Bayesian Modeling of Biomolecular Assemblies with Cryo-EM Maps

Bayesian methods in integrative structure modeling

There is a growing interest in characterizing the structure and dynamics of large biomolecular assemblies and their interactions within the cellular environment. A diverse array of experimental techniques allows us to study biomolecular systems on a variety of length and time scales. These techniques range from imaging with light, X-rays or electrons, to spectroscopic methods, cross-linking mass spectrometry and functional genomics approaches, and are complemented by AI-assisted protein structure prediction methods. A challenge is to integrate all of these data into a model of the system and its functional dynamics. This review focuses on Bayesian approaches to integrative structure modeling. We sketch the principles of Bayesian inference, highlight recent applications to integrative modeling and conclude with a discussion of current challenges and future perspectives.

Gentle and fast all-atom model refinement to cryo-EM densities via a maximum likelihood approach

Better detectors and automated data collection have generated a flood of high-resolution cryo-EM maps, which in turn has renewed interest in improving methods for determining structure models corresponding to these maps. However, automatically fitting atoms to densities becomes difficult as their resolution increases and the refinement potential has a vast number of local minima. In practice, the problem becomes even more complex when one also wants to achieve a balance between a good fit of atom positions to the map, while also establishing good stereochemistry or allowing protein secondary structure to change during fitting. Here, we present a solution to this challenge using a maximum likelihood approach by formulating the problem as identifying the structure most likely to have produced the observed density map. This allows us to derive new types of smooth refinement potential-based on relative entropy-in combination with a novel adaptive force scaling algorithm to allow balancing of force-field and density-based potentials. In a low-noise scenario, as expected from modern cryo-EM data, the relative-entropy based refinement potential outperforms alternatives, and the adaptive force scaling appears to aid all existing refinement potentials. The method is available as a component in the GROMACS molecular simulation toolkit.

MDSPACE: Extracting Continuous Conformational Landscapes from Cryo-EM Single Particle Datasets Using 3D-to-2D Flexible Fitting based on Molecular Dynamics Simulation

This article presents an original approach for extracting atomic-resolution landscapes of continuous conformational variability of biomolecular complexes from cryo electron microscopy (cryo-EM) single particle images. This approach is based on a new 3D-to-2D flexible fitting method, which uses molecular dynamics (MD) simulation and is embedded in an iterative conformational-landscape refinement scheme. This new approach is referred to as MDSPACE, which stands for Molecular Dynamics simulation for Single Particle Analysis of Continuous Conformational hEterogeneity. The article describes the MDSPACE approach and shows its performance using synthetic and experimental datasets.

NMR-Based Configurational Assignments of Natural Products: Gibbs Sampling and Bayesian Inference Using Floating Chirality Distance Geometry Calculations

Floating chirality restrained distance geometry (fc-rDG) calculations are used to directly evolve structures from NMR data such as NOE-derived intramolecular distances or anisotropic residual dipolar couplings (RDCs). In contrast to evaluating pre-calculated structures against NMR restraints, multiple configurations (diastereomers) and conformations are generated automatically within the experimental limits. In this report, we show that the "unphysical" rDG pseudo energies defined from NMR violations bear statistical significance, which allows assigning probabilities to configurational assignments made that are fully compatible with the method of Bayesian inference. These "diastereomeric differentiabilities" then even become almost independent of the actual values of the force constants used to model the restraints originating from NOE or RDC data.

Bayesian Random Tomography of Particle Systems

Random tomography is a common problem in imaging science and refers to the task of reconstructing a three-dimensional volume from two-dimensional projection images acquired in unknown random directions. We present a Bayesian approach to random tomography. At the center of our approach is a meshless representation of the unknown volume as a mixture of spherical Gaussians. Each Gaussian can be interpreted as a particle such that the unknown volume is represented by a particle cloud. The particle representation allows us to speed up the computation of projection images and to represent a large variety of structures accurately and efficiently. We develop Markov chain Monte Carlo algorithms to infer the particle positions as well as the unknown orientations. Posterior sampling is challenging due to the high dimensionality and multimodality of the posterior distribution. We tackle these challenges by using Hamiltonian Monte Carlo and a global rotational sampling strategy. We test the approach on various simulated and real datasets.

Bayesian Inference: The Comprehensive Approach to Analyzing Single-Molecule Experiments

Biophysics experiments performed at single-molecule resolution provide exceptional insight into the structural details and dynamic behavior of biological systems. However, extracting this information from the corresponding experimental data unequivocally requires applying a biophysical model. In this review, we discuss how to use probability theory to apply these models to single-molecule data. Many current single-molecule data analysis methods apply parts of probability theory, sometimes unknowingly, and thus miss out on the full set of benefits provided by this self-consistent framework. The full application of probability theory involves a process called Bayesian inference that fully accounts for the uncertainties inherent to single-molecule experiments. Additionally, using Bayesian inference provides a scientifically rigorous method of incorporating information from multiple experiments into a single analysis and finding the best biophysical model for an experiment without the risk of overfitting the data. These benefits make the Bayesian approach ideal for analyzing any type of single-molecule experiment.

