Data assimilation in operator algebras.

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

Time-resolved serial femtosecond crystallography at the European XFEL.

The European XFEL (EuXFEL) is a 3.4-km long X-ray source, which produces femtosecond, ultrabrilliant and spatially coherent X-ray pulses at megahertz (MHz) repetition rates. This X-ray source has been designed to enable the observation of ultrafast processes with near-atomic spatial resolution. Time-resolved crystallographic investigations on biological macromolecules belong to an important class of experiments that explore fundamental and functional structural displacements in these molecules. Due to the unusual MHz X-ray pulse structure at the EuXFEL, these experiments are challenging. Here, we demonstrate how a biological reaction can be followed on ultrafast timescales at the EuXFEL. We investigate the picosecond time range in the photocycle of photoactive yellow protein (PYP) with MHz X-ray pulse rates. We show that difference electron density maps of excellent quality can be obtained. The results connect the previously explored femtosecond PYP dynamics to timescales accessible at synchrotrons. This opens the door to a wide range of time-resolved studies at the EuXFEL.

Conformational landscape of a virus by single-particle X-ray scattering.

Using a manifold-based analysis of experimental diffraction snapshots from an X-ray free electron laser, we determine the three-dimensional structure and conformational landscape of the PR772 virus to a detector-limited resolution of 9 nm. Our results indicate that a single conformational coordinate controls reorganization of the genome, growth of a tubular structure from a portal vertex and release of the genome. These results demonstrate that single-particle X-ray scattering has the potential to shed light on key biological processes.

Propagation of Conformational Coordinates Across Angular Space in Mapping the Continuum of States from Cryo-EM Data by Manifold Embedding.

Recent approaches to the study of biological molecules employ manifold learning to single-particle cryo-EM data sets to map the continuum of states of a molecule into a low-dimensional space spanned by eigenvectors or "conformational coordinates". This is done separately for each projection direction (PD) on an angular grid. One important step in deriving a consolidated map of occupancies, from which the free energy landscape of the molecule can be derived, is to propagate the conformational coordinates from a given choice of "anchor PD" across the entire angular space. Even when one eigenvector dominates, its sign might invert from one PD to the next. The propagation of the second eigenvector is particularly challenging when eigenvalues of the second and third eigenvector are closely matched, leading to occasional inversions in their ranking as we move across the angular grid. In the absence of a computational approach, this propagation across the angular space has been done thus far "by hand" using visual clues, thus greatly limiting the general use of the technique. In this work we have developed a method that is able to solve the propagation problem computationally, by using optical flow and a probabilistic graphical model. We demonstrate its utility by selected examples.

Enzyme intermediates captured "on the fly" by mix-and-inject serial crystallography.

BACKGROUND: Ever since the first atomic structure of an enzyme was solved, the discovery of the mechanism and dynamics of reactions catalyzed by biomolecules has been the key goal for the understanding of the molecular processes that drive life on earth. Despite a large number of successful methods for trapping reaction intermediates, the direct observation of an ongoing reaction has been possible only in rare and exceptional cases. RESULTS: Here, we demonstrate a general method for capturing enzyme catalysis "in action" by mix-and-inject serial crystallography (MISC). Specifically, we follow the catalytic reaction of the Mycobacterium tuberculosis ß-lactamase with the third-generation antibiotic ceftriaxone by time-resolved serial femtosecond crystallography. The results reveal, in near atomic detail, antibiotic cleavage and inactivation from 30 ms to 2 s. CONCLUSIONS: MISC is a versatile and generally applicable method to investigate reactions of biological macromolecules, some of which are of immense biological significance and might be, in addition, important targets for structure-based drug design. With megahertz X-ray pulse rates expected at the Linac Coherent Light Source II and the European X-ray free-electron laser, multiple, finely spaced time delays can be collected rapidly, allowing a comprehensive description of biomolecular reactions in terms of structure and kinetics from the same set of X-ray data.

Continuous changes in structure mapped by manifold embedding of single-particle data in cryo-EM.

