Topology representing neural networks reconcile biomolecular shape, structure, and dynamics

Numerical geometry of map and model assessment

We are describing best practices and assessment strategies for the atomic interpretation of cryo-electron microscopy (cryo-EM) maps. Multiscale numerical geometry strategies in the Situs package and in secondary structure detection software are currently evolving due to the recent increases in cryo-EM resolution. Criteria that aim to predict the accuracy of fitted atomic models at low (worse than 8Å) and medium (4-8 Å) resolutions remain challenging. However, a high level of confidence in atomic models can be achieved by combining such criteria. The observed errors are due to map-model discrepancies and due to the effect of imperfect global docking strategies. Extending the earlier motion capture approach developed for flexible fitting, we use simulated fiducials (pseudoatoms) at varying levels of coarse-graining to track the local drift of structural features. We compare three tracking approaches: naïve vector quantization, a smoothly deformable model, and a tessellation of the structure into rigid Voronoi cells, which are fitted using a multi-fragment refinement approach. The lowest error is an upper bound for the (small) discrepancy between the crystal structure and the EM map due to different conditions in their structure determination. When internal features such as secondary structures are visible in medium-resolution EM maps, it is possible to extend the idea of point-based fiducials to more complex geometric representations such as helical axes, strands, and skeletons. We propose quantitative strategies to assess map-model pairs when such secondary structure patterns are prominent.

iMODFIT: efficient and robust flexible fitting based on vibrational analysis in internal coordinates

Here, we employed the collective motions extracted from Normal Mode Analysis (NMA) in internal coordinates (torsional space) for the flexible fitting of atomic-resolution structures into electron microscopy (EM) density maps. The proposed methodology was validated using a benchmark of simulated cases, highlighting its robustness over the full range of EM resolutions and even over coarse-grained representations. A systematic comparison with other methods further showcased the advantages of this proposed methodology, especially at medium to lower resolutions. Using this method, computational costs and potential overfitting problems are naturally reduced by constraining the search in low-frequency NMA space, where covalent geometry is implicitly maintained. This method also effectively captures the macromolecular changes of a representative set of experimental test cases. We believe that this novel approach will extend the currently available EM hybrid methods to the atomic-level interpretation of large conformational changes and their functional implications.

Conventions and workflows for using Situs

Situs is a modular program package for the multi-scale modeling of atomic resolution structures and low-resolution biophysical data from electron microscopy, tomography or small-angle X-ray scattering. This article provides an overview of recent developments in the Situs package, with an emphasis on workflows and conventions that are important for practical applications. The modular design of the programs facilitates scripting in the bash shell that allows specific programs to be combined in creative ways that go beyond the original intent of the developers. Several scripting-enabled functionalities, such as flexible transformations of data type, the use of symmetry constraints or the creation of two-dimensional projection images, are described. The processing of low-resolution biophysical maps in such workflows follows not only first principles but often relies on implicit conventions. Situs conventions related to map formats, resolution, correlation functions and feature detection are reviewed and summarized. The compatibility of the Situs workflow with CCP4 conventions and programs is discussed.

Using Sculptor and Situs for simultaneous assembly of atomic components into low-resolution shapes

We describe an integrated software system called Sculptor that combines visualization capabilities with molecular modeling algorithms for the analysis of multi-scale data sets. Sculptor features extensive special purpose visualization techniques that are based on modern GPU programming and are capable of representing complex molecular assemblies in real-time. The integration of graphics and modeling offers several advantages. The user interface not only eases the usually steep learning curve of pure algorithmic techniques, but it also permits instant analysis and post-processing of results, as well as the integration of results from external software. Here, we implemented an interactive peak-selection strategy that enables the user to explore a preliminary score landscape generated by the colors tool of Situs. The interactive placement of components, one at a time, is advantageous for low-resolution or ambiguously shaped maps, which are sometimes difficult to interpret by the fully automatic peak selection of colors. For the subsequent refinement of the preliminary models resulting from both interactive and automatic peak selection, we have implemented a novel simultaneous multi-body docking in Sculptor and Situs that softly enforces shape complementarities between components using the normalization of the cross-correlation coefficient. The proposed techniques are freely available in Situs version 2.6 and Sculptor version 2.0.

