrin take place, emerges within our time resolution of ~200 fs.These results are similar to those observed in Mb.On the other hand, PCN-224(Fe) lacks the progress of the heme doming between 0.6 and 3 ps observed in Mb.Another TR-SFX study proposed the formation of the high spin state in the heme group, characterized by positive DED surrounding iron in a cubic shape, and the occurrence of thermal effects, with negative DED at the iron position surrounded by a shell of positive DED5 .A DED map corresponding to the high-spin state formation was not observed, but the SADED of Ihot exhibits a similar feature due to the thermal effects observed in Mb.

Table 2 . Crystallographic parameters and complete experimental and refinement statistics for TR-SFX data from PCN-224(Fe)-CO for time delays from 0.4 ps to 1.85 ps.
Z, formula units in unit cell; ρ, density; μ, absorption coefficient; frames, number of recorded images including both hit and empty ones; crystals, number of indexed images; data, number of unique structure factors in refinement Supplementary