Computation of conformational transitions in proteins by virtual atom molecular mechanics as validated in application to adenylate kinase

Tackling Hysteresis in Conformational Sampling: How to Be Forgetful with MEMENTO

The structure of proteins has long been recognized to hold the key to understanding and engineering their function, and rapid advances in structural biology and protein structure prediction are now supplying researchers with an ever-increasing wealth of structural information. Most of the time, however, structures can only be determined in free energy minima, one at a time. While conformational flexibility may thus be inferred from static end-state structures, their interconversion mechanisms─a central ambition of structural biology─are often beyond the scope of direct experimentation. Given the dynamical nature of the processes in question, many studies have attempted to explore conformational transitions using molecular dynamics (MD). However, ensuring proper convergence and reversibility in the predicted transitions is extremely challenging. In particular, a commonly used technique to map out a path from a starting to a target conformation called steered MD (SMD) can suffer from starting-state dependence (hysteresis) when combined with techniques such as umbrella sampling (US) to compute the free energy profile of a transition. Here, we study this problem in detail on conformational changes of increasing complexity. We also present a new, history-independent approach that we term "MEMENTO" (Morphing End states by Modelling Ensembles with iNdependent TOpologies) to generate paths that alleviate hysteresis in the construction of conformational free energy profiles. MEMENTO utilizes template-based structure modelling to restore physically reasonable protein conformations based on coordinate interpolation (morphing) as an ensemble of plausible intermediates, from which a smooth path is picked. We compare SMD and MEMENTO on well-characterized test cases (the toy peptide deca-alanine and the enzyme adenylate kinase) before discussing its use in more complicated systems (the kinase P38α and the bacterial leucine transporter LeuT). Our work shows that for all but the simplest systems SMD paths should not in general be used to seed umbrella sampling or related techniques, unless the paths are validated by consistent results from biased runs in opposite directions. MEMENTO, on the other hand, performs well as a flexible tool to generate intermediate structures for umbrella sampling. We also demonstrate that extended end-state sampling combined with MEMENTO can aid the discovery of collective variables on a case-by-case basis.

Physics of biomolecular recognition and conformational dynamics

Biomolecular recognition usually leads to the formation of binding complexes, often accompanied by large-scale conformational changes. This process is fundamental to biological functions at the molecular and cellular levels. Uncovering the physical mechanisms of biomolecular recognition and quantifying the key biomolecular interactions are vital to understand these functions. The recently developed energy landscape theory has been successful in quantifying recognition processes and revealing the underlying mechanisms. Recent studies have shown that in addition to affinity, specificity is also crucial for biomolecular recognition. The proposed physical concept of intrinsic specificity based on the underlying energy landscape theory provides a practical way to quantify the specificity. Optimization of affinity and specificity can be adopted as a principle to guide the evolution and design of molecular recognition. This approach can also be used in practice for drug discovery using multidimensional screening to identify lead compounds. The energy landscape topography of molecular recognition is important for revealing the underlying flexible binding or binding-folding mechanisms. In this review, we first introduce the energy landscape theory for molecular recognition and then address four critical issues related to biomolecular recognition and conformational dynamics: (1) specificity quantification of molecular recognition; (2) evolution and design in molecular recognition; (3) flexible molecular recognition; (4) chromosome structural dynamics. The results described here and the discussions of the insights gained from the energy landscape topography can provide valuable guidance for further computational and experimental investigations of biomolecular recognition and conformational dynamics.

Generating conformational transition paths with low potential-energy barriers for proteins

The knowledge of conformational transition paths in proteins can be useful for understanding protein mechanisms. Recently, we have introduced the As-Rigid-As-Possible (ARAP) interpolation method, for generating interpolation paths between two protein conformations. The method was shown to preserve well the rigidity of the initial conformation along the path. However, because the method is totally geometry-based, the generated paths may be inconsistent because the atom interactions are ignored. Therefore, in this article, we would like to introduce a new method to generate conformational transition paths with low potential-energy barriers for proteins. The method is composed of three processing stages. First, ARAP interpolation is used for generating an initial path. Then, the path conformations are enhanced by a clash remover. Finally, Nudged Elastic Band, a path-optimization method, is used to produce a low-energy path. Large energy reductions are found in the paths obtained from the method than in those obtained from the ARAP interpolation method alone. The results also show that ARAP interpolation is a good candidate for generating an initial path because it leads to lower potential-energy paths than two other common methods for path interpolation.