Cryo-electron microscopy, when combined with single-particle reconstruction, is a powerful method for studying macromolecular structure. Recent developments in detector technology have pushed the resolution into a range comparable to that of X-ray crystallography. However, cryo-EM is able to separate and thus recover the structure of each of several discrete structures present in the sample. For the more general case involving continuous structural changes, a novel technique employing manifold embedding has been recently demonstrated. Potentially, the entire work-cycle of a molecular machine may be observed as it passes through a continuum of states, and its free-energy landscape may be mapped out. This technique will be outlined and discussed in the context of its application to a large single-particle dataset of yeast ribosomes.

Trajectories of the ribosome as a Brownian nanomachine.

A Brownian machine, a tiny device buffeted by the random motions of molecules in the environment, is capable of exploiting these thermal motions for many of the conformational changes in its work cycle. Such machines are now thought to be ubiquitous, with the ribosome, a molecular machine responsible for protein synthesis, increasingly regarded as prototypical. Here we present a new analytical approach capable of determining the free-energy landscape and the continuous trajectories of molecular machines from a large number of snapshots obtained by cryogenic electron microscopy. We demonstrate this approach in the context of experimental cryogenic electron microscope images of a large ensemble of nontranslating ribosomes purified from yeast cells. The free-energy landscape is seen to contain a closed path of low energy, along which the ribosome exhibits conformational changes known to be associated with the elongation cycle. Our approach allows model-free quantitative analysis of the degrees of freedom and the energy landscape underlying continuous conformational changes in nanomachines, including those important for biological function.

Energy landscapes from cryo-EM snapshots: a benchmarking study.

Biomolecules undergo continuous conformational motions, a subset of which are functionally relevant. Understanding, and ultimately controlling biomolecular function are predicated on the ability to map continuous conformational motions, and identify the functionally relevant conformational trajectories. For equilibrium and near-equilibrium processes, function proceeds along minimum-energy pathways on one or more energy landscapes, because higher-energy conformations are only weakly occupied. With the growing interest in identifying functional trajectories, the need for reliable mapping of energy landscapes has become paramount. In response, various data-analytical tools for determining structural variability are emerging. A key question concerns the veracity with which each data-analytical tool can extract functionally relevant conformational trajectories from a collection of single-particle cryo-EM snapshots. Using synthetic data as an independently known ground truth, we benchmark the ability of four leading algorithms to determine biomolecular energy landscapes and identify the functionally relevant conformational paths on these landscapes. Such benchmarking is essential for systematic progress toward atomic-level movies of continuous biomolecular function.

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

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.

Heterogeneity in M. tuberculosis ß-lactamase inhibition by Sulbactam.

For decades, researchers have elucidated essential enzymatic functions on the atomic length scale by tracing atomic positions in real-time. Our work builds on possibilities unleashed by mix-and-inject serial crystallography (MISC) at X-ray free electron laser facilities. In this approach, enzymatic reactions are triggered by mixing substrate or ligand solutions with enzyme microcrystals. Here, we report in atomic detail (between 2.2 and 2.7 Å resolution) by room-temperature, time-resolved crystallography with millisecond time-resolution (with timepoints between 3 ms and 700 ms) how the Mycobacterium tuberculosis enzyme BlaC is inhibited by sulbactam (SUB). Our results reveal ligand binding heterogeneity, ligand gating, cooperativity, induced fit, and conformational selection all from the same set of MISC data, detailing how SUB approaches the catalytic clefts and binds to the enzyme noncovalently before reacting to a trans-enamine. This was made possible in part by the application of singular value decomposition to the MISC data using a program that remains functional even if unit cell parameters change up to 3 Å during the reaction.

The symmetries of image formation by scattering. I. Theoretical framework.

We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, and the remotest corners of the universe. Here, we present an approach to image formation based on the symmetry properties of operations in three-dimensional space. Augmented with graph-theoretic means, this approach can recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. This is critical for the burgeoning field of structure recovery by X-ray Free Electron Lasers, as well as the more established electron microscopic techniques, including cryo-electron microscopy of biological systems. In a subsequent paper, we demonstrate the recovery of structure and dynamics from experimental, ultralow-signal random sightings of systems with X-rays, electrons, and photons, with no orientational or timing information.

The symmetries of image formation by scattering. II. Applications.