Coarse-Grained Representations of Large Biomolecular Complexes from Low-Resolution Structural Data

High-resolution atomistic structures of many large biomolecular complexes have not yet been solved by experiments, such as X-ray crystallography or NMR. Often however low-resolution information is obtained by alternative techniques, such as cryo-electron microscopy or small-angle X-ray scattering. Coarse-grained (CG) models are an appropriate choice to computationally study these complexes given the limited resolution experimental data. One of the important questions therefore is how to define CG representations from these low-resolution density maps. This work provides a space-based essential dynamics coarse-graining (ED-CG) method to define a CG representation from a density map without detailed knowledge of its underlying atomistic structure and primary sequence information. This method is demonstrated on G-actin (both the atomic structure and its density map). It is then applied to the density maps of the Escherichia coli 70S ribosome and the microtubule. The results indicate that the method can define highly CG models that still preserve functionally important dynamics of large biomolecular complexes.

Defining coarse-grained representations of large biomolecules and biomolecular complexes from elastic network models

Coarse-grained (CG) models of large biomolecular complexes enable simulations of these systems over long timescales that are not accessible for atomistic molecular dynamics (MD) simulations. A systematic methodology, called essential dynamics coarse-graining (ED-CG), has been developed for defining coarse-grained sites in a large biomolecule. The method variationally determines the CG sites so that key dynamic domains in the protein are preserved in the CG representation. The original ED-CG method relies on a principal component analysis (PCA) of a MD trajectory. However, for many large proteins and multi-protein complexes such an analysis may not converge or even be possible. This work develops a new ED-CG scheme using an elastic network model (ENM) of the protein structure. In this procedure, the low-frequency normal modes obtained by ENM are used to define dynamic domains and to define the CG representation accordingly. The method is then applied to several proteins, such as the HIV-1 CA protein dimer, ATP-bound G-actin, and the Arp2/3 complex. Numerical results show that ED-CG with ENM (ENM-ED-CG) is much faster than ED-CG with PCA because no MD is necessary. The ENM-ED-CG models also capture functional essential dynamics of the proteins almost as well as those using full MD with PCA. Therefore, the ENM-ED-CG method may be better suited to coarse-grain a very large biomolecule or biomolecular complex that is too computationally expensive to be simulated by conventional MD, or when a high resolution atomic structure is not even available.

Using Situs for the integration of multi-resolution structures

Situs is a modular and widely used software package for the integration of biophysical data across the spatial resolution scales. It has been developed over the last decade with a focus on bridging the resolution gap between atomic structures, coarse-grained models, and volumetric data from low-resolution biophysical origins, such as electron microscopy, tomography, or small-angle scattering. Structural models can be created and refined with various flexible and rigid body docking strategies. The software consists of multiple, stand-alone programs for the format conversion, analysis, visualization, manipulation, and assembly of 3D data sets. The programs have been ported to numerous platforms in both serial and shared memory parallel architectures and can be combined in various ways for specific modeling applications. The modular design facilitates the updating of individual programs and the development of novel application workflows. This review provides an overview of the Situs package as it exists today with an emphasis on functionality and workflows supported by version 2.5.

Bend-twist-stretch model for coarse elastic network simulation of biomolecular motion

The empirical harmonic potential function of elastic network models (ENMs) is augmented by three- and four-body interactions as well as by a parameter-free connection rule. In the new bend-twist-stretch (BTS) model the complexity of the parametrization is shifted from the spatial level of detail to the potential function, enabling an arbitrary coarse graining of the network. Compared to distance cutoff-based Hookean springs, the approach yields a more stable parametrization of coarse-grained ENMs for biomolecular dynamics. Traditional ENMs give rise to unbounded zero-frequency vibrations when (pseudo)atoms are connected to fewer than three neighbors. A large cutoff is therefore chosen in an ENM (about twice the average nearest-neighbor distance), resulting in many false-positive connections that reduce the spatial detail that can be resolved. More importantly, the required three-neighbor connectedness also limits the coarse graining, i.e., the network must be dense, even in the case of low-resolution structures that exhibit few spatial features. The new BTS model achieves such coarse graining by extending the ENM potential to include three-and four-atom interactions (bending and twisting, respectively) in addition to the traditional two-atom stretching. Thus, the BTS model enables reliable modeling of any three-dimensional graph irrespective of the atom connectedness. The additional potential terms were parametrized using continuum elastic theory of elastic rods, and the distance cutoff was replaced by a competitive Hebb connection rule, setting all free parameters in the model. We validate the approach on a carbon-alpha representation of adenylate kinase and illustrate its use with electron microscopy maps of E. coli RNA polymerase, E. coli ribosome, and eukaryotic chaperonin containing T-complex polypeptide 1, which were difficult to model with traditional ENMs. For adenylate kinase, we find excellent reproduction (>90% overlap) of the ENM modes and B factors when BTS is applied to the carbon-alpha representation as well as to coarser descriptions. For the volumetric maps, coarse BTS yields similar motions (70%-90% overlap) to those obtained from significantly denser representations with ENM. Our Python-based algorithms of ENM and BTS implementations are freely available.