Mapping the Dynamics Landscape of Conformational Transitions in Enzyme: The Adenylate Kinase Case

Conformational transition describes the essential dynamics and mechanism of enzymes in pursuing their various functions. The fundamental and practical challenge to researchers is to quantitatively describe the roles of large-scale dynamic transitions for regulating the catalytic processes. In this study, we tackled this challenge by exploring the pathways and free energy landscape of conformational changes in adenylate kinase (AdK), a key ubiquitous enzyme for cellular energy homeostasis. Using explicit long-timescale (up to microseconds) molecular dynamics and bias-exchange metadynamics simulations, we determined at the atomistic level the intermediate conformational states and mapped the transition pathways of AdK in the presence and absence of ligands. There is clearly chronological operation of the functional domains of AdK. Specifically in the ligand-free AdK, there is no significant energy barrier in the free energy landscape separating the open and closed states. Instead there are multiple intermediate conformational states, which facilitate the rapid transitions of AdK. In the ligand-bound AdK, the closed conformation is energetically most favored with a large energy barrier to open it up, and the conformational population prefers to shift to the closed form coupled with transitions. The results suggest a perspective for a hybrid of conformational selection and induced fit operations of ligand binding to AdK. These observations, depicted in the most comprehensive and quantitative way to date, to our knowledge, emphasize the underlying intrinsic dynamics of AdK and reveal the sophisticated conformational transitions of AdK in fulfilling its enzymatic functions. The developed methodology can also apply to other proteins and biomolecular systems.

Morphing methods to visualize coarse-grained protein dynamics

Morphing was initially developed as a cinematic effect, where one image is seamlessly transformed into another image. The technique was widely adopted by biologists to visualize the transition between protein conformational states, generating an interpolated pathway from an initial to a final protein structure. Geometric morphing seeks to create visually suggestive movies that illustrate structural changes between conformations but do not necessarily represent a biologically relevant pathway, while minimum energy path (MEP) interpolations aim at describing the true transition state between the crystal structure minima in the energy landscape.

Elucidating the ensemble of functionally-relevant transitions in protein systems with a robotics-inspired method

Background

Many proteins tune their biological function by transitioning between different functional states, effectively acting as dynamic molecular machines. Detailed structural characterization of transition trajectories is central to understanding the relationship between protein dynamics and function. Computational approaches that build on the Molecular Dynamics framework are in principle able to model transition trajectories at great detail but also at considerable computational cost. Methods that delay consideration of dynamics and focus instead on elucidating energetically-credible conformational paths connecting two functionally-relevant structures provide a complementary approach. Effective sampling-based path planning methods originating in robotics have been recently proposed to produce conformational paths. These methods largely model short peptides or address large proteins by simplifying conformational space.

Methods

We propose a robotics-inspired method that connects two given structures of a protein by sampling conformational paths. The method focuses on small- to medium-size proteins, efficiently modeling structural deformations through the use of the molecular fragment replacement technique. In particular, the method grows a tree in conformational space rooted at the start structure, steering the tree to a goal region defined around the goal structure. We investigate various bias schemes over a progress coordinate for balance between coverage of conformational space and progress towards the goal. A geometric projection layer promotes path diversity. A reactive temperature scheme allows sampling of rare paths that cross energy barriers.

Results and conclusions

Experiments are conducted on small- to medium-size proteins of length up to 214 amino acids and with multiple known functionally-relevant states, some of which are more than 13Å apart of each-other. Analysis reveals that the method effectively obtains conformational paths connecting structural states that are significantly different. A detailed analysis on the depth and breadth of the tree suggests that a soft global bias over the progress coordinate enhances sampling and results in higher path diversity. The explicit geometric projection layer that biases the exploration away from over-sampled regions further increases coverage, often improving proximity to the goal by forcing the exploration to find new paths. The reactive temperature scheme is shown effective in increasing path diversity, particularly in difficult structural transitions with known high-energy barriers.

