Discrete kinetic models from funneled energy landscape simulations

Reassessing the exon-foldon correspondence using frustration analysis

Protein folding and evolution are intimately linked phenomena. Here, we revisit the concept of exons as potential protein folding modules across a set of 38 abundant and conserved protein families. Taking advantage of genomic exon-intron organization and extensive protein sequence data, we explore exon boundary conservation and assess the foldon-like behavior of exons using energy landscape theoretic measurements. We found deviations in the exon size distribution from exponential decay indicating selection in evolution. We show that when taken together there is a pronounced tendency to independent foldability for segments corresponding to the more conserved exons, supporting the idea of exon-foldon correspondence. While 45% of the families follow this general trend when analyzed individually, there are some families for which other stronger functional determinants, such as preserving frustrated active sites, may be acting. We further develop a systematic partitioning of protein domains using exon boundary hotspots, showing that minimal common exons correspond with uninterrupted alpha and/or beta elements for the majority of the families but not for all of them.

Evolution and folding of repeat proteins

Repeat proteins are made with tandem copies of similar amino acid stretches that fold into elongated architectures. These proteins constitute excellent model systems to investigate how evolution relates to structure, folding, and function. Here, we propose a scheme to map evolutionary information at the sequence level to a coarse-grained model for repeat-protein folding and use it to investigate the folding of thousands of repeat proteins. We model the energetics by a combination of an inverse Potts-model scheme with an explicit mechanistic model of duplications and deletions of repeats to calculate the evolutionary parameters of the system at the single-residue level. These parameters are used to inform an Ising-like model that allows for the generation of folding curves, apparent domain emergence, and occupation of intermediate states that are highly compatible with experimental data in specific case studies. We analyzed the folding of thousands of natural Ankyrin repeat proteins and found that a multiplicity of folding mechanisms are possible. Fully cooperative all-or-none transitions are obtained for arrays with enough sequence-similar elements and strong interactions between them, while noncooperative element-by-element intermittent folding arose if the elements are dissimilar and the interactions between them are energetically weak. Additionally, we characterized nucleation-propagation and multidomain folding mechanisms. We show that the global stability and cooperativity of the repeating arrays can be predicted from simple sequence scores.

Comparing the Aggregation Free Energy Landscapes of Amyloid Beta(1-42) and Amyloid Beta(1-40)

Using a predictive coarse-grained protein force field, we compute and compare the free energy landscapes and relative stabilities of amyloid-β protein (1-42) and amyloid-β protein (1-40) in their monomeric and oligomeric forms up to the octamer. At the same concentration, the aggregation free energy profile of Aβ42 is more downhill, with a computed solubility that is about 10 times smaller than that of Aβ40. At a concentration of 40 μM, the clear free energy barrier between the pre-fibrillar tetramer form and the fibrillar pentamer in the Aβ40 aggregation landscape disappears for Aβ42, suggesting that the Aβ42 tetramer has a more diverse structural range. To further compare the landscapes, we develop a cluster analysis based on the structural similarity between configurations and use it to construct an oligomerization map that captures the paths of easy interconversion between different but structurally similar states of oligomers for both species. A taxonomy of the oligomer species based on β-sheet stacking topologies is proposed. The comparison of the two oligomerization maps highlights several key differences in the landscapes that can be attributed to the two additional C-terminal residues that Aβ40 lacks. In general, the two terminal residues strongly stabilize the oligomeric structures for Aβ42 relative to Aβ40, and greatly facilitate the conversion from pre-fibrillar trimers to fibrillar tetramers.

Repeat proteins challenge the concept of structural domains

Structural domains are believed to be modules within proteins that can fold and function independently. Some proteins show tandem repetitions of apparent modular structure that do not fold independently, but rather co-operate in stabilizing structural forms that comprise several repeat-units. For many natural repeat-proteins, it has been shown that weak energetic links between repeats lead to the breakdown of co-operativity and the appearance of folding sub-domains within an apparently regular repeat array. The quasi-1D architecture of repeat-proteins is crucial in detailing how the local energetic balances can modulate the folding dynamics of these proteins, which can be related to the physiological behaviour of these ubiquitous biological systems.