We show that the symmetries of image formation by scattering enable graph-theoretic manifold-embedding techniques to extract structural and timing information from simulated and experimental snapshots at extremely low signal. The approach constitutes a physically-based, computationally efficient, and noise-robust route to analyzing the large and varied datasets generated by existing and emerging methods for studying structure and dynamics by scattering. We demonstrate three-dimensional structure recovery from X-ray diffraction and cryo-electron microscope image snapshots of unknown orientation, the latter at 12 times lower dose than currently in use. We also show that ultra-low-signal, random sightings of dynamically evolving systems can be sequenced into high quality movies to reveal their evolution. Our approach offers a route to recovering timing information in time-resolved experiments, and extracting 3D movies from two-dimensional random sightings of dynamic systems.

Energy landscape of the SARS-CoV-2 reveals extensive conformational heterogeneity.

Cryo-electron microscopy (cryo-EM) has produced a number of structural models of the SARS-CoV-2 spike, already prompting biomedical outcomes. However, these reported models and their associated electrostatic potential maps represent an unknown admixture of conformations stemming from the underlying energy landscape of the spike protein. As with any protein, some of the spike's conformational motions are expected to be biophysically relevant, but cannot be interpreted only by static models. Using experimental cryo-EM images, we present the energy landscape of the glycosylated spike protein, and identify the diversity of low-energy conformations in the vicinity of its open (so called 1RBD-up) state. The resulting atomic refinement reveal global and local molecular rearrangements that cannot be inferred from an average 1RBD-up cryo-EM model. Here we report varied degrees of "openness" in global conformations of the 1RBD-up state, not revealed in the single-model interpretations of the density maps, together with conformations that overlap with the reported models. We discover how the glycan shield contributes to the stability of these low-energy conformations. Five out of six binding sites we analyzed, including those for engaging ACE2, therapeutic mini-proteins, linoleic acid, two different kinds of antibodies, switch conformations between their known apo- and holo-conformations, even when the global spike conformation is 1RBD-up. This apo-to-holo switching is reminiscent of a conformational preequilibrium. We found only one binding site, namely that of AB-C135 remains in apo state within all the sampled free energy-minimizing models, suggesting an induced fit mechanism for the docking of this antibody to the spike.

Dynamics retrieval from stochastically weighted incomplete data by low-pass spectral analysis.

Time-resolved serial femtosecond crystallography (TR-SFX) provides access to protein dynamics on sub-picosecond timescales, and with atomic resolution. Due to the nature of the experiment, these datasets are often highly incomplete and the measured diffracted intensities are affected by partiality. To tackle these issues, one established procedure is that of splitting the data into time bins, and averaging the multiple measurements of equivalent reflections within each bin. This binning and averaging often involve a loss of information. Here, we propose an alternative approach, which we call low-pass spectral analysis (LPSA). In this method, the data are projected onto the subspace defined by a set of trigonometric functions, with frequencies up to a certain cutoff. This approach attenuates undesirable high-frequency features and facilitates retrieving the underlying dynamics. A time-lagged embedding step can be included prior to subspace projection to improve the stability of the results with respect to the parameters involved. Subsequent modal decomposition allows to produce a low-rank description of the system's evolution. Using a synthetic time-evolving model with incomplete and partial observations, we analyze the LPSA results in terms of quality of the retrieved signal, as a function of the parameters involved. We compare the performance of LPSA to that of a range of other sophisticated data analysis techniques. We show that LPSA allows to achieve excellent dynamics reconstruction at modest computational cost. Finally, we demonstrate the superiority of dynamics retrieval by LPSA compared to time binning and merging, which is, to date, the most commonly used method to extract dynamical information from TR-SFX data.

The case for data science in experimental chemistry: examples and recommendations.

The physical sciences community is increasingly taking advantage of the possibilities offered by modern data science to solve problems in experimental chemistry and potentially to change the way we design, conduct and understand results from experiments. Successfully exploiting these opportunities involves considerable challenges. In this Expert Recommendation, we focus on experimental co-design and its importance to experimental chemistry. We provide examples of how data science is changing the way we conduct experiments, and we outline opportunities for further integration of data science and experimental chemistry to advance these fields. Our recommendations include establishing stronger links between chemists and data scientists; developing chemistry-specific data science methods; integrating algorithms, software and hardware to 'co-design' chemistry experiments from inception; and combining diverse and disparate data sources into a data network for chemistry research.