Development of free-energy-based models for chaperonin containing TCP-1 mediated folding of actin

A free-energy-based approach is used to describe the mechanism through which chaperonin-containing TCP-1 (CCT) folds the filament-forming cytoskeletal protein actin, which is one of its primary substrates. The experimental observations on the actin folding and unfolding pathways are collated and then re-examined from this perspective, allowing us to determine the position of the CCT intervention on the actin free-energy folding landscape. The essential role for CCT in actin folding is to provide a free-energy contribution from its ATP cycle, which drives actin to fold from a stable, trapped intermediate I3, to a less stable but now productive folding intermediate I2. We develop two hypothetical mechanisms for actin folding founded upon concepts established for the bacterial type I chaperonin GroEL and extend them to the much more complex CCT system of eukaryotes. A new model is presented in which CCT facilitates free-energy transfer through direct coupling of the nucleotide hydrolysis cycle to the phases of actin substrate maturation.

Three-dimensional reconstruction of tarantula myosin filaments suggests how phosphorylation may regulate myosin activity

Muscle contraction involves the interaction of the myosin heads of the thick filaments with actin subunits of the thin filaments. Relaxation occurs when this interaction is blocked by molecular switches on these filaments. In many muscles, myosin-linked regulation involves phosphorylation of the myosin regulatory light chains (RLCs). Electron microscopy of vertebrate smooth muscle myosin molecules (regulated by phosphorylation) has provided insight into the relaxed structure, revealing that myosin is switched off by intramolecular interactions between its two heads, the free head and the blocked head. Three-dimensional reconstruction of frozen-hydrated specimens revealed that this asymmetric head interaction is also present in native thick filaments of tarantula striated muscle. Our goal in this study was to elucidate the structural features of the tarantula filament involved in phosphorylation-based regulation. A new reconstruction revealed intra- and intermolecular myosin interactions in addition to those seen previously. To help interpret the interactions, we sequenced the tarantula RLC and fitted an atomic model of the myosin head that included the predicted RLC atomic structure and an S2 (subfragment 2) crystal structure to the reconstruction. The fitting suggests one intramolecular interaction, between the cardiomyopathy loop of the free head and its own S2, and two intermolecular interactions, between the cardiac loop of the free head and the essential light chain of the blocked head and between the Leu305-Gln327 interaction loop of the free head and the N-terminal fragment of the RLC of the blocked head. These interactions, added to those previously described, would help switch off the thick filament. Molecular dynamics simulations suggest how phosphorylation could increase the helical content of the RLC N-terminus, weakening these interactions, thus releasing both heads and activating the thick filament.

Biomolecular pleiomorphism probed by spatial interpolation of coarse models

In low resolution structures of biological assemblies one can often observe conformational deviations that require a flexible rearrangement of structural domains fitted at the atomic level. We are evaluating interpolation methods for the flexible alignment of atomic models based on coarse models. Spatial interpolation is well established in image-processing and visualization to describe the overall deformation or warping of an object or an image. Combined with a coarse representation of the biological system by feature vectors, such methods can provide a flexible approximation of the molecular structure. We have compared three well-known interpolation techniques and evaluated the results by comparing them with constrained molecular dynamics. One method, inverse distance weighting interpolation, consistently produced models that were nearly indistinguishable on the alpha carbon level from the molecular dynamics results. The method is simple to apply and enables flexing of structures by non-expert modelers. This is useful for the basic interpretation of volumetric data in biological applications such as electron microscopy. The method can be used as a general interpretation tool for sparsely sampled motions derived from coarse models.

Damped-dynamics flexible fitting

In fitting atomic structures into EM maps, it often happens that the map corresponds to a different conformation of the structure. We have developed a new methodology to handle these situations that preserves the covalent geometry of the structure and allows the modeling of large deformations. The first goal is achieved by working in generalized coordinates (positional and internal coordinates), and the second by avoiding harmonic potentials. Instead, we use dampers (shock absorbers) between every pair of atoms, combined with a force field that attracts the atomic structure toward incompletely occupied regions of the EM map. The trajectory obtained by integrating the resulting equations of motion converges to a conformation that, in our validation cases, was very close to the target atomic structure. Compared to current methods, our approach is more efficient and robust against wrong solutions and to overfitting, and does not require user intervention or subjective decisions. Applications to the computation of transition pathways between known conformers, homology and loop modeling, as well as protein docking, are also discussed.