Improved cryoEM-Guided Iterative Molecular Dynamics--Rosetta Protein Structure Refinement Protocol for High Precision Protein Structure Prediction

Many excellent methods exist that incorporate cryo-electron microscopy (cryoEM) data to constrain computational protein structure prediction and refinement. Previously, it was shown that iteration of two such orthogonal sampling and scoring methods – Rosetta and molecular dynamics (MD) simulations – facilitated exploration of conformational space in principle. Here, we go beyond a proof-of-concept study and address significant remaining limitations of the iterative MD–Rosetta protein structure refinement protocol. Specifically, all parts of the iterative refinement protocol are now guided by medium-resolution cryoEM density maps, and previous knowledge about the native structure of the protein is no longer necessary. Models are identified solely based on score or simulation time. All four benchmark proteins showed substantial improvement through three rounds of the iterative refinement protocol. The best-scoring final models of two proteins had sub-Ångstrom RMSD to the native structure over residues in secondary structure elements. Molecular dynamics was most efficient in refining secondary structure elements and was thus highly complementary to the Rosetta refinement which is most powerful in refining side chains and loop regions.

Network representation of conformational transitions between hidden intermediates of Rd-apocytochrome b562

The folding kinetics of Rd-apocytochrome b562 is two-state, but native-state hydrogen exchange experiments show that there are discrete partially unfolded (PUF) structures in equilibrium with the native state. These PUF structures are called hidden intermediates because they are not detected in kinetic experiments and they exist after the rate-limiting step. Structures of the mimics of hidden intermediates of Rd-apocytochrome b562 are resolved by NMR. Based upon their relative stability and structural features, the folding mechanism was proposed to follow a specific pathway (unfolded → rate-limiting transition state → PUF1 → PUF2 → native). Investigating the roles of equilibrium PUF structures in folding kinetics and their interrelationship not only deepens our understanding of the details of folding mechanism but also provides guides in protein design and prevention of misfolding. We performed molecular dynamics simulations starting from a hidden intermediate and the native state of Rd-apocytochrome b562 in explicit solvent, for a total of 37.18 μs mainly with Anton. We validated our simulations by detailed comparison with experimental data and other computations. We have verified that we sampled the post rate-limiting transition state region only. Markov state model was used to analyze the simulation results. We replace the specific pathway model with a network model. Transition-path theory was employed to calculate the net effective flux from the most unfolded state towards the most folded state in the network. The proposed sequential folding pathway via PUF1 then more stable, more native-like PUF2 is one of the routes in our network, but it is not dominant. The dominant path visits PUF2 without going through PUF1. There is also a route from PUF1 directly to the most folded state in the network without visiting PUF2. Our results indicate that the PUF states are not necessarily sequential in the folding. The major routes predicted in our network are testable by future experiments such as single molecule experiment.

Capturing coevolutionary signals inrepeat proteins

BACKGROUND

The analysis of correlations of amino acid occurrences in globular domains has led to the development of statistical tools that can identify native contacts - portions of the chains that come to close distance in folded structural ensembles. Here we introduce a direct coupling analysis for repeat proteins - natural systems for which the identification of folding domains remains challenging.

RESULTS

We show that the inherent translational symmetry of repeat protein sequences introduces a strong bias in the pair correlations at precisely the length scale of the repeat-unit. Equalizing for this bias in an objective way reveals true co-evolutionary signals from which local native contacts can be identified. Importantly, parameter values obtained for all other interactions are not significantly affected by the equalization. We quantify the robustness of the procedure and assign confidence levels to the interactions, identifying the minimum number of sequences needed to extract evolutionary information in several repeat protein families.

CONCLUSIONS

The overall procedure can be used to reconstruct the interactions at distances larger than repeat-pairs, identifying the characteristics of the strongest couplings in each family, and can be applied to any system that appears translationally symmetric.

Kinetics of protein-ligand unbinding: Predicting pathways, rates, and rate-limiting steps

The ability to predict the mechanisms and the associated rate constants of protein-ligand unbinding is of great practical importance in drug design. In this work we demonstrate how a recently introduced metadynamics-based approach allows exploration of the unbinding pathways, estimation of the rates, and determination of the rate-limiting steps in the paradigmatic case of the trypsin-benzamidine system. Protein, ligand, and solvent are described with full atomic resolution. Using metadynamics, multiple unbinding trajectories that start with the ligand in the crystallographic binding pose and end with the ligand in the fully solvated state are generated. The unbinding rate k off is computed from the mean residence time of the ligand. Using our previously computed binding affinity we also obtain the binding rate k on. Both rates are in agreement with reported experimental values. We uncover the complex pathways of unbinding trajectories and describe the critical rate-limiting steps with unprecedented detail. Our findings illuminate the role played by the coupling between subtle protein backbone fluctuations and the solvation by water molecules that enter the binding pocket and assist in the breaking of the shielded hydrogen bonds. We expect our approach to be useful in calculating rates for general protein-ligand systems and a valid support for drug design.