Architecture of the flexible tail tube of bacteriophage SPP1

Bacteriophage SPP1 is a double-stranded DNA virus of the Siphoviridae family that infects the bacterium Bacillus subtilis. This family of phages features a long, flexible, non-contractile tail that has been difficult to characterize structurally. Here, we present the atomic structure of the tail tube of phage SPP1. Our hybrid structure is based on the integration of structural restraints from solid-state nuclear magnetic resonance (NMR) and a density map from cryo-EM. We show that the tail tube protein gp17.1 organizes into hexameric rings that are stacked by flexible linker domains and, thus, form a hollow flexible tube with a negatively charged lumen suitable for the transport of DNA. Additionally, we assess the dynamics of the system by combining relaxation measurements with variances in density maps.

Bacteriophages of the Siphoviridae family have a long, flexible, non-contractile tail that has been difficult to characterize structurally. Here, the authors present the atomic structure of the tail tube of one of these phages, showing a hollow flexible tube formed by hexameric rings stacked by flexible linkers.

Hyper-Molecules: on the Representation and Recovery of Dynamical Structures for Applications in Flexible Macro-Molecules in Cryo-EM

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for obtaining 3-D reconstructions of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. These molecules are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori; one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in reconstructing rigid molecules based on homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the "hyper-molecule" theoretical framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for reconstructing such heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a preliminary prototype implementation, applied to synthetic data.

Integrating cryo-EM and NMR data

Single-particle cryo-electron microscopy (cryo-EM) is increasingly used as a technique to determine the atomic structure of challenging biological systems. Recent advances in microscope engineering, electron detection, and image processing have allowed the structural determination of bigger and more flexible targets than possible with the complementary techniques X-ray crystallography and NMR spectroscopy. However, there exist many biological targets for which atomic resolution cannot be currently achieved with cryo-EM, making unambiguous determination of the protein structure impossible. Although determining the structure of large biological systems using solely NMR is often difficult, highly complementary experimental atomic-level data for each molecule can be derived from the spectra, and used in combination with cryo-EM data. We review here strategies with which both techniques can be synergistically combined, in order to reach detail and understanding unattainable by each technique acting alone; and the types of biological systems for which such an approach would be desirable.

Use of Computational Biochemistry for Elucidating Molecular Mechanisms of Nitric Oxide Synthase

Nitric oxide (NO) is an essential signaling molecule in the regulation of multiple cellular processes. It is endogenously synthesized by NO synthase (NOS) as the product of L-arginine oxidation to L-citrulline, requiring NADPH, molecular oxygen, and a pterin cofactor. Two NOS isoforms are constitutively present in cells, nNOS and eNOS, and a third is inducible (iNOS). Despite their biological relevance, the details of their complex structural features and reactivity mechanisms are still unclear. In this review, we summarized the contribution of computational biochemistry to research on NOS molecular mechanisms. We described in detail its use in studying aspects of structure, dynamics and reactivity. We also focus on the numerous outstanding questions in the field that could benefit from more extensive computational investigations.

Advances in coarse-grained modeling of macromolecular complexes

Recent progress in coarse-grained (CG) molecular modeling and simulation has facilitated an influx of computational studies on biological macromolecules and their complexes. Given the large separation of length-scales and time-scales that dictate macromolecular biophysics, CG modeling and simulation are well-suited to bridge the microscopic and mesoscopic or macroscopic details observed from all-atom molecular simulations and experiments, respectively. In this review, we first summarize recent innovations in the development of CG models, which broadly include structure-based, knowledge-based, and dynamics-based approaches. We then discuss recent applications of different classes of CG models to explore various macromolecular complexes. Finally, we conclude with an outlook for the future in this ever-growing field of biomolecular modeling.

Bayesian Weighing of Electron Cryo-Microscopy Data for Integrative Structural Modeling

Cryo-electron microscopy (cryo-EM) has become a mainstream technique for determining the structures of complex biological systems. However, accurate integrative structural modeling has been hampered by the challenges in objectively weighing cryo-EM data against other sources of information due to the presence of random and systematic errors, as well as correlations, in the data. To address these challenges, we introduce a Bayesian scoring function that efficiently and accurately ranks alternative structural models of a macromolecular system based on their consistency with a cryo-EM density map as well as other experimental and prior information. The accuracy of this approach is benchmarked using complexes of known structure and illustrated in three applications: the structural determination of the GroEL/GroES, RNA polymerase II, and exosome complexes. The approach is implemented in the open-source Integrative Modeling Platform (http://integrativemodeling.org), thus enabling integrative structure determination by combining cryo-EM data with other sources of information.

CryoEM-based hybrid modeling approaches for structure determination

Recent advances in cryo-electron microscopy (cryoEM) have dramatically improved the resolutions at which vitrified biological specimens can be studied, revealing new structural and mechanistic insights over a broad range of spatial scales. Bolstered by these advances, much effort has been directed toward the development of hybrid modeling methodologies for the construction and refinement of high-fidelity atomistic models from cryoEM data. In this brief review, we will survey the key elements of cryoEM-based hybrid modeling, providing an overview of available computational tools and strategies as well as several recent applications.