Combining efficient conformational sampling with a deformable elastic network model facilitates structure refinement at low resolution

Structural studies of large proteins and protein assemblies are a difficult and pressing challenge in molecular biology. Experiments often yield only low-resolution or sparse data that are not sufficient to fully determine atomistic structures. We have developed a general geometry-based algorithm that efficiently samples conformational space under constraints imposed by low-resolution density maps obtained from electron microscopy or X-ray crystallography experiments. A deformable elastic network (DEN) is used to restrain the sampling to prior knowledge of an approximate structure. The DEN restraints dramatically reduce over-fitting, especially at low resolution. Cross-validation is used to optimally weight the structural information and experimental data. Our algorithm is robust even for noise-added density maps and has a large radius of convergence for our test case. The DEN restraints can also be used to enhance reciprocal space simulated annealing refinement.

Using surface envelopes to constrain molecular modeling

Molecular density information (as measured by electron microscopic reconstructions or crystallographic density maps) can be a powerful source of information for molecular modeling. Molecular density constrains models by specifying where atoms should and should not be. Low-resolution density information can often be obtained relatively quickly, and there is a need for methods that use it effectively. We have previously described a method for scoring molecular models with surface envelopes to discriminate between plausible and implausible fits. We showed that we could successfully filter out models with the wrong shape based on this discrimination power. Ideally, however, surface information should be used during the modeling process to constrain the conformations that are sampled. In this paper, we describe an extension of our method for using shape information during computational modeling. We use the envelope scoring metric as part of an objective function in a global optimization that also optimizes distances and angles while avoiding collisions. We systematically tested surface representations of proteins (using all nonhydrogen heavy atoms) with different abundance of distance information and showed that the root mean square deviation (RMSD) of models built with envelope information is consistently improved, particularly in data sets with relatively small sets of short-range distances.

Divergent substrate-binding mechanisms reveal an evolutionary specialization of eukaryotic prefoldin compared to its archaeal counterpart

Prefoldin (PFD) is a molecular chaperone that stabilizes and then delivers unfolded proteins to a chaperonin for facilitated folding. The PFD hexamer has undergone an evolutionary change in subunit composition, from two PFDalpha and four PFDbeta subunits in archaea to six different subunits (two alpha-like and four beta-like subunits) in eukaryotes. Here, we show by electron microscopy that PFD from the archaeum Pyrococcus horikoshii (PhPFD) selectively uses an increasing number of subunits to interact with nonnative protein substrates of larger sizes. PhPFD stabilizes unfolded proteins by interacting with the distal regions of the chaperone tentacles, a mechanism different from that of eukaryotic PFD, which encapsulates its substrate inside the cavity. This suggests that although the fundamental functions of archaeal and eukaryal PFD are conserved, their mechanism of substrate interaction have diverged, potentially reflecting a narrower range of substrates stabilized by the eukaryotic PFD.

Unfolding energetics of G-alpha-actin: a discrete intermediate can be re-folded to the native state by CCT

Nascent actin requires interactions with the highly conserved and essential eukaryotic chaperonin-containing TCP-1 (CCT) for its correct folding to the native state in vivo. Biochemical and structural analysis of the interaction between actin and CCT has been studied extensively but the underlying energetics and kinetics of the CCT-dependent actin folding process are not understood. We investigated the unfolding and folding pathways of actin, using stopped flow fluorescence and biochemical techniques. By using very low concentrations of actin, taking account of temperature and ATP concentration dependences we were able to determine accurately the activation energy of unfolding to a stable intermediate, I(3). Use of the fluorescent calcium chelator Quin-2 and consideration of the ATP concentration dependence on the unfolding rate has allowed the intrinsic kinetics to be linked to the accepted reaction scheme for actin denaturation. A free energy of -28.7(+/-0.2) kJ mol(-1) was determined for the loss of ATP from Ca-free G-actin, in good agreement with previous studies. Understanding the K(eq) value for this step then allowed the temperature dependence of the unfolding reaction of co-factor-free actin to be evaluated, yielding an activation energy for the unfolding of G-actin of 81.3(+/-3.3) kJ mol(-1). By chemical coupling of the extrinsic probe, Alexa Fluor 488 to cysteine 374 of native alpha-actin, we were able to follow the binding and folding of I(3) by CCT, observing for the first time, in vitro re-folding of EDTA-denatured G-actin. The high value of the activation energy between native actin and a non-native folding intermediate (I(3)) is characteristic of a partially folded, molten globule state expected to contain partial secondary structure.