Folding pathway of a multidomain protein depends on its topology of domain connectivity

How do the folding mechanisms of multidomain proteins depend on protein topology? We addressed this question by developing an Ising-like structure-based model and applying it for the analysis of free-energy landscapes and folding kinetics of an example protein, Escherichia coli dihydrofolate reductase (DHFR). DHFR has two domains, one comprising discontinuous N- and C-terminal parts and the other comprising a continuous middle part of the chain. The simulated folding pathway of DHFR is a sequential process during which the continuous domain folds first, followed by the discontinuous domain, thereby avoiding the rapid decrease in conformation entropy caused by the association of the N- and C-terminal parts during the early phase of folding. Our simulated results consistently explain the observed experimental data on folding kinetics and predict an off-pathway structural fluctuation at equilibrium. For a circular permutant for which the topological complexity of wild-type DHFR is resolved, the balance between energy and entropy is modulated, resulting in the coexistence of the two folding pathways. This coexistence of pathways should account for the experimentally observed complex folding behavior of the circular permutant.

Frustration in biomolecules

Biomolecules are the prime information processing elements of living matter. Most of these inanimate systems are polymers that compute their own structures and dynamics using as input seemingly random character strings of their sequence, following which they coalesce and perform integrated cellular functions. In large computational systems with finite interaction-codes, the appearance of conflicting goals is inevitable. Simple conflicting forces can lead to quite complex structures and behaviors, leading to the concept of frustration in condensed matter. We present here some basic ideas about frustration in biomolecules and how the frustration concept leads to a better appreciation of many aspects of the architecture of biomolecules, and especially how biomolecular structure connects to function by means of localized frustration. These ideas are simultaneously both seductively simple and perilously subtle to grasp completely. The energy landscape theory of protein folding provides a framework for quantifying frustration in large systems and has been implemented at many levels of description. We first review the notion of frustration from the areas of abstract logic and its uses in simple condensed matter systems. We discuss then how the frustration concept applies specifically to heteropolymers, testing folding landscape theory in computer simulations of protein models and in experimentally accessible systems. Studying the aspects of frustration averaged over many proteins provides ways to infer energy functions useful for reliable structure prediction. We discuss how frustration affects folding mechanisms. We review here how the biological functions of proteins are related to subtle local physical frustration effects and how frustration influences the appearance of metastable states, the nature of binding processes, catalysis and allosteric transitions. In this review, we also emphasize that frustration, far from being always a bad thing, is an essential feature of biomolecules that allows dynamics to be harnessed for function. In this way, we hope to illustrate how Frustration is a fundamental concept in molecular biology.

Learning To Fold Proteins Using Energy Landscape Theory

This review is a tutorial for scientists interested in the problem of protein structure prediction, particularly those interested in using coarse-grained molecular dynamics models that are optimized using lessons learned from the energy landscape theory of protein folding. We also present a review of the results of the AMH/AMC/AMW/AWSEM family of coarse-grained molecular dynamics protein folding models to illustrate the points covered in the first part of the article. Accurate coarse-grained structure prediction models can be used to investigate a wide range of conceptual and mechanistic issues outside of protein structure prediction; specifically, the paper concludes by reviewing how AWSEM has in recent years been able to elucidate questions related to the unusual kinetic behavior of artificially designed proteins, multidomain protein misfolding, and the initial stages of protein aggregation.