Energy Landscape of the SARS-CoV-2 Reveals Extensive Conformational Heterogeneity.

Cryo-electron microscopy (cryo-EM) has produced a number of structural models of the SARS-CoV-2 spike, already prompting biomedical outcomes. However, these reported models and their associated electrostatic potential maps represent an unknown admixture of conformations stemming from the underlying energy landscape of the spike protein. As for any protein, some of the spike's conformational motions are expected to be biophysically relevant, but cannot be interpreted only by static models. Using experimental cryo-EM images, we present the energy landscape of the spike protein conformations, and identify molecular rearrangements along the most-likely conformational path in the vicinity of the open (so called 1RBD-up) state. The resulting global and local atomic refinements reveal larger movements than those expected by comparing the reported 1RBD-up and 1RBD-down cryo-EM models. Here we report greater degrees of "openness" in global conformations of the 1RBD-up state, not revealed in the single-model interpretations of the density maps, together with conformations that overlap with the reported models. We discover how the glycan shield contributes to the stability of these conformations along the minimum free-energy pathway. A local analysis of seven key binding pockets reveals that six out them, including those for engaging ACE2, therapeutic mini-proteins, linoleic acid, two different kinds of antibodies, and protein-glycan interaction sites, switch conformations between their known apo- and holo-conformations, even when the global spike conformation is 1RBD-up. This is reminiscent of a conformational pre-equilibrium. We found only one binding pocket, namely antibody AB-C135 to remain closed along the entire minimum free energy path, suggesting an induced fit mechanism for this enzyme.

Selecting XFEL single-particle snapshots by geometric machine learning.

A promising new route for structural biology is single-particle imaging with an X-ray Free-Electron Laser (XFEL). This method has the advantage that the samples do not require crystallization and can be examined at room temperature. However, high-resolution structures can only be obtained from a sufficiently large number of diffraction patterns of individual molecules, so-called single particles. Here, we present a method that allows for efficient identification of single particles in very large XFEL datasets, operates at low signal levels, and is tolerant to background. This method uses supervised Geometric Machine Learning (GML) to extract low-dimensional feature vectors from a training dataset, fuse test datasets into the feature space of training datasets, and separate the data into binary distributions of "single particles" and "non-single particles." As a proof of principle, we tested simulated and experimental datasets of the Coliphage PR772 virus. We created a training dataset and classified three types of test datasets: First, a noise-free simulated test dataset, which gave near perfect separation. Second, simulated test datasets that were modified to reflect different levels of photon counts and background noise. These modified datasets were used to quantify the predictive limits of our approach. Third, an experimental dataset collected at the Stanford Linear Accelerator Center. The single-particle identification for this experimental dataset was compared with previously published results and it was found that GML covers a wide photon-count range, outperforming other single-particle identification methods. Moreover, a major advantage of GML is its ability to retrieve single particles in the presence of structural variability.

A glycan gate controls opening of the SARS-CoV-2 spike protein.

SARS-CoV-2 infection is controlled by the opening of the spike protein receptor binding domain (RBD), which transitions from a glycan-shielded 'down' to an exposed 'up' state to bind the human angiotensin-converting enzyme 2 receptor and infect cells. While snapshots of the 'up' and 'down' states have been obtained by cryo-electron microscopy and cryo-electron tomagraphy, details of the RBD-opening transition evade experimental characterization. Here over 130 µs of weighted ensemble simulations of the fully glycosylated spike ectodomain allow us to characterize more than 300 continuous, kinetically unbiased RBD-opening pathways. Together with ManifoldEM analysis of cryo-electron microscopy data and biolayer interferometry experiments, we reveal a gating role for the N-glycan at position N343, which facilitates RBD opening. Residues D405, R408 and D427 also participate. The atomic-level characterization of the glycosylated spike activation mechanism provided herein represents a landmark study for ensemble pathway simulations and offers a foundation for understanding the fundamental mechanisms of SARS-CoV-2 viral entry and infection.