Structural plasticity and conformational transitions of HIV envelope glycoprotein gp120

HIV envelope glycoproteins undergo large-scale conformational changes as they interact with cellular receptors to cause the fusion of viral and cellular membranes that permits viral entry to infect targeted cells. Conformational dynamics in HIV gp120 are also important in masking conserved receptor epitopes from being detected for effective neutralization by the human immune system. Crystal structures of HIV gp120 and its complexes with receptors and antibody fragments provide high-resolution pictures of selected conformational states accessible to gp120. Here we describe systematic computational analyses of HIV gp120 plasticity in such complexes with CD4 binding fragments, CD4 mimetic proteins, and various antibody fragments. We used three computational approaches: an isotropic elastic network analysis of conformational plasticity, a full atomic normal mode analysis, and simulation of conformational transitions with our coarse-grained virtual atom molecular mechanics (VAMM) potential function. We observe collective sub-domain motions about hinge points that coordinate those motions, correlated local fluctuations at the interfacial cavity formed when gp120 binds to CD4, and concerted changes in structural elements that form at the CD4 interface during large-scale conformational transitions to the CD4-bound state from the deformed states of gp120 in certain antibody complexes.

A generic force field for protein coarse-grained molecular dynamics simulation

Coarse-grained (CG) force fields have become promising tools for studies of protein behavior, but the balance of speed and accuracy is still a challenge in the research of protein coarse graining methodology. In this work, 20 CG beads have been designed based on the structures of amino acid residues, with which an amino acid can be represented by one or two beads, and a CG solvent model with five water molecules was adopted to ensure the consistence with the protein CG beads. The internal interactions in protein were classified according to the types of the interacting CG beads, and adequate potential functions were chosen and systematically parameterized to fit the energy distributions. The proposed CG force field has been tested on eight proteins, and each protein was simulated for 1000 ns. Even without any extra structure knowledge of the simulated proteins, the Cα root mean square deviations (RMSDs) with respect to their experimental structures are close to those of relatively short time all atom molecular dynamics simulations. However, our coarse grained force field will require further refinement to improve agreement with and persistence of native-like structures. In addition, the root mean square fluctuations (RMSFs) relative to the average structures derived from the simulations show that the conformational fluctuations of the proteins can be sampled.

Complex intramolecular mechanics of G-actin--an elastic network study

Systematic numerical investigations of conformational motions in single actin molecules were performed by employing a simple elastic-network (EN) model of this protein. Similar to previous investigations for myosin, we found that G-actin essentially behaves as a strain sensor, responding by well-defined domain motions to mechanical perturbations. Several sensitive residues within the nucleotide-binding pocket (NBP) could be identified, such that the perturbation of any of them can induce characteristic flattening of actin molecules and closing of the cleft between their two mobile domains. Extending the EN model by introduction of a set of breakable links which become effective only when two domains approach one another, it was observed that G-actin can possess a metastable state corresponding to a closed conformation and that a transition to this state can be induced by appropriate perturbations in the NBP region. The ligands were roughly modeled as a single particle (ADP) or a dimer (ATP), which were placed inside the NBP and connected by elastic links to the neighbors. Our approximate analysis suggests that, when ATP is present, it stabilizes the closed conformation of actin. This may play an important role in the explanation why, in the presence of ATP, the polymerization process is highly accelerated.

Event detection and sub-state discovery from biomolecular simulations using higher-order statistics: application to enzyme adenylate kinase

Biomolecular simulations at millisecond and longer time-scales can provide vital insights into functional mechanisms. Because post-simulation analyses of such large trajectory datasets can be a limiting factor in obtaining biological insights, there is an emerging need to identify key dynamical events and relating these events to the biological function online, that is, as simulations are progressing. Recently, we have introduced a novel computational technique, quasi-anharmonic analysis (QAA) (Ramanathan et al., PLoS One 2011;6:e15827), for partitioning the conformational landscape into a hierarchy of functionally relevant sub-states. The unique capabilities of QAA are enabled by exploiting anharmonicity in the form of fourth-order statistics for characterizing atomic fluctuations. In this article, we extend QAA for analyzing long time-scale simulations online. In particular, we present HOST4MD--a higher-order statistical toolbox for molecular dynamics simulations, which (1) identifies key dynamical events as simulations are in progress, (2) explores potential sub-states, and (3) identifies conformational transitions that enable the protein to access those sub-states. We demonstrate HOST4MD on microsecond timescale simulations of the enzyme adenylate kinase in its apo state. HOST4MD identifies several conformational events in these simulations, revealing how the intrinsic coupling between the three subdomains (LID, CORE, and NMP) changes during the simulations. Further, it also identifies an inherent asymmetry in the opening/closing of the two binding sites. We anticipate that HOST4MD will provide a powerful and extensible framework for detecting biophysically relevant conformational coordinates from long time-scale simulations.