A disorder-induced domino-like destabilization mechanism governs the folding and functional dynamics of the repeat protein IκBα

The stability of the repeat protein IκBα, a transcriptional inhibitor in mammalian cells, is critical in the functioning of the NF-κB signaling module implicated in an array of cellular processes, including cell growth, disease, immunity and apoptosis. Structurally, IκBα is complex, with both ordered and disordered regions, thus posing a challenge to the available computational protocols to model its conformational behavior. Here, we introduce a simple procedure to model disorder in systems that undergo binding-induced folding that involves modulation of the contact map guided by equilibrium experimental observables in combination with an Ising-like Wako-Saitô-Muñoz-Eaton model. This one-step procedure alone is able to reproduce a variety of experimental observables, including ensemble thermodynamics (scanning calorimetry, pre-transitions, m-values) and kinetics (roll-over in chevron plot, intermediates and their identity), and is consistent with hydrogen-deuterium exchange measurements. We further capture the intricate distance-dynamics between the domains as measured by single-molecule FRET by combining the model predictions with simple polymer physics arguments. Our results reveal a unique mechanism at work in IκBα folding, wherein disorder in one domain initiates a domino-like effect partially destabilizing neighboring domains, thus highlighting the effect of symmetry-breaking at the level of primary sequences. The offshoot is a multi-state and a dynamic conformational landscape that is populated by increasingly partially folded ensembles upon destabilization. Our results provide, in a straightforward fashion, a rationale to the promiscuous binding and short intracellular half-life of IκBα evolutionarily engineered into it through repeats with variable stabilities and expand the functional repertoire of disordered regions in proteins.

Funneling and frustration in the energy landscapes of some designed and simplified proteins

We explore the similarities and differences between the energy landscapes of proteins that have been selected by nature and those of some proteins designed by humans. Natural proteins have evolved to function as well as fold, and this is a source of energetic frustration. The sequence of Top7, on the other hand, was designed with architecture alone in mind using only native state stability as the optimization criterion. Its topology had not previously been observed in nature. Experimental studies show that the folding kinetics of Top7 is more complex than the kinetics of folding of otherwise comparable naturally occurring proteins. In this paper, we use structure prediction tools, frustration analysis, and free energy profiles to illustrate the folding landscapes of Top7 and two other proteins designed by Takada. We use both perfectly funneled (structure-based) and predictive (transferable) models to gain insight into the role of topological versus energetic frustration in these systems and show how they differ from those found for natural proteins. We also study how robust the folding of these designs would be to the simplification of the sequences using fewer amino acid types. Simplification using a five amino acid type code results in comparable quality of structure prediction to the full sequence in some cases, while the two-letter simplification scheme dramatically reduces the quality of structure prediction.

Detecting repetitions and periodicities in proteins by tiling the structural space

The notion of energy landscapes provides conceptual tools for understanding the complexities of protein folding and function. Energy landscape theory indicates that it is much easier to find sequences that satisfy the “Principle of Minimal Frustration” when the folded structure is symmetric (Wolynes, P. G. Symmetry and the Energy Landscapes of Biomolecules. Proc. Natl. Acad. Sci. U.S.A.1996, 93, 14249–14255). Similarly, repeats and structural mosaics may be fundamentally related to landscapes with multiple embedded funnels. Here we present analytical tools to detect and compare structural repetitions in protein molecules. By an exhaustive analysis of the distribution of structural repeats using a robust metric, we define those portions of a protein molecule that best describe the overall structure as a tessellation of basic units. The patterns produced by such tessellations provide intuitive representations of the repeating regions and their association toward higher order arrangements. We find that some protein architectures can be described as nearly periodic, while in others clear separations between repetitions exist. Since the method is independent of amino acid sequence information, we can identify structural units that can be encoded by a variety of distinct amino acid sequences.

Stepwise protein folding at near amino acid resolution by hydrogen exchange and mass spectrometry

The kinetic folding of ribonuclease H was studied by hydrogen exchange (HX) pulse labeling with analysis by an advanced fragment separation mass spectrometry technology. The results show that folding proceeds through distinct intermediates in a stepwise pathway that sequentially incorporates cooperative native-like structural elements to build the native protein. Each step is seen as a concerted transition of one or more segments from an HX-unprotected to an HX-protected state. Deconvolution of the data to near amino acid resolution shows that each step corresponds to the folding of a secondary structural element of the native protein, termed a "foldon." Each folded segment is retained through subsequent steps of foldon addition, revealing a stepwise buildup of the native structure via a single dominant pathway. Analysis of the pertinent literature suggests that this model is consistent with experimental results for many proteins and some current theoretical results. Two biophysical principles appear to dictate this behavior. The principle of cooperativity determines the central role of native-like foldon units. An interaction principle termed "sequential stabilization" based on native-like interfoldon interactions orders the pathway.