Biomolecular dynamics: order-disorder transitions and energy landscapes

While the energy landscape theory of protein folding is now a widely accepted view for understanding how relatively weak molecular interactions lead to rapid and cooperative protein folding, such a framework must be extended to describe the large-scale functional motions observed in molecular machines. In this review, we discuss (1) the development of the energy landscape theory of biomolecular folding, (2) recent advances toward establishing a consistent understanding of folding and function and (3) emerging themes in the functional motions of enzymes, biomolecular motors and other biomolecular machines. Recent theoretical, computational and experimental lines of investigation have provided a very dynamic picture of biomolecular motion. In contrast to earlier ideas, where molecular machines were thought to function similarly to macroscopic machines, with rigid components that move along a few degrees of freedom in a deterministic fashion, biomolecular complexes are only marginally stable. Since the stabilizing contribution of each atomic interaction is on the order of the thermal fluctuations in solution, the rigid body description of molecular function must be revisited. An emerging theme is that functional motions encompass order-disorder transitions and structural flexibility provides significant contributions to the free energy. In this review, we describe the biological importance of order-disorder transitions and discuss the statistical-mechanical foundation of theoretical approaches that can characterize such transitions.

Structural distributions from single-molecule measurements as a tool for molecular mechanics

A mechanical view provides an attractive alternative for predicting the behavior of complex systems since it circumvents the resource-intensive requirements of atomistic models; however, it remains extremely challenging to characterize the mechanical responses of a system at the molecular level. Here, the structural distribution is proposed to be an effective means to extracting the molecular mechanical properties. End-to-end distance distributions for a series of short poly-L-proline peptides with the sequence P(n)CG(3)K-biotin (n = 8, 12, 15 and 24) were used to experimentally illustrate this new approach. High-resolution single-molecule Förster-type resonance energy transfer (FRET) experiments were carried out and the conformation-resolving power was characterized and discussed in the context of the conventional constant-time binning procedure for FRET data analysis. It was shown that the commonly adopted theoretical polymer models-including the worm-like chain, the freely jointed chain, and the self-avoiding chain-could not be distinguished by the averaged end-to-end distances, but could be ruled out using the molecular details gained by conformational distribution analysis because similar polymers of different sizes could respond to external forces differently. Specifically, by fitting the molecular conformational distribution to a semi-flexible polymer model, the effective persistence lengths for the series of short poly-L-proline peptides were found to be size-dependent with values of ~190 Å, ~67 Å, ~51 Å, and ~76 Å for n = 8, 12, 15, and 24, respectively. A comprehensive computational modeling was carried out to gain further insights for this surprising discovery. It was found that P(8) exists as the extended all-trans isomaer whereas P(12) and P(15) predominantly contained one proline residue in the cis conformation. P(24) exists as a mixture of one-cis (75%) and two-cis (25%) isomers where each isomer contributes to an experimentally resolvable conformational mode. This work demonstrates the resolving power of the distribution-based approach, and the capacity of integrating high-resolution single-molecule FRET experiments with molecular modeling to reveal detailed structural information about the conformation of molecules on the length scales relevant to the study of biological molecules.

Protein folding pathways and state transitions described by classical equations of motion of an elastic network model

Protein topology defined by the matrix of residue contacts has proved to be a fruitful basis for the study of protein dynamics. The widely implemented coarse-grained elastic network model of backbone fluctuations has been used to describe crystallographic temperature factors, allosteric couplings, and some aspects of the folding pathway. In the present study, we develop a model of protein dynamics based on the classical equations of motion of a damped network model (DNM) that describes the folding path from a completely unfolded state to the native conformation through a single-well potential derived purely from the native conformation. The kinetic energy gained through the collapse of the protein chain is dissipated through a friction term in the equations of motion that models the water bath. This approach is completely general and sufficiently fast that it can be applied to large proteins. Folding pathways for various proteins of different classes are described and shown to correlate with experimental observations and molecular dynamics and Monte Carlo simulations. Allosteric transitions between alternative protein structures are also modeled within the DNM through an asymmetric double-well potential.

Integrating ion mobility mass spectrometry with molecular modelling to determine the architecture of multiprotein complexes

Current challenges in the field of structural genomics point to the need for new tools and technologies for obtaining structures of macromolecular protein complexes. Here, we present an integrative computational method that uses molecular modelling, ion mobility-mass spectrometry (IM-MS) and incomplete atomic structures, usually from X-ray crystallography, to generate models of the subunit architecture of protein complexes. We begin by analyzing protein complexes using IM-MS, and by taking measurements of both intact complexes and sub-complexes that are generated in solution. We then examine available high resolution structural data and use a suite of computational methods to account for missing residues at the subunit and/or domain level. High-order complexes and sub-complexes are then constructed that conform to distance and connectivity constraints imposed by IM-MS data. We illustrate our method by applying it to multimeric protein complexes within the Escherichia coli replisome: the sliding clamp, (β₂), the γ complex (γ₃δδ′), the DnaB helicase (DnaB₆) and the Single-Stranded Binding Protein (SSB₄).

Multiscale coarse-graining of the protein energy landscape

A variety of coarse-grained (CG) models exists for simulation of proteins. An outstanding problem is the construction of a CG model with physically accurate conformational energetics rivaling all-atom force fields. In the present work, atomistic simulations of peptide folding and aggregation equilibria are force-matched using multiscale coarse-graining to develop and test a CG interaction potential of general utility for the simulation of proteins of arbitrary sequence. The reduced representation relies on multiple interaction sites to maintain the anisotropic packing and polarity of individual sidechains. CG energy landscapes computed from replica exchange simulations of the folding of Trpzip, Trp-cage and adenylate kinase resemble those of other reduced representations; non-native structures are observed with energies similar to those of the native state. The artifactual stabilization of misfolded states implies that non-native interactions play a deciding role in deviations from ideal funnel-like cooperative folding. The role of surface tension, backbone hydrogen bonding and the smooth pairwise CG landscape is discussed. Ab initio folding aside, the improved treatment of sidechain rotamers results in stability of the native state in constant temperature simulations of Trpzip, Trp-cage, and the open to closed conformational transition of adenylate kinase, illustrating the potential value of the CG force field for simulating protein complexes and transitions between well-defined structural states.

Biological function originates from the dynamical motions of proteins in response to cellular stimuli. Protein dynamics arise from physical interactions that are well-predicted by detailed atomistic simulations. In order to examine large protein complexes on long timescales of biological importance, however, coarse-grained simulation approaches are needed to complement experiment. Previous coarse-grained models have proved successful for investigations involving a given protein's native structure, including protein folding and structure prediction. We construct a model capable of simulating proteins regardless of their sequence or structure. The present coarse-grained model was, however, developed rigorously from the underlying atomistic forces as opposed to knowledge-based or ad hoc parameterizations. Examination of the model predictions on various accessible timescales reveals successes and limitations of the model. While functionally relevant conformational transitions can be studied, the coarse-grained representation has some difficulty with the ab initio folding of the peptide chain into its proper structure. Our observations highlight the complex molecular nature of a protein's underlying energy landscape, offering rigorous insight into the information missing in reduced representations of the peptide chain. With these caveats in mind, the physical interaction–based, coarse-grained model will find application in simulations of a wide variety of proteins and continue to guide future coarse-graining efforts.

A force field for virtual atom molecular mechanics of proteins

Activities of many biological macromolecules involve large conformational transitions for which crystallography can specify atomic details of alternative end states, but the course of transitions is often beyond the reach of computations based on full-atomic potential functions. We have developed a coarse-grained force field for molecular mechanics calculations based on the virtual interactions of C alpha atoms in protein molecules. This force field is parameterized based on the statistical distribution of the energy terms extracted from crystallographic data, and it is formulated to capture features dependent on secondary structure and on residue-specific contact information. The resulting force field is applied to energy minimization and normal mode analysis of several proteins. We find robust convergence in minimizations to low energies and energy gradients with low degrees of structural distortion, and atomic fluctuations calculated from the normal mode analyses correlate well with the experimental B-factors obtained from high-resolution crystal structures. These findings suggest that the virtual atom force field is a suitable tool for various molecular mechanics applications on large macromolecular systems undergoing large conformational changes.